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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{111 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{333 x_{n}^{2}}{1000} + \cos{\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{111 (5.0000000000)^{3}}{1000} + \sin{\left((5.0000000000) \right)} + 9}{- \frac{333 (5.0000000000)^{2}}{1000} + \cos{\left((5.0000000000) \right)}} = 4.2745082461\end{equation*} \begin{equation*}x_{2} = (4.2745082461) - \frac{- \frac{111 (4.2745082461)^{3}}{1000} + \sin{\left((4.2745082461) \right)} + 9}{- \frac{333 (4.2745082461)^{2}}{1000} + \cos{\left((4.2745082461) \right)}} = 4.1861761633\end{equation*} \begin{equation*}x_{3} = (4.1861761633) - \frac{- \frac{111 (4.1861761633)^{3}}{1000} + \sin{\left((4.1861761633) \right)} + 9}{- \frac{333 (4.1861761633)^{2}}{1000} + \cos{\left((4.1861761633) \right)}} = 4.1849852842\end{equation*} \begin{equation*}x_{4} = (4.1849852842) - \frac{- \frac{111 (4.1849852842)^{3}}{1000} + \sin{\left((4.1849852842) \right)} + 9}{- \frac{333 (4.1849852842)^{2}}{1000} + \cos{\left((4.1849852842) \right)}} = 4.1849850689\end{equation*} \begin{equation*}x_{5} = (4.1849850689) - \frac{- \frac{111 (4.1849850689)^{3}}{1000} + \sin{\left((4.1849850689) \right)} + 9}{- \frac{333 (4.1849850689)^{2}}{1000} + \cos{\left((4.1849850689) \right)}} = 4.1849850689\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{111 x^{3}}{1000} - 9$ using $x_0=5$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{111 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{333 x_{n}^{2}}{1000} + \cos{\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{111 (5.0000000000)^{3}}{1000} + \sin{\left((5.0000000000) \right)} + 9}{- \frac{333 (5.0000000000)^{2}}{1000} + \cos{\left((5.0000000000) \right)}} = 4.2745082461\end{equation*}
\begin{equation*}x_{2} =  (4.2745082461) - \frac{- \frac{111 (4.2745082461)^{3}}{1000} + \sin{\left((4.2745082461) \right)} + 9}{- \frac{333 (4.2745082461)^{2}}{1000} + \cos{\left((4.2745082461) \right)}} = 4.1861761633\end{equation*}
\begin{equation*}x_{3} =  (4.1861761633) - \frac{- \frac{111 (4.1861761633)^{3}}{1000} + \sin{\left((4.1861761633) \right)} + 9}{- \frac{333 (4.1861761633)^{2}}{1000} + \cos{\left((4.1861761633) \right)}} = 4.1849852842\end{equation*}
\begin{equation*}x_{4} =  (4.1849852842) - \frac{- \frac{111 (4.1849852842)^{3}}{1000} + \sin{\left((4.1849852842) \right)} + 9}{- \frac{333 (4.1849852842)^{2}}{1000} + \cos{\left((4.1849852842) \right)}} = 4.1849850689\end{equation*}
\begin{equation*}x_{5} =  (4.1849850689) - \frac{- \frac{111 (4.1849850689)^{3}}{1000} + \sin{\left((4.1849850689) \right)} + 9}{- \frac{333 (4.1849850689)^{2}}{1000} + \cos{\left((4.1849850689) \right)}} = 4.1849850689\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{111 x^{3}}{1000} - 9 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B111%20x%5E%7B3%7D%7D%7B1000%7D%20-%209%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= \frac{111 x^{3}}{1000} - 9 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{111 x^{3}}{1000} - 9 " />  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>

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{111 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{333 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20x_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20x_%7Bn%7D%5E%7B2%7D%7D%7B1000%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{111 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{333 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{111 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{333 x_{n}^{2}}{1000} + \cos{\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{111 (5.0000000000)^{3}}{1000} + \sin{\left((5.0000000000) \right)} + 9}{- \frac{333 (5.0000000000)^{2}}{1000} + \cos{\left((5.0000000000) \right)}} = 4.2745082461 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%285.