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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \sin{\left(x \right)}= \frac{x^{3}}{100} - 9\) using \(\displaystyle x_0=9\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{100} + \sin{\left(x_{n} \right)} + 9}{- \frac{3 x_{n}^{2}}{100} + \cos{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 9\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (9.0000000000) - \frac{- \frac{(9.0000000000)^{3}}{100} + \sin{\left((9.0000000000) \right)} + 9}{- \frac{3 (9.0000000000)^{2}}{100} + \cos{\left((9.0000000000) \right)}} = 9.6351498801\end{equation*} \begin{equation*}x_{2} = (9.6351498801) - \frac{- \frac{(9.6351498801)^{3}}{100} + \sin{\left((9.6351498801) \right)} + 9}{- \frac{3 (9.6351498801)^{2}}{100} + \cos{\left((9.6351498801) \right)}} = 9.5942992861\end{equation*} \begin{equation*}x_{3} = (9.5942992861) - \frac{- \frac{(9.5942992861)^{3}}{100} + \sin{\left((9.5942992861) \right)} + 9}{- \frac{3 (9.5942992861)^{2}}{100} + \cos{\left((9.5942992861) \right)}} = 9.5942142678\end{equation*} \begin{equation*}x_{4} = (9.5942142678) - \frac{- \frac{(9.5942142678)^{3}}{100} + \sin{\left((9.5942142678) \right)} + 9}{- \frac{3 (9.5942142678)^{2}}{100} + \cos{\left((9.5942142678) \right)}} = 9.5942142674\end{equation*} \begin{equation*}x_{5} = (9.5942142674) - \frac{- \frac{(9.5942142674)^{3}}{100} + \sin{\left((9.5942142674) \right)} + 9}{- \frac{3 (9.5942142674)^{2}}{100} + \cos{\left((9.5942142674) \right)}} = 9.5942142674\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{x^{3}}{100} - 9$ using $x_0=9$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{100} + \sin{\left(x_{n} \right)} + 9}{- \frac{3 x_{n}^{2}}{100} + \cos{\left(x_{n} \right)}} \end{equation*}
Using $x_0 = 9$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} = (9.0000000000) - \frac{- \frac{(9.0000000000)^{3}}{100} + \sin{\left((9.0000000000) \right)} + 9}{- \frac{3 (9.0000000000)^{2}}{100} + \cos{\left((9.0000000000) \right)}} = 9.6351498801\end{equation*}
\begin{equation*}x_{2} = (9.6351498801) - \frac{- \frac{(9.6351498801)^{3}}{100} + \sin{\left((9.6351498801) \right)} + 9}{- \frac{3 (9.6351498801)^{2}}{100} + \cos{\left((9.6351498801) \right)}} = 9.5942992861\end{equation*}
\begin{equation*}x_{3} = (9.5942992861) - \frac{- \frac{(9.5942992861)^{3}}{100} + \sin{\left((9.5942992861) \right)} + 9}{- \frac{3 (9.5942992861)^{2}}{100} + \cos{\left((9.5942992861) \right)}} = 9.5942142678\end{equation*}
\begin{equation*}x_{4} = (9.5942142678) - \frac{- \frac{(9.5942142678)^{3}}{100} + \sin{\left((9.5942142678) \right)} + 9}{- \frac{3 (9.5942142678)^{2}}{100} + \cos{\left((9.5942142678) \right)}} = 9.5942142674\end{equation*}
\begin{equation*}x_{5} = (9.5942142674) - \frac{- \frac{(9.5942142674)^{3}}{100} + \sin{\left((9.5942142674) \right)} + 9}{- \frac{3 (9.5942142674)^{2}}{100} + \cos{\left((9.5942142674) \right)}} = 9.5942142674\end{equation*}
