tag:blogger.com,1999:blog-13690723416946399022024-03-04T22:13:18.736-08:00Belajar MatematikaMatematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.comBlogger26125tag:blogger.com,1999:blog-1369072341694639902.post-84049074839731785352019-09-12T20:53:00.003-07:002019-09-12T20:53:44.060-07:00Integral<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pengertian Integral</h2>
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Integral merupakan bentuk operasi <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">matematika</span> yang menjadi kebalikan (invers) dari operasi turunan dan limit dari jumlah atau suatu luas daerah tertentu. Berdasarkan pengertian tersebut ada dua hal yang dilakukan dalam integral sehingga dikategorikan menjadi 2 jenis integral. Pertama, integral sebagai invers/ kebalikan dari turunan disebut sebagai Integral Tak Tentu. Kedua, integral sebagai limit dari jumlah atau suatu luas daerah tertentu disebut integral tentu.</div>
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<img alt="integral - pengertian rumus trigonometri" class="aligncenter size-full wp-image-2187 jetpack-lazy-image jetpack-lazy-image--handled" height="150" src="https://i0.wp.com/www.studiobelajar.com/wp-content/uploads/2018/06/integral-pengertian-rumus-trigonometri.png?resize=150%2C150&ssl=1" style="border: 0px; font: inherit; height: auto; margin: 0px auto; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="150" /></div>
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Integral Tak Tentu</h2>
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Integral tak tentu seperti sebelumnya dijelaskan merupakan invers/kebalikan dari turunan. Turunan dari suatu fungsi, jika diintegralkan akan menghasilkan fungsi itu sendiri. Perhatikanlah contoh turunan-turunan dalam fungsi aljabar berikut ini:</div>
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<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Turunan dari fungsi aljabar y = x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> adalah y<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">I</span> = 3x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Turunan dari fungsi aljabar y = x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> + 8 adalah y<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">I</span> = 3x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Turunan dari fungsi aljabar y = x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> + 17 adalah y<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">I</span> = 3x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Turunan dari fungsi aljabar y = x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> – 6 adalah y<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">I</span> = 3x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span></li>
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Seperti yang sudah dipelajari dalam materi turunan, variabel dalam suatu fungsi mengalami penurunan pangkat. Berdasarkan contoh tersebut, diketahui bahwa ada banyak fungsi yang memiliki hasil turunan yang sama yaitu y<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">I </span>= 3x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span>. Fungsi dari variabel x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> ataupun fungsi dari variabel x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> yang ditambah atau dikurang suatu bilangan (misal contoh: +8, +17, atau -6) memiliki turunan yang sama. Jika turunan tersebut dintegralkan, seharusnya adalah menjadi fungsi-fungsi awal sebelum diturunkan. Namun, dalam kasus tidak diketahui fungsi awal dari suatu turunan, maka hasil integral dari turunan tersebut dapat ditulis:</div>
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f(x) = y = x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">3</span> + C</div>
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Dengan nilai C bisa berapapun. Notasi C ini disebut sebagai konstanta integral. Integral tak tentu dari suatu fungsi dinotasikan sebagai:</div>
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<img alt="\int f(x) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+f%28x%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int f(x) dx" /></div>
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Pada notasi tersebut dapat dibaca integral terhadap x”. notasi disebut integran. Secara umum integral dari fungsi f(x) adalah penjumlahan F(x) dengan C atau:</div>
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<img alt="\int f(x) dx = F(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+f%28x%29+dx+%3D+F%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int f(x) dx = F(x)" /></div>
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Karena integral dan turunan berkaitan, maka rumus integral dapat diperoleh dari rumusan penurunan. Jika turunan:</div>
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<img alt="\frac{d}{dx}\frac{a}{(n+1)}x^{(n+1)} = ax^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7Bd%7D%7Bdx%7D%5Cfrac%7Ba%7D%7B%28n%2B1%29%7Dx%5E%7B%28n%2B1%29%7D+%3D+ax%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{d}{dx}\frac{a}{(n+1)}x^{(n+1)} = ax^n" /></div>
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Maka rumus integral aljabar diperoleh:</div>
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<img alt="\int ax^n dx = \frac{a}{(n+1)}x^{n+1} + C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+ax%5En+dx+%3D+%5Cfrac%7Ba%7D%7B%28n%2B1%29%7Dx%5E%7Bn%2B1%7D+%2B+C&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int ax^n dx = \frac{a}{(n+1)}x^{n+1} + C" /></div>
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dengan syarat <img alt="n \neq 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=n+%5Cneq+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="n \neq 1" />.</div>
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Sebagai contoh lihatlah integral aljabar fungsi-fungsi berikut:</div>
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<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int 4x^3dx=\frac{4}{(3+1)}x^{(3+1)}+ C = x^4 + C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+4x%5E3dx%3D%5Cfrac%7B4%7D%7B%283%2B1%29%7Dx%5E%7B%283%2B1%29%7D%2B+C+%3D+x%5E4+%2B+C&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int 4x^3dx=\frac{4}{(3+1)}x^{(3+1)}+ C = x^4 + C" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int \frac{1}{x^3}dx = \int x^{-3} dx = \frac{1}{(-3+1)}x^{-3+1}+C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Cfrac%7B1%7D%7Bx%5E3%7Ddx+%3D+%5Cint+x%5E%7B-3%7D+dx+%3D+%5Cfrac%7B1%7D%7B%28-3%2B1%29%7Dx%5E%7B-3%2B1%7D%2BC&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \frac{1}{x^3}dx = \int x^{-3} dx = \frac{1}{(-3+1)}x^{-3+1}+C" /><br style="margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;" /><img alt="= -\frac{1}{2}x^{-2}+C = -\frac{1}{2x^2}+C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B-2%7D%2BC+%3D+-%5Cfrac%7B1%7D%7B2x%5E2%7D%2BC&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= -\frac{1}{2}x^{-2}+C = -\frac{1}{2x^2}+C" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int 4x^3 - 3x^2 dx = \frac{4}{(3+1)} x^{(3+1)} + \frac{3}{(2+1)}x^{(2+1)}+C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+4x%5E3+-+3x%5E2+dx+%3D+%5Cfrac%7B4%7D%7B%283%2B1%29%7D+x%5E%7B%283%2B1%29%7D+%2B+%5Cfrac%7B3%7D%7B%282%2B1%29%7Dx%5E%7B%282%2B1%29%7D%2BC&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int 4x^3 - 3x^2 dx = \frac{4}{(3+1)} x^{(3+1)} + \frac{3}{(2+1)}x^{(2+1)}+C" /><br style="margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;" /><img alt="= x^4+x^3+C" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+x%5E4%2Bx%5E3%2BC&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= x^4+x^3+C" /></li>
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Integral Trigonometri</h2>
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Integral juga bisa dioperasikan pada fungsi <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">trigonometri</span>. Pengoperasian integral trigonometri juga dilakukan dengan konsep yang sama pada pada integral aljabar yaitu kebalikan dari penurunan. Sehingga dapat simpulkan bahwa:</div>
<table style="background-color: #f9f9f9; border-collapse: collapse; border-spacing: 0px; border: 0px; color: black; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: auto; word-break: normal;"><tbody style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">No.</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">Fungsi f(x) = y</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">Turunan <img alt="\frac{dy}{dx}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7Bdy%7D%7Bdx%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{dy}{dx}" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271">Integral</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">1</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = sin x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">cos x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"> <img alt="\int \cos x dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ccos+x+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \cos x dx" />= sin x</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">2</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = cos x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">– sin x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"><img alt="\int \sin x dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Csin+x+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \sin x dx" /> = – cos x</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">3</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = tan x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">sec<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span> x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"><img alt="\int \sec^2 x dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Csec%5E2+x+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \sec^2 x dx" /> = tan x</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">4</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = cot x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">– csc<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span> x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"><img alt="\int \csc^2 x dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ccsc%5E2+x+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \csc^2 x dx" /> = – cot x</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">5</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = sec x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">tan x . sec x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"><img alt="\int \tan x . \sec x d" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ctan+x+.+%5Csec+x+d&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \tan x . \sec x d" /> = sec x</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="48">6</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="138">y = csc x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="156">-.cot x . csc x</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="271"><img alt="\int \cot x . \csc x dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ccot+x+.+%5Ccsc+x+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \cot x . \csc x dx" /> = – csc x</td></tr>
</tbody></table>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Selain rumus dasar diatas, ada rumus lain yang bisa digunakan pada pengoperasian integral trigonometri yaitu:</div>
<table style="background-color: #f9f9f9; border-collapse: collapse; border-spacing: 0px; border: 0px; color: black; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: 624px; word-break: normal;"><tbody style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150">Fungsi f(x) = y</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">Turunan <img alt="\frac{dy}{dx}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7Bdy%7D%7Bdx%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{dy}{dx}" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294">Integral</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150"><img alt="y = \frac{1}{a} \sin(ax+b)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Cfrac%7B1%7D%7Ba%7D+%5Csin%28ax%2Bb%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \frac{1}{a} \sin(ax+b)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">cos (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"><img alt="\int \cos (ax+b) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ccos+%28ax%2Bb%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \cos (ax+b) dx" /> = <img alt="\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{1}{a}" /> sin (ax + b) + C</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150"> <img alt="y = - \frac{1}{a} \cos (ax + b)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D%C2%A0-+%5Cfrac%7B1%7D%7Ba%7D+%5Ccos+%28ax+%2B+b%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = - \frac{1}{a} \cos (ax + b)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">sin (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"><img alt="\int \sin (ax+b) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Csin+%28ax%2Bb%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \sin (ax+b) dx" /> = <img alt="-\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="-\frac{1}{a}" /> cos (ax + b) + C</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150">y = <img alt="\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{1}{a}" /> tan (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">sec<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span> (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"> <img alt="\int \sec^2(ax+b)dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Csec%5E2%28ax%2Bb%29dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \sec^2(ax+b)dx" />= <img alt="\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{1}{a}" /> tan (ax + b) + C</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150">y = <img alt="-\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="-\frac{1}{a}" /> cot (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">csc<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">2</span> (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"><img alt="\int \csc^2(ax+b) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+%5Ccsc%5E2%28ax%2Bb%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int \csc^2(ax+b) dx" /> = <img alt="- \frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-+%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="- \frac{1}{a}" /> cot (ax + b)</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150">y = <img alt="-\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="-\frac{1}{a}" /> sec (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">tan (ax + b) . sec (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"><img alt="\int" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int" /> (ax+b) . sec(ax + b) dx= <img alt="\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{1}{a}" /> sec (ax + b) + C</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="150">y = <img alt="-\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="-\frac{1}{a}" /> csc (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="180">cot (ax + b) . csc (ax + b)</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;" width="294"><img alt="\int" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int" /> cot (ax + b) . csc (ax + b) dx = <img alt="-\frac{1}{a}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=-%5Cfrac%7B1%7D%7Ba%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="-\frac{1}{a}" /> csc (ax + b)</td></tr>
</tbody></table>
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Sifat-sifat dari integral yaitu:</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int k. f(x) dx=k.\int f(x)dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+k.+f%28x%29+dx%3Dk.%5Cint+f%28x%29dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int k. f(x) dx=k.\int f(x)dx" /> (dengan k adalah konstanta)</li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int f(x)+g(x)dx =\int(x)dx+\int g(x) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+f%28x%29%2Bg%28x%29dx+%3D%5Cint%28x%29dx%2B%5Cint+g%28x%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int f(x)+g(x)dx =\int(x)dx+\int g(x) dx" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\int f(x) - g(x)dx = \int f(x)dx-\int g(x) dx" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cint+f%28x%29+-+g%28x%29dx+%3D+%5Cint+f%28x%29dx-%5Cint+g%28x%29+dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\int f(x) - g(x)dx = \int f(x)dx-\int g(x) dx" /></li>
</ul>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com3tag:blogger.com,1999:blog-1369072341694639902.post-36622620832144589192019-09-12T20:52:00.000-07:002019-09-12T20:52:27.712-07:00Turunan<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pengertian Turunan</h2>
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Turunan adalah pengukuran terhadap bagaimana fungsi berubah seiring perubahan nilai yang dimasukan, atau secara umum turunan menunjukkan bagaimana suatu besaran berubah akibat perubahan besaran lainnya. Proses dalam menemukan turunan disebut diferensiasi.</div>
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Pada fungsi y = f(x), turunan dari variabel y terhadap variabel x dinotasikan dengan <img alt="\frac{dy}{dx}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7Bdy%7D%7Bdx%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{dy}{dx}" /> atau <img alt="\frac{df(x)}{dx}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7Bdf%28x%29%7D%7Bdx%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{df(x)}{dx}" /> atau y’ dan didefinisikan sebagai:</div>
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<img alt="f'(x) =\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D%5Clim%5Climits_%7Bh%5Cto+0%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) =\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}" /></div>
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</div>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Rumus-rumus Turunan Fungsi Aljabar</h2>
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Dengan definisi turunan akan dicari rumus-rumus turunan fungsi aljabar yang terdiri dari fungsi pangkat <img alt="f(x) = x^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+x%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = x^n" />, hasil kali fungsi f(x) = u(x) . v(x), hasil pembagian fungsi <img alt="f(x) = \frac{u(x)}{v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%5Cfrac%7Bu%28x%29%7D%7Bv%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = \frac{u(x)}{v(x)}" />, dan pangkat dari fungsi <img alt="f(x) = (u(x))^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%28u%28x%29%29%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = (u(x))^n" />.</div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
1. Rumus turunan fungsi pangkat <img alt="f(x) = x^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+x%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = x^n" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Fungsi berbentuk pangkat turunannya dapat menggunakan rumus <img alt="f'(x) = \lim \limits_{h \to 0}\frac{f(x+h)-f(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Clim+%5Climits_%7Bh+%5Cto+0%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \lim \limits_{h \to 0}\frac{f(x+h)-f(x)}{h}" /> sebagai:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \lim \limits_{h \to 0}\frac{(x+h)^n - (x)^n}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Clim+%5Climits_%7Bh+%5Cto+0%7D%5Cfrac%7B%28x%2Bh%29%5En+-+%28x%29%5En%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \lim \limits_{h \to 0}\frac{(x+h)^n - (x)^n}{h}" /></div>
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<img alt="= \lim \limits_{h \to 0}\frac{\sum^n_{i=0}C^n_ix^{n-i}h^i-x^n}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh+%5Cto+0%7D%5Cfrac%7B%5Csum%5En_%7Bi%3D0%7DC%5En_ix%5E%7Bn-i%7Dh%5Ei-x%5En%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h \to 0}\frac{\sum^n_{i=0}C^n_ix^{n-i}h^i-x^n}{h}" /></div>
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<img alt="= \lim_{h \to 0}\frac{C^n_0x^n+C^n_1x^{n-1}h+C^n_2x^{n-2}h^2+\cdots+C^n_nh^n-x^n}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim_%7Bh+%5Cto+0%7D%5Cfrac%7BC%5En_0x%5En%2BC%5En_1x%5E%7Bn-1%7Dh%2BC%5En_2x%5E%7Bn-2%7Dh%5E2%2B%5Ccdots%2BC%5En_nh%5En-x%5En%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim_{h \to 0}\frac{C^n_0x^n+C^n_1x^{n-1}h+C^n_2x^{n-2}h^2+\cdots+C^n_nh^n-x^n}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}\frac{x^n+nx^{n-1}h+\frac{n(n-1)}{2!}x^{n-2}h^2+\cdots+h^n-x^n}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bx%5En%2Bnx%5E%7Bn-1%7Dh%2B%5Cfrac%7Bn%28n-1%29%7D%7B2%21%7Dx%5E%7Bn-2%7Dh%5E2%2B%5Ccdots%2Bh%5En-x%5En%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}\frac{x^n+nx^{n-1}h+\frac{n(n-1)}{2!}x^{n-2}h^2+\cdots+h^n-x^n}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}\frac{nx^{n-1}h+\frac{n(n-1)}{2!}x^{n-2}h^2+\cdots+h^n}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bnx%5E%7Bn-1%7Dh%2B%5Cfrac%7Bn%28n-1%29%7D%7B2%21%7Dx%5E%7Bn-2%7Dh%5E2%2B%5Ccdots%2Bh%5En%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}\frac{nx^{n-1}h+\frac{n(n-1)}{2!}x^{n-2}h^2+\cdots+h^n}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}(nx^{n-1}+\frac{n(n-1)}{2!}x^{n-2}h+\cdots+h^{n-1})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%28nx%5E%7Bn-1%7D%2B%5Cfrac%7Bn%28n-1%29%7D%7B2%21%7Dx%5E%7Bn-2%7Dh%2B%5Ccdots%2Bh%5E%7Bn-1%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}(nx^{n-1}+\frac{n(n-1)}{2!}x^{n-2}h+\cdots+h^{n-1})" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= nx^{n-1}+0+0+\cdots+0=nx^{n-1}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+nx%5E%7Bn-1%7D%2B0%2B0%2B%5Ccdots%2B0%3Dnx%5E%7Bn-1%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= nx^{n-1}+0+0+\cdots+0=nx^{n-1}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jadi rumus turunan fungsi pangkat adalah:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x ) = nx^{n-1}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x+%29+%3D+nx%5E%7Bn-1%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x ) = nx^{n-1}" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
2. Rumus turunan hasil kali fungsi <img alt="f(x) = u(x) \cdot v(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+u%28x%29+%5Ccdot+v%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = u(x) \cdot v(x)" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Fungsi f(x) yang terbentuk dari perkalian fungsi u(x) dan v(x), turunannya didapat dengan:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \lim \limits_{h\to0}\frac{f(x+h)-f(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \lim \limits_{h\to0}\frac{f(x+h)-f(x)}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{h\to0}=\frac{u(x+h)v(x+h)-u(x)v(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bh%5Cto0%7D%3D%5Cfrac%7Bu%28x%2Bh%29v%28x%2Bh%29-u%28x%29v%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{h\to0}=\frac{u(x+h)v(x+h)-u(x)v(x)}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{h\to0}=\frac{u(x+h)v(x+h)-u(x+h)v(x)+u(x+h)v(x)-u(x)v(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bh%5Cto0%7D%3D%5Cfrac%7Bu%28x%2Bh%29v%28x%2Bh%29-u%28x%2Bh%29v%28x%29%2Bu%28x%2Bh%29v%28x%29-u%28x%29v%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{h\to0}=\frac{u(x+h)v(x+h)-u(x+h)v(x)+u(x+h)v(x)-u(x)v(x)}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\lim\limits_{h\to0}\frac{[u(x+h)v(x+h)-u(x+h)v(x)]+[u(x+h)v(x)-u(x)v(x)]}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B%5Bu%28x%2Bh%29v%28x%2Bh%29-u%28x%2Bh%29v%28x%29%5D%2B%5Bu%28x%2Bh%29v%28x%29-u%28x%29v%28x%29%5D%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\lim\limits_{h\to0}\frac{[u(x+h)v(x+h)-u(x+h)v(x)]+[u(x+h)v(x)-u(x)v(x)]}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}\frac{u(x+h)[v(x+h)-v(x)]}{h}+\lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%2Bh%29%5Bv%28x%2Bh%29-v%28x%29%5D%7D%7Bh%7D%2B%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B%5Bu%28x%2Bh%29-u%28x%29%5Dv%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}\frac{u(x+h)[v(x+h)-v(x)]}{h}+\lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}u(x+h) \cdot \lim \limits_{h\to0}\frac{v(x+h)-v(x)}{h}+\lim \limits_{h\to0}\frac{u(x+h)-u(x)}{h}.\lim_{h\to0}v(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7Du%28x%2Bh%29+%5Ccdot+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bv%28x%2Bh%29-v%28x%29%7D%7Bh%7D%2B%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%2Bh%29-u%28x%29%7D%7Bh%7D.%5Clim_%7Bh%5Cto0%7Dv%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}u(x+h) \cdot \lim \limits_{h\to0}\frac{v(x+h)-v(x)}{h}+\lim \limits_{h\to0}\frac{u(x+h)-u(x)}{h}.\lim_{h\to0}v(x)" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= u(x+0) \cdot v'(x)+u'(x) \cdot v(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+u%28x%2B0%29+%5Ccdot+v%27%28x%29%2Bu%27%28x%29+%5Ccdot+v%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= u(x+0) \cdot v'(x)+u'(x) \cdot v(x)" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="u'(x).v(x)+u(x).v'(x)\overset{atau}{\rightarrow}u'.v+u.v'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=u%27%28x%29.v%28x%29%2Bu%28x%29.v%27%28x%29%5Coverset%7Batau%7D%7B%5Crightarrow%7Du%27.v%2Bu.v%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="u'(x).v(x)+u(x).v'(x)\overset{atau}{\rightarrow}u'.v+u.v'" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jadi rumus turunan fungsinya adalah:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x)=u'v+uv'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29%3Du%27v%2Buv%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x)=u'v+uv'" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
3. Rumus turunan fungsi pembagian <img alt="f(x)=\frac{u(x)}{v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29%3D%5Cfrac%7Bu%28x%29%7D%7Bv%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x)=\frac{u(x)}{v(x)}" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \lim \limits_{h\to0}\frac{f(x+h)-f(x)}{h}\overset{menjadi}{\rightarrow}\lim \limits_{h\to0}\frac{\frac{u(x+h)}{v(x+h)} - \frac{u(x)}{v(x)}}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%5Coverset%7Bmenjadi%7D%7B%5Crightarrow%7D%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B%5Cfrac%7Bu%28x%2Bh%29%7D%7Bv%28x%2Bh%29%7D+-+%5Cfrac%7Bu%28x%29%7D%7Bv%28x%29%7D%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \lim \limits_{h\to0}\frac{f(x+h)-f(x)}{h}\overset{menjadi}{\rightarrow}\lim \limits_{h\to0}\frac{\frac{u(x+h)}{v(x+h)} - \frac{u(x)}{v(x)}}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
