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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{3 x_{n}^{3}}{250} + \sin{\left(x_{n} \right)} + 7}{- \frac{9 x_{n}^{2}}{250} + \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{3 (9.0000000000)^{3}}{250} + \sin{\left((9.0000000000) \right)} + 7}{- \frac{9 (9.0000000000)^{2}}{250} + \cos{\left((9.0000000000) \right)}} = 8.6509443308\end{equation*} \begin{equation*}x_{2} = (8.6509443308) - \frac{- \frac{3 (8.6509443308)^{3}}{250} + \sin{\left((8.6509443308) \right)} + 7}{- \frac{9 (8.6509443308)^{2}}{250} + \cos{\left((8.6509443308) \right)}} = 8.6303435217\end{equation*} \begin{equation*}x_{3} = (8.6303435217) - \frac{- \frac{3 (8.6303435217)^{3}}{250} + \sin{\left((8.6303435217) \right)} + 7}{- \frac{9 (8.6303435217)^{2}}{250} + \cos{\left((8.6303435217) \right)}} = 8.6302603173\end{equation*} \begin{equation*}x_{4} = (8.6302603173) - \frac{- \frac{3 (8.6302603173)^{3}}{250} + \sin{\left((8.6302603173) \right)} + 7}{- \frac{9 (8.6302603173)^{2}}{250} + \cos{\left((8.6302603173) \right)}} = 8.6302603159\end{equation*} \begin{equation*}x_{5} = (8.6302603159) - \frac{- \frac{3 (8.6302603159)^{3}}{250} + \sin{\left((8.6302603159) \right)} + 7}{- \frac{9 (8.6302603159)^{2}}{250} + \cos{\left((8.6302603159) \right)}} = 8.6302603159\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{3 x^{3}}{250} - 7$ using $x_0=9$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{3 x_{n}^{3}}{250} + \sin{\left(x_{n} \right)} + 7}{- \frac{9 x_{n}^{2}}{250} + \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{3 (9.0000000000)^{3}}{250} + \sin{\left((9.0000000000) \right)} + 7}{- \frac{9 (9.0000000000)^{2}}{250} + \cos{\left((9.0000000000) \right)}} = 8.6509443308\end{equation*}
\begin{equation*}x_{2} =  (8.6509443308) - \frac{- \frac{3 (8.6509443308)^{3}}{250} + \sin{\left((8.6509443308) \right)} + 7}{- \frac{9 (8.6509443308)^{2}}{250} + \cos{\left((8.6509443308) \right)}} = 8.6303435217\end{equation*}
\begin{equation*}x_{3} =  (8.6303435217) - \frac{- \frac{3 (8.6303435217)^{3}}{250} + \sin{\left((8.6303435217) \right)} + 7}{- \frac{9 (8.6303435217)^{2}}{250} + \cos{\left((8.6303435217) \right)}} = 8.6302603173\end{equation*}
\begin{equation*}x_{4} =  (8.6302603173) - \frac{- \frac{3 (8.6302603173)^{3}}{250} + \sin{\left((8.6302603173) \right)} + 7}{- \frac{9 (8.6302603173)^{2}}{250} + \cos{\left((8.6302603173) \right)}} = 8.6302603159\end{equation*}
\begin{equation*}x_{5} =  (8.6302603159) - \frac{- \frac{3 (8.6302603159)^{3}}{250} + \sin{\left((8.6302603159) \right)} + 7}{- \frac{9 (8.6302603159)^{2}}{250} + \cos{\left((8.6302603159) \right)}} = 8.6302603159\end{equation*}
}

\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}

HTML for Canvas

<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{3 x^{3}}{250} - 7 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B3%20x%5E%7B3%7D%7D%7B250%7D%20-%207%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= \frac{3 x^{3}}{250} - 7 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{3 x^{3}}{250} - 7 " />  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>

