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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{5 x_{n}^{3}}{8} + \sin{\left(x_{n} \right)} + 4}{- \frac{15 x_{n}^{2}}{8} + \cos{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 1\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (1.0000000000) - \frac{- \frac{5 (1.0000000000)^{3}}{8} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{15 (1.0000000000)^{2}}{8} + \cos{\left((1.0000000000) \right)}} = 4.1591206034\end{equation*} \begin{equation*}x_{2} = (4.1591206034) - \frac{- \frac{5 (4.1591206034)^{3}}{8} + \sin{\left((4.1591206034) \right)} + 4}{- \frac{15 (4.1591206034)^{2}}{8} + \cos{\left((4.1591206034) \right)}} = 2.8903962244\end{equation*} \begin{equation*}x_{3} = (2.8903962244) - \frac{- \frac{5 (2.8903962244)^{3}}{8} + \sin{\left((2.8903962244) \right)} + 4}{- \frac{15 (2.8903962244)^{2}}{8} + \cos{\left((2.8903962244) \right)}} = 2.2384657382\end{equation*} \begin{equation*}x_{4} = (2.2384657382) - \frac{- \frac{5 (2.2384657382)^{3}}{8} + \sin{\left((2.2384657382) \right)} + 4}{- \frac{15 (2.2384657382)^{2}}{8} + \cos{\left((2.2384657382) \right)}} = 2.0162880030\end{equation*} \begin{equation*}x_{5} = (2.0162880030) - \frac{- \frac{5 (2.0162880030)^{3}}{8} + \sin{\left((2.0162880030) \right)} + 4}{- \frac{15 (2.0162880030)^{2}}{8} + \cos{\left((2.0162880030) \right)}} = 1.9888766831\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{5 x^{3}}{8} - 4$ using $x_0=1$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{5 x_{n}^{3}}{8} + \sin{\left(x_{n} \right)} + 4}{- \frac{15 x_{n}^{2}}{8} + \cos{\left(x_{n} \right)}}  \end{equation*}
Using $x_0 = 1$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} =  (1.0000000000) - \frac{- \frac{5 (1.0000000000)^{3}}{8} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{15 (1.0000000000)^{2}}{8} + \cos{\left((1.0000000000) \right)}} = 4.1591206034\end{equation*}
\begin{equation*}x_{2} =  (4.1591206034) - \frac{- \frac{5 (4.1591206034)^{3}}{8} + \sin{\left((4.1591206034) \right)} + 4}{- \frac{15 (4.1591206034)^{2}}{8} + \cos{\left((4.1591206034) \right)}} = 2.8903962244\end{equation*}
\begin{equation*}x_{3} =  (2.8903962244) - \frac{- \frac{5 (2.8903962244)^{3}}{8} + \sin{\left((2.8903962244) \right)} + 4}{- \frac{15 (2.8903962244)^{2}}{8} + \cos{\left((2.8903962244) \right)}} = 2.2384657382\end{equation*}
\begin{equation*}x_{4} =  (2.2384657382) - \frac{- \frac{5 (2.2384657382)^{3}}{8} + \sin{\left((2.2384657382) \right)} + 4}{- \frac{15 (2.2384657382)^{2}}{8} + \cos{\left((2.2384657382) \right)}} = 2.0162880030\end{equation*}
\begin{equation*}x_{5} =  (2.0162880030) - \frac{- \frac{5 (2.0162880030)^{3}}{8} + \sin{\left((2.0162880030) \right)} + 4}{- \frac{15 (2.0162880030)^{2}}{8} + \cos{\left((2.0162880030) \right)}} = 1.9888766831\end{equation*}
}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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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{5 x^{3}}{8} - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B5%20x%5E%7B3%7D%7D%7B8%7D%20-%204%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= \frac{5 x^{3}}{8} - 4 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{5 x^{3}}{8} - 4 " />  using  <img class="equation_image" title=" \displaystyle x_0=1 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D1%20" alt="LaTeX:  \displaystyle x_0=1 " data-equation-content=" \displaystyle x_0=1 " /> . </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{5 x_{n}^{3}}{8} + \sin{\left(x_{n} \right)} + 4}{- \frac{15 x_{n}^{2}}{8} + \cos{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20x_%7Bn%7D%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20x_%7Bn%7D%5E%7B2%7D%7D%7B8%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{5 x_{n}^{3}}{8} + \sin{\left(x_{n} \right)} + 4}{- \frac{15 x_{n}^{2}}{8} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{5 x_{n}^{3}}{8} + \sin{\left(x_{n} \right)} + 4}{- \frac{15 x_{n}^{2}}{8} + \cos{\left(x_{n} \right)}}   " /> 
Using  <img class="equation_image" title=" \displaystyle x_0 = 1 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%201%20" alt="LaTeX:  \displaystyle x_0 = 1 " data-equation-content=" \displaystyle x_0 = 1 " />  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} =  (1.0000000000) - \frac{- \frac{5 (1.0000000000)^{3}}{8} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{15 (1.0000000000)^{2}}{8} + \cos{\left((1.0000000000) \right)}} = 4.1591206034 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%281.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20%281.0000000000%29%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20%281.0000000000%29%5E%7B2%7D%7D%7B8%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%7D%20%3D%204.1591206034%20" alt="LaTeX:  x_{1} =  (1.0000000000) - \frac{- \frac{5 (1.0000000000)^{3}}{8} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{15 (1.0000000000)^{2}}{8} + \cos{\left((1.0000000000) \right)}} = 4.1591206034 " data-equation-content=" x_{1} =  (1.0000000000) - \frac{- \frac{5 (1.0000000000)^{3}}{8} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{15 (1.0000000000)^{2}}{8} + \cos{\left((1.0000000000) \right)}} = 4.1591206034 " /> 