sehingga</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \lim \limits_{h\to0}\frac{u(x+h)v(x)-u(x)v(x+h)}{h \cdot v(x+h)v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%2Bh%29v%28x%29-u%28x%29v%28x%2Bh%29%7D%7Bh+%5Ccdot+v%28x%2Bh%29v%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \lim \limits_{h\to0}\frac{u(x+h)v(x)-u(x)v(x+h)}{h \cdot v(x+h)v(x)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\lim \limits_{h\to0}\frac{u(x+h)v(x)-u(x)v(x)-u(x)v(x+h)+u(x)v(x)}{h.v(x+h)v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%2Bh%29v%28x%29-u%28x%29v%28x%29-u%28x%29v%28x%2Bh%29%2Bu%28x%29v%28x%29%7D%7Bh.v%28x%2Bh%29v%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\lim \limits_{h\to0}\frac{u(x+h)v(x)-u(x)v(x)-u(x)v(x+h)+u(x)v(x)}{h.v(x+h)v(x)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)-u(x)[v(x+h)-v(x)]}{h.v(x+h)v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B%5Bu%28x%2Bh%29-u%28x%29%5Dv%28x%29-u%28x%29%5Bv%28x%2Bh%29-v%28x%29%5D%7D%7Bh.v%28x%2Bh%29v%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)-u(x)[v(x+h)-v(x)]}{h.v(x+h)v(x)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)}{h \cdot v(x+h)v(x)} - \lim \limits_{h\to0}\frac{u(x)[v(x+h)-v(x)]}{h \cdot v(x+h)v(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B%5Bu%28x%2Bh%29-u%28x%29%5Dv%28x%29%7D%7Bh+%5Ccdot+v%28x%2Bh%29v%28x%29%7D+-+%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%29%5Bv%28x%2Bh%29-v%28x%29%5D%7D%7Bh+%5Ccdot+v%28x%2Bh%29v%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{h\to0}\frac{[u(x+h)-u(x)]v(x)}{h \cdot v(x+h)v(x)} - \lim \limits_{h\to0}\frac{u(x)[v(x+h)-v(x)]}{h \cdot v(x+h)v(x)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\lim \limits_{h\to0}\frac{u(x+h)-u(x)}{h}.\lim\limits_{h\to0}\frac{v(x)}{v(x+h)v(x)}- \lim\limits_{h\to0}\frac{u(x)}{v(x+h)v(x)}.\lim\limits_{h\to0}\frac{v(x+h)-v(x)}{h}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Clim+%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%2Bh%29-u%28x%29%7D%7Bh%7D.%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bv%28x%29%7D%7Bv%28x%2Bh%29v%28x%29%7D-+%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bu%28x%29%7D%7Bv%28x%2Bh%29v%28x%29%7D.%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7Bv%28x%2Bh%29-v%28x%29%7D%7Bh%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\lim \limits_{h\to0}\frac{u(x+h)-u(x)}{h}.\lim\limits_{h\to0}\frac{v(x)}{v(x+h)v(x)}- \lim\limits_{h\to0}\frac{u(x)}{v(x+h)v(x)}.\lim\limits_{h\to0}\frac{v(x+h)-v(x)}{h}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= u'(x).\frac{v(x)}{v(x+0)v(x)}-\frac{u(x)}{v(x+0)v(x)} \cdot v'(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+u%27%28x%29.%5Cfrac%7Bv%28x%29%7D%7Bv%28x%2B0%29v%28x%29%7D-%5Cfrac%7Bu%28x%29%7D%7Bv%28x%2B0%29v%28x%29%7D+%5Ccdot+v%27%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= u'(x).\frac{v(x)}{v(x+0)v(x)}-\frac{u(x)}{v(x+0)v(x)} \cdot v'(x)" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\frac{u'(x)v(x)}{v(x)v(x)}-\frac{u(x)v'(x)}{v(x)v(x)} \rightarrow\frac{u'(x)v(x)-u(x)v'(x)}{(u(x))^2} \rightarrow \frac{u'v-uv'}{v^2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Cfrac%7Bu%27%28x%29v%28x%29%7D%7Bv%28x%29v%28x%29%7D-%5Cfrac%7Bu%28x%29v%27%28x%29%7D%7Bv%28x%29v%28x%29%7D+%5Crightarrow%5Cfrac%7Bu%27%28x%29v%28x%29-u%28x%29v%27%28x%29%7D%7B%28u%28x%29%29%5E2%7D+%5Crightarrow+%5Cfrac%7Bu%27v-uv%27%7D%7Bv%5E2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\frac{u'(x)v(x)}{v(x)v(x)}-\frac{u(x)v'(x)}{v(x)v(x)} \rightarrow\frac{u'(x)v(x)-u(x)v'(x)}{(u(x))^2} \rightarrow \frac{u'v-uv'}{v^2}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jadi rumus turunan fungsinya adalah</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \frac{u'v-uv'}{v^2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7Bu%27v-uv%27%7D%7Bv%5E2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{u'v-uv'}{v^2}" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
4. Rumus turunan pangkat dari fungsi <img alt="f(x)=(u(x))^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29%3D%28u%28x%29%29%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x)=(u(x))^n" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Ingat jika <img alt="f(x) = x^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+x%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = x^n" />, maka:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x)=\frac{df(x)}{dx}= \frac{dx^n}{dx} = nx^n-1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29%3D%5Cfrac%7Bdf%28x%29%7D%7Bdx%7D%3D+%5Cfrac%7Bdx%5En%7D%7Bdx%7D+%3D+nx%5En-1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x)=\frac{df(x)}{dx}= \frac{dx^n}{dx} = nx^n-1" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Karena <img alt="f(x) = (u(x))^n=u^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%28u%28x%29%29%5En%3Du%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = (u(x))^n=u^n" />, maka:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \frac{df(x)}{dx} = \frac{du^n}{dx} \cdot \frac{du}{du}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7Bdf%28x%29%7D%7Bdx%7D+%3D+%5Cfrac%7Bdu%5En%7D%7Bdx%7D+%5Ccdot+%5Cfrac%7Bdu%7D%7Bdu%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{df(x)}{dx} = \frac{du^n}{dx} \cdot \frac{du}{du}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Atau</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \frac{du^n}{du} \cdot \frac{du}{dx} = nu^{n-1} \cdot u'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7Bdu%5En%7D%7Bdu%7D+%5Ccdot+%5Cfrac%7Bdu%7D%7Bdx%7D+%3D+nu%5E%7Bn-1%7D+%5Ccdot+u%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{du^n}{du} \cdot \frac{du}{dx} = nu^{n-1} \cdot u'" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jadi rumus turunan fungsinya adalah:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = nu^(n-1) \cdot u'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+nu%5E%28n-1%29+%5Ccdot+u%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = nu^(n-1) \cdot u'" /></div>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Rumus-rumus Turunan Trigonometri</h2>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Dengan menggunakan definisi turunan, dapat diperoleh rumus-rumus turunan <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">trigonometri</span> berikut: (dengan u dan v masing-masing fungsi dari x)</div>
<ol style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style-image: initial; list-style-position: outside; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="margin: 0px; outline: 0px; overflow-wrap: normal; padding: 2px 0px; word-break: normal;"><ol style="border: 0px; line-height: inherit; list-style-image: initial; list-style-position: outside; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sin x \rightarrow y' = \cos x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csin+x+%5Crightarrow+y%27+%3D+%5Ccos+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sin x \rightarrow y' = \cos x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cos x \rightarrow y' = - \sin x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccos+x+%5Crightarrow+y%27+%3D+-+%5Csin+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cos x \rightarrow y' = - \sin x" /></li>
</ol>
</li>
</ol>
<ol style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style-image: initial; list-style-position: outside; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \tan x \rightarrow y' = \sec^2 x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ctan+x+%5Crightarrow+y%27+%3D+%5Csec%5E2+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \tan x \rightarrow y' = \sec^2 x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cot x \rightarrow y' = - \csc^2 x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccot+x+%5Crightarrow+y%27+%3D+-+%5Ccsc%5E2+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cot x \rightarrow y' = - \csc^2 x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sec x \rightarrow y'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csec+x+%5Crightarrow+y%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sec x \rightarrow y'" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \csc x \rightarrow - \csc \times \cot x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccsc+x+%5Crightarrow+-+%5Ccsc+%5Ctimes+%5Ccot+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \csc x \rightarrow - \csc \times \cot x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sin^n x y' = n \sin^{n-1} \times \cos x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csin%5En+x+y%27+%3D+n+%5Csin%5E%7Bn-1%7D+%5Ctimes+%5Ccos+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sin^n x y' = n \sin^{n-1} \times \cos x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cos^nx \rightarrow y' = -n \cos^{n-1} \times \sin x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccos%5Enx+%5Crightarrow+y%27+%3D+-n+%5Ccos%5E%7Bn-1%7D+%5Ctimes+%5Csin+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cos^nx \rightarrow y' = -n \cos^{n-1} \times \sin x" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sin u \rightarrow y' = u' \cos u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csin+u+%5Crightarrow+y%27+%3D+u%27+%5Ccos+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sin u \rightarrow y' = u' \cos u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cos u \rightarrow y' = - u' \sin u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccos+u+%5Crightarrow+y%27+%3D+-+u%27+%5Csin+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cos u \rightarrow y' = - u' \sin u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \tan u \rightarrow y' = u' \sec^2 u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ctan+u+%5Crightarrow+y%27+%3D+u%27+%5Csec%5E2+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \tan u \rightarrow y' = u' \sec^2 u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cot u \rightarrow y' =-u' \csc^2u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccot+u+%5Crightarrow+y%27+%3D-u%27+%5Ccsc%5E2u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cot u \rightarrow y' =-u' \csc^2u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sec u \rightarrow y' = u' \sec u \tan u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csec+u+%5Crightarrow+y%27+%3D+u%27+%5Csec+u+%5Ctan+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sec u \rightarrow y' = u' \sec u \tan u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \csc u \rightarrow y' = -u' \csc u \cot u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccsc+u+%5Crightarrow+y%27+%3D+-u%27+%5Ccsc+u+%5Ccot+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \csc u \rightarrow y' = -u' \csc u \cot u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \sin^nu \rightarrow y' = n.u' \sin^{n-1} \cos u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Csin%5Enu+%5Crightarrow+y%27+%3D+n.u%27+%5Csin%5E%7Bn-1%7D+%5Ccos+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \sin^nu \rightarrow y' = n.u' \sin^{n-1} \cos u" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="y = \cos ^nu \rightarrow y'= -n \cdot u' cos^{n-1}u \cdot \sin u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+%5Ccos+%5Enu+%5Crightarrow+y%27%3D+-n+%5Ccdot+u%27+cos%5E%7Bn-1%7Du+%5Ccdot+%5Csin+u&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = \cos ^nu \rightarrow y'= -n \cdot u' cos^{n-1}u \cdot \sin u" /></li>
</ol>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Aplikasi Turunan</h2>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
1. Menentukan gradien garis singgung suatu kurva</h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Gradien garis singgung (m) pada suatu kurva y = f(x) dirumuskan sebagai:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="m = y' = f'(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=m+%3D+y%27+%3D+f%27%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="m = y' = f'(x)" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">Persamaan garis singgung</span> pada suatu kurva y = f(x) di titik singgung <img alt="(x_1, y_1)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%28x_1%2C+y_1%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="(x_1, y_1)" /> dirumuskan sebagai:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="y - y_1 = m(x - x_1) \rightarrow m = f'(x_1)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+-+y_1+%3D+m%28x+-+x_1%29+%5Crightarrow+m+%3D+f%27%28x_1%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y - y_1 = m(x - x_1) \rightarrow m = f'(x_1)" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
2. Menentukan interval fungsi naik dan fungsi turun</h3>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Syarat interval fungsi naik <img alt="\rightarrow f'(x) > 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%27%28x%29+%3E+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f'(x) > 0" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Syarat interval fungsi turun <img alt="\rightarrow f'(x) < 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%27%28x%29+%3C+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f'(x) < 0" /></li>
</ul>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
3. Menentukan nilai stasioner suatu fungsi dan jenisnya</h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika fungsi y = f(x) kontinu dan diferensiabel di x = a dan f'(x) = 0, maka fungsi memiliki nilai statisioner di x = a. Jenis nilai stasioner dari fungsi y = f(x) dapat berupa nilai balik minimum, nilai balik maksimum, atau nilai belok. Jenis nilai stasioner ini bisa ditentukan dengan menggunakan turunan kedua dari fungsi tersebut.</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Nilai maksimum <img alt="\rightarrow f'(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%27%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f'(x) = 0" /> dan <img alt="\rightarrow f"(x) < 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%22%28x%29+%3C+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f"(x) < 0" /></li>
</ul>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika <img alt="f'(x_1) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1) = 0" /> dan <img alt="f'(x_1) < 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29+%3C+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1) < 0" />, maka <img alt="f'(x_1)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1)" /> adalah nilai balik maksimum dari fungsi y = f(x) dan titik <img alt="(x_1 f(x))" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%28x_1+f%28x%29%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="(x_1 f(x))" /> adalah titik balik maksimum dari kurva y = f(x).</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Nilai minimum <img alt="\rightarrow f'(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%27%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f'(x) = 0" /> dan <img alt="f"(x) > 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%22%28x%29+%3E+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f"(x) > 0" /></li>
</ul>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika <img alt="f'(x_1) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1) = 0" /> dan <img alt="f'(x_1) > 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29+%3E+0+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1) > 0" />, maka <img alt="f(x_1)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x_1%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x_1)" /> adalah nilai balik minimum dari fungsi <img alt="y = f(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y+%3D+f%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y = f(x)" /> dan titik <img alt="(x_1f(x))" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%28x_1f%28x%29%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="(x_1f(x))" /> adalah titik balik minimum dari kurva y = f(x).</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Nilai belok <img alt="\rightarrow f'(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+f%27%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow f'(x) = 0" /> dan <img alt="f"(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%22%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f"(x) = 0" /></li>
</ul>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika <img alt="f'(x_1) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x_1%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x_1) = 0" /> dan <img alt="f''(x_1 = 0)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%27%28x_1+%3D+0%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f''(x_1 = 0)" />, maka <img alt="f(x_1)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x_1%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x_1)" /> adalah nilai belok dari fungsi y = f(x) dan titik <img alt="(x_1f(x))" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%28x_1f%28x%29%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="(x_1f(x))" /> adalah titik belok dari kurva y = f(x).</div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
4. Menyelesaikan soal limit berbentuk tak tentu <img alt="\frac{0}{0}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B0%7D%7B0%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{0}{0}" /> atau <img alt="\frac{\infty}{\infty}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Cinfty%7D%7B%5Cinfty%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{\infty}{\infty}" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika <img alt="\lim \limits_{x\to a}\frac{f(x)}{g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+a%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to a}\frac{f(x)}{g(x)}" /> merupakan limit berbentuk tak tentu <img alt="\frac{0}{0}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B0%7D%7B0%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{0}{0}" /> atau <img alt="\frac{\infty}{\infty}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Cinfty%7D%7B%5Cinfty%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{\infty}{\infty}" />, maka penyelesaiannya dapat menggunakan turunan, yaitu f(x) dan g(x) masing-masing diturunkan.</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim\limits_{x\to a}\frac{f(x)}{g(x)} =\lim\limits_{x\to a}\frac{f'(x)}{g'(x)} = \frac{f'(a)}{g'(a)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim%5Climits_%7Bx%5Cto+a%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D+%3D%5Clim%5Climits_%7Bx%5Cto+a%7D%5Cfrac%7Bf%27%28x%29%7D%7Bg%27%28x%29%7D+%3D+%5Cfrac%7Bf%27%28a%29%7D%7Bg%27%28a%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim\limits_{x\to a}\frac{f(x)}{g(x)} =\lim\limits_{x\to a}\frac{f'(x)}{g'(x)} = \frac{f'(a)}{g'(a)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Jika dengan turunan pertama sudah dihasilkan bentuk tertentu, maka bentuk tertentu itu adalah penyelesaiannya. Tetapi jika dengan turunan pertama masih dihasilkan bentuk tak tentu, maka masing-masing f(x) dan f(x) diturunkan lagi sampai diperoleh hasil berbentuk tertentu. Cara penyelesaian seperti ini disebut Dalil L’hopital.</div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
5. Menentukan rumus kecepatan dan percepatan</h3>
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Jika rumus atau persamaan posisi gerak suatu benda sebagai fungsi waktu diketahui yaitu s = f(t), maka rumus kecepatan dan kecepatannya dapat ditentukan yaitu:</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Rumus kecepatan <img alt="\rightarrow v = s' = f'(t)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+v+%3D+s%27+%3D+f%27%28t%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow v = s' = f'(t)" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Rumus percepatan <img alt="\rightarrow a = s' = f"(t)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Crightarrow+a+%3D+s%27+%3D+f%22%28t%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\rightarrow a = s' = f"(t)" /></li>
</ul>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal Turunan Fungsi dan Pembahasan</h2>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal 1 – Turunan Fungsi Aljabar</h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Turunan pertama dari <img alt="f(x) = 4 \sqrt{2x^3 - 1}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+4+%5Csqrt%7B2x%5E3+-+1%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = 4 \sqrt{2x^3 - 1}" /> adalah</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pembahasan 1:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Soal ini merupakan fungsi yang berbentuk y = <img alt="au^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=au%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="au^n" /> yang dapat diselesaikan dengan menggunakan rumus <img alt="y' = n \cdot a \cdot u^{n-1} \cdot u'" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y%27+%3D+n+%5Ccdot+a+%5Ccdot+u%5E%7Bn-1%7D+%5Ccdot+u%27&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y' = n \cdot a \cdot u^{n-1} \cdot u'" />. Maka:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f(x) = 4 \sqrt{2x^3-1} = 4(2x^3-1)^{\frac{1}{2}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+4+%5Csqrt%7B2x%5E3-1%7D+%3D+4%282x%5E3-1%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = 4 \sqrt{2x^3-1} = 4(2x^3-1)^{\frac{1}{2}}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Sehingga turunannya:</div>
<div class="code-block code-block-6" style="background-color: #f9f9f9; border: 0px; clear: both; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 8px auto; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f'(x) = \frac{1}{2} \cdot 4(2x^3-1)^{-\frac{1}{2}} \cdot 6x^2" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7B1%7D%7B2%7D+%5Ccdot+4%282x%5E3-1%29%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D+%5Ccdot+6x%5E2&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{1}{2} \cdot 4(2x^3-1)^{-\frac{1}{2}} \cdot 6x^2" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=2(2x^3-1) \cdot 6x^2" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D2%282x%5E3-1%29+%5Ccdot+6x%5E2&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=2(2x^3-1) \cdot 6x^2" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= 12x^2(2x^3-1)^{-\frac{1}{2}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+12x%5E2%282x%5E3-1%29%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= 12x^2(2x^3-1)^{-\frac{1}{2}}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \frac{12x^2}{(2x^3-1)^{\frac{1}{2}}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Cfrac%7B12x%5E2%7D%7B%282x%5E3-1%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \frac{12x^2}{(2x^3-1)^{\frac{1}{2}}}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\frac{12^2}{\sqrt{2x^3-1}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Cfrac%7B12%5E2%7D%7B%5Csqrt%7B2x%5E3-1%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\frac{12^2}{\sqrt{2x^3-1}}" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal 2 – Turunan Fungsi Trigonometri</h3>
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Tentukan turunan pertama dari</div>
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<img alt="f(x) = \frac{6}{\sqrt[3]{\sin (3x-\frac{\pi}{5})}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%5Cfrac%7B6%7D%7B%5Csqrt%5B3%5D%7B%5Csin+%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = \frac{6}{\sqrt[3]{\sin (3x-\frac{\pi}{5})}}" /></div>
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Pembahasan 2:</div>
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Untuk menyelesaikan soal ini menggunakan rumus campuran yaitu <img alt="f'(x) = \frac{u'v-uv'}{v^2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7Bu%27v-uv%27%7D%7Bv%5E2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{u'v-uv'}{v^2}" /> dan juga <img alt="y' = n \cdot u' \sin^{n-1}u \cdot \cos u" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=y%27+%3D+n+%5Ccdot+u%27+%5Csin%5E%7Bn-1%7Du+%5Ccdot+%5Ccos+u+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="y' = n \cdot u' \sin^{n-1}u \cdot \cos u" />. Sehingga:</div>
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<img alt="f(x) = \frac{6}{\sqrt[3]{sin(3x-\frac{\pi}{5})}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%5Cfrac%7B6%7D%7B%5Csqrt%5B3%5D%7Bsin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = \frac{6}{\sqrt[3]{sin(3x-\frac{\pi}{5})}}" /></div>
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<img alt="f(x) = \frac{6}{(sin(3x-\frac{\pi}{5}))^{\frac{1}{2}}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+%5Cfrac%7B6%7D%7B%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = \frac{6}{(sin(3x-\frac{\pi}{5}))^{\frac{1}{2}}}" /></div>
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<img alt="f'(x) = \frac{0 - 6 \cdot 3 \cdot \frac{1}{3}(\sin (3x - \frac{\pi}{5}))^{- \frac{2}{3}} \cdot \cos (3x - \frac{\pi}{5})}{(\sin (3x - \frac{\pi}{5}))^\frac{2}{3}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7B0+-+6+%5Ccdot+3+%5Ccdot+%5Cfrac%7B1%7D%7B3%7D%28%5Csin+%283x+-+%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B-+%5Cfrac%7B2%7D%7B3%7D%7D+%5Ccdot+%5Ccos+%283x+-+%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7B%28%5Csin+%283x+-+%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%5Cfrac%7B2%7D%7B3%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{0 - 6 \cdot 3 \cdot \frac{1}{3}(\sin (3x - \frac{\pi}{5}))^{- \frac{2}{3}} \cdot \cos (3x - \frac{\pi}{5})}{(\sin (3x - \frac{\pi}{5}))^\frac{2}{3}}" /></div>
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<img alt="f'(x) = \frac{-6(sin(3x-\frac{\pi}{5}))^{-\frac{2}{3}}.cos(3x-\frac{\pi}{5})}{(sin(3x-\frac{\pi}{5}))^{\frac{2}{3}}}. \frac{(sin(3x-\frac{\pi}{5}))^{\frac{1}{3}}}{(sin(3x-\frac{\pi}{5}))^{-\frac{1}{3}}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7B-6%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B-%5Cfrac%7B2%7D%7B3%7D%7D.cos%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7B%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D.+%5Cfrac%7B%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D%7D%7B%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{-6(sin(3x-\frac{\pi}{5}))^{-\frac{2}{3}}.cos(3x-\frac{\pi}{5})}{(sin(3x-\frac{\pi}{5}))^{\frac{2}{3}}}. \frac{(sin(3x-\frac{\pi}{5}))^{\frac{1}{3}}}{(sin(3x-\frac{\pi}{5}))^{-\frac{1}{3}}}" /></div>
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<img alt="f'(x) = \frac{-6(sin(3x-\frac{\pi}{5}))^{-1} cos(3x-\frac{\pi}{5})}{\sqrt[3]{sin(3x-\frac{\pi}{5}})}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7B-6%28sin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%29%5E%7B-1%7D+cos%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7B%5Csqrt%5B3%5D%7Bsin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%7D%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{-6(sin(3x-\frac{\pi}{5}))^{-1} cos(3x-\frac{\pi}{5})}{\sqrt[3]{sin(3x-\frac{\pi}{5}})}" /></div>
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<img alt="f'(x) = \frac{-6cot(3x-\frac{\pi}{5})}{\sqrt[3]{sin(3x-\frac{\pi}{5})}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+%5Cfrac%7B-6cot%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7B%5Csqrt%5B3%5D%7Bsin%283x-%5Cfrac%7B%5Cpi%7D%7B5%7D%29%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = \frac{-6cot(3x-\frac{\pi}{5})}{\sqrt[3]{sin(3x-\frac{\pi}{5})}}" /></div>
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Contoh Soal 3 – Aplikasi Turunan</h3>
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Tentukan nilai maksimum dari <img alt="f(x) = x^3 - 6x^2 + 9x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%28x%29+%3D+x%5E3+-+6x%5E2+%2B+9x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f(x) = x^3 - 6x^2 + 9x" /> pada interval -1 ≤ x ≤ 3.</div>
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Pembahasan 3:</div>
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Ingat syarat nilai fungsi f(x) maksimum adalah <img alt="f'(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = 0" /> dan <img alt="f"(x) < 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%22%28x%29+%3C+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f"(x) < 0" /> maka:</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="f_{max}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f_%7Bmax%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f_{max}" /> jika <img alt="f'(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f%27%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f'(x) = 0" /></li>
</ul>
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<img alt="3x^2 - 12x + 9 = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=3x%5E2+-+12x+%2B+9+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="3x^2 - 12x + 9 = 0" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="x^2 - 4x + 3 = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x%5E2+-+4x+%2B+3+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x^2 - 4x + 3 = 0" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="(x - 1)(x - 3) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%28x+-+1%29%28x+-+3%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="(x - 1)(x - 3) = 0" /></div>
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dan <img alt="x_1 = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x_1+%3D+1+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x_1 = 1" /> dan <img alt="x_2 = 3" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x_2+%3D+3&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x_2 = 3" /></div>
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<img alt="f_{max} = f(1) = 1^3 - 6.1^2 + 9.1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f_%7Bmax%7D+%3D+f%281%29+%3D+1%5E3+-+6.1%5E2+%2B+9.1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f_{max} = f(1) = 1^3 - 6.1^2 + 9.1" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="f_{max} = 4" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=f_%7Bmax%7D+%3D+4&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="f_{max} = 4" /></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-72744744334415169952019-09-12T20:50:00.002-07:002019-09-12T20:50:22.569-07:00Limit Fungsi<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pengertian Limit Fungsi</h2>
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Limit merupakan sebuah konsep <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">matematika</span> dimana sesuatu dikatakan “hampir” atau “mendekati” nilai suatu bilangan tertentu. Limit dapat berupa sebuah fungsi yang kodomainnya “hampir” atau “mendekati” nilai suatu bilangan asli tertentu.</div>
<div class="wp-caption aligncenter" style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px auto; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: 310px; word-break: normal;">
</div>
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</div>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Limit Fungsi Aljabar</h2>
