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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \cos{\left(x \right)}= \frac{9 x^{3}}{250} - 6\) using \(\displaystyle x_0=5\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 6}{- \frac{27 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 5\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 6}{- \frac{27 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 6.0244598552\end{equation*} \begin{equation*}x_{2} = (6.0244598552) - \frac{- \frac{9 (6.0244598552)^{3}}{250} + \cos{\left((6.0244598552) \right)} + 6}{- \frac{27 (6.0244598552)^{2}}{250} - \sin{\left((6.0244598552) \right)}} = 5.7775187984\end{equation*} \begin{equation*}x_{3} = (5.7775187984) - \frac{- \frac{9 (5.7775187984)^{3}}{250} + \cos{\left((5.7775187984) \right)} + 6}{- \frac{27 (5.7775187984)^{2}}{250} - \sin{\left((5.7775187984) \right)}} = 5.7557861087\end{equation*} \begin{equation*}x_{4} = (5.7557861087) - \frac{- \frac{9 (5.7557861087)^{3}}{250} + \cos{\left((5.7557861087) \right)} + 6}{- \frac{27 (5.7557861087)^{2}}{250} - \sin{\left((5.7557861087) \right)}} = 5.7556234554\end{equation*} \begin{equation*}x_{5} = (5.7556234554) - \frac{- \frac{9 (5.7556234554)^{3}}{250} + \cos{\left((5.7556234554) \right)} + 6}{- \frac{27 (5.7556234554)^{2}}{250} - \sin{\left((5.7556234554) \right)}} = 5.7556234464\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\cos{\left(x \right)}= \frac{9 x^{3}}{250} - 6$ using $x_0=5$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 6}{- \frac{27 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} \end{equation*}
Using $x_0 = 5$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 6}{- \frac{27 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 6.0244598552\end{equation*}
\begin{equation*}x_{2} = (6.0244598552) - \frac{- \frac{9 (6.0244598552)^{3}}{250} + \cos{\left((6.0244598552) \right)} + 6}{- \frac{27 (6.0244598552)^{2}}{250} - \sin{\left((6.0244598552) \right)}} = 5.7775187984\end{equation*}
\begin{equation*}x_{3} = (5.7775187984) - \frac{- \frac{9 (5.7775187984)^{3}}{250} + \cos{\left((5.7775187984) \right)} + 6}{- \frac{27 (5.7775187984)^{2}}{250} - \sin{\left((5.7775187984) \right)}} = 5.7557861087\end{equation*}
\begin{equation*}x_{4} = (5.7557861087) - \frac{- \frac{9 (5.7557861087)^{3}}{250} + \cos{\left((5.7557861087) \right)} + 6}{- \frac{27 (5.7557861087)^{2}}{250} - \sin{\left((5.7557861087) \right)}} = 5.7556234554\end{equation*}
\begin{equation*}x_{5} = (5.7556234554) - \frac{- \frac{9 (5.7556234554)^{3}}{250} + \cos{\left((5.7556234554) \right)} + 6}{- \frac{27 (5.7556234554)^{2}}{250} - \sin{\left((5.7556234554) \right)}} = 5.7556234464\end{equation*}
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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 \cos{\left(x \right)}= \frac{9 x^{3}}{250} - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B9%20x%5E%7B3%7D%7D%7B250%7D%20-%206%20" alt="LaTeX: \displaystyle \cos{\left(x \right)}= \frac{9 x^{3}}{250} - 6 " data-equation-content=" \displaystyle \cos{\left(x \right)}= \frac{9 x^{3}}{250} - 6 " /> using <img class="equation_image" title=" \displaystyle x_0=5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D5%20" alt="LaTeX: \displaystyle x_0=5 " data-equation-content=" \displaystyle x_0=5 " /> . </p> </p><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 6}{- \frac{27 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20x_%7Bn%7D%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20x_%7Bn%7D%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%7D%20%20%20" alt="LaTeX: x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 6}{- \frac{27 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{250} + \cos{\left(x_{n} \right)} + 6}{- \frac{27 x_{n}^{2}}{250} - \sin{\left(x_{n} \right)}} " />
Using <img class="equation_image" title=" \displaystyle x_0 = 5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%205%20" alt="LaTeX: \displaystyle x_0 = 5 " data-equation-content=" \displaystyle x_0 = 5 " /> 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} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 6}{- \frac{27 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 6.0244598552 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%285.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%285.0000000000%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20%285.0000000000%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%7D%20%3D%206.0244598552%20" alt="LaTeX: x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 6}{- \frac{27 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 6.0244598552 " data-equation-content=" x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{250} + \cos{\left((5.0000000000) \right)} + 6}{- \frac{27 (5.0000000000)^{2}}{250} - \sin{\left((5.0000000000) \right)}} = 6.0244598552 " />