 <img class="equation_image" title=" x_{2} =  (4.1591206034) - \frac{- \frac{5 (4.1591206034)^{3}}{8} + \sin{\left((4.1591206034) \right)} + 4}{- \frac{15 (4.1591206034)^{2}}{8} + \cos{\left((4.1591206034) \right)}} = 2.8903962244 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%284.1591206034%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20%284.1591206034%29%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28%284.1591206034%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20%284.1591206034%29%5E%7B2%7D%7D%7B8%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.1591206034%29%20%5Cright%29%7D%7D%20%3D%202.8903962244%20" alt="LaTeX:  x_{2} =  (4.1591206034) - \frac{- \frac{5 (4.1591206034)^{3}}{8} + \sin{\left((4.1591206034) \right)} + 4}{- \frac{15 (4.1591206034)^{2}}{8} + \cos{\left((4.1591206034) \right)}} = 2.8903962244 " data-equation-content=" x_{2} =  (4.1591206034) - \frac{- \frac{5 (4.1591206034)^{3}}{8} + \sin{\left((4.1591206034) \right)} + 4}{- \frac{15 (4.1591206034)^{2}}{8} + \cos{\left((4.1591206034) \right)}} = 2.8903962244 " /> 
 <img class="equation_image" title=" x_{3} =  (2.8903962244) - \frac{- \frac{5 (2.8903962244)^{3}}{8} + \sin{\left((2.8903962244) \right)} + 4}{- \frac{15 (2.8903962244)^{2}}{8} + \cos{\left((2.8903962244) \right)}} = 2.2384657382 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.8903962244%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20%282.8903962244%29%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28%282.8903962244%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20%282.8903962244%29%5E%7B2%7D%7D%7B8%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.8903962244%29%20%5Cright%29%7D%7D%20%3D%202.2384657382%20" alt="LaTeX:  x_{3} =  (2.8903962244) - \frac{- \frac{5 (2.8903962244)^{3}}{8} + \sin{\left((2.8903962244) \right)} + 4}{- \frac{15 (2.8903962244)^{2}}{8} + \cos{\left((2.8903962244) \right)}} = 2.2384657382 " data-equation-content=" x_{3} =  (2.8903962244) - \frac{- \frac{5 (2.8903962244)^{3}}{8} + \sin{\left((2.8903962244) \right)} + 4}{- \frac{15 (2.8903962244)^{2}}{8} + \cos{\left((2.8903962244) \right)}} = 2.2384657382 " /> 
 <img class="equation_image" title=" x_{4} =  (2.2384657382) - \frac{- \frac{5 (2.2384657382)^{3}}{8} + \sin{\left((2.2384657382) \right)} + 4}{- \frac{15 (2.2384657382)^{2}}{8} + \cos{\left((2.2384657382) \right)}} = 2.0162880030 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.2384657382%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20%282.2384657382%29%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28%282.2384657382%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20%282.2384657382%29%5E%7B2%7D%7D%7B8%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.2384657382%29%20%5Cright%29%7D%7D%20%3D%202.0162880030%20" alt="LaTeX:  x_{4} =  (2.2384657382) - \frac{- \frac{5 (2.2384657382)^{3}}{8} + \sin{\left((2.2384657382) \right)} + 4}{- \frac{15 (2.2384657382)^{2}}{8} + \cos{\left((2.2384657382) \right)}} = 2.0162880030 " data-equation-content=" x_{4} =  (2.2384657382) - \frac{- \frac{5 (2.2384657382)^{3}}{8} + \sin{\left((2.2384657382) \right)} + 4}{- \frac{15 (2.2384657382)^{2}}{8} + \cos{\left((2.2384657382) \right)}} = 2.0162880030 " /> 
 <img class="equation_image" title=" x_{5} =  (2.0162880030) - \frac{- \frac{5 (2.0162880030)^{3}}{8} + \sin{\left((2.0162880030) \right)} + 4}{- \frac{15 (2.0162880030)^{2}}{8} + \cos{\left((2.0162880030) \right)}} = 1.9888766831 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%282.0162880030%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B5%20%282.0162880030%29%5E%7B3%7D%7D%7B8%7D%20%2B%20%5Csin%7B%5Cleft%28%282.0162880030%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B15%20%282.0162880030%29%5E%7B2%7D%7D%7B8%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.0162880030%29%20%5Cright%29%7D%7D%20%3D%201.9888766831%20" alt="LaTeX:  x_{5} =  (2.0162880030) - \frac{- \frac{5 (2.0162880030)^{3}}{8} + \sin{\left((2.0162880030) \right)} + 4}{- \frac{15 (2.0162880030)^{2}}{8} + \cos{\left((2.0162880030) \right)}} = 1.9888766831 " data-equation-content=" x_{5} =  (2.0162880030) - \frac{- \frac{5 (2.0162880030)^{3}}{8} + \sin{\left((2.0162880030) \right)} + 4}{- \frac{15 (2.0162880030)^{2}}{8} + \cos{\left((2.0162880030) \right)}} = 1.9888766831 " /> 
</p> </p>