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Dalam pengoperasian limit fungsi aljabar, terdapat beberapa hukum atau teorema limit yang perlu diperhatikan. Jika k konstanta, <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">fungsi</span> f dan fungsi g adalah fungsi-fungsi memiliki nilai limit yang mendekati bilangan c, maka:</div>
<table style="background-color: #f9f9f9; border-collapse: collapse; border-spacing: 0px; border: 0px; color: black; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: auto; word-break: normal;"><tbody style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">No</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331">TEOREMA</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">1</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}k=k" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7Dk%3Dk&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}k=k" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">2</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}x = c" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7Dx+%3D+c&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}x = c" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">3</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="k \cdot \lim \limits_{x\to c}f(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=k+%5Ccdot+%5Clim+%5Climits_%7Bx%5Cto+c%7Df%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="k \cdot \lim \limits_{x\to c}f(x)" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">4</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x \to c}({f(x) + g(x)})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx+%5Cto+c%7D%28%7Bf%28x%29+%2B+g%28x%29%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x \to c}({f(x) + g(x)})" /> <img alt="= \lim_{x\to c}f(x) + \lim_{x\to c}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim_%7Bx%5Cto+c%7Df%28x%29+%2B+%5Clim_%7Bx%5Cto+c%7Dg%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim_{x\to c}f(x) + \lim_{x\to c}g(x)" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">5</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_x\to c({f(x) - g(x)}) = \lim_x\to c{f(x) - g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_x%5Cto+c%28%7Bf%28x%29+-+g%28x%29%7D%29+%3D+%5Clim_x%5Cto+c%7Bf%28x%29+-+g%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_x\to c({f(x) - g(x)}) = \lim_x\to c{f(x) - g(x)}" /> <img alt="= \lim_x{\to c}f(x) - \lim_{x \to c}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim_x%7B%5Cto+c%7Df%28x%29+-+%5Clim_%7Bx+%5Cto+c%7Dg%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim_x{\to c}f(x) - \lim_{x \to c}g(x)" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">6</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}({f(x) \cdot g(x)}) = \lim_{x\to c}f(x) \cdot \lim_{x\to c}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7D%28%7Bf%28x%29+%5Ccdot+g%28x%29%7D%29+%3D+%5Clim_%7Bx%5Cto+c%7Df%28x%29+%5Ccdot+%5Clim_%7Bx%5Cto+c%7Dg%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}({f(x) \cdot g(x)}) = \lim_{x\to c}f(x) \cdot \lim_{x\to c}g(x)" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">7</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"> <img alt="\lim \limits_{x\to c}\frac{f(x)}{g(x)} = \frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D+%3D+%5Cfrac%7B%5Clim_%7Bx%5Cto+c%7Df%28x%29%7D%7B%5Clim_%7Bx%5Cto+c%7Dg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}\frac{f(x)}{g(x)} = \frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">8</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}(f(x)^n) = (\lim_{x\to c}f(x))^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7D%28f%28x%29%5En%29+%3D+%28%5Clim_%7Bx%5Cto+c%7Df%28x%29%29%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}(f(x)^n) = (\lim_{x\to c}f(x))^n" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">9</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}(\sqrt[n]{f(x)}) = \sqrt[n]{\lim_{x\to c}f(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7D%28%5Csqrt%5Bn%5D%7Bf%28x%29%7D%29+%3D+%5Csqrt%5Bn%5D%7B%5Clim_%7Bx%5Cto+c%7Df%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}(\sqrt[n]{f(x)}) = \sqrt[n]{\lim_{x\to c}f(x)}" /></td></tr>
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Ada tiga metode dalam mengerjakan limit fungsi aljabar, yaitu:</div>
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1. Metode substitusi</h3>
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Metode paling mudah dengan menentukan hasil suatu limit fungsi adalah dengan mensubstitusi langsung nilai kedalam fungsi f(x). Syarat metode ini adalah jika hasil substitusi tidak membentuk nilai “tak tentu”. Contoh:</div>
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<img alt="\lim \limits_{x\to 3}\frac{x^2 - 4}{x + 2} = \frac{9 - 4}{3 + 2} = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+3%7D%5Cfrac%7Bx%5E2+-+4%7D%7Bx+%2B+2%7D+%3D+%5Cfrac%7B9+-+4%7D%7B3+%2B+2%7D+%3D+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 3}\frac{x^2 - 4}{x + 2} = \frac{9 - 4}{3 + 2} = 1" /></div>
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2. Metode pemfaktoran</h3>
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Jika pada metode substitusi menghasilkan suatu nilai bentuk tak tentu seperti:</div>
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∞, <img alt="\frac{0}{0}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B0%7D%7B0%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{0}{0}" />, <img alt="\frac{\infty}{\infty}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Cinfty%7D%7B%5Cinfty%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{\infty}{\infty}" />, 0 x∞, ∞ – ∞, 0<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">0</span>, ∞<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">0</span>, atau ∞<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">∞</span></div>
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maka fungsi tersebut harus difaktorkan terlebih dahulu sehingga bentuknya tidak menjadi bentuk tak tentu, baru kemudian bisa disubstitusikan <img alt="x\to c" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x%5Cto+c&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x\to c" />. Contoh:</div>
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<img alt="\lim \limits_{x\to 3}\frac{x^2 - 3x}{2x - 6} = \frac{x(x - 3)}{2(x - 3)} = \frac{3}{2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+3%7D%5Cfrac%7Bx%5E2+-+3x%7D%7B2x+-+6%7D+%3D+%5Cfrac%7Bx%28x+-+3%29%7D%7B2%28x+-+3%29%7D+%3D+%5Cfrac%7B3%7D%7B2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 3}\frac{x^2 - 3x}{2x - 6} = \frac{x(x - 3)}{2(x - 3)} = \frac{3}{2}" /></div>
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3. Metode perkalian dengan akar sekawan</h3>
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Metode ini digunakan jika pada metode substitusi langsung menghasilkan nilai limit yang irasional. Fungsi dikalikan dengan akar sekawannya agar bentuk limit tersebut tidak irasional, sehingga bisa dilakukan kembali substitusi langsung nilai <img alt="x\to c" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x%5Cto+c&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x\to c" />. Contoh:</div>
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<img alt="\lim \limits_{x\to -1}\frac{x +1}{1 - \sqrt{x + 2}} x (\frac{1 + \sqrt{x +2}}{1 + \sqrt{x + 2}}) = \frac{(x + 1)(1 + \sqrt{x+ 2})}{1 - (x + 2)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+-1%7D%5Cfrac%7Bx+%2B1%7D%7B1+-+%5Csqrt%7Bx+%2B+2%7D%7D+x+%28%5Cfrac%7B1+%2B+%5Csqrt%7Bx+%2B2%7D%7D%7B1+%2B+%5Csqrt%7Bx+%2B+2%7D%7D%29+%3D+%5Cfrac%7B%28x+%2B+1%29%281+%2B+%5Csqrt%7Bx%2B+2%7D%29%7D%7B1+-+%28x+%2B+2%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to -1}\frac{x +1}{1 - \sqrt{x + 2}} x (\frac{1 + \sqrt{x +2}}{1 + \sqrt{x + 2}}) = \frac{(x + 1)(1 + \sqrt{x+ 2})}{1 - (x + 2)}" /></div>
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<img alt="=\frac{(x + 1)(1+\sqrt{x+2})}{-x - 1} = \frac{(x+1)(1+\sqrt{x+2})}{-(x+1)} = -(1 + \sqrt{x + 2})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Cfrac%7B%28x+%2B+1%29%281%2B%5Csqrt%7Bx%2B2%7D%29%7D%7B-x+-+1%7D+%3D+%5Cfrac%7B%28x%2B1%29%281%2B%5Csqrt%7Bx%2B2%7D%29%7D%7B-%28x%2B1%29%7D+%3D+-%281+%2B+%5Csqrt%7Bx+%2B+2%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\frac{(x + 1)(1+\sqrt{x+2})}{-x - 1} = \frac{(x+1)(1+\sqrt{x+2})}{-(x+1)} = -(1 + \sqrt{x + 2})" /></div>
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<img alt="=-(1 + \sqrt{-(1) + 2}) = -(1 + 1) = -2" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D-%281+%2B+%5Csqrt%7B-%281%29+%2B+2%7D%29+%3D+-%281+%2B+1%29+%3D+-2&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=-(1 + \sqrt{-(1) + 2}) = -(1 + 1) = -2" /></div>
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Dalam pengoprasian limit fungsi aljabar, terkadang nilai x mendekati <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">tak berhingga</span> (∞), sehingga jika disubstitusikan fungsi menghasilkan nilai tak tentu. Dalam pengoperasian limitnya, terdapat beberapa hukum atau teorema limit yang perlu diperhatikan. Jika n adalah bilangan bulat, k konstanta, fungsi f dan fungsi g adalah fungsi-fungsi memiliki nilai limit yang mendekati bilangan c, maka:</div>
<table style="background-color: #f9f9f9; border-collapse: collapse; border-spacing: 0px; border: 0px; color: black; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: auto; word-break: normal;"><tbody style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">No</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331">TEOREMA</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156">SYARAT</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">1</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}k = k" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dk+%3D+k&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}k = k" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156">k adalah konstanta</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td rowspan="2" style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">2</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}x^n = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dx%5En+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}x^n = \infty" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}\frac{1}{x^n} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5En%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{1}{x^n} = 0" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td rowspan="2" style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">3</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to -\infty}\frac{1}{x^n} = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+-%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5En%7D+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to -\infty}\frac{1}{x^n} = \infty" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156">Jika n = genap</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to - \infty}\frac{1}{x^n}= - \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+-+%5Cinfty%7D%5Cfrac%7B1%7D%7Bx%5En%7D%3D+-+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to - \infty}\frac{1}{x^n}= - \infty" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156">Jika n = ganjil</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">4</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}k \cdot f(x) = k \cdot \lim \limits_{x\to \infty}f(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dk+%5Ccdot+f%28x%29+%3D+k+%5Ccdot+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}k \cdot f(x) = k \cdot \lim \limits_{x\to \infty}f(x)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156">k adalah konstanta</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">5</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}({f(x) + g(x)}) = \lim \limits_{x\to \infty} f(x) + \lim \limits_{x\to \infty}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%7Bf%28x%29+%2B+g%28x%29%7D%29+%3D+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D+f%28x%29+%2B+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}({f(x) + g(x)}) = \lim \limits_{x\to \infty} f(x) + \lim \limits_{x\to \infty}g(x)" /> <img alt="\lim \limits_{x\to \infty}{f(x) + g(x)} = \lim \limits_{x\to \infty} f(x) + \lim \limits_{x\to \infty}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%7Bf%28x%29+%2B+g%28x%29%7D+%3D+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D+f%28x%29+%2B+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}{f(x) + g(x)} = \lim \limits_{x\to \infty} f(x) + \lim \limits_{x\to \infty}g(x)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">6</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}({f(x) - g(x)})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%7Bf%28x%29+-+g%28x%29%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}({f(x) - g(x)})" /> <img alt="= \lim \limits_{x\to \infty}f(x) - \lim \limits_{x\to \infty}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29+-+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to \infty}f(x) - \lim \limits_{x\to \infty}g(x)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">7</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}({f(x) \cdot g(x)}) = \lim \limits_{x\to \infty}f(x) \cdot \lim \limits_{x\to \infty} g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%7Bf%28x%29+%5Ccdot+g%28x%29%7D%29+%3D+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29+%5Ccdot+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D+g%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}({f(x) \cdot g(x)}) = \lim \limits_{x\to \infty}f(x) \cdot \lim \limits_{x\to \infty} g(x)" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">8</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}\frac{f(x)}{g(x)} = \frac{\lim \limits_{x\to \infty}f(x)}{\lim \limits_{x\to \infty}g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D+%3D+%5Cfrac%7B%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29%7D%7B%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{f(x)}{g(x)} = \frac{\lim \limits_{x\to \infty}f(x)}{\lim \limits_{x\to \infty}g(x)}" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">9</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}(f(x)^n) = (\lim \limits_{x\to c} f(x))^n" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=+%5Clim+%5Climits_%7Bx%5Cto+c%7D%28f%28x%29%5En%29+%3D+%28%5Clim+%5Climits_%7Bx%5Cto+c%7D+f%28x%29%29%5En&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}(f(x)^n) = (\lim \limits_{x\to c} f(x))^n" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">10</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to c}(\sqrt[n]{f(x)}) = \sqrt[n]{\lim \limits_{x\to c}f(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+c%7D%28%5Csqrt%5Bn%5D%7Bf%28x%29%7D%29+%3D+%5Csqrt%5Bn%5D%7B%5Clim+%5Climits_%7Bx%5Cto+c%7Df%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to c}(\sqrt[n]{f(x)}) = \sqrt[n]{\lim \limits_{x\to c}f(x)}" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td rowspan="2" style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="34">11</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"> <img alt="\lim \limits_{x\to \infty}\frac{1}{f(x)} = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7B1%7D%7Bf%28x%29%7D+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{1}{f(x)} = \infty" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"><img alt="\lim \limits_{x\to \infty}f(x) = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}f(x) = 0" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="331"><img alt="\lim \limits_{x\to \infty}\frac{1}{f(x)} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7B1%7D%7Bf%28x%29%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{1}{f(x)} = 0" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="156"><img alt="\lim \limits_{x\to \infty}f(x) = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}f(x) = \infty" /></td></tr>
</tbody></table>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<br /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Ada dua metode dalam mengerjakan limit fungsi aljabar bentuk tak berhingga:</div>
<ol style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style-image: initial; list-style-position: outside; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Membagi dengan pangkat tertinggi</li>
</ol>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Metode ini digunakan pada limit fungsi bentuk <img alt="\lim \limits_{x\to \infty}\frac{f(x)}{g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{f(x)}{g(x)}" />. Metode ini dapat dikerjakan dengan membagi pembilang f(x) dan penyebut g(x) dengan variabel x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">n</span> berpangkat tertinggi yang ada dalam fungsi f(x) dan g(x). Setelahnya, baru dapat disubstitusi dengan <img alt="x\to \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x%5Cto+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x\to \infty" />. Contoh:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{x\to \infty}\frac{4x-1}{x^2+2} = \frac{\frac{4x}{x^2} - \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{2}{x^2}} = \frac{\frac{4}{x} - \frac{1}{x^2}}{1+\frac{2}{x^2}} = \frac{0}{1} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%5Cfrac%7B4x-1%7D%7Bx%5E2%2B2%7D+%3D+%5Cfrac%7B%5Cfrac%7B4x%7D%7Bx%5E2%7D+-+%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7B%5Cfrac%7Bx%5E2%7D%7Bx%5E2%7D+%2B+%5Cfrac%7B2%7D%7Bx%5E2%7D%7D+%3D+%5Cfrac%7B%5Cfrac%7B4%7D%7Bx%7D+-+%5Cfrac%7B1%7D%7Bx%5E2%7D%7D%7B1%2B%5Cfrac%7B2%7D%7Bx%5E2%7D%7D+%3D+%5Cfrac%7B0%7D%7B1%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}\frac{4x-1}{x^2+2} = \frac{\frac{4x}{x^2} - \frac{1}{x^2}}{\frac{x^2}{x^2} + \frac{2}{x^2}} = \frac{\frac{4}{x} - \frac{1}{x^2}}{1+\frac{2}{x^2}} = \frac{0}{1} = 0" /></div>
<ol start="2" style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style-image: initial; list-style-position: outside; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">Mengalikan bentuk sekawan</li>
</ol>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Metode ini digunakan pada limit fungsi bentuk <img alt="\lim \limits_{x\to \infty}f(x) - \lim \limits_{x\to \infty}g(x)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29+-+%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}f(x) - \lim \limits_{x\to \infty}g(x)" />. Metode ini dapat diselesaikan dengan perkalian bentuk sekawan:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\frac{\lim \limits_{x\to \infty}f(x)+\lim \limits_{x\to \infty}g(x)}{\lim \limits_{x\to \infty}f(x)+\lim \limits_{x \to \infty}g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29%2B%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Dg%28x%29%7D%7B%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7Df%28x%29%2B%5Clim+%5Climits_%7Bx+%5Cto+%5Cinfty%7Dg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{\lim \limits_{x\to \infty}f(x)+\lim \limits_{x\to \infty}g(x)}{\lim \limits_{x\to \infty}f(x)+\lim \limits_{x \to \infty}g(x)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
kemudian dilanjutkan pembagian dengan metode pertama yaitu membagi dengan pangkat tertinggi. Contoh:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{x\to \infty}(\sqrt{x^2 + 4x - 5 - \sqrt{x^2 -x -2}})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%5Csqrt%7Bx%5E2+%2B+4x+-+5+-+%5Csqrt%7Bx%5E2+-x+-2%7D%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}(\sqrt{x^2 + 4x - 5 - \sqrt{x^2 -x -2}})" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\lim \limits_{n\to \infty}(\sqrt{x^2 + 4x -5 - \sqrt{x^2 - x - 2}})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Clim+%5Climits_%7Bn%5Cto+%5Cinfty%7D%28%5Csqrt%7Bx%5E2+%2B+4x+-5+-+%5Csqrt%7Bx%5E2+-+x+-+2%7D%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\lim \limits_{n\to \infty}(\sqrt{x^2 + 4x -5 - \sqrt{x^2 - x - 2}})" /> <img alt="x \frac{(\sqrt{x^2+4x-5 + \sqrt{x^2 - x - 2}})}{(\sqrt{x^2+4x-5} + \sqrt{x^2-x-2})}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=x+%5Cfrac%7B%28%5Csqrt%7Bx%5E2%2B4x-5+%2B+%5Csqrt%7Bx%5E2+-+x+-+2%7D%7D%29%7D%7B%28%5Csqrt%7Bx%5E2%2B4x-5%7D+%2B+%5Csqrt%7Bx%5E2-x-2%7D%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="x \frac{(\sqrt{x^2+4x-5 + \sqrt{x^2 - x - 2}})}{(\sqrt{x^2+4x-5} + \sqrt{x^2-x-2})}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{n\to \infty}\frac{((x^2+4x-5) - (x^2-x-2))}{(\sqrt{x^2+4x-5} + \sqrt{x^2-x-2})}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bn%5Cto+%5Cinfty%7D%5Cfrac%7B%28%28x%5E2%2B4x-5%29+-+%28x%5E2-x-2%29%29%7D%7B%28%5Csqrt%7Bx%5E2%2B4x-5%7D+%2B+%5Csqrt%7Bx%5E2-x-2%7D%29%7D+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{n\to \infty}\frac{((x^2+4x-5) - (x^2-x-2))}{(\sqrt{x^2+4x-5} + \sqrt{x^2-x-2})}" /> <img alt="= \lim \limits_{n\to \infty}frac{5x-3}{(\sqrt{x^2+4x-5})+\sqrt{x^2-x-2}}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bn%5Cto+%5Cinfty%7Dfrac%7B5x-3%7D%7B%28%5Csqrt%7Bx%5E2%2B4x-5%7D%29%2B%5Csqrt%7Bx%5E2-x-2%7D%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{n\to \infty}frac{5x-3}{(\sqrt{x^2+4x-5})+\sqrt{x^2-x-2}}" /></div>
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Kemudian pembilang dan penyebut dibagi x pangkat tertinggi yaitu x<span style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">1</span>:</div>
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<img alt="\lim \limits_{n\to \infty}\frac{\frac{5x}{x}-\frac{3}{x}}{(\sqrt{\frac{x^2+4x-5}{x^2}}+\sqrt{\frac{x^2-x-2}{x^2}})}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bn%5Cto+%5Cinfty%7D%5Cfrac%7B%5Cfrac%7B5x%7D%7Bx%7D-%5Cfrac%7B3%7D%7Bx%7D%7D%7B%28%5Csqrt%7B%5Cfrac%7Bx%5E2%2B4x-5%7D%7Bx%5E2%7D%7D%2B%5Csqrt%7B%5Cfrac%7Bx%5E2-x-2%7D%7Bx%5E2%7D%7D%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{n\to \infty}\frac{\frac{5x}{x}-\frac{3}{x}}{(\sqrt{\frac{x^2+4x-5}{x^2}}+\sqrt{\frac{x^2-x-2}{x^2}})}" /> <img alt="= \lim \limits_{n\to \infty}\frac{5-\frac{3}{x}}{(\sqrt{1+\frac{4}{x}-\frac{5}{x^2}}+ \sqrt{1-\frac{1}{x}-\frac{2}{x^2}})}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bn%5Cto+%5Cinfty%7D%5Cfrac%7B5-%5Cfrac%7B3%7D%7Bx%7D%7D%7B%28%5Csqrt%7B1%2B%5Cfrac%7B4%7D%7Bx%7D-%5Cfrac%7B5%7D%7Bx%5E2%7D%7D%2B+%5Csqrt%7B1-%5Cfrac%7B1%7D%7Bx%7D-%5Cfrac%7B2%7D%7Bx%5E2%7D%7D%29%7D+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{n\to \infty}\frac{5-\frac{3}{x}}{(\sqrt{1+\frac{4}{x}-\frac{5}{x^2}}+ \sqrt{1-\frac{1}{x}-\frac{2}{x^2}})}" /></div>
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<img alt="= \frac{(-0}{(1+1)} = \frac{5}{2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Cfrac%7B%28-0%7D%7B%281%2B1%29%7D+%3D+%5Cfrac%7B5%7D%7B2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \frac{(-0}{(1+1)} = \frac{5}{2}" /></div>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Limit Fungsi Trigonometri</h2>
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Limit juga dapat digunakan pada fungsi <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">trigonometri</span>. Penyelesaiannya sama dengan fungsi limit aljabar. Namun, agar mengerti penjalasan selanjutnya harus mengerti terlebih dahulu konsep dari trigonometri. Penyelesaian dalam limit fungsi ini dalam trigonometri bisa dilakukan dengan melakukan perubahan-perubahan bentuk sinus, cosinus, dan tangen.</div>
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Ada tiga bentuk umum dalam limit fungsi trigonometri, yaitu bentuk :</div>
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1. Bentuk <img alt="\lim_{x\to c}f(x) = f(c)" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto+c%7Df%28x%29+%3D+f%28c%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim_{x\to c}f(x) = f(c)" /></h3>
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Pada bentuk ini, limit dari fungsi trigonometri f(x) merupakan hasil dari substitusi nilai c ke dalam x dari trigonometri. Contoh :</div>
<table style="background-color: #f9f9f9; border-collapse: collapse; border-spacing: 0px; border: 0px; color: black; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; width: auto; word-break: normal;"><tbody style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="49">No.</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="126">CONTOH</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="132">NILAI LIMIT</td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="49">1</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="126"><img alt="\lim \limits_{x\to ^\pi/_4} \sin 2x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5E%5Cpi%2F_4%7D+%5Csin+2x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to ^\pi/_4} \sin 2x" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="132"> <img alt="\sin \frac{\pi}{2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Csin+%5Cfrac%7B%5Cpi%7D%7B2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\sin \frac{\pi}{2}" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="49">2</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="126"><img alt="\lim \limits_{x\to \pi} \cos \frac{1}{2}x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cpi%7D+%5Ccos+%5Cfrac%7B1%7D%7B2%7Dx&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \pi} \cos \frac{1}{2}x" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="132"><img alt="cos\frac{1}{2}\pi" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=cos%5Cfrac%7B1%7D%7B2%7D%5Cpi&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="cos\frac{1}{2}\pi" /></td></tr>
<tr style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="49">3</td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="126"><img alt="\lim \limits_{x\to ^\pi/_2} \tan x" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5E%5Cpi%2F_2%7D+%5Ctan+x&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to ^\pi/_2} \tan x" /></td><td style="border-collapse: collapse; border: 1px solid rgb(0, 0, 0); font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;" width="132"> <img alt="\tan\frac{\pi}{2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Ctan%5Cfrac%7B%5Cpi%7D%7B2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\tan\frac{\pi}{2}" /></td></tr>
</tbody></table>
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Jika c = 0, maka rumus limit-limit trigonometrinya adalah sebagai berikut :</div>
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<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0} \sin x = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D+%5Csin+x+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0} \sin x = 0" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0} \cos x = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D+%5Ccos+x+%3D+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0} \cos x = 1" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0} \cos x = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D+%5Ccos+x+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0} \cos x = 0" /></li>
</ul>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
2. Bentuk <img alt="\lim_{x\to c}\frac{f(x)}{g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto+c%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim_{x\to c}\frac{f(x)}{g(x)}" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pada bentuk ini, limit diperoleh dari perbandingan 2 trigonometri berbeda. Kedua trigonometri tersebut jika langsung disubstitusi dengan nilai c menghasilkan f(c) = 0 dan g(c) = 0. Sehingga, nilai limit trigonometri tersebut menjadi bilangan tak tentu<img alt="\frac{0}{0}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Cfrac%7B0%7D%7B0%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\frac{0}{0}" />. Penyelesaiannya sama dengan limit fungsi aljabar yaitu pemfaktoran. Contoh bentuk ini yaitu:</div>
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<img alt="\lim \limits_{x\to ^\pi/_2}\frac{\sin 2x}{cosx} = \frac{2 \cos x \sin x}{cosx} = 2 \sin x = 2 \sin \frac{\pi}{2} = 2" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5E%5Cpi%2F_2%7D%5Cfrac%7B%5Csin+2x%7D%7Bcosx%7D+%3D+%5Cfrac%7B2+%5Ccos+x+%5Csin+x%7D%7Bcosx%7D+%3D+2+%5Csin+x+%3D+2+%5Csin+%5Cfrac%7B%5Cpi%7D%7B2%7D+%3D+2&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to ^\pi/_2}\frac{\sin 2x}{cosx} = \frac{2 \cos x \sin x}{cosx} = 2 \sin x = 2 \sin \frac{\pi}{2} = 2" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