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\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use Newton's method to find the first 5 approximations of the solution to the equation <img class="equation_image" title=" \displaystyle \sin{\left(x \right)}= \frac{x^{3}}{100} - 9 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B100%7D%20-%209%20" alt="LaTeX: \displaystyle \sin{\left(x \right)}= \frac{x^{3}}{100} - 9 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{x^{3}}{100} - 9 " /> using <img class="equation_image" title=" \displaystyle x_0=9 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D9%20" alt="LaTeX: \displaystyle x_0=9 " data-equation-content=" \displaystyle x_0=9 " /> . </p> </p><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{100} + \sin{\left(x_{n} \right)} + 9}{- \frac{3 x_{n}^{2}}{100} + \cos{\left(x_{n} \right)}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7Bx_%7Bn%7D%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20x_%7Bn%7D%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%7D%20%20%20" alt="LaTeX: x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{100} + \sin{\left(x_{n} \right)} + 9}{- \frac{3 x_{n}^{2}}{100} + \cos{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{100} + \sin{\left(x_{n} \right)} + 9}{- \frac{3 x_{n}^{2}}{100} + \cos{\left(x_{n} \right)}} " />
Using <img class="equation_image" title=" \displaystyle x_0 = 9 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%209%20" alt="LaTeX: \displaystyle x_0 = 9 " data-equation-content=" \displaystyle x_0 = 9 " /> and <img class="equation_image" title=" \displaystyle n = 0,1,2,3, " src="/equation_images/%20%5Cdisplaystyle%20n%20%3D%200%2C1%2C2%2C3%2C%20" alt="LaTeX: \displaystyle n = 0,1,2,3, " data-equation-content=" \displaystyle n = 0,1,2,3, " /> and <img class="equation_image" title=" \displaystyle 4 " src="/equation_images/%20%5Cdisplaystyle%204%20" alt="LaTeX: \displaystyle 4 " data-equation-content=" \displaystyle 4 " /> gives:
<img class="equation_image" title=" x_{1} = (9.0000000000) - \frac{- \frac{(9.0000000000)^{3}}{100} + \sin{\left((9.0000000000) \right)} + 9}{- \frac{3 (9.0000000000)^{2}}{100} + \cos{\left((9.0000000000) \right)}} = 9.6351498801 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%289.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%289.0000000000%29%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28%289.0000000000%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20%289.0000000000%29%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.0000000000%29%20%5Cright%29%7D%7D%20%3D%209.6351498801%20" alt="LaTeX: x_{1} = (9.0000000000) - \frac{- \frac{(9.0000000000)^{3}}{100} + \sin{\left((9.0000000000) \right)} + 9}{- \frac{3 (9.0000000000)^{2}}{100} + \cos{\left((9.0000000000) \right)}} = 9.6351498801 " data-equation-content=" x_{1} = (9.0000000000) - \frac{- \frac{(9.0000000000)^{3}}{100} + \sin{\left((9.0000000000) \right)} + 9}{- \frac{3 (9.0000000000)^{2}}{100} + \cos{\left((9.0000000000) \right)}} = 9.6351498801 " />
<img class="equation_image" title=" x_{2} = (9.6351498801) - \frac{- \frac{(9.6351498801)^{3}}{100} + \sin{\left((9.6351498801) \right)} + 9}{- \frac{3 (9.6351498801)^{2}}{100} + \cos{\left((9.6351498801) \right)}} = 9.5942992861 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%289.6351498801%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%289.6351498801%29%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28%289.6351498801%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20%289.6351498801%29%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.6351498801%29%20%5Cright%29%7D%7D%20%3D%209.5942992861%20" alt="LaTeX: x_{2} = (9.6351498801) - \frac{- \frac{(9.6351498801)^{3}}{100} + \sin{\left((9.6351498801) \right)} + 9}{- \frac{3 (9.6351498801)^{2}}{100} + \cos{\left((9.6351498801) \right)}} = 9.5942992861 " data-equation-content=" x_{2} = (9.6351498801) - \frac{- \frac{(9.6351498801)^{3}}{100} + \sin{\left((9.6351498801) \right)} + 9}{- \frac{3 (9.6351498801)^{2}}{100} + \cos{\left((9.6351498801) \right)}} = 9.5942992861 " />