HTML for Canvas

<p> <p>Using the formula for Newton's method gives
 <img class="equation_image" title=" x_{n+1} =  x_{n} - \frac{- \frac{3 x_{n}^{3}}{250} + \sin{\left(x_{n} \right)} + 7}{- \frac{9 x_{n}^{2}}{250} + \cos{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20x_%7Bn%7D%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20x_%7Bn%7D%5E%7B2%7D%7D%7B250%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{3 x_{n}^{3}}{250} + \sin{\left(x_{n} \right)} + 7}{- \frac{9 x_{n}^{2}}{250} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{3 x_{n}^{3}}{250} + \sin{\left(x_{n} \right)} + 7}{- \frac{9 x_{n}^{2}}{250} + \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{3 (9.0000000000)^{3}}{250} + \sin{\left((9.0000000000) \right)} + 7}{- \frac{9 (9.0000000000)^{2}}{250} + \cos{\left((9.0000000000) \right)}} = 8.6509443308 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%289.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20%289.0000000000%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28%289.0000000000%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20%289.0000000000%29%5E%7B2%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%289.0000000000%29%20%5Cright%29%7D%7D%20%3D%208.6509443308%20" alt="LaTeX:  x_{1} =  (9.0000000000) - \frac{- \frac{3 (9.0000000000)^{3}}{250} + \sin{\left((9.0000000000) \right)} + 7}{- \frac{9 (9.0000000000)^{2}}{250} + \cos{\left((9.0000000000) \right)}} = 8.6509443308 " data-equation-content=" x_{1} =  (9.0000000000) - \frac{- \frac{3 (9.0000000000)^{3}}{250} + \sin{\left((9.0000000000) \right)} + 7}{- \frac{9 (9.0000000000)^{2}}{250} + \cos{\left((9.0000000000) \right)}} = 8.6509443308 " /> 
 <img class="equation_image" title=" x_{2} =  (8.6509443308) - \frac{- \frac{3 (8.6509443308)^{3}}{250} + \sin{\left((8.6509443308) \right)} + 7}{- \frac{9 (8.6509443308)^{2}}{250} + \cos{\left((8.6509443308) \right)}} = 8.6303435217 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%288.6509443308%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20%288.6509443308%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28%288.6509443308%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20%288.6509443308%29%5E%7B2%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%288.6509443308%29%20%5Cright%29%7D%7D%20%3D%208.6303435217%20" alt="LaTeX:  x_{2} =  (8.6509443308) - \frac{- \frac{3 (8.6509443308)^{3}}{250} + \sin{\left((8.6509443308) \right)} + 7}{- \frac{9 (8.6509443308)^{2}}{250} + \cos{\left((8.6509443308) \right)}} = 8.6303435217 " data-equation-content=" x_{2} =  (8.6509443308) - \frac{- \frac{3 (8.6509443308)^{3}}{250} + \sin{\left((8.6509443308) \right)} + 7}{- \frac{9 (8.6509443308)^{2}}{250} + \cos{\left((8.6509443308) \right)}} = 8.6303435217 " /> 
 <img class="equation_image" title=" x_{3} =  (8.6303435217) - \frac{- \frac{3 (8.6303435217)^{3}}{250} + \sin{\left((8.6303435217) \right)} + 7}{- \frac{9 (8.6303435217)^{2}}{250} + \cos{\left((8.6303435217) \right)}} = 8.6302603173 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%288.6303435217%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20%288.6303435217%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28%288.6303435217%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20%288.6303435217%29%5E%7B2%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%288.6303435217%29%20%5Cright%29%7D%7D%20%3D%208.6302603173%20" alt="LaTeX:  x_{3} =  (8.6303435217) - \frac{- \frac{3 (8.6303435217)^{3}}{250} + \sin{\left((8.6303435217) \right)} + 7}{- \frac{9 (8.6303435217)^{2}}{250} + \cos{\left((8.6303435217) \right)}} = 8.6302603173 " data-equation-content=" x_{3} =  (8.6303435217) - \frac{- \frac{3 (8.6303435217)^{3}}{250} + \sin{\left((8.6303435217) \right)} + 7}{- \frac{9 (8.6303435217)^{2}}{250} + \cos{\left((8.6303435217) \right)}} = 8.6302603173 " /> 
 <img class="equation_image" title=" x_{4} =  (8.6302603173) - \frac{- \frac{3 (8.6302603173)^{3}}{250} + \sin{\left((8.6302603173) \right)} + 7}{- \frac{9 (8.6302603173)^{2}}{250} + \cos{\left((8.6302603173) \right)}} = 8.6302603159 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%288.6302603173%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20%288.6302603173%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28%288.6302603173%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20%288.6302603173%29%5E%7B2%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%288.6302603173%29%20%5Cright%29%7D%7D%20%3D%208.6302603159%20" alt="LaTeX:  x_{4} =  (8.6302603173) - \frac{- \frac{3 (8.6302603173)^{3}}{250} + \sin{\left((8.6302603173) \right)} + 7}{- \frac{9 (8.6302603173)^{2}}{250} + \cos{\left((8.6302603173) \right)}} = 8.6302603159 " data-equation-content=" x_{4} =  (8.6302603173) - \frac{- \frac{3 (8.6302603173)^{3}}{250} + \sin{\left((8.6302603173) \right)} + 7}{- \frac{9 (8.6302603173)^{2}}{250} + \cos{\left((8.6302603173) \right)}} = 8.6302603159 " /> 
 <img class="equation_image" title=" x_{5} =  (8.6302603159) - \frac{- \frac{3 (8.6302603159)^{3}}{250} + \sin{\left((8.6302603159) \right)} + 7}{- \frac{9 (8.6302603159)^{2}}{250} + \cos{\left((8.6302603159) \right)}} = 8.6302603159 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%288.6302603159%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B3%20%288.6302603159%29%5E%7B3%7D%7D%7B250%7D%20%2B%20%5Csin%7B%5Cleft%28%288.6302603159%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B9%20%288.6302603159%29%5E%7B2%7D%7D%7B250%7D%20%2B%20%5Ccos%7B%5Cleft%28%288.6302603159%29%20%5Cright%29%7D%7D%20%3D%208.6302603159%20" alt="LaTeX:  x_{5} =  (8.6302603159) - \frac{- \frac{3 (8.6302603159)^{3}}{250} + \sin{\left((8.6302603159) \right)} + 7}{- \frac{9 (8.6302603159)^{2}}{250} + \cos{\left((8.6302603159) \right)}} = 8.6302603159 " data-equation-content=" x_{5} =  (8.6302603159) - \frac{- \frac{3 (8.6302603159)^{3}}{250} + \sin{\left((8.6302603159) \right)} + 7}{- \frac{9 (8.6302603159)^{2}}{250} + \cos{\left((8.6302603159) \right)}} = 8.6302603159 " /> 
</p> </p>