3. Bentuk <img alt="\lim_{x\to 0}\frac{f(x)}{g(x)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto+0%7D%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim_{x\to 0}\frac{f(x)}{g(x)}" /></h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pada bentuk ini, limit diperoleh dari perbandingan antara trigonometri dan fungsi aljabar. Jika disubstitusikan langsung akan menghaslikan bilangan tak tentu. Pada bentuk ini dikerjakan dengan konsep <span style="color: blue; font-family: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; word-break: normal;">turunan</span>. Bentuk rumus dasar limit ini adalah:</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{x}{\sin x} = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Csin+x%7D%7Bx%7D+%3D+%5Clim_%7Bx%5Cto+0%7D%5Cfrac%7Bx%7D%7B%5Csin+x%7D+%3D+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{x}{\sin x} = 1" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{x}{\cos x} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7Bx%7D%7B%5Ccos+x%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{x}{\cos x} = 0" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\cos x}{x} = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Ccos+x%7D%7Bx%7D+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\cos x}{x} = \infty" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\tan x}{x} = \lim_{x\to}\frac{x}{\tan x} = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Ctan+x%7D%7Bx%7D+%3D+%5Clim_%7Bx%5Cto%7D%5Cfrac%7Bx%7D%7B%5Ctan+x%7D+%3D+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\tan x}{x} = \lim_{x\to}\frac{x}{\tan x} = 1" /></li>
</ul>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Berdasarkan rumus dasar diataas, jika dikembangkan menjadi rumus-rumus berikut:</div>
<div class="code-block code-block-6" style="background-color: #f9f9f9; border: 0px; clear: both; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin: 8px auto; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
</div>
<ul style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; list-style: outside square; margin: 0px 0px 20px 30px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\sin ax}{bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Csin+ax%7D%7Bbx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\sin ax}{bx} = \frac{a}{b}" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{ax}{\sin bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7Bax%7D%7B%5Csin+bx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{ax}{\sin bx} = \frac{a}{b}" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{ax}{\cos bx} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7Bax%7D%7B%5Ccos+bx%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{ax}{\cos bx} = 0" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\cos ax}{bx} = \infty" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Ccos+ax%7D%7Bbx%7D+%3D+%5Cinfty&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\cos ax}{bx} = \infty" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\tan ax}{bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Ctan+ax%7D%7Bbx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\tan ax}{bx} = \frac{a}{b}" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{ax}{\tan bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7Bax%7D%7B%5Ctan+bx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{ax}{\tan bx} = \frac{a}{b}" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\sin ax}{\tan bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Csin+ax%7D%7B%5Ctan+bx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\sin ax}{\tan bx} = \frac{a}{b}" /></li>
<li style="border: 0px; font: inherit; margin: 0px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;"><img alt="\lim \limits_{x\to 0}\frac{\tan ax}{\sin bx} = \frac{a}{b}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%5Cfrac%7B%5Ctan+ax%7D%7B%5Csin+bx%7D+%3D+%5Cfrac%7Ba%7D%7Bb%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}\frac{\tan ax}{\sin bx} = \frac{a}{b}" /></li>
</ul>
<h2 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 23px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal Limit Fungsi dan Pembahasan</h2>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal Limit 1</h3>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Tentukanlah nilai dari <img alt="\lim_{x\to 2}(\frac{6-x}{x^2-4} - \frac{1}{x-2})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim_%7Bx%5Cto+2%7D%28%5Cfrac%7B6-x%7D%7Bx%5E2-4%7D+-+%5Cfrac%7B1%7D%7Bx-2%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim_{x\to 2}(\frac{6-x}{x^2-4} - \frac{1}{x-2})" /> (UAN 2002)</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pembahasan 1 :</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{x\to 2}(\frac{6-x}{x^2} - \frac{1}{x-2}) = \lim \limits_{x\to 2}(\frac{6-x}{x^2-4} - \frac{x+2}{(x-2)(x+2)})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+2%7D%28%5Cfrac%7B6-x%7D%7Bx%5E2%7D+-+%5Cfrac%7B1%7D%7Bx-2%7D%29+%3D+%5Clim+%5Climits_%7Bx%5Cto+2%7D%28%5Cfrac%7B6-x%7D%7Bx%5E2-4%7D+-+%5Cfrac%7Bx%2B2%7D%7B%28x-2%29%28x%2B2%29%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 2}(\frac{6-x}{x^2} - \frac{1}{x-2}) = \lim \limits_{x\to 2}(\frac{6-x}{x^2-4} - \frac{x+2}{(x-2)(x+2)})" /> <img alt="= \lim \limits_{x\to 2}\frac{(6-x) - (x+2)}{(x-2)(x+2)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+2%7D%5Cfrac%7B%286-x%29+-+%28x%2B2%29%7D%7B%28x-2%29%28x%2B2%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to 2}\frac{(6-x) - (x+2)}{(x-2)(x+2)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=\lim \limits_{x\to 2}\frac{4-2x}{(x-2)(x+2)} = \lim \limits_{x\to 2}\frac{-2(x-2)}{(x-2)(x+2)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%5Clim+%5Climits_%7Bx%5Cto+2%7D%5Cfrac%7B4-2x%7D%7B%28x-2%29%28x%2B2%29%7D+%3D+%5Clim+%5Climits_%7Bx%5Cto+2%7D%5Cfrac%7B-2%28x-2%29%7D%7B%28x-2%29%28x%2B2%29%7D+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=\lim \limits_{x\to 2}\frac{4-2x}{(x-2)(x+2)} = \lim \limits_{x\to 2}\frac{-2(x-2)}{(x-2)(x+2)}" /> <img alt="= \lim \limits_{x\to 2}\frac{-2}{(x+2)}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+2%7D%5Cfrac%7B-2%7D%7B%28x%2B2%29%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to 2}\frac{-2}{(x+2)}" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=-\frac{2}{4} = -\frac{1}{2}" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D-%5Cfrac%7B2%7D%7B4%7D+%3D+-%5Cfrac%7B1%7D%7B2%7D&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=-\frac{2}{4} = -\frac{1}{2}" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal Limit 2</h3>
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Tentukanlah nilai dari <img alt="\lim \limits_{x\to \infty}(\frac{\sqrt{4x+x^2} - \sqrt{2+x^2}}{3x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%5Cfrac%7B%5Csqrt%7B4x%2Bx%5E2%7D+-+%5Csqrt%7B2%2Bx%5E2%7D%7D%7B3x%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}(\frac{\sqrt{4x+x^2} - \sqrt{2+x^2}}{3x})" /> (UN 2009)</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Pembahasan 2:</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{x\to \infty}(\frac{\sqrt{4x+x^2}-\sqrt{2+x^2}}{3x}) = \lim_{x\to \infty}(\frac{\sqrt{\frac{4}{x^2}+\frac{x^2}{x^2}}-\sqrt{\frac{2}{x^2}+\frac{x^2}{x^2}}}{\frac{3x}{x}})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+%5Cinfty%7D%28%5Cfrac%7B%5Csqrt%7B4x%2Bx%5E2%7D-%5Csqrt%7B2%2Bx%5E2%7D%7D%7B3x%7D%29+%3D+%5Clim_%7Bx%5Cto+%5Cinfty%7D%28%5Cfrac%7B%5Csqrt%7B%5Cfrac%7B4%7D%7Bx%5E2%7D%2B%5Cfrac%7Bx%5E2%7D%7Bx%5E2%7D%7D-%5Csqrt%7B%5Cfrac%7B2%7D%7Bx%5E2%7D%2B%5Cfrac%7Bx%5E2%7D%7Bx%5E2%7D%7D%7D%7B%5Cfrac%7B3x%7D%7Bx%7D%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to \infty}(\frac{\sqrt{4x+x^2}-\sqrt{2+x^2}}{3x}) = \lim_{x\to \infty}(\frac{\sqrt{\frac{4}{x^2}+\frac{x^2}{x^2}}-\sqrt{\frac{2}{x^2}+\frac{x^2}{x^2}}}{\frac{3x}{x}})" /> <img alt="= \lim_{x\to \infty}(\frac{\sqrt{\frac{4}{x^2}+1}-\sqrt{\frac{2}{x^2}+1}}{3})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim_%7Bx%5Cto+%5Cinfty%7D%28%5Cfrac%7B%5Csqrt%7B%5Cfrac%7B4%7D%7Bx%5E2%7D%2B1%7D-%5Csqrt%7B%5Cfrac%7B2%7D%7Bx%5E2%7D%2B1%7D%7D%7B3%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim_{x\to \infty}(\frac{\sqrt{\frac{4}{x^2}+1}-\sqrt{\frac{2}{x^2}+1}}{3})" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=(\frac{1-1}{3}) = \frac{0}{3} = 0" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%28%5Cfrac%7B1-1%7D%7B3%7D%29+%3D+%5Cfrac%7B0%7D%7B3%7D+%3D+0&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=(\frac{1-1}{3}) = \frac{0}{3} = 0" /></div>
<h3 style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 18px; font-weight: normal; line-height: 1.6; margin: 0px 0px 10px; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: baseline; word-break: normal;">
Contoh Soal Limit 3</h3>
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Tentukanlah nilai dari <img alt="\lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{1- \cos 2x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%2B+%5Csin+x+%5Ctan+x%7D%7B1-+%5Ccos+2x%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{1- \cos 2x})" /> (SPMB 2002)</div>
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Pembahasan 3 :</div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="\lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{1- \cos 2x}) = \lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{2sin^2x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%2B+%5Csin+x+%5Ctan+x%7D%7B1-+%5Ccos+2x%7D%29+%3D+%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%2B+%5Csin+x+%5Ctan+x%7D%7B2sin%5E2x%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="\lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{1- \cos 2x}) = \lim \limits_{x\to 0}(\frac{x^2+ \sin x \tan x}{2sin^2x})" /> <img alt="= \lim \limits_{x\to 0}(\frac{x^2}{2 \sin^2x}+\frac{\sin x \tan x}{2sin^2x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%7D%7B2+%5Csin%5E2x%7D%2B%5Cfrac%7B%5Csin+x+%5Ctan+x%7D%7B2sin%5E2x%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to 0}(\frac{x^2}{2 \sin^2x}+\frac{\sin x \tan x}{2sin^2x})" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="= \lim \limits_{x\to 0}(\frac{x^2}{2sin^2x}+\frac{\sin x \tan x}{2 \sin^2x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%7D%7B2sin%5E2x%7D%2B%5Cfrac%7B%5Csin+x+%5Ctan+x%7D%7B2+%5Csin%5E2x%7D%29+&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to 0}(\frac{x^2}{2sin^2x}+\frac{\sin x \tan x}{2 \sin^2x})" /> <img alt="= \lim \limits_{x\to 0}(\frac{x^2}{2 \sin^2x}+\frac{\tan x}{2 \sin x})" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D+%5Clim+%5Climits_%7Bx%5Cto+0%7D%28%5Cfrac%7Bx%5E2%7D%7B2+%5Csin%5E2x%7D%2B%5Cfrac%7B%5Ctan+x%7D%7B2+%5Csin+x%7D%29&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="= \lim \limits_{x\to 0}(\frac{x^2}{2 \sin^2x}+\frac{\tan x}{2 \sin x})" /></div>
<div style="background-color: #f9f9f9; border: 0px; font-family: arial, Arial, Helvetica, sans-serif; font-size: 15px; line-height: inherit; margin-bottom: 20px; outline: 0px; overflow-wrap: normal; padding: 0px; text-align: center; vertical-align: baseline; word-break: normal;">
<img alt="=(\frac{1}{2(1^2)}+\frac{1}{2(1)}) = \frac{1}{2}+\frac{1}{2} = 1" class="latex jetpack-lazy-image jetpack-lazy-image--handled" src="https://s0.wp.com/latex.php?latex=%3D%28%5Cfrac%7B1%7D%7B2%281%5E2%29%7D%2B%5Cfrac%7B1%7D%7B2%281%29%7D%29+%3D+%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B2%7D+%3D+1&bg=f9f9f9&fg=000000&s=0" style="border: 0px; font: inherit; height: auto; margin: 0px; max-width: 100%; outline: 0px; overflow-wrap: normal; padding: 0px; vertical-align: middle; word-break: normal;" title="=(\frac{1}{2(1^2)}+\frac{1}{2(1)}) = \frac{1}{2}+\frac{1}{2} = 1" /></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-9701083128583606022017-10-07T08:42:00.003-07:002017-10-07T08:42:39.041-07:00MENGUBAH DESIMAL KE PECAHAN BIASA“Bagaimana cara mengubah pecahan desimal ke pecahan biasa?”<br />
“Tinggal lihat berapa angka di belakang koma,”<br />
<br />
1 angka di belakang koma maka menjadi per 10<br />
<br />
Contoh:<br />
<br />
0,5 = 5/10<br />
0,3 = 3/10<br />
1,2 = 12/10 = 1 2/10 = 1 1/5<br />
<br />
2 angka di belakang koma maka menjadi per 100<br />
<br />
Contoh:<br />
<br />
0,75 = 75/100 = 3/4<br />
0,25 = 25/100 = 1/4<br />
4,25 = 425/100 = 4 25/100 = 4 1/4<br />
<br />
3 angka di belakang koma maka menjadi per 1000, dan seterusnya.<br />
<br />
“katanya ada juga pecahan desimal yang panjang tidak terbatas. Bagaimana maksudnya?” tanya Al.<br />
<br />
Misal:<br />
0,3333333….. = ?<br />
<br />
“Mari sedikit kita gunakan teknik aljabar,” .<br />
<br />
0,333… = a<br />
<br />
Kalikan dengan 10 maka menghasilkan 10a.<br />
<br />
10a = 0,333… x 10 = 3,333….<br />
<br />
Kurangkan 10a – a = 9a<br />
<br />
9a = 3,333…. – 0,333…<br />
9a = 3<br />
a = 3/9 = 1/3<br />
<br />
Jadi, a = 1/3 = 0,333333…..Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-60997859654982201732017-09-07T15:01:00.002-07:002019-09-21T20:42:48.565-07:00Download Kumpulan Rumus Cepat Matematika SMA Lengkap PDF<h2 style="background: rgb(255, 255, 255); font-family: sans-serif; margin: 0px; outline: 0px; padding: 0px; text-align: justify; vertical-align: baseline;">
Download Kumpulan Rumus Cepat Matematika SMA Lengkap PDF</h2>
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Koleksi Rumus Cepat Matematika SMA / SMK terbaru lengkap - Banyak yang bilang bahwa pelajaran matematika itu adalah pelajaran yang paling menakutkan. Matematika seolah menjadi momok bagi para siswa SMA maupun SMK Sederajat. Ada banyak sekali cara yang bisa digunakan untuk mempelajari matematika dengan suasana yang menyenangkan. Karena sebenarnya matematika itu tidaklah sulit dan menakutkan. Semua sisi kehidupan pastilah berkaitan dengan hitungan, angka atau bilangan. Disitulah pelajaran matematika diterapkan. Disekolah pasti juga akan menghadapi ujian, baik Ujian Nasional Online, Ujian Sekolah dan Ujian Tengah Semester atau UTS SMK / SMA. Ada beberapa cara praktis agar anda bisa mengerjakan soal matematika dengan cepat. Berhitung dan berfikir secara logis menggunakan logika yang benar. </div>
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<a href="https://autoratio.com/5JCzudjLEj">Download Rumus Matematika SMA/SMK Lengkap</a></h3>
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Sekarang ini soal latihan dan beberapa contoh materi pelajaran matematika bisa dengan mudah didownload di Internet. Dari beberapa pengamatan banyak sekali pelajar SMA yang mencari rumus cepat matematika yang bisa di download, kemudian cara cepat kerjakan matematika, kumpulan rumus cepat menghitung. Misalnya mengitung perkalian, akar pangkat, rumus matematika Integral. Berdasarkan hal itulah kami memberikan dan mengumpulkan File Kumpulan rumus cepat matematika dasar yang bisa dipelajari dengan mudah tanpa anda harus membeli buku.</div>
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<a href="http://yoalizer.com/1t7R" target="_blank">Tips dan Trik Berhitung Cepat Matematika SMA/SMK</a></h4>
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Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-635150364584535352017-09-06T14:59:00.002-07:002017-09-06T14:59:08.686-07:00kumpulan soal UN<div 10px="" 14px="" border-box="" box-sizing:="" class="MsoNormal" font-family:="" font-size:="" geneva="" margin-bottom:="" sans-serif="" tahoma="" verdana="">
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<br /></div>
P<a href="http://yoalizer.com/1Mc7" target="_blank">aket 1</a><br />
<div class="MsoNormal">
<a href="http://yoalizer.com/1Mfn" target="_blank">Paket 2</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1MgZ" target="_blank">Paket 3</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1Mjq" target="_blank">Paket 7</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1MkM" target="_blank">Paket 8</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1Mkv" target="_blank">Paket 9</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1Mpx" target="_blank">Paket 16</a><o:p></o:p></div>
<div class="MsoNormal">
<a href="https://drive.google.com/file/d/0B2Dy4k5WfMl4bEZIRzYyRVYwMUU/view?usp=sharing" target="_blank">Paket 17</a><o:p></o:p></div>
<div class="MsoNormal">
<a href="http://yoalizer.com/1Mz3" target="_blank">Paket 18</a><o:p></o:p></div>
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<a href="http://yoalizer.com/1N1t" target="_blank">Paket 19</a><o:p></o:p></div>
<a href="http://yoalizer.com/1NAT" target="_blank">Paket 20</a><br />
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<o:p></o:p></div>
<br />
<br />
kunci jawab paket 1 - 20 (<a href="http://yoalizer.com/1NpC" target="_blank">download</a>)<br />
<br />
Soal OSTN<br />
<br />
<a href="http://yoalizer.com/1FnK" target="_blank">download</a>Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-81445130019090703482015-07-08T17:19:00.002-07:002017-09-08T02:56:39.405-07:00Barisan Geometri<b>Barisan Geometri Soal dan Pembahasan</b>
- Setelah pada kesempatan yang lalu kita telah membahas barisan dan
deret aritmatika, saatnya kita ke materi selanjutnya yaitu geometri.
Langsung saja sobat, mari kita langsung ke materi saja.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="106" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkcq99moWhD8LtM_75Szl0vjMTO9bV5bUOxYotJOdWyqxAqXhldLypt6pwxk8NiSZDGl9DqYKKOm4ZVUyqQl_0RW8dQ_tdlMt7W02qn_c2hQraYsyjR4K94zAzT_gAvYXlNT1wnymgrU_t/s1600/BarisanGeo.png" width="320" /></div>
<br />
<b><a href="http://belajar-mate-matika.blogspot.co.id/">Barisan Geometri</a></b><br />
<br />
Barisan geometri merupakan barisan yang suku-sukunya diperoleh dengan
mengalikan satu bilangan tetap ke suku sebelumnya. Bilangan tetap itu
sering disebut sebagai pembanding atau rasio yang dilambangkan dengan <b><i>r</i></b>.<br />
<br />
Barisan <span style="font-family: "calibri" , sans-serif; font-size: 14pt; line-height: 115%;"><i>U</i></span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">1</sub><span style="font-family: "calibri" , sans-serif; font-size: 14pt; line-height: 115%;">
, <i>U</i></span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">2</sub><span style="font-family: "calibri" , sans-serif; font-size: 14pt; line-height: 115%;"> , <i>U</i></span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">3</sub><span style="font-family: "calibri" , sans-serif; font-size: 14pt; line-height: 115%;"> , <i>U</i></span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">4</sub><span style="font-family: "calibri" , sans-serif; font-size: 14pt; line-height: 115%;"> , ….. , <i>U</i></span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">n</sub> disebut sebagai barisan geometri jika memenuhi<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpqKwMaC_x7r_-uR-Mjhkdzujk5g7nug9M34qi_aLBcbk1Tum_gf3aXukDievMnfhd2UmoAuiQgdbhUddTk8Xfen9pABuFREP5nelTaw_0O3LuQUCssbRY7Xg1Y67f-lB8ZwNZmqFq_U9j/s1600/Rasio.png" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpqKwMaC_x7r_-uR-Mjhkdzujk5g7nug9M34qi_aLBcbk1Tum_gf3aXukDievMnfhd2UmoAuiQgdbhUddTk8Xfen9pABuFREP5nelTaw_0O3LuQUCssbRY7Xg1Y67f-lB8ZwNZmqFq_U9j/s1600/Rasio.png" /></a><br />
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Contoh barisan geometri : 7, 21, 63, 189, ....<br />
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<b><i><span style="color: red;">Rumus Suku ke-n</span></i></b><br />
<br />
Jika suku pertama ( U1 ) dari suatu barisan geometri disimbolkan dengan a
, maka rumus suku ke-n barisan geometri dapat ditentukan sebagai
berikut:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiner1SGZ-zTRFRQQzCVZjpEwtUe5GKSk1fSOeM8Rf8f90uCxs42cbfVI_Dxd_2tgU06io1B2zP0cPtoPEEWr4mdOiN37x5asG7PIOaWqdUiMrf1XaP0_BLov_yJzNd7LDoaUiHq3WIoeED/s1600/DG.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiner1SGZ-zTRFRQQzCVZjpEwtUe5GKSk1fSOeM8Rf8f90uCxs42cbfVI_Dxd_2tgU06io1B2zP0cPtoPEEWr4mdOiN37x5asG7PIOaWqdUiMrf1XaP0_BLov_yJzNd7LDoaUiHq3WIoeED/s1600/DG.png" /></a></div>
<br />
Dari pernyataan diatas, dapat ditarik kesimpulan bahwa rumus suku ke-n dari suatu barisan geometri adalah<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj77-ddC2Nyd7RH-SBfxLnYFx-Tfxm3ljm31BySNCU4HivMWBnHQyPR0pizvelt-vcuoc6HKqgMZFLVqBYK9Ldp7u_BO8qRWcNMbpFEvXUBxr9snc628hEE6LeRYxrMxXfBU3nZIjM3n8Z1/s1600/RumusDG.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj77-ddC2Nyd7RH-SBfxLnYFx-Tfxm3ljm31BySNCU4HivMWBnHQyPR0pizvelt-vcuoc6HKqgMZFLVqBYK9Ldp7u_BO8qRWcNMbpFEvXUBxr9snc628hEE6LeRYxrMxXfBU3nZIjM3n8Z1/s1600/RumusDG.png" /></a></div>
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Dimana r adalah rasio atau pembanding yang dapat dicari dengan cara berikut:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisEP5sRdI4unHoAWNEv5YlqoTwzb2o9Q-zE4DOJjlBfKvs_0uDzHfre4_sqFU-GOHVXDidfeD26Kqr5SGXMrbmd3Kachf2Vt7gPJB3ylGLZCs6x1z7o1rWYI_fcN7BeJIn6VpVt6GV1ekE/s1600/RumusR.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisEP5sRdI4unHoAWNEv5YlqoTwzb2o9Q-zE4DOJjlBfKvs_0uDzHfre4_sqFU-GOHVXDidfeD26Kqr5SGXMrbmd3Kachf2Vt7gPJB3ylGLZCs6x1z7o1rWYI_fcN7BeJIn6VpVt6GV1ekE/s1600/RumusR.png" /></a></div>
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<br />
Contoh Soal :<br />
<br />
<ol>
<li>Tentukan suku ke tujuh dari barisan geometri 3, 6, 12, .....!</li>
<li>Tentukan Rumus Suku ke-n dari barisan 48 , 24 , 12 , ……!</li>
<li>Dari barisan geometri diketahui bahwa U3 = 4 dan U9 = 256, maka tentukan U12!</li>
<li>Tiga buah bilangan membentuk barisan geometri. Jumlah ketiga
bilangan tersebut sama dengan 35, sedangkan hasil kali ketiga bilangan
itu sama dengan 1.000. Maka tentukan barisan geometri tersebut!</li>
</ol>
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<div>
Jawab :</div>
<div>
</div>
<div>
1. Dari Barisan 3, 6, 12, ... didapat <b><i>a</i> = </b>3 dan <i style="font-weight: bold;">r</i> = 6/3 = 2 sehingga,</div>
<pre style="background-color: #f9f9f9; border: 1px solid rgb(221, 221, 221); margin-bottom: 1.5em; margin-top: 1.5em; overflow: auto; padding: 1em; text-align: justify; word-wrap: normal;"><span style="color: #555555; font-family: "calibri" , sans-serif;"><span style="font-size: 15px; line-height: 16.8666667938232px;"><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">n</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">a.r</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">n-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">3.2</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">7-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 3<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">.2</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">6</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 3.64
<b><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 115%;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 192</b>
</span></span></pre>
2. Dari barisan 48, 24, 12, .... didapat <i style="font-weight: bold;">a</i> = 48 dan <i style="font-weight: bold;">r</i> = 24/48 = 1/2 sehingga,<br />
<pre style="background-color: #f9f9f9; border: 1px solid rgb(221, 221, 221); margin-bottom: 1.5em; margin-top: 1.5em; overflow: auto; padding: 1em; text-align: justify; word-wrap: normal;"><span style="color: #555555; font-family: "calibri" , sans-serif;"><span style="font-size: 15px; line-height: 16.8666667938232px;"><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">a.r</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">48.(1/2)</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">48.(</span><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">(2</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">-1</sup><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">)</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">n</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 3<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">.16.</span><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">2</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">1</sup><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">-n</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 3.<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">2</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">4</sup><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">.2</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">1-n</sup>
<b><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">7</sub><span class="Apple-tab-span" style="white-space: pre;"> </span>= 3.<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">2</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">5-n</sup></b>
</span></span></pre>
3. Pertama, kita jabarkan terlebih dahulu <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">3</sub> dan <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">9</sub> kemudian kita cari nilai <b><i>rasionya</i></b><br />
<pre style="background-color: #f9f9f9; border: 1px solid rgb(221, 221, 221); margin-bottom: 1.5em; margin-top: 1.5em; overflow: auto; padding: 1em; text-align: justify; word-wrap: normal;"><span style="color: #555555; font-family: "calibri" , sans-serif;"><span style="font-size: 15px; line-height: 16.8666667938232px;"><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">3 </sub><i style="font-weight: bold;">= </i>4 <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">→ </span><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">a.r</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">2</sup><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;"> </sub><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">= 4</span>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">9 </sub><i style="font-weight: bold;">= </i>256 <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">→ </span><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">a.r</span><sup style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">8</sup><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;"> </sub><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">= 256</span>
</span></span></pre>
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Kemudian substitusikan untuk mencari <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;"><b>U</b></span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;"><b>1</b> </sub>atau <b><i>a</i></b>!<br />
<pre style="background-color: #f9f9f9; border: 1px solid rgb(221, 221, 221); margin-bottom: 1.5em; margin-top: 1.5em; overflow: auto; padding: 1em; text-align: justify; word-wrap: normal;"><span style="color: #555555; font-family: "calibri" , sans-serif;"><span style="font-size: 15px; line-height: 16.8666667938232px;"><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">→ </span><span style="font-family: "calibri" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">a.r<sup>2</sup> = 4</span>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">→ </span><span style="font-family: "calibri" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">a.2<sup>2</sup> = 4</span>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 115%;">→ </span><span style="font-family: "calibri" , "sans-serif"; font-size: 12.0pt; line-height: 115%;">a = 1</span>
</span></span></pre>
Next, cari nilai <b><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-family: Calibri, sans-serif; line-height: 15.3333320617676px;">12</sub></b> dengan menggunakan rumus umum barisan geometri!<br />
<div class="MsoListParagraphCxSpFirst">
<div class="MsoListParagraphCxSpFirst">
<pre style="background-color: #f9f9f9; border: 1px solid rgb(221, 221, 221); margin-bottom: 1.5em; margin-top: 1.5em; overflow: auto; padding: 1em; text-align: justify; word-wrap: normal;"><span style="color: #555555; font-family: "calibri" , sans-serif;"><span style="font-size: 15px; line-height: 16.8666667938232px;"><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="line-height: 15.3333320617676px;">12</sub><span class="Apple-tab-span"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">a.r</span><sup style="line-height: 15.3333320617676px;">n-1</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="line-height: 15.3333320617676px;">12</sub><span class="Apple-tab-span"> </span>= <span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">1.2</span><sup style="line-height: 15.3333320617676px;">11</sup>
<span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="line-height: 15.3333320617676px;">12</sub><span class="Apple-tab-span"> </span>= </span><span style="line-height: 18.3999996185303px;">1.2048</span><span style="font-size: 15px; line-height: 16.8666667938232px;">
</span><b><span style="font-family: "calibri" , sans-serif; font-size: 12pt; line-height: 18.3999996185303px;">U</span><sub style="font-size: 15px; line-height: 15.3333320617676px;">12</sub><span class="Apple-tab-span" style="font-size: 15px; line-height: 16.8666667938232px;"> </span><span style="font-size: 15px; line-height: 16.8666667938232px;">= 2048</span></b></span></pre>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_5n9QtYGQPWpI82FFRTOosm63s-VhF_Jr1vn3iXRyxnhTyjlMwDOh1lUislVplc1WULATvhk4sR03H-RIrEJBVT6p4nBrtgqPksssMg2EsAAsL0_CL_9aZH8nZzv8_hYoABg5Y-Ro0aL6/s1600/Jawaban+no+4.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_5n9QtYGQPWpI82FFRTOosm63s-VhF_Jr1vn3iXRyxnhTyjlMwDOh1lUislVplc1WULATvhk4sR03H-RIrEJBVT6p4nBrtgqPksssMg2EsAAsL0_CL_9aZH8nZzv8_hYoABg5Y-Ro0aL6/s1600/Jawaban+no+4.png" /></a></div>
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Mungkin itu saja sobat informasi yang bisa saya berikan tentang <b>Barisan Geometri <a href="http://belajar-mate-matika.blogspot.co.id/">Matematika</a></b> semoga bermanfaat.
