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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- x_{n}^{3} + \sin{\left(x_{n} \right)} + 8}{- 3 x_{n}^{2} + \cos{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 3\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (3.0000000000) - \frac{- (3.0000000000)^{3} + \sin{\left((3.0000000000) \right)} + 8}{- 3 (3.0000000000)^{2} + \cos{\left((3.0000000000) \right)}} = 2.3262277582\end{equation*} \begin{equation*}x_{2} = (2.3262277582) - \frac{- (2.3262277582)^{3} + \sin{\left((2.3262277582) \right)} + 8}{- 3 (2.3262277582)^{2} + \cos{\left((2.3262277582) \right)}} = 2.0980887335\end{equation*} \begin{equation*}x_{3} = (2.0980887335) - \frac{- (2.0980887335)^{3} + \sin{\left((2.0980887335) \right)} + 8}{- 3 (2.0980887335)^{2} + \cos{\left((2.0980887335) \right)}} = 2.0709853054\end{equation*} \begin{equation*}x_{4} = (2.0709853054) - \frac{- (2.0709853054)^{3} + \sin{\left((2.0709853054) \right)} + 8}{- 3 (2.0709853054)^{2} + \cos{\left((2.0709853054) \right)}} = 2.0706164538\end{equation*} \begin{equation*}x_{5} = (2.0706164538) - \frac{- (2.0706164538)^{3} + \sin{\left((2.0706164538) \right)} + 8}{- 3 (2.0706164538)^{2} + \cos{\left((2.0706164538) \right)}} = 2.0706163859\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= x^{3} - 8$ using $x_0=3$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- x_{n}^{3} + \sin{\left(x_{n} \right)} + 8}{- 3 x_{n}^{2} + \cos{\left(x_{n} \right)}}  \end{equation*}
Using $x_0 = 3$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} =  (3.0000000000) - \frac{- (3.0000000000)^{3} + \sin{\left((3.0000000000) \right)} + 8}{- 3 (3.0000000000)^{2} + \cos{\left((3.0000000000) \right)}} = 2.3262277582\end{equation*}
\begin{equation*}x_{2} =  (2.3262277582) - \frac{- (2.3262277582)^{3} + \sin{\left((2.3262277582) \right)} + 8}{- 3 (2.3262277582)^{2} + \cos{\left((2.3262277582) \right)}} = 2.0980887335\end{equation*}
\begin{equation*}x_{3} =  (2.0980887335) - \frac{- (2.0980887335)^{3} + \sin{\left((2.0980887335) \right)} + 8}{- 3 (2.0980887335)^{2} + \cos{\left((2.0980887335) \right)}} = 2.0709853054\end{equation*}
\begin{equation*}x_{4} =  (2.0709853054) - \frac{- (2.0709853054)^{3} + \sin{\left((2.0709853054) \right)} + 8}{- 3 (2.0709853054)^{2} + \cos{\left((2.0709853054) \right)}} = 2.0706164538\end{equation*}
\begin{equation*}x_{5} =  (2.0706164538) - \frac{- (2.0706164538)^{3} + \sin{\left((2.0706164538) \right)} + 8}{- 3 (2.0706164538)^{2} + \cos{\left((2.0706164538) \right)}} = 2.0706163859\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)}= x^{3} - 8 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20x%5E%7B3%7D%20-%208%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= x^{3} - 8 " data-equation-content=" \displaystyle \sin{\left(x \right)}= x^{3} - 8 " />  using  <img class="equation_image" title=" \displaystyle x_0=3 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D3%20" alt="LaTeX:  \displaystyle x_0=3 " data-equation-content=" \displaystyle x_0=3 " /> . </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{- x_{n}^{3} + \sin{\left(x_{n} \right)} + 8}{- 3 x_{n}^{2} + \cos{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20x_%7Bn%7D%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20x_%7Bn%7D%5E%7B2%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{- x_{n}^{3} + \sin{\left(x_{n} \right)} + 8}{- 3 x_{n}^{2} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- x_{n}^{3} + \sin{\left(x_{n} \right)} + 8}{- 3 x_{n}^{2} + \cos{\left(x_{n} \right)}}   " /> 
Using  <img class="equation_image" title=" \displaystyle x_0 = 3 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%203%20" alt="LaTeX:  \displaystyle x_0 = 3 " data-equation-content=" \displaystyle x_0 = 3 " />  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} =  (3.0000000000) - \frac{- (3.0000000000)^{3} + \sin{\left((3.0000000000) \right)} + 8}{- 3 (3.0000000000)^{2} + \cos{\left((3.0000000000) \right)}} = 2.3262277582 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%283.0000000000%29%20-%20%5Cfrac%7B-%20%283.0000000000%29%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20%283.0000000000%29%5E%7B2%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%7D%20%3D%202.3262277582%20" alt="LaTeX:  x_{1} =  (3.0000000000) - \frac{- (3.0000000000)^{3} + \sin{\left((3.0000000000) \right)} + 8}{- 3 (3.0000000000)^{2} + \cos{\left((3.0000000000) \right)}} = 2.3262277582 " data-equation-content=" x_{1} =  (3.0000000000) - \frac{- (3.0000000000)^{3} + \sin{\left((3.0000000000) \right)} + 8}{- 3 (3.0000000000)^{2} + \cos{\left((3.0000000000) \right)}} = 2.3262277582 " /> 