<img class="equation_image" title=" x_{2} = (6.0244598552) - \frac{- \frac{9 (6.0244598552)^{3}}{250} + \cos{\left((6.0244598552) \right)} + 6}{- \frac{27 (6.0244598552)^{2}}{250} - \sin{\left((6.0244598552) \right)}} = 5.7775187984 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%286.0244598552%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%286.0244598552%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%286.0244598552%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20%286.0244598552%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%286.0244598552%29%20%5Cright%29%7D%7D%20%3D%205.7775187984%20" alt="LaTeX: x_{2} = (6.0244598552) - \frac{- \frac{9 (6.0244598552)^{3}}{250} + \cos{\left((6.0244598552) \right)} + 6}{- \frac{27 (6.0244598552)^{2}}{250} - \sin{\left((6.0244598552) \right)}} = 5.7775187984 " data-equation-content=" x_{2} = (6.0244598552) - \frac{- \frac{9 (6.0244598552)^{3}}{250} + \cos{\left((6.0244598552) \right)} + 6}{- \frac{27 (6.0244598552)^{2}}{250} - \sin{\left((6.0244598552) \right)}} = 5.7775187984 " />
<img class="equation_image" title=" x_{3} = (5.7775187984) - \frac{- \frac{9 (5.7775187984)^{3}}{250} + \cos{\left((5.7775187984) \right)} + 6}{- \frac{27 (5.7775187984)^{2}}{250} - \sin{\left((5.7775187984) \right)}} = 5.7557861087 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%285.7775187984%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%285.7775187984%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.7775187984%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20%285.7775187984%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%285.7775187984%29%20%5Cright%29%7D%7D%20%3D%205.7557861087%20" alt="LaTeX: x_{3} = (5.7775187984) - \frac{- \frac{9 (5.7775187984)^{3}}{250} + \cos{\left((5.7775187984) \right)} + 6}{- \frac{27 (5.7775187984)^{2}}{250} - \sin{\left((5.7775187984) \right)}} = 5.7557861087 " data-equation-content=" x_{3} = (5.7775187984) - \frac{- \frac{9 (5.7775187984)^{3}}{250} + \cos{\left((5.7775187984) \right)} + 6}{- \frac{27 (5.7775187984)^{2}}{250} - \sin{\left((5.7775187984) \right)}} = 5.7557861087 " />
<img class="equation_image" title=" x_{4} = (5.7557861087) - \frac{- \frac{9 (5.7557861087)^{3}}{250} + \cos{\left((5.7557861087) \right)} + 6}{- \frac{27 (5.7557861087)^{2}}{250} - \sin{\left((5.7557861087) \right)}} = 5.7556234554 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%285.7557861087%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%285.7557861087%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.7557861087%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20%285.7557861087%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%285.7557861087%29%20%5Cright%29%7D%7D%20%3D%205.7556234554%20" alt="LaTeX: x_{4} = (5.7557861087) - \frac{- \frac{9 (5.7557861087)^{3}}{250} + \cos{\left((5.7557861087) \right)} + 6}{- \frac{27 (5.7557861087)^{2}}{250} - \sin{\left((5.7557861087) \right)}} = 5.7556234554 " data-equation-content=" x_{4} = (5.7557861087) - \frac{- \frac{9 (5.7557861087)^{3}}{250} + \cos{\left((5.7557861087) \right)} + 6}{- \frac{27 (5.7557861087)^{2}}{250} - \sin{\left((5.7557861087) \right)}} = 5.7556234554 " />
<img class="equation_image" title=" x_{5} = (5.7556234554) - \frac{- \frac{9 (5.7556234554)^{3}}{250} + \cos{\left((5.7556234554) \right)} + 6}{- \frac{27 (5.7556234554)^{2}}{250} - \sin{\left((5.7556234554) \right)}} = 5.7556234464 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%285.7556234554%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%285.7556234554%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.7556234554%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B27%20%285.7556234554%29%5E%7B2%7D%7D%7B250%7D%20-%20%5Csin%7B%5Cleft%28%285.7556234554%29%20%5Cright%29%7D%7D%20%3D%205.7556234464%20" alt="LaTeX: x_{5} = (5.7556234554) - \frac{- \frac{9 (5.7556234554)^{3}}{250} + \cos{\left((5.7556234554) \right)} + 6}{- \frac{27 (5.7556234554)^{2}}{250} - \sin{\left((5.7556234554) \right)}} = 5.7556234464 " data-equation-content=" x_{5} = (5.7556234554) - \frac{- \frac{9 (5.7556234554)^{3}}{250} + \cos{\left((5.7556234554) \right)} + 6}{- \frac{27 (5.7556234554)^{2}}{250} - \sin{\left((5.7556234554) \right)}} = 5.7556234464 " />
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