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-7498385258658768702014-10-10T21:59:00.003-07:002014-10-10T21:59:37.816-07:00Cara Menghafal Trigonometri Sudut Istimewa<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 24px; margin-bottom: 12px; padding: 0px; text-align: justify;">
Sobat hitung, berikut ini ada Cara Menghafal <strong style="margin: 0px; padding: 0px;">trigonometri sudut istimewa</strong> yang bisa memudahkan sobat untuk menghafal nilai sin, cos, dan tan dari sudut-sudut istimewa.</div>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 24px; margin-bottom: 12px; padding: 0px; text-align: justify;">
<span style="margin: 0px; padding: 0px; text-decoration: underline;">Cara Menghafal trigonometri sudut istimewa dengan Konsep Tangan</span></div>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 24px; margin-bottom: 12px; padding: 0px;">
<a href="http://rumushitung.com/wp-content/uploads/2012/12/cara-cepat-menghafal-trigonometri-sudut-istimewa.png" style="color: #006ce0; margin: 0px; padding: 0px; text-decoration: none;"><img alt="cara cepat menghafal trigonometri sudut istimewa" class=" wp-image-583 aligncenter" height="349" src="http://rumushitung.com/wp-content/uploads/2012/12/cara-cepat-menghafal-trigonometri-sudut-istimewa.png" style="border: 0px; clear: both; display: block; height: auto; margin: 5px auto; max-width: 100%; padding: 0px;" title="cara cepat menghafal trigonometri sudut istimewa" width="400" /></a><br style="margin: 0px; padding: 0px;" />Cara menggunakannya,</div>
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<li style="line-height: 24px; margin: 0px; padding: 3px;">Rumus kita gunakan sebagai rumus dasar menentukan nilai <a href="http://id.wikipedia.org/wiki/Trigonometri" style="color: #006ce0; margin: 0px; padding: 0px; text-decoration: none;">trigonometri sudut istimewa</a> adalah “1/2 akar n” dengan n adalah angka-angka di jari tangan.</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Untuk Sin x menggunakan angka dengan background HIJAU yang searah dengan jarum jam, dan Cos x Backgroud Kuning yang berlawanan dengan arah jarum jam.</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Sudut Istimewa mulai dari 0 di kelingking sampai 90 di jempol</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Untuk mencari trigonometri (sin cos tan) sudut istimewa kita tinggal memasukkan nilai n pada rumus yang ada d</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Untuk Mendapatkan Nilai tangen trigonometri sudut istimewa tinggal membagi nilai Sin dengan nilai Cos yang telah sobat temukan.</li>
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<blockquote style="background-color: white; background-image: url(http://rumushitung.com/wp-content/themes/newswp/images/quote.gif); background-position: 0% 0%; background-repeat: no-repeat; font-family: Arial, Helvetica, sans-serif; font-size: 13px; font-style: italic; line-height: 18px; margin: 10px 5px 10px 0px; padding: 0px 0px 0px 50px;">
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tan x = sin x/ cos x</div>
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Bingngun? Mari kita lihat contoh berikut</div>
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Sin 90, Lihat warna hijau, jari telunjuk n= 4 —-> sin 90 = 1/2 x akar 4 = 1/2 x 2 = 1<br style="margin: 0px; padding: 0px;" />Cos 60, Lihat warna kuning, jari telunjuk n =1 makan Cos 60 = 1/2 akar 1 = 1/2</div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-83870336661803594872014-10-10T21:57:00.001-07:002014-10-10T21:57:39.174-07:00Aplikasi Android Rumus Tabung<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 24px; margin-bottom: 12px; padding: 0px; text-align: justify;">
Dear all sobat hitung, kali ini kami akan berbagi aplikasi gratis yang bisa digunakan buat menghitung luas dan volume tabung secara praktis. Tidak hanya itu, aplikasi tersebut juga berisi <a href="http://rumushitung.com/2013/06/17/sifat-sifat-bangun-ruang-lengkap/" style="color: #006ce0; margin: 0px; padding: 0px; text-decoration: none;" title="Sifat-Sifat Bangun Ruang Lengkap">materi bangun ruang</a>tabung mulai dari sifat-sifat tabung, rumus-rumusnya, dan tak ketinggalan contoh soal dan pembahasannya. Apa saja yang ada di dalam aplikasi rumus tabung ini? Berikut detailnya.</div>
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<a href="http://rumushitung.com/wp-content/uploads/2014/09/splash.png" style="color: #006ce0; margin: 0px; padding: 0px; text-decoration: none;"><img alt="aplikasi rumus tabung" class="aligncenter wp-image-4344" height="359" src="http://rumushitung.com/wp-content/uploads/2014/09/splash.png" style="border: 0px; clear: both; display: block; height: auto; margin: 5px auto; max-width: 100%; padding: 0px;" width="270" /></a></div>
<table border="0" cellpadding="2" cellspacing="2" style="background-color: white; color: black; font-family: Arial, Helvetica, sans-serif; font-size: 13px; height: 116px; line-height: 18px; margin: 0px; padding: 0px; width: 456px;"><tbody style="margin: 0px; padding: 0px;">
<tr style="margin: 0px; padding: 0px;"><td style="margin: 0px; padding: 0px; width: 156px;">Nama Aplikasi</td><td style="margin: 0px; padding: 0px; width: 4px;">:</td><td style="margin: 0px; padding: 0px; width: 274px;">Rumus Tabung Versi 1.0</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="margin: 0px; padding: 0px; width: 156px;">Ukuran file APK</td><td style="margin: 0px; padding: 0px; width: 4px;">:</td><td style="margin: 0px; padding: 0px; width: 274px;">2 MB</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="margin: 0px; padding: 0px; width: 156px;">Minumum Android</td><td style="margin: 0px; padding: 0px; width: 4px;">:</td><td style="margin: 0px; padding: 0px; width: 274px;">Android Froyo</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="margin: 0px; padding: 0px; width: 156px;">Status</td><td style="margin: 0px; padding: 0px; width: 4px;">:</td><td style="margin: 0px; padding: 0px; width: 274px;">Gratis</td></tr>
<tr style="margin: 0px; padding: 0px;"><td style="margin: 0px; padding: 0px; width: 156px;">Link Download</td><td style="margin: 0px; padding: 0px; width: 4px;">:</td><td style="margin: 0px; padding: 0px; width: 274px;"> <a href="https://play.google.com/store/apps/details?id=com.donbull.rumustabung" target="_blank"><img alt="download aplikasi kalkulator berat bayi ideal" class="wp-image-4335 alignnone" height="41" src="http://rumushitung.com/wp-content/uploads/2014/09/download-aplikasi-kalkulator-berat-bayi-ideal.png" style="border: 0px; height: auto; margin: 0px; max-width: 100%; padding: 0px;" width="123" /></a></td></tr>
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<span style="margin: 0px; padding: 0px;">Fitur Aplikasi</span></h3>
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<span style="color: white; font-weight: bold; margin: 0px; padding: 0px;">Apa itu Tabung</span></div>
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Menu ini berisikan deskripsi singkat tentang apa itu tabung, ciri-cirinya, dan sejarah singkatnya</div>
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<span style="font-weight: bold; margin: 0px; padding: 0px;">Rumus Tabung</span></div>
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Sobat bisa menemukan materi tentang semua rumus tabung mulai dari rumus luas, rumus volume, dan rumus selimut.</div>
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Kalkulator Tabung</div>
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Di dalamnya sobat akan menemukan 2 buah kalkulator praktis masing-masing kalkulator untuk menghitung volume tabung dan yang kedua kalkulator untuk menghitung luas tabung. Cara menggunakannya cukup mudah. Sobat tinggal memasukkan nilai jari-jari dan juga tinggi kemudian tab di tombol hitung maka nilai luas atau volume akan keluar secara otomatis.</div>
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<span style="font-weight: bold; margin: 0px; padding: 0px;">Contoh Soal</span></div>
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Menu ini berisi contoh-contoh soal variatif seputar rumus tabung dengan disertai penjelasan mudah, pembahasan singkat, serta mudah dimengerti.</div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-74138001699152459142014-10-10T21:55:00.001-07:002017-10-07T08:49:04.865-07:00Cara Mudah Mengubah Desimal Ke Pecahan Biasa Dan Campuran<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 24px; margin-bottom: 12px; padding: 0px; text-align: justify;">
Coba sobat perhatikan perkalian angka di bawah ini,</div>
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<tr style="margin: 0px; padding: 0px;"><td style="background-color: #ffff33; margin: 0px; padding: 0px; text-align: center;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><span style="font-weight: bold; margin: 0px; padding: 0px;">16 x 0,125</span></big></big></big></big></td></tr>
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Cukup lama tidak kira-kira sobat mengerjakan soal di atas? Bagi kebanyakan kalian mungkin akan cukup lama mengalikan 16 dengan bilangan desimal 0,125. Sebenarnya dengan sedikit mengubah angka desimal ke bentuk pecahan biasa sobat bisa dengan mudah mengerjakan soal di atas. Lakukan perubahan sebagai berikut:,</div>
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<tr style="margin: 0px; padding: 0px;"><td style="background-color: #ffff33; margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><span style="font-weight: bold; margin: 0px; padding: 0px;">=16 x 0,125<br style="margin: 0px; padding: 0px;" />= <span style="margin: 0px; padding: 0px; text-decoration: line-through;">16</span> x 1/<span style="margin: 0px; padding: 0px; text-decoration: line-through;">8</span><br style="margin: 0px; padding: 0px;" />= 2</span></big></big></big></td></tr>
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Taraaa ternyata dengan mengubah angka desimal 0,125 ke bentuk pecahan biasa 1/8 bisa membantu sobat lebih cepat menghitung. Itulah salah satu manfaat mengubah desimal ke bentuk pecahan biasa. Adan beberapa angka yang familiar yang bisa diubah ke bentuk pecahan untuk mempermudah penghitung seperti 0,5 = 1/2, 0,25 = 1/4, dan 0,75 = 3/4, Pertanyaannya bagaimana cara mudahnya mengubah 0,125 menjadi bentukk 1/8? temukan jawabannya di bawah ini</div>
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<a href="https://belajar-mate-matika.blogspot.co.id/2017/10/mengubah-desimal-ke-pecahan-biasa.html">Cara Mudah Mengubah Desimal ke Pecahan Biasa</a></h3>
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Cara paling mudah mengubah bilangan desimal dengan menjadi pecahan biasa adalah dengan mengubahnya ke bentuk pecahan dengan penyebut kelipatan 10. Kemudian sederhanakan hingga ketemu pecahan yang paling sederha. Misalnya</div>
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<big style="margin: 0px; padding: 0px;">0,5 = 5/10 ruas atas dan bawah dibagi 5 hasilnya = 1/2</big><br />
<big style="margin: 0px; padding: 0px;">0,25 = 25/100 masing masing dibagi 25 ketemu = 1/4</big><br />
<big style="margin: 0px; padding: 0px;">0,45 = 45/100 = 9/20</big><br />
<big style="margin: 0px; padding: 0px;">0,125 = 125 /1000 = 25/200 = 5/40 = 1/8</big></div>
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Kalau dibuat langkah-langkanya,</div>
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<li style="line-height: 24px; margin: 0px; padding: 3px;">Hitung jumlah angka dibelakang koma, angka nol di dibelakang tidak usah dihitung<br style="margin: 0px; padding: 0px;" />misal 0,225 → jumlah angka di belakang koma = 3<br style="margin: 0px; padding: 0px;" />misal ),350 → jumlah angka dibelakang koma = 2 (angka nol di belakang tidak usah dianggap)</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Hilangkan tanda koma dan angka nol di depan koma kemudian setelah dihilangkan jadikan bilangan tersebut manjadi pembilang pecahan (ingat pembilang letakknya di atas)<br style="margin: 0px; padding: 0px;" />misal 0,13 → setelah dihilangkan → 13 → kita jadikan pembilang</li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Bagi bilangan yang sobat dapat di nomor dua dengan angka satu yang dirangkai dengan angka nol-sebanyak angka yang didapat pada angka satu lalu sederhanakan.Misalnya 0,75<br style="margin: 0px; padding: 0px;" />→ angka dibelakang koma 2 (langkah 1)<br style="margin: 0px; padding: 0px;" />→ dihilangkan koma dan nol didepannya menjadi 75 → pembilang (langkah 2)<br style="margin: 0px; padding: 0px;" />→ bagi dengan 75 dengan angka 1 diikuti 2 bual 0 jadi 75/100 = 3/4 (pembilang dan penyebut masing-masing dibagi 25)</li>
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Bagaimana Mengubah Desimal ke Pecahan Campuran?</h3>
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Caranya hampir sama dengan mengubah desimal ke pecahan biasa. Contoh ada bilangan 1,15</div>
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→ 1,15 kita pisahkan bilangan desimal dari bilangan bulat<br />
→ 1,15 = 1 + 0,15<br />
kita cari pecahan dari 0,15<br />
0,15 = 15/100 = 3/20<br />
→ 1,15 = 1 + 0,15 = 1 + 3/20 = 1 <small style="margin: 0px; padding: 0px;">3/20</small></div>
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Bagaiman Kak Kalau Bilangan Desimal Tak Terhingga?</h3>
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Untuk mengubah bilangan desimal tak hingga ke bentuk pecahan ada triknya. Sobat pasti pernah menjumpai angka seperti ini</div>
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<tr style="margin: 0px; padding: 0px;"><td style="background-color: #66ff99; margin: 0px; padding: 0px; text-align: center;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><big style="margin: 0px; padding: 0px;"><span style="font-weight: bold; margin: 0px; padding: 0px;">0,777…</span></big></big></big></td></tr>
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Bagimana mengubah 0,777…. dst ke bentuk pecahan? Sobat bisa menggunakan aljabar sederhana. Berikut cara</div>
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<li style="line-height: 24px; margin: 0px; padding: 3px;">Sobat misalkan 0,777 sebagai x<br style="margin: 0px; padding: 0px;" /><span style="color: #cc0000; font-family: "comic sans ms"; font-weight: bold; margin: 0px; padding: 0px;">0,777… → x</span></li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Kemudaian buat persamaan<br style="margin: 0px; padding: 0px;" />jika <span style="color: #cc0000; font-family: "comic sans ms"; font-weight: bold; margin: 0px; padding: 0px;">x = 0,777….</span> maka <span style="color: #cc0000; font-family: "comic sans ms"; font-style: italic; font-weight: bold; margin: 0px; padding: 0px;">10x = 7,777…</span></li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Kita kurangkan 10 x dengan x<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6h7sq_0zooyH_q5JSSbBhIUpo-ZXqT9-m2ZWjDiJpqDT1D62m7lDgGe6baNr-6YAwCGwziCnPTTzDYTcJ3Jn1Edzu-3BTBbMYdBfsLRiyIMZKdjSnQBWuwl6kZz9mxDSd4X-MYuAos466/s1600/mengubah-desimal-ke-pecahan-biasa.png" imageanchor="1" style="font-family: "Times New Roman"; font-size: medium; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="178" data-original-width="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6h7sq_0zooyH_q5JSSbBhIUpo-ZXqT9-m2ZWjDiJpqDT1D62m7lDgGe6baNr-6YAwCGwziCnPTTzDYTcJ3Jn1Edzu-3BTBbMYdBfsLRiyIMZKdjSnQBWuwl6kZz9mxDSd4X-MYuAos466/s1600/mengubah-desimal-ke-pecahan-biasa.png" /></a></li>
<li style="line-height: 24px; margin: 0px; padding: 3px;">Taraaaa kita dapatkan hasilnya bilangan desimal tak hingga 0,777… pecahan desimalnya dalah 7/9. Kalau sobat belum yakin silahkan dicocokan di calculator HP masing-masing</li>
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Yang perlu sobat perhatikan, permisalan yang digunakan akan tergantung pada jumlah angka yang berulang</div>
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jika cuma satu seperti 0,777…. maka dimisalkan 10 x<br />
jika 2 angka seperti 0,242424… maka sobat misalkan 100 x<br />
jika 3 angka seperti desimal 0,123123123… maka sobat misalkan saja 1000 x</div>
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Yuk kita buktikan bilangan desimal yang terakhir</div>
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<span style="color: #cc0000; font-family: "comic sans ms"; font-weight: bold; margin: 0px; padding: 0px;">0,123123… → x</span><br />
<span style="color: #cc0000; font-family: "comic sans ms"; font-weight: bold; margin: 0px; padding: 0px;">123,123123…. → 1000x</span></div>
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Bagimana sobat, cukup mudah kan untuk mengubah bilangan desimal ke pecahan biasa maupun campuran. Semoga bermanfaat. So pasti tetap semangat belajarnya. Biar tambah semangatnya silahkan coba cari pecahan biasa dari bilangan desimal 0,6464…</div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-47943563076681920512014-10-08T19:36:00.003-07:002017-10-02T04:26:53.122-07:008 Rumus Cepat Matematika Saat Berhitung<h1 class="post-title entry-title" style="box-sizing: border-box; font-family: 'Droid Sans', Arial, Verdana, sans-serif; font-size: 21px; font-weight: normal; margin: 0px 0px 5px; padding: 0px;">
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<span class="timestamp-link" style="box-sizing: border-box;">00.44</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><a href="https://belajar-mate-matika.blogspot.co.id/2017/09/download-kumpulan-rumus-cepat.html">Rumus Cepat Matematika</a> dalam menghitung tentu amat berguna bagi Anda semua. Ada banyak sistem perhitungan dalam Matematika yang lebih cepat serta belum tentu kita ketahui. Melalui artikel ini, kami akan membagi 10 cara cepat matematika saat menghitung. Diharapkan, melalui berbagai macam Rumus Cepat Matematika berikut, Anda bisa mempunya sistem kalkulasi yang lebih baik dari sebelumnya.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Trik Sebelas</span></h2>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><a href="https://belajar-mate-matika.blogspot.co.id/2017/09/download-kumpulan-rumus-cepat.html">Rumus Cepat Matematika</a> pertama dalam menghitung adalah trik sebelas. Beberapa dari Anda mungkin amat familiar dengan trik yang satu ini. Trik sebelas merupakan sebuah trik yang mempermudah kita mencetak hasil dari suatu angka terhadap perkalian angka sebelas. Oke, langsung saja Anda lihat demonstrasi trik ampuh yang satu ini.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Ambil dua angka yang Anda inginkan (ingat, hanya dua digit). Tambahkan kedua angka tersebut, kemudian taruh di tengah. Dalam contoh ini, kami menggunakan 45. Jadi, 45 dikalikan 11 sama dengan:</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">45 = 4 (4+5) 5 = 495</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Sedangkan, apabila jumlah angka yang dihasilkan sama dengan dua digit, maka jumlahkan angka yang ada didepan dengan angka awal yang ada di paling kiri. Lihat contohnya,</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">89 = 8 (8+9) 9 = 8 (17) 9 = (8+1) 7 9 = 979</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">97 = 9 (9+7) 7 = 9 (16) 7 = (9+1) 6 7 = 1067</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Bagaimana, mudah bukan?</span></div>
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<o:p style="box-sizing: border-box;"><span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></o:p></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Kuadrat Lima</span></h2>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Jika Anda ingin melakukan perkalian kuadrat terhadap dua digit angka yang diakhiri dengan angka 5, maka solusinya adalah :</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Kalikan angka depan dengan hasil angka tersebut ditambah satu. Kemudian, taruh angka 25 dibelakang. Contoh lebih jelasnya sebagai berikut,</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">45<sup style="box-sizing: border-box;">2 </sup>= 4 x (4+1) & 25 = 4x5 & 25 = 2025</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">55<sup style="box-sizing: border-box;">2</sup> = 5 x (5+1) & 25 = 5x6 & 25 = 3025</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Silakan coba sendiri..</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Perkalian Lima</span></h2>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Banyak orang yang berpikir terlalu keras ketika menghadapi jumlah angka begitu banyak, kemudian dikalikan dengan 5. Ketika Anda mempunyai kalkulator, maka tidak masalah tentunya. Tetapi jika kalkulator tidak ada dan Anda harus tetap menghitungnya, maka cara cepat matematika yang satu ini tentu bisa membantu.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">46789846 x 5</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">56735691 x 5</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Bagi angka tersebut dengan dua, kemudian jika hasilnya penuh, maka tambahkan angka nol dibelakang. Namun, apabila hasilnya adalah desimal, maka tambahkan angka lima dibelakang dan hilangkan angka setelah koma.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">46789846 x 5 = 46789846 : 2 = 23394923 (karena angkanya penuh, maka tambahkan 0) = 233949230</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">56735691 x 5 = 56735691 : 2 = 28367845.5 (karena angkanya desimal, maka hilangkan angka dibelakang koma dan tambahkan angka 5) = 283678455</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Silakan coba..</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Perkalian Sembilan</span></h2>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Ketika Anda mengalikan bilangan antara 1 hingga 9 dengan angka 9, maka tekniknya begitu mudah. Kami ambil contoh 7x9. Genggam semua jari Anda, dan buka jari ketujuh dari kiri. Hitung angka sebelum jari ketujuh, maka ada angka 6 bukan? Kemudian hitung angka setelah jari ketujuh yang Anda buka tadi, maka ada angka 3 bukan? Taruh angka 6 dikiri kemudian taruh angka 3 dibagian kanan.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Contoh lagi, misalkan 8x9. Genggam semua jari Anda, buka jari kedelapan dan hitung angka sebelum jari kedelapan, ada 7 angka (7 jari). Kemudian, hitung angka sesudah jari kedelapan, yaitu 2 angka (2 jari). Letakkan 7 dikiri dan 2 dikanan, maka hasilnya adalah 72.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Sangat mudah dan terus terbukti.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Perkalian Empat</span></h2>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><a href="https://belajar-mate-matika.blogspot.co.id/2017/09/download-kumpulan-rumus-cepat.html">Rumus Cepat Matematika</a> yang satu ini, hanya mampu terjadi pada beberapa kasus, sementara kasus lainnya tidak. Contoh,</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">58 x 4</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">58 x 2 = 116 x 2 = 232 (terbukti)</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Kalikan angka dengan dua, kemudian kalikan kembali hasilnya dengan dua.</span></div>
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<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">64 x 4</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">64 x 2 = 128 x 2 = 256 (terbukti)</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<o:p style="box-sizing: border-box;"><span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></o:p></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<br style="box-sizing: border-box;" /></div>
<h2 style="box-sizing: border-box; font-size: 24px; font-weight: normal; margin: 0px 0px 5px; padding: 0px;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Pencarian 15%</span></h2>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Ketika Anda ingin mendapatkan 15% dari sebuah angka, maka lakukan cara yang lebih cepat saat kalkulator Anda tidak ada di tangan. Ambil 10 persen dari kedua angka kemudian salah satu hasilnya dibagi dengan dua. Pada akhirnya, tambahkan kedua hasil tersebut.</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">1500 x 15% = (1500 x 10%) + ((1500 x 10%) : 2) = 150 + 75 = 225</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">4600 x 15% = (4600 x 10%) + ((4600 x 10%) : 2) = 460 + 230 = 690</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<br style="box-sizing: border-box;" /></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<h2 style="box-sizing: border-box; font-size: 24px; font-weight: normal; margin: 0px 0px 5px; padding: 0px;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Memudahkan Perkalian</span></h2>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Ketika Anda mempunyai angka dengan perkalian yang cukup sulit, maka dengan mudah Anda bisa mengkalikan serta membagi angka tersebut dengan angka dua untuk memudahkan pencarian hasil. </span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">32 x 125 =</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">16 x 250 =</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">8 x 500 =</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">4 x 1000 = 4,000</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Perhatikan, angka 32 terus dibagi dua hingga menjadi angka 4. Kemudian angka 125 dikalikan dua hingga menjadi angka 1.000. Hasilnya, mudah ditebak yaitu 4.000.</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<br style="box-sizing: border-box;" /></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<h2 style="box-sizing: border-box; font-size: 24px; font-weight: normal; margin: 0px 0px 5px; padding: 0px;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Pembagian Lima</span></h2>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Anda mempunyai angka yang ingin dibagi 5? Caranya sangatlah mudah, cukup kalikan angka tersebut dengan dua, kemudian ubah ke angka desimal (satu angka dibelakang koma).</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;"><br style="box-sizing: border-box;" /></span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">3952 : 5</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">3952 x 2 = 7904 = 790,4 (dijadikan decimal dengan satu angka dibelakang koma)</span></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<br style="box-sizing: border-box;" /></div>
<div class="MsoNormal" style="box-sizing: border-box; text-align: justify;">
<span style="box-sizing: border-box; font-family: "verdana" , sans-serif;">Itulah delapan Rumus Cepat Matematika saat berhitung yang semoga dapat meningkatkan pengetahuan Anda.</span></div>
</div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com1tag:blogger.com,1999:blog-1369072341694639902.post-65791647035210729632014-10-08T19:31:00.001-07:002014-10-08T19:31:43.041-07:00Trik Menghitung Pembagian<br />
Berikut ada beberapa trik yang cukup menarik hanya dengan menggunakan model perkalian dan kemudian dengan membaginya dengan kelipatan angka 10. Perhatikan penjelasan berikut ini :<br />
<br />
<ol>
<li>Untuk membagi bilangan yang dibagi 125, caranya yaitu kalikan bilangan tersebut dengan 8, kemudian bagilah dengan 1000.misalnya; 7000/125 = (7000 x 8)/1000 = 56.</li>