 <img class="equation_image" title=" x_{2} =  (2.3262277582) - \frac{- (2.3262277582)^{3} + \sin{\left((2.3262277582) \right)} + 8}{- 3 (2.3262277582)^{2} + \cos{\left((2.3262277582) \right)}} = 2.0980887335 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%282.3262277582%29%20-%20%5Cfrac%7B-%20%282.3262277582%29%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28%282.3262277582%29%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20%282.3262277582%29%5E%7B2%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.3262277582%29%20%5Cright%29%7D%7D%20%3D%202.0980887335%20" alt="LaTeX:  x_{2} =  (2.3262277582) - \frac{- (2.3262277582)^{3} + \sin{\left((2.3262277582) \right)} + 8}{- 3 (2.3262277582)^{2} + \cos{\left((2.3262277582) \right)}} = 2.0980887335 " data-equation-content=" x_{2} =  (2.3262277582) - \frac{- (2.3262277582)^{3} + \sin{\left((2.3262277582) \right)} + 8}{- 3 (2.3262277582)^{2} + \cos{\left((2.3262277582) \right)}} = 2.0980887335 " /> 
 <img class="equation_image" title=" x_{3} =  (2.0980887335) - \frac{- (2.0980887335)^{3} + \sin{\left((2.0980887335) \right)} + 8}{- 3 (2.0980887335)^{2} + \cos{\left((2.0980887335) \right)}} = 2.0709853054 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.0980887335%29%20-%20%5Cfrac%7B-%20%282.0980887335%29%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28%282.0980887335%29%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20%282.0980887335%29%5E%7B2%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.0980887335%29%20%5Cright%29%7D%7D%20%3D%202.0709853054%20" alt="LaTeX:  x_{3} =  (2.0980887335) - \frac{- (2.0980887335)^{3} + \sin{\left((2.0980887335) \right)} + 8}{- 3 (2.0980887335)^{2} + \cos{\left((2.0980887335) \right)}} = 2.0709853054 " data-equation-content=" x_{3} =  (2.0980887335) - \frac{- (2.0980887335)^{3} + \sin{\left((2.0980887335) \right)} + 8}{- 3 (2.0980887335)^{2} + \cos{\left((2.0980887335) \right)}} = 2.0709853054 " /> 
 <img class="equation_image" title=" x_{4} =  (2.0709853054) - \frac{- (2.0709853054)^{3} + \sin{\left((2.0709853054) \right)} + 8}{- 3 (2.0709853054)^{2} + \cos{\left((2.0709853054) \right)}} = 2.0706164538 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.0709853054%29%20-%20%5Cfrac%7B-%20%282.0709853054%29%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28%282.0709853054%29%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20%282.0709853054%29%5E%7B2%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.0709853054%29%20%5Cright%29%7D%7D%20%3D%202.0706164538%20" alt="LaTeX:  x_{4} =  (2.0709853054) - \frac{- (2.0709853054)^{3} + \sin{\left((2.0709853054) \right)} + 8}{- 3 (2.0709853054)^{2} + \cos{\left((2.0709853054) \right)}} = 2.0706164538 " data-equation-content=" x_{4} =  (2.0709853054) - \frac{- (2.0709853054)^{3} + \sin{\left((2.0709853054) \right)} + 8}{- 3 (2.0709853054)^{2} + \cos{\left((2.0709853054) \right)}} = 2.0706164538 " /> 
 <img class="equation_image" title=" x_{5} =  (2.0706164538) - \frac{- (2.0706164538)^{3} + \sin{\left((2.0706164538) \right)} + 8}{- 3 (2.0706164538)^{2} + \cos{\left((2.0706164538) \right)}} = 2.0706163859 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%282.0706164538%29%20-%20%5Cfrac%7B-%20%282.0706164538%29%5E%7B3%7D%20%2B%20%5Csin%7B%5Cleft%28%282.0706164538%29%20%5Cright%29%7D%20%2B%208%7D%7B-%203%20%282.0706164538%29%5E%7B2%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.0706164538%29%20%5Cright%29%7D%7D%20%3D%202.0706163859%20" alt="LaTeX:  x_{5} =  (2.0706164538) - \frac{- (2.0706164538)^{3} + \sin{\left((2.0706164538) \right)} + 8}{- 3 (2.0706164538)^{2} + \cos{\left((2.0706164538) \right)}} = 2.0706163859 " data-equation-content=" x_{5} =  (2.0706164538) - \frac{- (2.0706164538)^{3} + \sin{\left((2.0706164538) \right)} + 8}{- 3 (2.0706164538)^{2} + \cos{\left((2.0706164538) \right)}} = 2.0706163859 " /> 
</p> </p>