<img class="equation_image" title=" x_{3} = (9.5942992861) - \frac{- \frac{(9.5942992861)^{3}}{100} + \sin{\left((9.5942992861) \right)} + 9}{- \frac{3 (9.5942992861)^{2}}{100} + \cos{\left((9.5942992861) \right)}} = 9.5942142678 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%289.5942992861%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%289.5942992861%29%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28%289.5942992861%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20%289.5942992861%29%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.5942992861%29%20%5Cright%29%7D%7D%20%3D%209.5942142678%20" alt="LaTeX: x_{3} = (9.5942992861) - \frac{- \frac{(9.5942992861)^{3}}{100} + \sin{\left((9.5942992861) \right)} + 9}{- \frac{3 (9.5942992861)^{2}}{100} + \cos{\left((9.5942992861) \right)}} = 9.5942142678 " data-equation-content=" x_{3} = (9.5942992861) - \frac{- \frac{(9.5942992861)^{3}}{100} + \sin{\left((9.5942992861) \right)} + 9}{- \frac{3 (9.5942992861)^{2}}{100} + \cos{\left((9.5942992861) \right)}} = 9.5942142678 " />
<img class="equation_image" title=" x_{4} = (9.5942142678) - \frac{- \frac{(9.5942142678)^{3}}{100} + \sin{\left((9.5942142678) \right)} + 9}{- \frac{3 (9.5942142678)^{2}}{100} + \cos{\left((9.5942142678) \right)}} = 9.5942142674 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%289.5942142678%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%289.5942142678%29%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28%289.5942142678%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20%289.5942142678%29%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.5942142678%29%20%5Cright%29%7D%7D%20%3D%209.5942142674%20" alt="LaTeX: x_{4} = (9.5942142678) - \frac{- \frac{(9.5942142678)^{3}}{100} + \sin{\left((9.5942142678) \right)} + 9}{- \frac{3 (9.5942142678)^{2}}{100} + \cos{\left((9.5942142678) \right)}} = 9.5942142674 " data-equation-content=" x_{4} = (9.5942142678) - \frac{- \frac{(9.5942142678)^{3}}{100} + \sin{\left((9.5942142678) \right)} + 9}{- \frac{3 (9.5942142678)^{2}}{100} + \cos{\left((9.5942142678) \right)}} = 9.5942142674 " />
<img class="equation_image" title=" x_{5} = (9.5942142674) - \frac{- \frac{(9.5942142674)^{3}}{100} + \sin{\left((9.5942142674) \right)} + 9}{- \frac{3 (9.5942142674)^{2}}{100} + \cos{\left((9.5942142674) \right)}} = 9.5942142674 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%289.5942142674%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%289.5942142674%29%5E%7B3%7D%7D%7B100%7D%20%2B%20%5Csin%7B%5Cleft%28%289.5942142674%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B3%20%289.5942142674%29%5E%7B2%7D%7D%7B100%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.5942142674%29%20%5Cright%29%7D%7D%20%3D%209.5942142674%20" alt="LaTeX: x_{5} = (9.5942142674) - \frac{- \frac{(9.5942142674)^{3}}{100} + \sin{\left((9.5942142674) \right)} + 9}{- \frac{3 (9.5942142674)^{2}}{100} + \cos{\left((9.5942142674) \right)}} = 9.5942142674 " data-equation-content=" x_{5} = (9.5942142674) - \frac{- \frac{(9.5942142674)^{3}}{100} + \sin{\left((9.5942142674) \right)} + 9}{- \frac{3 (9.5942142674)^{2}}{100} + \cos{\left((9.5942142674) \right)}} = 9.5942142674 " />
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