<li>Untuk membagi bilangan yang dibagi 50, caranya; kalikan 2 dan bagi 100. contoh ; 300/50 = (300 x 2)/100 = 6.</li>
<li>Untuk membagi bilangan yang dibagi 500, kalikan 2 kemudian bagi 1000. contoh; 7500/500 = (7500×2)/1000 = 15.</li>
<li>Untuk membagi bilangan yang dibagi 5, kalikan 2 kemudian bagi 10. contoh; 35/5 = (35×2)/10 = 7.</li>
<li>Untuk membagi bilangan yang dibagi 25, kalikan 4 kemudian bagi 100. contoh; 3700/25 = (3700×4)/100= 148.</li>
<li>Untuk bilangan yang dibagi dengan 250, kalikan 4 kemudian bagi 1000</li>
<li>Untuk bilangan yang dibagi dengan 16 2/3, kalikan 6 kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 33 1/3, kalikan 3 kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 166 2/3, kalikan 6 kemudian bagi 1000.</li>
<li>Untuk bilangan yang dibagi dengan 333 1/3, kalikan 3 kemudian bagi 1000.</li>
<li>Untuk bilangan yang dibagi dengan 6 2/3, kalikan 15 kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 66 2/3, kalikan 15 kemudian bagi 1000.</li>
<li>Untuk bilangan yang dibagi dengan 8 1/3, kalikan12 kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 83 1/3, kalikan dengan12 kemudian bagi 1000.</li>
<li>Untuk bilangan yang dibagi dengan 6 1/4, kalikan dengan16 kemudian bagi dengan100.</li>
<li>Untuk bilangan yang dibagi dengan 62 1/2, kalikan dengan16 kemudian bagi engan1000.</li>
<li>Untuk bilangan yang dibagi dengan 18 3/4, kalikan dengan16, bagi 3, kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 37 1/2, kalikan dengan8 bagi 3, kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 87 1/2 bagi 7, kalikan 8 kemudian bagi 100.</li>
<li>Untuk bilangan yang dibagi dengan 75 bagi dengan 3, kalikan 4 kemudian bagi 100.</li>
</ol>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-72645612936596237352013-10-13T23:12:00.003-07:002013-10-13T23:12:25.379-07:00Cara Mencari KPK dan FPB dengan Excel<b>Cara Mencari KPK dan FPB dengan Excel - </b>Mencari KPK (Kelipatan
Persekutuan Terkecil) dan FBP (Faktor Persekutuan Terkecil) merupakan
pelajaran matematika SD, masih ingat gak? Ada beberapa cara yang bisa
kita tempuh untuk mencarinya baik secara manual maupun dengan bantuan
sofware komputer semacam Microsoft Excel. <br />
<br />
Memanfaatkan Microsoft Excel yang pastinya ada di komputer kita untuk
mencari KPK dan FPB merupakan cara yang sangat praktis dan mudah. Namun
perlu di ingat, walaupun dengan bantuan excel menjadi sangat mudah,
jangan sampai kita malas untuk belajar secara manual. Karena bagaimana
pun siswa harus tahu cara menghitungnya secara manual dan paham tentang
konsep dasar KPK dan FPB. Jadi penggunaan software semacam ini bisa kita
jadikan alat untuk mengecek apakah sudah benar pekerjaan kita atau
belum..okey...<br />
<br />
Nah..sekarang langsung aja kita praktekin bagaimana cara Mencari KPK dan FPB dengan Excel.<br />
Misalkan ada soal nih..<br />
<br />
<i><b>Tentukan nilai KPK dan FPB dari 24, 30, dan 36 !</b></i><br />
<br />
Untuk memecahkan persoalan tersebut dengan bantuan Microsoft Excel, langkah-langkahnya adalah:<br />
1. Buka dulu microsoft excel di komputer kita<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFUsob4GtH5XwfY9fXGNvI3fMDUfonA8l8xbqHM3OHNaCuXlEzCfyffPqf45LbbvOcv1KQcsVdb03jzxo9zFAp2KdmqxFxyBWVkMpYtDEHUGTtM1IcSrs-bcBHMhUrkF_eREyie9YVTuM/s1600/mencari+KPK+dan+FPB.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFUsob4GtH5XwfY9fXGNvI3fMDUfonA8l8xbqHM3OHNaCuXlEzCfyffPqf45LbbvOcv1KQcsVdb03jzxo9zFAp2KdmqxFxyBWVkMpYtDEHUGTtM1IcSrs-bcBHMhUrkF_eREyie9YVTuM/s320/mencari+KPK+dan+FPB.png" width="211" /></a></div>
<br />
2. Setelah M. Excel kebuka, sekarang tulis aja fungsi berikut di salah satu kotak yang ada.<br />
3. Untuk mencari KPK, fungsi yang digunakan yaitu <span style="color: blue;">LCM</span> (<i>Least Common Multiple of Integer)</i><br />
<i> Ketik</i><br />
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJLmRZR1AROVhYImgWcbk_tZfX0veUbOBPyaH1xLNmRACpIwbm7vQq5WH1Xmfz9hGN7rQJJUSHMOLlrGFPl09UVP-FMrapU4BXsHZOSJ-D73UqMBMDD5R6i55yngqznujCzodMnRGeBWI/s1600/mencari+KPK+dan+FPB+2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJLmRZR1AROVhYImgWcbk_tZfX0veUbOBPyaH1xLNmRACpIwbm7vQq5WH1Xmfz9hGN7rQJJUSHMOLlrGFPl09UVP-FMrapU4BXsHZOSJ-D73UqMBMDD5R6i55yngqznujCzodMnRGeBWI/s1600/mencari+KPK+dan+FPB+2.png" /> </a></div>
<div class="separator" style="clear: both; text-align: left;">
Kemudian klik <i>Enter ,</i> hasilnya :</div>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0Ndjt9-LpoEd3FVmXE1ubpPH0XnSyT1NCU8p0hm3V7OkvwyXa2TwDQm5Ng9OJXzfr-rhlWBP8NzS0CNLpblF0BGlBAeKKb5uPaKkJtxxKml2KqFhZj_HBrC1EyOqp-9OtK-_UxGa6BfA/s1600/mencari+KPK+dan+FPB+3.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0Ndjt9-LpoEd3FVmXE1ubpPH0XnSyT1NCU8p0hm3V7OkvwyXa2TwDQm5Ng9OJXzfr-rhlWBP8NzS0CNLpblF0BGlBAeKKb5uPaKkJtxxKml2KqFhZj_HBrC1EyOqp-9OtK-_UxGa6BfA/s1600/mencari+KPK+dan+FPB+3.png" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
Jadi, KPK dari <i><b>24, 30, dan 36 </b></i>adalah <b>360</b></div>
<div style="text-align: center;">
<span style="font-size: x-small;"><span class="Apple-style-span" style="font-family: 'Courier New',Courier,monospace;"><b><span class="Apple-style-span" style="color: blue;"><br /></span></b></span></span></div>
<br />
4. Untuk mencari FPB, fungsi yang digunakan yaitu <span style="color: blue;">GCD</span><i> </i>(<i>Greatest Common Divisor</i>)<br />
<i> </i> <i>Ketik<br />
</i>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigcMkVsxwjwWaJBwzST5mF-OZpABV3mOQSdfK6nHmflb5e8PMWkKi9GiTxo51SMbrI76909Tdo1FES-INMh6xdDIoyQmbg1irzHwYDvu0ELLUZcazR-GTPHVfS2rOLxDOrLGyT_dLbS9o/s1600/mencari+KPK+dan+FPB+4.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigcMkVsxwjwWaJBwzST5mF-OZpABV3mOQSdfK6nHmflb5e8PMWkKi9GiTxo51SMbrI76909Tdo1FES-INMh6xdDIoyQmbg1irzHwYDvu0ELLUZcazR-GTPHVfS2rOLxDOrLGyT_dLbS9o/s1600/mencari+KPK+dan+FPB+4.png" /></a></div>
<br />
<div class="" style="clear: both; text-align: left;">
Kemudian klik <i>Enter ,</i> hasilnya :</div>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh98Hf_eaLeyzUzLCqQoqS0zXifB0iooDQpVmZgPhAcpWjkxC6Ypx9hOt4T8d8lQXLoiGFxnd5TkYE1hEIjd_YDQ3TRGeaoyjJNPLIXpujVI9kqKUs-fa7C3DJ_wTu14zjHWF5MiA9CZ7M/s1600/mencari+KPK+dan+FPB+5.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh98Hf_eaLeyzUzLCqQoqS0zXifB0iooDQpVmZgPhAcpWjkxC6Ypx9hOt4T8d8lQXLoiGFxnd5TkYE1hEIjd_YDQ3TRGeaoyjJNPLIXpujVI9kqKUs-fa7C3DJ_wTu14zjHWF5MiA9CZ7M/s1600/mencari+KPK+dan+FPB+5.png" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
Jadi, FPB dari <i><b>24, 30, dan 36 </b></i>adalah <b>360</b></div>
<div class="separator" style="clear: both; text-align: left;">
</div>
<div class="separator" style="clear: both; text-align: left;">
Selesai,,,,,,, Mudah bukan?? </div>
<div class="separator" style="clear: both; text-align: left;">
</div>
Semoga artikel tentang Cara Mencari KPK dan FPB dengan Excel ini bermanfaat<br />Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-49352670449842066152013-01-15T21:58:00.003-08:002013-01-15T21:58:45.961-08:00Mainan Peninggalan Pythagoras<div align="justify">
<span style="font-family: Arial, Helvetica, sans-serif;"><br />
Semasa kecil, Pythagoras pernah menyusun kerikil dalam bentuk segi-tiga
dengan jumlah kerikil yang berbeda namun berurutan:</span></div>
<div align="justify">
<span style="font-family: Arial, Helvetica, sans-serif;">1
= 1<br />
1 + 2 = 3<br />
1 + 2 + 3 = 6<br />
1 + 2 + 3 + 4 = 10<br />
1 + 2 + 3 + 4 + 5 = 15</span></div>
<div align="justify">
<br /></div>
<div align="justify">
<span style="font-family: Arial, Helvetica, sans-serif;">Dengan
menjumlah 2 angka yang bersebelahan akan ditemukan hasil suatu bilangan
yang dikuadratkan:</span></div>
<div align="justify">
<span style="font-family: Arial, Helvetica, sans-serif;">
1 + 3 = 4 (2 x 2)<br />
3 + 6 = 9 (3 x 3)<br />
6 + 10 = 16 (4 x 4)<br />
10 + 15 = 25 (5 x 5)</span></div>
<div align="justify">
<br /></div>
<div align="justify">
<span style="font-family: Arial, Helvetica, sans-serif;">“Mainan”
ini ternyata memicu terjadinya rumus Pythagoras yang terkenal: <br />
a² + b² = c². Seorang guru memberi tebakan “mainan”
ini kepada Galileo sehingga akhirnya Galileo tertarik untuk menekuni
matematika, sebagai alat untuk menjelaskan alam semesta (kosmologi).</span></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-37978870747187022772013-01-15T21:44:00.002-08:002013-01-15T21:44:22.880-08:00Aljabar iseng<br />Menghadapi problem yang ruwet adalah biasa, namun jika kembali pada problem aljabar sederhana terkadang justru menyulitkan. Cobalah dengan hati dingin dan lapang dada menjawab problem:<br /><br />x² = x.<br /><br />Mana mungkin bilangan dikuadrat sama dengan bilangan sama yang tidak dikuadratkan. Rasanya salah tulis.<br /><br />
<br />
<b>Jawaban:</b><br />Jadikan ke dalam satu ruas sehingga menjadi persamaan x² - x = 0, dengan menggunakan rumus abc dapat diperoleh x1=0 dan x2=1Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-80108313920576285722013-01-15T19:19:00.002-08:002013-01-15T22:01:47.585-08:00Tutorial Geogebra<embed allowfullscreen="true" allowscriptaccess="always" height="400" src="https://www.box.com/embed/rcrwg1lgm5u5r5b.swf" type="application/x-shockwave-flash" width="466" wmode="opaque"></embed>Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-21033488730166867282012-11-23T19:20:00.000-08:002012-11-23T19:20:31.651-08:00Menentukan akar-akar persamaan polinomial dengan GeoGebra<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Bagaimana
cara menentukan akar-akar persamaan polinomial ini?</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>4</sup>
– 4x<sup>3</sup> – x<sup>2</sup> + 16x – 12 = 0</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Penyelesaian:</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Kalau
kita tulis akar-akar polinomial itu adalah p, q, r, dan s, maka
menurut teorema vieta berlaku</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x</i><sup><i>4</i></sup><i>
– (p+q+r+s)x</i><sup><i>3</i></sup><i> + (pq + pr + ps + qr + qs +
rs)x</i><sup><i>2</i></sup><i> – (pqr + pqs + prs + qrs)x +
(pqrs)=0</i>.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Ini
artinya</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>p
+ q + r + s = 4,</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>pq
+ pr + ps + qr + qs + rs = – 1,</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>pqr
+ pqs + prs + qrs = – 16, </i><span style="font-style: normal;">dan</span></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>pqrs
= – 12</i>.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Nah,
yang akan kita lihat adalah pada pqrs nya atau pada koefisien
berderajat paling kecil, lebih mudahnya adalah biasanya yang paling
belakang dari polinomial itu. Pada persamaan itu nilai yang akan
menjadi patokan adalah – 12. Karena 12 itu adalah hasil kali dari
akar-akarnya, maka ada kemungkinan akar-akar polinomialnya adalah
faktor dari 12. Sekarang kita sebutkan faktor-faktor dari 12, yaitu
1, 2, 3, 4, 6, dan 12, itu juga berlaku untuk bilangan negatifnya.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Langkah
selanjutnya adalah menggunakan aturan Horner.</span></div>
<table cellpadding="3" cellspacing="0">
<colgroup><col width="43*"></col>
<col width="43*"></col>
<col width="43*"></col>
<col width="43*"></col>
<col width="43*"></col>
<col width="43*"></col>
</colgroup><tbody>
<tr valign="TOP">
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="JUSTIFY">
</div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>4</sup></i></span></div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>3</sup></i></span></div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>2</sup>
</i></span>
</div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>1</sup>
</i></span>
</div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: 1px solid #000000; padding: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>0</sup>
</i></span>
</div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="JUSTIFY">
<span style="font-family: Liberation Serif,serif;">koefisien</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">4</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">16</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0.1cm; padding-top: 0cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">12</span></div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="JUSTIFY">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
</div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">3</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">4</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0.1cm; padding-top: 0cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">12</span></div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1.55pt solid #000000; border-left: 1.55pt solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="RIGHT">
<span style="font-family: Liberation Serif,serif;"><i>h(x) </i>=</span></div>
</td>
<td style="border-bottom: 1.55pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1.55pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">3</span></div>
</td>
<td style="border-bottom: 1.55pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="17%"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">4</span></div>
</td>
<td style="border-bottom: 1.55pt solid #000000; border-left: 1px solid #000000; border-right: 1.55pt solid #000000; border-top: 1.55pt solid #000000; padding: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">12</span></div>
</td>
<td style="border: 1.55pt solid #000000; padding: 0.1cm;" width="17%"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">0</span></div>
</td>
</tr>
</tbody></table>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Ya,
sisanya nol. Berarti dugaan kita benar. 1 adalah faktor dari
polinomial itu. Berarti 1 adalah salah satu akar persamaan polinomial
itu. Sekarang kita punya hasil bagi h(x)= x<sup>3 </sup>– 3x<sup>2</sup>
– 4x + 12. </span>
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Secara
lengkap boleh kita tulis seperti ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>4</sup>
– 4x<sup>3</sup> – x<sup>2</sup> + 16x – 12 = (x – 1)(x<sup>3
</sup>– 3x<sup>2</sup> – 4x + 12)</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">mungkin
2 adalah akar yang lain. Siapa tau kan? Kita coba saja lagi dengan
Horner. Kita pecah lagi h(x) yang telah kita dapat.</span></div>
<table cellpadding="3" cellspacing="0" style="width: 451px;">
<colgroup><col width="83"></col>
<col width="84"></col>
<col width="84"></col>
<col width="85"></col>
<col width="84"></col>
</colgroup><tbody>
<tr valign="TOP">
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="83"><div align="JUSTIFY">
</div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>3</sup></i></span></div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>2</sup></i></span></div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1px solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="85"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>1</sup>
</i></span>
</div>
</td>
<td bgcolor="#c0c0c0" style="border-bottom: 1.30pt solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: 1px solid #000000; padding: 0.1cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>0</sup>
</i></span>
</div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="83"><div align="JUSTIFY">
<span style="font-family: Liberation Serif,serif;">koefisien</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="84"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">3</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="85"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">4</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0.1cm; padding-top: 0cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">12</span></div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="83"><div align="JUSTIFY">
<span style="font-family: Liberation Serif,serif;">2</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="84"><div align="CENTER">
</div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">2</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0cm;" width="85"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">2</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: 1px solid #000000; border-top: none; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0.1cm; padding-top: 0cm;" width="84"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">12</span></div>
</td>
</tr>
<tr valign="TOP">
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="83"><div align="RIGHT">
<span style="font-family: Liberation Serif,serif;"><i>h(x) =</i></span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="84"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">1</span></div>
</td>
<td style="border-bottom: 1px solid #000000; border-left: 1px solid #000000; border-right: none; border-top: 1.55pt solid #000000; padding-bottom: 0.1cm; padding-left: 0.1cm; padding-right: 0cm; padding-top: 0.1cm;" width="85"><div align="CENTER">
– <span style="font-family: Liberation Serif,serif;">6</span></div>
</td>
<td style="border-bottom: 1.80pt solid #000000; border-left: 1.80pt solid #000000; border-right: 1.80pt solid #000000; border-top: 1.55pt solid #000000; padding: 0.1cm;" width="84"><div align="CENTER">
<span style="font-family: Liberation Serif,serif;">0</span></div>
</td>
</tr>
</tbody></table>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Benar
sekali! :D berarti 2 juga akar persamaan polinomial itu. Kita
dapatkan <i>h(x)= </i><i>x</i><sup><i>2</i></sup><i>– x – 6</i>.
Sekarang kita punya bentuk menarik dari polinomial yang tadi menjadi
seperti ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="http://www.blogger.com/blogger.g?blogID=1855132236564101894" name="__DdeLink__364_1784494344"></a>
<span style="font-family: Liberation Serif,serif;"><i>x</i><sup><i>4</i></sup><i> –
4x</i><sup><i>3</i></sup><i> – x</i><sup><i>2</i></sup><i> + 16x –
12 </i><i>= (x – 1)(x – 2)(x</i><sup><i>2</i></sup><i>– x –
6)</i>.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Pastinya
dengan sangat mudah kita dapat memfaktorkan bentuk <i>h(x)</i>
terakhir itu menjadi seperti ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x</i><sup><i>2</i></sup><i>–
x – 6 = (x – 3)(x + 2)</i>. Sehingga secara lengkap persamaan
polinomial tadi dapat kita ubah menjadi seperti ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>4</sup>
– 4x<sup>3</sup> – x<sup>2</sup> + 16x – 12 = 0</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
⇔ <span style="font-family: Liberation Serif,serif;"><i>(x
– 1)(x – 2)(x – 3)(x + 2) = 0</i>.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Jadi
akar-akar persamaan polinomial itu adalah <i>x</i><sub><i>1</i></sub><i>
</i>= – 2, <i>x</i><sub><i>2</i></sub> = 1, <i>x</i><sub><i>3</i></sub>
= 2, dan <i>x</i><sub><i>4</i></sub> = 3.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Sekarang
kita kerjakan dengan geogebra.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Buka
geogebranya, kemudian kita masukkan polinomialnya pada input, (tanpa
= 0) seperti ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjC-xG8zd2OVrAk3116wHiaeQmCXvhGxatIHZ0cnua8SrKNYlVeM7Po2OCSe8KfZEP-RXppJQ6qjn38PtJ4DyLSjP-9jNo5ReqV0C-0bhLYVihnfycCZNt9PSkilCkqXkTQ8WC4jTOsXqg/s1600/01.+input.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjC-xG8zd2OVrAk3116wHiaeQmCXvhGxatIHZ0cnua8SrKNYlVeM7Po2OCSe8KfZEP-RXppJQ6qjn38PtJ4DyLSjP-9jNo5ReqV0C-0bhLYVihnfycCZNt9PSkilCkqXkTQ8WC4jTOsXqg/s320/01.+input.png" width="320" /></a><br />
<span style="font-family: Liberation Serif,serif;">Tekan
enter untuk melihat hasilnya, setelah disesuaikan hasilnya seperti
ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFo_Bjm4anufMJ_aHSyFrcBS5HvCdx34UvJrk083BW5YLQdIsOuRGjG7s1Z4G9EiTmghe67KVNthMFyzof_dy_syj0QkoGWW7nU-tsyD109st-A2uI31sEWYzgHJeybsVscvAkVlvn9lg/s1600/02.+muncul+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFo_Bjm4anufMJ_aHSyFrcBS5HvCdx34UvJrk083BW5YLQdIsOuRGjG7s1Z4G9EiTmghe67KVNthMFyzof_dy_syj0QkoGWW7nU-tsyD109st-A2uI31sEWYzgHJeybsVscvAkVlvn9lg/s320/02.+muncul+grafik.png" width="320" /></a><br />
<span style="font-family: Liberation Serif,serif;">Akar
persamaan artinya nilai x berapa saja sehingga polinomialnya itu
nilainya nol? Kalimat itu berarti kapan (untuk x berapa saja) grafik
itu berpotongan dengan sumbu X? Kita bisa mengetahuinya dengan sangat
mudah dengan cara begini. </span>
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">1.
Pilih intersect two object (perpotongan dua objek) </span>
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWDvnxKIc7MEY7lFKNFazDP6OxLUm6Dtg6DsROTkC5OkUy8L8otvdTUMYm8IiStBMXH17nVCd67TCGzpJE2RAFVbcj7rjt5AUeNuzEb3evgTXdxmPTnjzC1paSLH0xJKTQjVo8aweN2M8/s1600/03.+intersect.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWDvnxKIc7MEY7lFKNFazDP6OxLUm6Dtg6DsROTkC5OkUy8L8otvdTUMYm8IiStBMXH17nVCd67TCGzpJE2RAFVbcj7rjt5AUeNuzEb3evgTXdxmPTnjzC1paSLH0xJKTQjVo8aweN2M8/s1600/03.+intersect.png" /></a><br />
<span style="font-family: Liberation Serif,serif;">2.
kemudian klik grafik dan klik sumbu X.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijmU8cgc3DTDdzu_y6F6rV9Db3NR8C5v5pNojtVficmn643WGSpcPaOAs5uwnIi-1KKZ4h4XHEUqmj-h2QLPozZCST_5qmH6uGikrSfsUSehHFEtd6SWcD8C7kC3aj1dWO0RMyvyixkl8/s1600/04.+klik+potong.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijmU8cgc3DTDdzu_y6F6rV9Db3NR8C5v5pNojtVficmn643WGSpcPaOAs5uwnIi-1KKZ4h4XHEUqmj-h2QLPozZCST_5qmH6uGikrSfsUSehHFEtd6SWcD8C7kC3aj1dWO0RMyvyixkl8/s1600/04.+klik+potong.png" /></a><br />
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">3.
Seketika muncul titik perpotongan grafik dan sumbu X. Itulah akar
persamaan polinomial</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x<sup>4</sup>
– 4x<sup>3</sup> – x<sup>2</sup> + 16x – 12 = 0.</i></span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgznOT5rXcEHzkvlryuvgR0Km-lFcc-cLyMr6ANs0udlHzU9C6I7Uo_SGE5OW-B0Ml6LJMCmEe6Wh9lxfQ9e1SjdZo-u_VowKCddl9ylanu-pOwlX9EuYmrg0GPgB-K8fQnV8Fp3llsdfE/s1600/05.+titik+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgznOT5rXcEHzkvlryuvgR0Km-lFcc-cLyMr6ANs0udlHzU9C6I7Uo_SGE5OW-B0Ml6LJMCmEe6Wh9lxfQ9e1SjdZo-u_VowKCddl9ylanu-pOwlX9EuYmrg0GPgB-K8fQnV8Fp3llsdfE/s320/05.+titik+grafik.png" width="320" /></a><span style="font-family: Liberation Serif,serif;"><i> </i></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Di
sisi aljabar (sebelah kiri) akan terlihat koordinat titik A, B, C,
dan D seperti ini.</span><br />
<span style="font-family: Liberation Serif,serif;"> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRzCG-ucXI1YBNCFok_43tP4JFTp4IaboPL8gurP23mnsAIGlFHYG7uG-4ZGhInFFlrH2LCXZWknBv1nFys-OpQ50SMRUQLfc0LkTMjUClfCGL87ECuEIYDkcoB-j2fLCTrN-RtN4a2ZY/s1600/06.+titik+aljabar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRzCG-ucXI1YBNCFok_43tP4JFTp4IaboPL8gurP23mnsAIGlFHYG7uG-4ZGhInFFlrH2LCXZWknBv1nFys-OpQ50SMRUQLfc0LkTMjUClfCGL87ECuEIYDkcoB-j2fLCTrN-RtN4a2ZY/s1600/06.+titik+aljabar.png" /></a></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Mudah
sekali bukan? Akar-akarnya adalah A= – 2, <i>B</i> = 1, C = 2, dan
<i>D</i> = 3.</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i><b>Polinomial
dengan akarnya berupa bilangan irrasional.</b></i></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Sekarang bagaimana kalau
persamaan polinomialnya seperti ini? Masih bisakah kita
menyelesaikannya? Tentu saja bisa.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<a href="http://www.blogger.com/blogger.g?blogID=1855132236564101894" name="__DdeLink__369_1784494344"></a>
<span style="font-family: Liberation Serif,serif;"><i>x⁴ </i><span style="font-style: normal;">–</span><i>
2x³ </i><span style="font-style: normal;">–</span><i> x² + 6x </i><span style="font-style: normal;">–</span><i>
6 = 0</i></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Penyelesaian:</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Kita lihat, faktornya 6 adalah
1, 2, 3, dan 6. akan tetapi kalau kita masukkan bilangan-bilangan
itu, tidak menghasilkan nol. Bagaimana ini? Apa yang harus kita
lakukan?</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Jangan panik. Kita cek dulu
dengan geogebra. Masukkan polinomial itu pada kotak input kemudian
tekan enter.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYzgtFKV9n0aDhO7jZRGN7ETP4TA9j1sRdoX88kYDfBYeCYJM2FV2Se5h0BAMocYbjQAb9IKWVsBBdcDwAWQGm00F2oIz_936dpnK5AC2tRwA1ddIoREBGf4rLRt6jVAJSe7MEwttJ-iY/s1600/07.+input+lagi.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYzgtFKV9n0aDhO7jZRGN7ETP4TA9j1sRdoX88kYDfBYeCYJM2FV2Se5h0BAMocYbjQAb9IKWVsBBdcDwAWQGm00F2oIz_936dpnK5AC2tRwA1ddIoREBGf4rLRt6jVAJSe7MEwttJ-iY/s1600/07.+input+lagi.png" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Hasilnya
adalah seperti ini.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3dSvfkYFS8qZ2eUEfIauJlKeJXqQvyKBFLSpD_WCtzMmzremZQ1SzXg967ZApVDGkidQRlfJ4v8S1HujpEzNM50hqV4epoZwURSBTB2RrMEGUwruesTV5hxIuTXVJuu64j_46ArpR5yg/s1600/08.+jadi+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3dSvfkYFS8qZ2eUEfIauJlKeJXqQvyKBFLSpD_WCtzMmzremZQ1SzXg967ZApVDGkidQRlfJ4v8S1HujpEzNM50hqV4epoZwURSBTB2RrMEGUwruesTV5hxIuTXVJuu64j_46ArpR5yg/s320/08.+jadi+grafik.png" width="320" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Untuk
mengetahui akar-akar polinomialnya, kita cari titik potong antara
grafik itu dengan sumbu X. Caranya dengan intersect two object, pilih
grafiknya, kemudian pilih sumbu X.</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn9wuWTxF-ZBPpPRZw65rIFHatEx1E4_XTHKTsYx-VXldm8Ff_83g7X5_vsg9bWdMJ_WuYwj58pXLMBX_iQ9uY4GI9s32M1b0wd31Z0LMMYEFHBKt6jFVvZhktFRdoOrEsya0k_kt7NH8/s1600/09.+potong.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn9wuWTxF-ZBPpPRZw65rIFHatEx1E4_XTHKTsYx-VXldm8Ff_83g7X5_vsg9bWdMJ_WuYwj58pXLMBX_iQ9uY4GI9s32M1b0wd31Z0LMMYEFHBKt6jFVvZhktFRdoOrEsya0k_kt7NH8/s320/09.+potong.png" width="320" /></a></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Hasilnya seperti ini.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-dFL5h52GozYnJCfMmiNVHTvoK4mVC0U3bSHO1YfRBDZFHaAyPusVeAHeX7stHa_wBv-oZ_smr-_AmVafKjn2a86shCm50qJnYEEqaQDMqIUEtSl44WYqcY3hB0tUvT30-zCzqof7PyU/s1600/10a.+ketemu+titiknya+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-dFL5h52GozYnJCfMmiNVHTvoK4mVC0U3bSHO1YfRBDZFHaAyPusVeAHeX7stHa_wBv-oZ_smr-_AmVafKjn2a86shCm50qJnYEEqaQDMqIUEtSl44WYqcY3hB0tUvT30-zCzqof7PyU/s1600/10a.+ketemu+titiknya+grafik.png" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Dan
di bagian aljabar kita lihat titiknya adalah seperti ini.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUMQy2tajKQm2d8PZ8nFhbFIGMXbcyxEJeIrT3sUwueM8fYvNqvwjU1vWt_JYGJsiCH7epes8xpmwp7f1fGJqXQYVG0CYdOILNUlpXgKqjU4mX_YAerwwPtbaBVCTrikfbBTMzJO8a6A/s1600/10b.+ketemu+titiknya+aljabar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUMQy2tajKQm2d8PZ8nFhbFIGMXbcyxEJeIrT3sUwueM8fYvNqvwjU1vWt_JYGJsiCH7epes8xpmwp7f1fGJqXQYVG0CYdOILNUlpXgKqjU4mX_YAerwwPtbaBVCTrikfbBTMzJO8a6A/s1600/10b.+ketemu+titiknya+aljabar.png" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Akar-akarnya
adalah A = –<i> </i>1,73 dan B = 1,73. kok hasilnya aneh? Desimal
gitu sih? Pasti itu hasilnya adalah pembulatan. Kurang tepat dong..
Apalagi kalau nanti kita cek dengan Horner, x kita ganti dengan 1,73
mungkin tidak menghasilkan nol.</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Jangan
terburu-buru kecewa seperti itu, kawan, tidak baik. Sabar, orang
sabar akan disayang Allah. Mari kita menggunakan sarana yang ada di
geogebra untuk mengungkap apa yang terkandung di balik rahasia yang
ada. Kita pakai bantuan dari lingkaran. Klik ikon lingkaran </span><br />
<span style="font-family: Liberation Serif,serif;"> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB8enesYqP4PAVHAtelAMRVXqey89UfXOHpiCf1vm8iy8hDkbQNl8htPOuTZSZs06TL-kqQs4eYa7lv3BciiojA0hhoCpp_c270RUbj63PeiXLtVGRrmIJ1uztfaftgjOg5NBhhqGW_d4/s1600/11a.+ikon+lingkaran.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB8enesYqP4PAVHAtelAMRVXqey89UfXOHpiCf1vm8iy8hDkbQNl8htPOuTZSZs06TL-kqQs4eYa7lv3BciiojA0hhoCpp_c270RUbj63PeiXLtVGRrmIJ1uztfaftgjOg5NBhhqGW_d4/s1600/11a.+ikon+lingkaran.png" /></a><span style="font-family: Liberation Serif,serif;"> </span><br />
<span style="font-family: Liberation Serif,serif;">(<i>circle
with center through point</i>: lingkaran dengan pusat tertentu dan
melewati titik tertentu). Kemudian klik pada titik pusat O(0,0) dan
klik titik B. Apa yang terjadi?</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDWtGs261Vo0dAJsArrDH0PcgQN2e17ldOq5OmasM4fwqo3vU0g80IrO3jvNXARydZy2mMv4ibhVqmyJsvTshx6P577BggIPJZVQuuOIo7Yu9ivc76MjoI7PllN2QzXXwkrbnruO7REqk/s1600/11b.+grafik+lingkaran.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDWtGs261Vo0dAJsArrDH0PcgQN2e17ldOq5OmasM4fwqo3vU0g80IrO3jvNXARydZy2mMv4ibhVqmyJsvTshx6P577BggIPJZVQuuOIo7Yu9ivc76MjoI7PllN2QzXXwkrbnruO7REqk/s1600/11b.+grafik+lingkaran.png" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Terbentuk
satu lingkaran dengan pusat O(0,0) dan berjari-jari berapa? Kita
lihat pada bagian aljabarnya, seperti ini.</span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcWO2eW8pC7CsCxWKd4EnW5hcZfs1XYv-H3lxsJL9Dtz-HP8gh6-AJ1S7tWUu2_gpF3LetymPZLWepILhRCicG4CPGkXx4wqzhB87MnI-HOYzyOVX_8OUWEXA95SGwcSU7H0-YSs2mP7Q/s1600/11c.+lingkaran+aljabar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcWO2eW8pC7CsCxWKd4EnW5hcZfs1XYv-H3lxsJL9Dtz-HP8gh6-AJ1S7tWUu2_gpF3LetymPZLWepILhRCicG4CPGkXx4wqzhB87MnI-HOYzyOVX_8OUWEXA95SGwcSU7H0-YSs2mP7Q/s1600/11c.+lingkaran+aljabar.png" /></a><span style="font-family: Liberation Serif,serif;"> </span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Diperoleh
persamaan lingkaran <i>x² +y² = 3.</i></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Lho, itu kan lingkaran dengan
pusat (0,0) dan jari-jarinya <span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Sekarang aku tau nih.. Berarti
titik A = </span><span style="font-family: Liberation Serif,serif;"><span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span></span><span style="font-family: Liberation Serif,serif;"> dan
titik<i> </i></span><span style="font-family: Liberation Serif,serif;">B = <span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Artinya akar-akar persamaan
polinomial itu adalah </span><span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Ada satu lagi cara menarik yang
ditemukan oleh <b>Ajatoel Oelja</b>, seorang yang penuh energi dan
berselebrasi njungkel-njungkel, rol depan. Sangat unik pria yang satu
ini.</span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">Persamaan polinomial </span>
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
</i><span style="font-style: normal;">–</span><i> 2x³ </i><span style="font-style: normal;">–</span><i>
x² + 6x </i><span style="font-style: normal;">–</span><i> 6 = 0</i></span></div>
<div align="JUSTIFY" style="font-style: normal; line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;">bisa kita kerjakan dengan
mengelompokkan pangkat-pangkat yang selang-seling (pangkat genap
dengan pangkat genap: <i>x⁴, x², x</i><sup><i>0</i></sup> dan
pangkat ganjil dengan pangkat ganjil: <i>x</i><sup><i>3</i></sup><i>,
x</i><sup><i>1</i></sup><i>)</i> sehingga tercipta suasana yang sejuk
untuk dinikmati. Jadi persamaan polinomial itu kini menjadi seperti
ini.</span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
</i><span style="font-style: normal;">–</span><i> x² </i><span style="font-style: normal;">–</span><i>
6 </i><span style="font-style: normal;">–</span><i> 2x³ + 6x = 0</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
⇔
<span style="font-family: Liberation Serif,serif;"><i>(x⁴ </i><span style="font-style: normal;">–</span><i>
x² </i><span style="font-style: normal;">–</span><i> 6) </i><span style="font-style: normal;">–</span><i>
(2x³ </i><span style="font-style: normal;">–</span><i> 6x) = 0</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
⇔
<span style="font-family: Liberation Serif,serif;"><i>(x² + 2)(x² – 3) –
2x(x² – 3) = 0</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
⇔
<span style="font-family: Liberation Serif,serif;"><i>(x² – 2x + 2)(x² – 3) =
0.</i></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><span style="font-style: normal;">Jelas
bahwa </span><i>x² – 2x + 2 > 0 </i><span style="font-style: normal;">(positif).
Jadi </span><i>x² – 3</i><span style="font-style: normal;">
haruslah bernilai nol.</span></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><span style="font-style: normal;">Diperoleh
</span><i>x² – 3</i><span style="font-style: normal;"> = 0 </span></span>
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span><br />
<span class="MathJax_Preview"></span><span class="MathJax_Error">[Math Processing Error]</span> </div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
</div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><span style="font-style: normal;">kira-kira
saya sudah paham belum ya? Untuk mengujinya, silakan kerjakan soal
latihan di bawah ini dengan cara Ajatoel Oelja tadi.</span></span></div>
<div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><span style="font-style: normal;">Tentukan
akar-akar persamaan polinomial berikut ini.</span></span></div>
<ol>
<li><div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
– 2x³ + 3x² + 4x – 10 = 0</i></span></div>
</li>
<li><div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
– 2x³ + 6x – 9 = 0</i></span></div>
</li>
<li><div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
– 3x³ – x² + 15x – 20 = 0</i></span></div>
</li>
<li><div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
– x³ – 3x² + 6x – 18 = 0</i></span></div>
</li>
<li><div align="JUSTIFY" style="line-height: 150%; margin-bottom: 0cm;">
<span style="font-family: Liberation Serif,serif;"><i>x⁴
– 4x³ – x² + 28x – 42 = 0</i></span></div>
</li>
</ol>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-59801468389579199662012-11-23T19:16:00.003-08:002017-09-08T02:58:10.740-07:00Persamaan kuadrat dengan GeoGebra<a href="http://belajar-mate-matika.blogspot.com/2013/01/tutorial-geogebra.html" target="_blank">geogebra</a> mempunyai dua tampilan, sebelah kiri untuk aljabarnya, dan sebelah kanan untuk geometrinya.<br />
jika tampilan geogebra masih menggunakan bahasa inggris, kita bisa
menggantinya dengan bahasa indonesia dengan cara klik option, language, F
- L, Indonesian.<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_aJp7q-LQbAXFHPMMCxiQVoVlQ8fMS_y8Xh7SMepmXNLWbSAYrirDbBVXceup0C8nMKU-y_tVsDnr7hBmSML4PQM-qMeFnOEpMC-9XkGNmyukoAOqQOVIeJC3Ik5UUjt4wuPx5dw2Zkg/s1600/08+indonesia.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_aJp7q-LQbAXFHPMMCxiQVoVlQ8fMS_y8Xh7SMepmXNLWbSAYrirDbBVXceup0C8nMKU-y_tVsDnr7hBmSML4PQM-qMeFnOEpMC-9XkGNmyukoAOqQOVIeJC3Ik5UUjt4wuPx5dw2Zkg/s320/08+indonesia.png" width="263" /></a></div>
<br />
<br />
<br />
<br />
ok, kita mulai.<br />
sebagai contoh pertama, kita akan mencari bagaimana bentuk grafik dari persamaan kuadrat ini?<br />
x^2-7x+10=0<br />
bagaimana solusinya?<br />
pertama, masukkan x^2-7x+10 ke kotak masukan: (berada di kiri bawah) terus ditekan enter.<br />
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBKK8LOh6qNLbzU5H_WKKpgGghiAXHq8Zox6cT2QOzhRHJkeaxr3b7-umr9fkT4cKV_Ux7DkkYVz5yjSjiyRM1e8T0G2rwieqsFAz6CPuJ8e4MvTsVo2BfS-LApDWA0Pwh_NPjB43miFY/s1600/09+mas+dua.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBKK8LOh6qNLbzU5H_WKKpgGghiAXHq8Zox6cT2QOzhRHJkeaxr3b7-umr9fkT4cKV_Ux7DkkYVz5yjSjiyRM1e8T0G2rwieqsFAz6CPuJ8e4MvTsVo2BfS-LApDWA0Pwh_NPjB43miFY/s1600/09+mas+dua.png" /></a></div>
</div>
<br />
akan muncul pada jendela sebelah kiri (aljabar) f(x)=x^2-7x+10 dan pada
jendela sebelah kanan (geometri), kita bisa melihat gambar grafik dari
fungsi yang kita masukkan. di sini fungsi itu secara default (otomatis)
didefinisikan sebagai f.<br />
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDUDWu9eUGZcQG70DWj6i-YH6ZN7XIcU1_K60vHxjC10KAXNwIRaJzO1wxPkJuLd98JnQToDeXMfC3-pHCRcaeHuEhI5qZN2xuJWmnoJ939fVvcHKMss0KE21wUpO6VWx9mRaRH6CWhoA/s1600/09+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDUDWu9eUGZcQG70DWj6i-YH6ZN7XIcU1_K60vHxjC10KAXNwIRaJzO1wxPkJuLd98JnQToDeXMfC3-pHCRcaeHuEhI5qZN2xuJWmnoJ939fVvcHKMss0KE21wUpO6VWx9mRaRH6CWhoA/s320/09+grafik.png" width="320" /></a></div>
</div>
<br />
kemudian untuk mengetahui akar-akar persamaan kuadrat itu, ketikkan pada
kotak masukan: akar[f], kalau sudah menggunakan bahasa indonesia,
biasanya diketikkan 'aka' sudah muncul sendiri akar[], kalau masih
menggunakan bahasa inggris, silakan ketik root[f].<br />
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZXU93shYLhxGv2u6clWSj3DOqy2FWozi56Oo8HDZuqN_E0qvbdgucEnRtXXu9GUIqI0KhNXTeVYVUI95WCIIDKN3YnA7XTO7o-aaNuSiWX6XyjyGwXoNOkGlhcE3gRqyE25U6vVMI_eg/s1600/09+akarf.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZXU93shYLhxGv2u6clWSj3DOqy2FWozi56Oo8HDZuqN_E0qvbdgucEnRtXXu9GUIqI0KhNXTeVYVUI95WCIIDKN3YnA7XTO7o-aaNuSiWX6XyjyGwXoNOkGlhcE3gRqyE25U6vVMI_eg/s1600/09+akarf.png" /></a></div>
</div>
<br />
sekarang pada jendela aljabar, di bawah objek-objek terikat muncul
solusinya, yaitu A=(2,0) dan B=(5,0). ini berarti akar-akar persamaan
f(x)=0 adalah x1=2 dan x2=5.<br />
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_2KGWZWEsFWUULokDezzXgOROMa2txS7EY8MHcKT2CtLCYtLQkFVa_CFmsq79rX_EDUH-Jj9ZMAFaqX0Bs5mLGS6gfP-7dlnseqmiOt1BEDt6UvbPwhE4AK471E5ZgUG-07i8FexXw3c/s1600/09+aljabar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_2KGWZWEsFWUULokDezzXgOROMa2txS7EY8MHcKT2CtLCYtLQkFVa_CFmsq79rX_EDUH-Jj9ZMAFaqX0Bs5mLGS6gfP-7dlnseqmiOt1BEDt6UvbPwhE4AK471E5ZgUG-07i8FexXw3c/s1600/09+aljabar.png" /></a></div>
</div>
<div style="text-align: center;">
</div>
terlihat pada jendela geometri kedua akar itu, yaitu titik A dan B.<br />
<div style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiK3Lp6dyeaygIKezPJH6sStVwruDMysH6CuA2pdBJ7PDo0l2XyjOtouu21MzDJVbBiAG2pzelqQupUwJ6kX223Py0NJo6yB1bXUsstZYeAO5oB_Bncd8dvjtu4kfFjQAa9e0Xt7Dne7E/s1600/09+geometri.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiK3Lp6dyeaygIKezPJH6sStVwruDMysH6CuA2pdBJ7PDo0l2XyjOtouu21MzDJVbBiAG2pzelqQupUwJ6kX223Py0NJo6yB1bXUsstZYeAO5oB_Bncd8dvjtu4kfFjQAa9e0Xt7Dne7E/s1600/09+geometri.png" /></a>
</div>
<br />
ok. ilustrasi dan penyelesaiannya sudah kita dapatkan.<br />
sekarang hapus semuanya yuk..<br />
caranya, pastikan icon panah (di bawah tulisan berkas) terpilih.<br />
klik pada sembarang daerah kosong,<br />
tekan ctrl + a (menyeleksi semua objek di jendela)<br />
kemudian tekan tombol delete pada keyboard. (kalau di laptop 14" biasanya di sebelah pojok kanan atas).<br />
secara ajaib dan mengejutkan, semuanya bersih tanpa sisa.<br />
:D<br />
<br />
<i><b>episode ke dua</b></i><br />
sekarang bagaimana dengan persamaan kuadrat ini?<br />
x^2-8x+16=0<br />
yuk, kita ikuti langkahnya. masih ingat yang tadi kan?<br />
masukkan x^2-8x+16 ke kotak masukan, lalu tekan enter.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcYQY6L9UAMEjHRUjlkSkB1lD0OytecFukPoO8YUOjAxLiaigKcIKm1PwfZIdUglTfVu00YgsoZOzspBGVpV9xNfrhxn5fV2N2wVic07TeSyXQo5h-y11Qcu3Rlvr6OljvKtraQ8KMJ8M/s1600/10+satu.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcYQY6L9UAMEjHRUjlkSkB1lD0OytecFukPoO8YUOjAxLiaigKcIKm1PwfZIdUglTfVu00YgsoZOzspBGVpV9xNfrhxn5fV2N2wVic07TeSyXQo5h-y11Qcu3Rlvr6OljvKtraQ8KMJ8M/s1600/10+satu.png" /></a></div>
<br />
objek-objek bebas akan terlihat rumus fungsi f.<br />
<div style="text-align: center;">
<div style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDPQj49XYVk_mMkYhWmNX5ryQLUX-EASXhDFm5NDEsohgyk6gzRhxK2HTaSOrs7Cr4cKKoDSBsrht66JTGYZ75r2eULsazVM_-wbRwbaOfCxFzP9rWIAOgBXta_2x25PeYBHqsDwuc2cs/s1600/10+rumus.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDPQj49XYVk_mMkYhWmNX5ryQLUX-EASXhDFm5NDEsohgyk6gzRhxK2HTaSOrs7Cr4cKKoDSBsrht66JTGYZ75r2eULsazVM_-wbRwbaOfCxFzP9rWIAOgBXta_2x25PeYBHqsDwuc2cs/s1600/10+rumus.png" /></a> </div>
<div style="text-align: center;">
</div>
</div>
dan tampilan secara menyeluruh adalah seperti ini.<br />
<br />
<div style="text-align: center;">
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPaJjMM1RGvJBuUtzaOGJp51dmw5_nGyc5LuHAKIuj7ue66fEW_EWovusayLhyphenhyphenspqYEf7QaiI0pCbgtEIbvCrL0hCOkcVfFRiFOsx023Q99CYJJyI2cLhQ75tVt4oMcb20dxVfqMqBom4/s1600/10+grafik.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPaJjMM1RGvJBuUtzaOGJp51dmw5_nGyc5LuHAKIuj7ue66fEW_EWovusayLhyphenhyphenspqYEf7QaiI0pCbgtEIbvCrL0hCOkcVfFRiFOsx023Q99CYJJyI2cLhQ75tVt4oMcb20dxVfqMqBom4/s320/10+grafik.png" width="320" /></a></div>
</div>
</div>
<br />
ketikkan akar[f] pada kotak masukan, enter.<br />
hasilnya seperti ini.<br />
<div style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7OAnZQOi5-qYJWn1aMJXIsetVrcSdkxgn5uR8NkHzdQuviRxgG6PTU7mCWYi7j_-9nyErlN2TiFm2LxbMX2oheCu0raI14jm38N5T_BmPov6W3fveVmiqdosA6dFPaQmJhnSfhbdoJcE/s1600/10+akar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7OAnZQOi5-qYJWn1aMJXIsetVrcSdkxgn5uR8NkHzdQuviRxgG6PTU7mCWYi7j_-9nyErlN2TiFm2LxbMX2oheCu0raI14jm38N5T_BmPov6W3fveVmiqdosA6dFPaQmJhnSfhbdoJcE/s1600/10+akar.png" /></a></div>
<div style="text-align: left;">
</div>
<div style="text-align: left;">
gambar grafik f menyentuh sumbu x di satu titik saja. seperti ini nih..</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHM7L-igYa-KAm3rOwGnMdvnXP1QMypmDcwUqD3iicvDlulNxkniJK10ZXwj5oSgkpm1uu9ngyNWUnieh2ehNyvQhSiiz09WOGphCl9IJziZhtDWJiG54UKYynBcX9MF-D0djWJ25ab2E/s1600/10+akar2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHM7L-igYa-KAm3rOwGnMdvnXP1QMypmDcwUqD3iicvDlulNxkniJK10ZXwj5oSgkpm1uu9ngyNWUnieh2ehNyvQhSiiz09WOGphCl9IJziZhtDWJiG54UKYynBcX9MF-D0djWJ25ab2E/s1600/10+akar2.png" /></a></div>
</div>
<br />
hapus lagi semuanya.<br />
<br />
<i><b>episode ke tiga</b></i><br />
kalau fungsi ini bagaimana solusinya?<br />
x^2-6x+10=0<br />
gambar grafik fungsi terlihat naik dan tidak menyentuh sama sekali.<br />
<br />
<div style="text-align: center;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtAdVtXVoA6-dqTIACRwxn8s3qagioSY_JnDyK7_PKPd4HFnKbRDZEVy2Zl9ywdzoxU0O-klqDunlxmZBIeFO_J-EsmhC9uLskSLjQCwueKkklP3RCtcR3HvChcz7CT_iSM_6aRfw253s/s1600/11+takada.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtAdVtXVoA6-dqTIACRwxn8s3qagioSY_JnDyK7_PKPd4HFnKbRDZEVy2Zl9ywdzoxU0O-klqDunlxmZBIeFO_J-EsmhC9uLskSLjQCwueKkklP3RCtcR3HvChcz7CT_iSM_6aRfw253s/s1600/11+takada.png" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjI3jnG-N7KHdPRpY_XzHunIFZzG0LAuN-lioEuHz7m0Xdys8eouq9Xt3cvz88UOJl-hRVLPARXV88F4oMn-WpNvc4q84qdVhj04mkD4mvho4OAE4KeWBrdcvcBS1cVIW4glOS3YQGWIks/s1600/11+takada+total.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjI3jnG-N7KHdPRpY_XzHunIFZzG0LAuN-lioEuHz7m0Xdys8eouq9Xt3cvz88UOJl-hRVLPARXV88F4oMn-WpNvc4q84qdVhj04mkD4mvho4OAE4KeWBrdcvcBS1cVIW4glOS3YQGWIks/s320/11+takada+total.png" width="320" /></a></div>
</div>
<br />
dan jika kita ingin tahu berapa nilai x yang memenuhi persamaan itu maka
masukkan saja perintahnya pada kotak masukan: akar[f]. apa yang
terjadi? ya seperti ini.<br />
<div style="text-align: center;">
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-V68bE33xk9XDWMbYqgmrqVWgT-qUdsgYrh5aGWlB8luSmTescQ1RoV_eff27NWMEkl-MtwKbuSpejHbS06Sxl3xYLfY318JhX2OILtlvLEUWSQ3sytTf8EThoIm3pgv6HMtIj2XDikw/s1600/11+takada+akar.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-V68bE33xk9XDWMbYqgmrqVWgT-qUdsgYrh5aGWlB8luSmTescQ1RoV_eff27NWMEkl-MtwKbuSpejHbS06Sxl3xYLfY318JhX2OILtlvLEUWSQ3sytTf8EThoIm3pgv6HMtIj2XDikw/s1600/11+takada+akar.png" /></a></div>
</div>
<br />
<b>oh, sekarang aku tahu!!</b><br />
jadi grafik fungsi kuadrat itu ada 3 macam jenis,<br />
yang pertama bisa memotong sumbu x di dua titik,<br />
yang ke dua menyinggung sumbu x (di satu titik saja),<br />
dan yang ke tiga, bisa tidak menyentuh sumbu x sama sekali.<br />
<br />
ketika kita mengaitkan akar-akar persamaan kuadrat tadi dengan gambar grafik, ternyata pas.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSsqrfEjCFIceInZY8ilrS1IMsMkPdz5GBYRv97P5TeSQtL8omK6H14oQA6N4VOXnnMf9d_PBmkDK6vTC0Wtp9nASoHpq79GjF3FuC1CJ7J9HMRQ3cY16Qv9v9_iERxd2kdOarh-UeSLY/s1600/x12.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSsqrfEjCFIceInZY8ilrS1IMsMkPdz5GBYRv97P5TeSQtL8omK6H14oQA6N4VOXnnMf9d_PBmkDK6vTC0Wtp9nASoHpq79GjF3FuC1CJ7J9HMRQ3cY16Qv9v9_iERxd2kdOarh-UeSLY/s1600/x12.png" /></a></div>
nah, di dalam akar ini yang menentukan apakah persamaan kuadrat itu punya satu, dua, atau tidak ada sama sekali penyelesaian.<br />
kalau di dalam akar ini nilainya > 0 (baca: positif) maka nantinya akan ada dua, karena ada tanda + dan - di depannya.<br />
kalau di dalam akar nilainya = 0 (nol) maka ada satu saja akarnya, karena + akar nol = - akar nol = ya nol juga.<br />
trus kalau di dalam akar nilainya negatif, maka sementara ini tidak
boleh karena di bilangan riil tak ada akar negatif. mungkin kalau nanti
belajar bilangan kompleks ya ada nilainya. dan itu bisa kita lihat
sendiri grafik fungsi yang tidak ada penyelesaiannya itu tidak menyentuh
sama sekali sumbu x. di sumbu x tidak ada ketinggian, berarti fungsi f
nya sama dengan nol.Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com1tag:blogger.com,1999:blog-1369072341694639902.post-22099091650682813242012-11-23T19:12:00.000-08:002012-11-23T19:12:00.332-08:00Contoh penyelesaian soal dengan GeoGebraContoh soal:<br />
D adalah sebuah titik dalam segitiga sama sisi ABC yang memenuhi sudut
ADB=150 derajat. Buktikan bahwa segitiga yang dibentuk dengan mengambil
segmen CD, D'D, CD' sebagai tiga sisinya adalah segitiga siku-siku
dengan merotasikan segitiga ADB sebesar 60 derajat.<br />
<br />
Dengan menggunakan GeoGebra berikut langkah-langkahnya:<br />
Seperti yang diketahui diatas bahwa terdapat titik D dalam segitiga ABC sama sisi dengan sudut ADC sebesar 150 derajat.<br />
<br />
Langkah pertama kita buat titik A dengan menggunakan tool <b>New Point</b> kemudian membuat sudut titik D sebesar 150 derajat dengan menggunakan tool <b>Angel with Given Size</b>.<br />
<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijxo9Ecuu4u8mygI4CzrFEVpkNIy31Rf5qWxZkgh_w6pEBOrC7r-F_nEHhTe7remPvfR_hUd4LD1bDu-n0GLrnzkv5Q1m_OlHhyduaQUgC2lhGNIAr_glzCSrrnuLdxBD7JsWn5djvsmKl/s1600/gambar-1.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="207" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijxo9Ecuu4u8mygI4CzrFEVpkNIy31Rf5qWxZkgh_w6pEBOrC7r-F_nEHhTe7remPvfR_hUd4LD1bDu-n0GLrnzkv5Q1m_OlHhyduaQUgC2lhGNIAr_glzCSrrnuLdxBD7JsWn5djvsmKl/s320/gambar-1.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gambar 1</td></tr>
</tbody></table>
Ganti nama titik B dengan D sesuai yang diketahui diatas dengan mengklik dua kali titik tersebut lalu pilih rename.<br />
Langkah selanjutnya gunakan tool <b>Midpoint or Center</b> untuk mendapatkan titik B diantara titik D dan A' yang merupakan titik tengah dari DA' dan salah satu titik pada segitiga ABC.<br />
Selanjutnya membuat segitiga ABC dengan perpotongan dua lingkaran
dititik tengah A dan titik tengah B yang akan menghasilkan titik C. Tool
yang digunakan adalah <b>Circle with Center through point</b>.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNI-pF7j0rIy1BteWMrZOhox-yPAgrzu0C2jHf8ArB3txQouxqbpML9EbrzDRK49wFfkoCAx6ITrIEyM_JNJ2JBLlBlERkYfMBilW1pOlHWNUw3anEqnFkRQGEro31ucpE9q5xrd1vOu1_/s1600/gambar-2.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNI-pF7j0rIy1BteWMrZOhox-yPAgrzu0C2jHf8ArB3txQouxqbpML9EbrzDRK49wFfkoCAx6ITrIEyM_JNJ2JBLlBlERkYfMBilW1pOlHWNUw3anEqnFkRQGEro31ucpE9q5xrd1vOu1_/s320/gambar-2.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gambar 2</td></tr>
</tbody></table>
<br />
Untuk mendapatkan perpotongan di C pada dua lingkaran tersebut gunakan tool <b>Intersect Two Objects</b> dengan cara mengklik pada kedua garis yang berpotongan tersebut.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-MC7QpWDfUuOaVYUFmP7nJyZW_6l9sxZPeOzZEcejLE-LSld-G1rdflV3t_6xLhJmHhm6d9etgOiWYGvLsJdCZXHVqy5OpIpXkA6LOqbL3R8ERFa-m6U8N3JKEyytFdWT3NXaExef96J/s1600/gambar-3.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-MC7QpWDfUuOaVYUFmP7nJyZW_6l9sxZPeOzZEcejLE-LSld-G1rdflV3t_6xLhJmHhm6d9etgOiWYGvLsJdCZXHVqy5OpIpXkA6LOqbL3R8ERFa-m6U8N3JKEyytFdWT3NXaExef96J/s320/gambar-3.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gambar 3</td></tr>
</tbody></table>
Langkah selanjutnya adalah menyembunyikan bagian-bagian yang tidak perlu
dengan cara mengklik lingkaran penuh pada jendela kiri sehingga
lingkaran tersebut menjadi putih. Bila anda masih belum memahami
sebaiknya anda melihat dulu <a href="http://cinghedotcom.blogspot.com/2012/04/geogebra-pengenalan-menu-dalam-geogebra.html">pengenalan menu dalam GeoGebra</a> yang sudah saya bahas sebelumnya.<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
Selanjutnya kita akan menghubungkan ketiga titik ABC yang sudah dibuat
agar menjadi sebuah segitiga sama sisi dengan menggunakan tool Polygon
dengan cara mengklik titik A lalu titik B lalu titik C dan terakhir
titik A.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcCeMapyOiKDsisQPd15cGGeDi-TFSk5PZaNhdVerweWbTgWppax4PgPb6wbeI0rCeUoBLrOBjgK6JihffYGICJVQzsXgOoW1jKCONP_jMvTnZ9Q5nJgiX2g3zD1xcwcHXqkMO7B1-uNcF/s1600/gambar-4.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="211" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcCeMapyOiKDsisQPd15cGGeDi-TFSk5PZaNhdVerweWbTgWppax4PgPb6wbeI0rCeUoBLrOBjgK6JihffYGICJVQzsXgOoW1jKCONP_jMvTnZ9Q5nJgiX2g3zD1xcwcHXqkMO7B1-uNcF/s320/gambar-4.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gambar 4</td></tr>
</tbody></table>
Masih dengan tool yang sama ulangi langkah seperti di atas untuk membuat
segitiga ADB dengan mengklik dari A lalu B lalu D dan terakhir ke A<br />
<br />
Sekarang kita akan merotasi titik ADB sebesar 60 derajat seperti yang diketahui diatas dengan cara menggunakan tool <b>Rotate Object around point by Angle</b> yaitu dengan mengklik segitiga ADB lalu klik pada titik A maka akan menghasilkan rotasi yaitu segitiga AD'B.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHc7lwkyBQbu1pNGp5sTwTM2r27z9DvnMvmYlPdsvJ0OyRoIKtPUgGtbcMk2u8EmpMhe3CEbD-i0zHLpDhMK8sZTRvQQhce6R8N_UaYdXyWKyWnu98KIE41I4H-NLuOOlqLDiM2HYnDhTO/s1600/gambar-5.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHc7lwkyBQbu1pNGp5sTwTM2r27z9DvnMvmYlPdsvJ0OyRoIKtPUgGtbcMk2u8EmpMhe3CEbD-i0zHLpDhMK8sZTRvQQhce6R8N_UaYdXyWKyWnu98KIE41I4H-NLuOOlqLDiM2HYnDhTO/s320/gambar-5.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gambar 5</td></tr>
</tbody></table>
Langkah terakhir membuktikan segitiga CDD' merupakan segitiga siku-siku
dengan menghubungkan tiga titik C, D dan D' menggunakan tool <b>Polygon</b> agar terbentuk sebuah segitiga. Setelah itu gunakan tool <b>Angle </b>untuk melihat besar sudut pada segitiga CDD' dengan cara mengklik dari titik D lalu titik D' lalu titik C.<br />
<br />Berikut hasil akhirnya<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy72e3i0xqLDZMrkIUm0ClwBnSonybsvpfCMsO3Dh9T6_h3Fi645gQDWa_Y9RJKdQu1A-_0y-eL6BXHORTxZ3daNGa-1W8lJm3S66SZgUoX2xcQUtip6dwTRLP0WiEgLDrA5UQltbSiFg_/s1600/hasil-akhir.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="279" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy72e3i0xqLDZMrkIUm0ClwBnSonybsvpfCMsO3Dh9T6_h3Fi645gQDWa_Y9RJKdQu1A-_0y-eL6BXHORTxZ3daNGa-1W8lJm3S66SZgUoX2xcQUtip6dwTRLP0WiEgLDrA5UQltbSiFg_/s320/hasil-akhir.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Hasil akhir</td></tr>
</tbody></table>
Selamat mencoba...
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-80357628537264010452012-11-23T18:50:00.001-08:002012-11-23T18:50:26.876-08:00Cara menggunakan menu input GeoGebraLangsung saja kita buka program GeoGebra dan buat sebuah segitiga ABC.
Kenapa saya menggunakan Segitiga bukan bangun datar lainnya? karena
tutorial yang saya bahas kemarin banyak menggunakan segitiga sehingga
anda akan lebih mudah dalam mengaplikasikannya.<br />
Pada tutorial ini kita akan mencari sebuah Luas dari segitiga yang anda
buat nanti. Rumus untuk mencari Luas Segitiga adalah 1/2 x alas x
tinggi.<br />
Setelah anda membuat Segitiga ABC selanjutnya cari tinggi dari ABC dengan menggunakan <b>Tool perpendikuler line</b>. Alas yang saya gunakan adalah BC seperti gambar dibawah :<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjz4P-ecATJuuKhRBgOfNRQ4jXAB6e7ZRf3jpzYTvRXSCqOtIW2wgWzVhh4WmBYmPsvmckIPFz8Wpgcpb2vJUdNX4RV7Z46D3Kn3IOhXKTy7dVL7qUjPtOEPyFJgOja-r0_x7jPNMf0F_iw/s1600/Gambar-1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjz4P-ecATJuuKhRBgOfNRQ4jXAB6e7ZRf3jpzYTvRXSCqOtIW2wgWzVhh4WmBYmPsvmckIPFz8Wpgcpb2vJUdNX4RV7Z46D3Kn3IOhXKTy7dVL7qUjPtOEPyFJgOja-r0_x7jPNMf0F_iw/s1600/Gambar-1.jpg" /></a></div>
Setelah anda membuat garis lurus gunakan <b>Tool Intersec Two Object</b> untuk memberi perpotongan di D dan gunakan <b>Tool Segmeny between Two poin</b> untuk menarik garis dari titik A ke titik D. <br />
Sekarang masukkan input di dalam <b>Input Bar</b> yaitu <b>alas = segment [B,C]</b> dan <b>tinggi = segment [A,D] </b>dan akan keluar di <b>Tampilan Aljabar</b><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgmJgAMUvMQfWtDUUvyz1Dc7W8CfaUDVEJcOIteqr8Tr37vRzSElkpt_QJJF26S5j1WTR4xbCoiraFi3kVpeoUI2HUuMbBB3gNQYst2uTGUXTZLzEVOGf1T5zkkr7BJ-vG-cSqLqUv9mMG/s1600/Gambar-2.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgmJgAMUvMQfWtDUUvyz1Dc7W8CfaUDVEJcOIteqr8Tr37vRzSElkpt_QJJF26S5j1WTR4xbCoiraFi3kVpeoUI2HUuMbBB3gNQYst2uTGUXTZLzEVOGf1T5zkkr7BJ-vG-cSqLqUv9mMG/s1600/Gambar-2.jpg" /></a></div>
Langkah terakhir kita gunakan rumus diatas pada menu input kita tulis <b>L = 0.5*alas*tinggi</b><br />
<i>1/2 kita tulis 0.5 (gunakan titik bukan koma)</i><br />
<i>* (menyatakan perkalian)</i><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRu_jokyXevmKknMRf8a65QPhFd6Cd4z3nreOM6uJCtvlXykwJ8cT02Xuk2rPOT05o-WQ9ma-oPTi8-cBn94vnchFqD5C-dS-_pi0Y9o7FyBPNVAfApHi1MrWXLZADg6aRkqRVVZSTFbvk/s1600/Gambar-3.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRu_jokyXevmKknMRf8a65QPhFd6Cd4z3nreOM6uJCtvlXykwJ8cT02Xuk2rPOT05o-WQ9ma-oPTi8-cBn94vnchFqD5C-dS-_pi0Y9o7FyBPNVAfApHi1MrWXLZADg6aRkqRVVZSTFbvk/s1600/Gambar-3.jpg" /></a></div>
Jadi benar bahwa Luas dari segitiga ABC tersebut adalah 9.28 dan hasilnya sama dengan <b>poly1</b> yang merupakan gambar segitiga ABC yang kita buat dari awal tadi dan rumus segitiga tersebut terbukti kebenarannya.Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-64949953862719449782012-11-23T18:44:00.002-08:002012-11-23T18:44:35.127-08:00GeoGebra Software Pembelajaran Matematika Gratis<br />
<br />
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</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieWOpG7F3aHnvfNiVeSyZqKYSV-3X-tpgzubRfz_ce0rLlws6lhugbvqZKM5TPWKzDSfk8GpHKglIHohbsS3d3lN-WssQ_wdqAfUasD87VVu3EbIwXaufgKO0liPDozXwbk7w96vFHc-Rw/s1600/Geogebra.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieWOpG7F3aHnvfNiVeSyZqKYSV-3X-tpgzubRfz_ce0rLlws6lhugbvqZKM5TPWKzDSfk8GpHKglIHohbsS3d3lN-WssQ_wdqAfUasD87VVu3EbIwXaufgKO0liPDozXwbk7w96vFHc-Rw/s200/Geogebra.jpg" width="185" /></a></div>
Ada <b>Software Matematika</b> yang bagus untuk penunjang sebagai guru matematika, yaitu <b>GeoGebra</b>. Apa itu <b>GeoGebra</b>? <b>GeoGebra </b>adalah software gratis berbasis <b>pembelajaran matematika</b> yang dikembangkan oleh Markus Hohenwarter yang berasal dari Austria. <b>GeoGebra </b>bersifat dinamis dan interaktif untuk mendukung pembelajaran dan penyelesaian soal-soal matematika khususnya <b>Geometri</b>, <b>Aljabar </b>dan <b>Kalkulus</b>. Dengan <b>GeoGebra </b>kita
bisa membuat sebuah kontruksi titik yang berupa Point, Vektor,
Segmentasi ataupun Garis termasuk dengan fungsinya. Selain itu GeoGebra
juga memiliki kemampuan untuk menggunakan variable dengan angka, vektor
dan titik, menemukan turunan dan integral fungsi dan menawarkan perintah
akar. <b>GeoGebra </b>bisa digunakan dengan <b>Online </b>dan <b>Offline</b>, tetapi sebelumnya komputer anda harus terinstal Aplikasi JAVA dulu. Jika belum punya download <a href="http://www.mediafire.com/?i2586ksop94lx7s" target="_blank"><b>Disini</b></a>.<br />
<br />
Jika anda mempunyai sebuah website, <b>GeoGebra </b>juga bisa anda pasang dalam sebuah website, tetapi website anda harus mendukung Applet. File <b>HTML </b>yang
anda dapatkan dari hasil ekport dari GeoGebra dan Copy kan file HTML
tadi ke website anda dengan menambahkan awal <applet> dan diakhiri
</applet>.<br />
<br />
<b><i>Untuk kedepannya saya juga akan memposting cara menggunakan GeoGebra jadi terus ikuti blog saya ya...</i></b><br />
<br />
Untuk menggunakannya silahkan download file installernya secara offline dibawah ini:<br />
<b>GeoGebra 4.0.30.0</b> - <a href="http://www.mediafire.com/?mbu3vm590gmf8h9" rel="nofollow" target="_blank" title="Download GeoGebra Disini"><b>Download Here</b></a> <i>(Mediafire - 11Mb)</i>Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-16572899130667352442012-11-23T18:32:00.001-08:002012-11-23T18:32:33.340-08:00Mengalikan dengan 11<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Mengalikan bilangan dengan 11 adalah menjumlahkan
pasangan bilangan yang berdampingan. Contohnya 436 x 11 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> [4][4 + 3][3 + 6][6] </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=> </span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">4796</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Contoh lain: 3254 x 11 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> [3][3 +
2][2 + 5][5 + 4][4] => </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 35794</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAOl62slqz1kRiSucD2A6k1iObcgS5vo9pGAoyGeDQijZBxo6jWOX-kDXMrnD9rBHRkSFRokIw11yYIgSzIaOthhqnw3OnN00G_Bv5o-uoXx29nilhGJqnKquFhUNYXYrBfL16RXgMYkOf/s1600/3254.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAOl62slqz1kRiSucD2A6k1iObcgS5vo9pGAoyGeDQijZBxo6jWOX-kDXMrnD9rBHRkSFRokIw11yYIgSzIaOthhqnw3OnN00G_Bv5o-uoXx29nilhGJqnKquFhUNYXYrBfL16RXgMYkOf/s1600/3254.jpg" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWn3p9qA_bwANnhwnEjGIdSl7fmQUf-OEwJnJwcYzIgFn-VK15-B_-iUU16KwTrJbsgLwq9CjqPQDLoNpqJQQiF34sMjAGTmuFohaRaxzHkNHarTwOGNQjmFmxxgqK1c5gsHcsKxg6iLH2/s1600/436.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="166" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWn3p9qA_bwANnhwnEjGIdSl7fmQUf-OEwJnJwcYzIgFn-VK15-B_-iUU16KwTrJbsgLwq9CjqPQDLoNpqJQQiF34sMjAGTmuFohaRaxzHkNHarTwOGNQjmFmxxgqK1c5gsHcsKxg6iLH2/s200/436.jpg" width="200" /></a><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> </span><br />
<br />
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<span style="color: #333333; font-size: 12pt;">Kedua contoh di atas kebetulan hasil penjumlahan kedua
angka yang berdampingan berjumlah di bawah 10. Bagaimana jika hasil
penjumlahannya 10 atau lebih? Perhatikan contoh berikut:</span></div>
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</div>
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<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">4657 x 11 =></span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> [4][4+6][6+5][5+7][7] => </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">[4][10][11][12][7]</span></div>
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<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Dalam kasus ini, pindahkan digit pertama dari dua
digit yang ada dan tambahkan ke angka yang ada di kotak depan seperti berikut:
[4+1][0+1][1+1][2][7] </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 51227</span></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxZ4VZUtER6AWG4TfHAQn1b4kPbpsZPuXXJIUrgsAr_gjCUhGKH1vHcQHjAFdLhey6ZzVPF23IThclBJ3R1COVa1WZKi3pUHtAdMZW0zQlN9hPb9QErp1zVqKaT7Os5lOMg3DUBApuACkE/s1600/4657.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="135" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxZ4VZUtER6AWG4TfHAQn1b4kPbpsZPuXXJIUrgsAr_gjCUhGKH1vHcQHjAFdLhey6ZzVPF23IThclBJ3R1COVa1WZKi3pUHtAdMZW0zQlN9hPb9QErp1zVqKaT7Os5lOMg3DUBApuACkE/s200/4657.jpg" width="200" /></a></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Bagaimana cara menghitungnya di kepala? </span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Ketika kita mendengar atau melihat bilangan 4657, kita
tahu bahwa 4 + 6 = 10, 6 + 5 = 11, dan 5 + 7 = 12, maka yang kita lakukan di
kepala kita adalah:</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan pertama, karena 4 + 6 = 10, maka kita
menambah 4 dengan 1 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">= </span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">5, tersisa bilangan 0</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan kedua, karena 6 + 5 = 11 dan kita
menyimpan angka 0, maka kita menambah 0 dengan 1 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=</span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 1,
tersisa bilangan 1.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan ketiga, karena 5 + 7 = 12 dan kita
menyimpan 1, maka kita menambah 1 dengan 1 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=</span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 2, tersisa bilangan 2.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Sehingga kita mendapatkan bilangan 5, 1, 2, 2, dan
bilangan terakhir 7, menjadi 51227.</span></div>
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</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-u7C1ioZ6auPR9XoU2lKO-PryWIUEXame5u2RO9fEo562SL_8a-8rCmimw43yjit5qtnY8bk6_gl8MsUI5COQU9l-_HNJUhdTCkinn1IgXECNAlabwEO9RK8-rG0R2jfGYBBh_xgaobB/s1600/7715.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="134" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP-u7C1ioZ6auPR9XoU2lKO-PryWIUEXame5u2RO9fEo562SL_8a-8rCmimw43yjit5qtnY8bk6_gl8MsUI5COQU9l-_HNJUhdTCkinn1IgXECNAlabwEO9RK8-rG0R2jfGYBBh_xgaobB/s200/7715.jpg" width="200" /></a></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">7715 x 11 = ?</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Karena 7 + 7 = 14, 7 + 1 = 8, dan 1 + 5 = 6, maka:</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan pertama, karena 7 + 7 = 14, maka kita
menambah 7 dengan 1 = </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">8, tersisa bilangan 4</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan kedua, karena 7 + 1 = 8 (kurang dari
10) dan kita menyimpan angka 4, maka kita menambah 4 dengan 0 =</span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 4, tersisa bilangan 8.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan ketiga, karena 1 + 5 = 6 (kurang dari
10) dan kita menyimpan 8, maka kita menambah 8 dengan 0 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=</span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 8, tersisa bilangan 6.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Sehingga kita mendapatkan bilangan 8, 4, 8, 6, dan
bilangan terakhir 5, menjadi 84865.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyw8Ioq7rYmrei7t0wmjCq9xdP2fYV2Rb-441ShPvfxs4rKkmLQMXELPOvrmOtMG9_EjHXtJx4kbOQeQRxFzcIZW8cHixTm658xlVUQ9RHcBla3RljxtpuSZ7uVqrofrCriQ63nCCjoazj/s1600/999.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyw8Ioq7rYmrei7t0wmjCq9xdP2fYV2Rb-441ShPvfxs4rKkmLQMXELPOvrmOtMG9_EjHXtJx4kbOQeQRxFzcIZW8cHixTm658xlVUQ9RHcBla3RljxtpuSZ7uVqrofrCriQ63nCCjoazj/s200/999.jpg" width="200" /></a></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">999 x 11 = </span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Karena 9 + 9 = 18, dan 9 + 9 = 18, maka:</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan pertama, karena 9 + 9 = 18, maka kita
menambah 9 dengan 1 = </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">10, tersisa bilangan 8</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan kedua, karena 9 + 9 = 18 dan kita
menyimpan angka 8, maka kita menambah 8 dengan 1 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=</span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 9,
tersisa bilangan 8.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Sehingga kita mendapatkan bilangan 10, 9, 8, dan
bilangan terakhir 9, menjadi 10989.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh42QVOqYbwIYLlH84-72X_kInOMW2rsm2eFli0qzI0I5lW9b648QEY0GQ96hHAyWGSna416gY_IhGGYNm2x77PFyfuRcMpQ1lk6P-uYLiui3zNwa9JGu1pyvKiSyJWxhdzeL5qhHVNVbFT/s1600/828.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh42QVOqYbwIYLlH84-72X_kInOMW2rsm2eFli0qzI0I5lW9b648QEY0GQ96hHAyWGSna416gY_IhGGYNm2x77PFyfuRcMpQ1lk6P-uYLiui3zNwa9JGu1pyvKiSyJWxhdzeL5qhHVNVbFT/s200/828.jpg" width="200" /></a></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">828 x 11 = </span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Karena 8 + 2 = 10, dan 2 + 8 = 10, maka:</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan pertama, karena 8 + 2 = 10, maka kita
menambah 8 dengan 1 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">= </span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">9, tersisa bilangan 0</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Untuk bilangan kedua, karena 2 + 8 = 10 dan kita
menyimpan angka 0, maka kita menambah 0 dengan 1 =</span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 1,
tersisa bilangan 0.</span></div>
<span style="color: #333333; font-family: "Calibri","sans-serif"; font-size: 12.0pt; mso-ansi-language: IN; mso-ascii-theme-font: minor-latin; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-theme-font: minor-latin;">Sehingga kita mendapatkan
bilangan 9, 1, 0, dan bilangan terakhir 8, menjadi 9108.</span>Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-5132013998074441372012-10-17T00:28:00.001-07:002012-10-17T00:28:22.176-07:00Mengalikan dengan 9 dengan bilangan kembar<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">11 x 9 = 99</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">22 x 9 = ....? Kalikan 2 dengan 9 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=</span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 18, kemudian sisipkan angka 9 di tengah =></span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 198</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">33 x 9 =></span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 3 x 9 = 27, sisipkan angka 9 di tengah </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 297</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">44 x 9 = 396</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">77 x 9 = 693 ... dan seterusnya.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">111 x 9 =></span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;"></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 1 x 9 = 09, sisipkan 99 di tengah </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 999</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">222 x 9 </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 2 x 9 = 18, sisipkan 99 di tengah </span><span style="color: #333333; font-family: Wingdings; font-size: 12.0pt; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-char-type: symbol; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-symbol-font-family: Wingdings;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">=></span></span><span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"> 1998</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">555 x 9 = 4995</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">666 x 9 = 5994</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">888 x 9 = 7992</span></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-2823619421337187112012-10-17T00:27:00.001-07:002012-10-17T00:27:08.882-07:00Mengalikan dengan 9, atau 99, atau 999<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Karena 9 = 10 – 1, maka mengalikan suatu bilangan dengan
9 adalah sama persis dengan “<b>mengalikan dengan 10 – 1</b>”. Jadi, 9 x 9
adalah 9 x (10 – 1) = (9 x 10) – (90 x 1) = 90 – 9 = <b>81</b>.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Kita coba yang lain: 4 x 9 = 4 x (10 – 1) = (4 x 10) –
(4 x 1) = 40 – 4 = <b>36.</b></span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Kita persingkat prosesnya: 3 x 9 = (3 x 10) – (3 x 1)
= 30 – 3 = <b>27.</b></span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Maka:<span style="mso-spacerun: yes;"> </span>6 x 9 =
60 – 6 = 54</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"><span style="mso-spacerun: yes;"> </span>8 x
9 = 80 – 8 = 72</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;"><span style="mso-spacerun: yes;"> </span>2 x
9 = 20 – 2 = 18</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Kita coba yang lebih rumit: </span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">13 x 9 = 130 – 13 = 117</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">46 x 9 = 460 – 46 = 414</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">68 x 9 = 680 – 68 = 612</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Bagaimana jika dikalikan dengan 99? Itu berarti
dikalikan dengan (100 – 1)</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">13 x 99 = 1300 – 13 = 1287</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">46 x 99 = 4600 – 46 = 4554</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">68 x 99 = 6800 – 68 = 6732</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Dikalikan 999 = dikalikan (1000 – 1)</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">13 x 999 = 13000 – 13 = 12987</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">46 x 999 = 46000 – 46 = 45954</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">68 x 999 = 68000 – 68 = 67932</span></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0tag:blogger.com,1999:blog-1369072341694639902.post-22537260819553907882012-10-14T23:18:00.002-07:002012-10-14T23:18:23.507-07:00Mengalikan dengan 5, 25, atau 125<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Mengalikan dengan 5 sama saja dengan mengalikan suatu
bilangan dengan 10 kemudian membaginya dengan 2, karena 10 = 5 x 2.</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Contoh: 37 x 5 = (37 x 10)/2 = 370/2 = 185</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Contoh: 14 x 5 = (14 x 10)/2 = 140/2 = 70</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Bisa juga sebaliknya, membagi dulu dengan 2 baru
mengalikan dengan 10. Cara ini lebih mudah bila bilangan yang dikalikan dengan
5 adalah bilangan genap</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Contoh: 14 x 5 = (14/2) x 10 = 7 x 10 = 70</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Contoh: 66 x 5 = (66/2) x 10 = 33 x 10 = 330</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Mengalikan dengan 25 sama dengan mengalikan dengan 100
kemudian membaginya dengan 4, karena 100 = 25 x 4. Membagi dengan 4 sama dengan
membagi dengan 2 sebanyak dua kali</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">23 x 25 = (23 x 100)/4 = 2300/4 = (2300/2)/2 = 1150/2
= 575</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">48 x 25 = (48 x 100)/4 = 4800/4 = (4800/2)/2 = 2400/2
= 1200</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Jika bilangan yang dikali merupakan bilangan genap,
maka bisa dibagi terlebih dahulu dengan 4 baru dikalikan dengan 100. Membagi
dengan 4 sama dengan membagi dengan 2 sebanyak dua kali</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">48 x 25 = (48/4) x 100 = ((48/2)/2) x 100 = (24/2) x
100 = 12 x 100 = 1200</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">164 x 25 = ((164/2)/2) x 100 = (82/2) x 100 = 41 x 100
= 4100</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
</div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Mengalikan dengan 125 sama dengan mengalikan dengan
1000 kemudian membaginya dengan 8, karena 1000 = 125 x 8. Membagi dengan 8 sama
dengan membagi dengan 2 sebanyak tiga kali</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">37 x 125 = (37 x 1000)/8 = 37000/8 = ((37000/2)/2)/2 =
(18500/2)/2 = 9250/2 = 4625</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">48 x 125 = (48 x 1000)/8 = 48000/8 = ((48000/2)/2)/2 =
(24000/2)/2 = 12000/2 = 6000</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">Jika bilangan yang dikali merupakan bilangan genap,
maka bisa dibagi terlebih dahulu dengan 8 baru dikalikan dengan 1000. Membagi
dengan 8 sama dengan membagi dengan 2 sebanyak tiga kali</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">48 x 125 = (48/8) x 1000 = (((48/2)/2)/2) x 1000 =
((24/2)/2) x 1000 = (12/2) x 1000 = 6 x 1000 = 6000</span></div>
<div align="left" class="MsoNormal" style="margin-left: 0cm; text-align: left; text-indent: 0cm;">
<span style="color: #333333; font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: IN;">184 x 125 = (((184/2)/2)/2) x 1000 = ((92/2)/2) x 1000
= (46/2) x 1000 = 23 x 1000= 23000</span></div>
Matematikahttp://www.blogger.com/profile/02457970364409482535noreply@blogger.com0