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20%285.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20%285.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%7D%20%3D%204.2745082461%20" alt="LaTeX:  x_{1} =  (5.0000000000) - \frac{- \frac{111 (5.0000000000)^{3}}{1000} + \sin{\left((5.0000000000) \right)} + 9}{- \frac{333 (5.0000000000)^{2}}{1000} + \cos{\left((5.0000000000) \right)}} = 4.2745082461 " data-equation-content=" x_{1} =  (5.0000000000) - \frac{- \frac{111 (5.0000000000)^{3}}{1000} + \sin{\left((5.0000000000) \right)} + 9}{- \frac{333 (5.0000000000)^{2}}{1000} + \cos{\left((5.0000000000) \right)}} = 4.2745082461 " /> 
 <img class="equation_image" title=" x_{2} =  (4.2745082461) - \frac{- \frac{111 (4.2745082461)^{3}}{1000} + \sin{\left((4.2745082461) \right)} + 9}{- \frac{333 (4.2745082461)^{2}}{1000} + \cos{\left((4.2745082461) \right)}} = 4.1861761633 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%284.2745082461%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20%284.2745082461%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%284.2745082461%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20%284.2745082461%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.2745082461%29%20%5Cright%29%7D%7D%20%3D%204.1861761633%20" alt="LaTeX:  x_{2} =  (4.2745082461) - \frac{- \frac{111 (4.2745082461)^{3}}{1000} + \sin{\left((4.2745082461) \right)} + 9}{- \frac{333 (4.2745082461)^{2}}{1000} + \cos{\left((4.2745082461) \right)}} = 4.1861761633 " data-equation-content=" x_{2} =  (4.2745082461) - \frac{- \frac{111 (4.2745082461)^{3}}{1000} + \sin{\left((4.2745082461) \right)} + 9}{- \frac{333 (4.2745082461)^{2}}{1000} + \cos{\left((4.2745082461) \right)}} = 4.1861761633 " /> 
 <img class="equation_image" title=" x_{3} =  (4.1861761633) - \frac{- \frac{111 (4.1861761633)^{3}}{1000} + \sin{\left((4.1861761633) \right)} + 9}{- \frac{333 (4.1861761633)^{2}}{1000} + \cos{\left((4.1861761633) \right)}} = 4.1849852842 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%284.1861761633%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20%284.1861761633%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%284.1861761633%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20%284.1861761633%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.1861761633%29%20%5Cright%29%7D%7D%20%3D%204.1849852842%20" alt="LaTeX:  x_{3} =  (4.1861761633) - \frac{- \frac{111 (4.1861761633)^{3}}{1000} + \sin{\left((4.1861761633) \right)} + 9}{- \frac{333 (4.1861761633)^{2}}{1000} + \cos{\left((4.1861761633) \right)}} = 4.1849852842 " data-equation-content=" x_{3} =  (4.1861761633) - \frac{- \frac{111 (4.1861761633)^{3}}{1000} + \sin{\left((4.1861761633) \right)} + 9}{- \frac{333 (4.1861761633)^{2}}{1000} + \cos{\left((4.1861761633) \right)}} = 4.1849852842 " /> 
 <img class="equation_image" title=" x_{4} =  (4.1849852842) - \frac{- \frac{111 (4.1849852842)^{3}}{1000} + \sin{\left((4.1849852842) \right)} + 9}{- \frac{333 (4.1849852842)^{2}}{1000} + \cos{\left((4.1849852842) \right)}} = 4.1849850689 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%284.1849852842%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20%284.1849852842%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%284.1849852842%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20%284.1849852842%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.1849852842%29%20%5Cright%29%7D%7D%20%3D%204.1849850689%20" alt="LaTeX:  x_{4} =  (4.1849852842) - \frac{- \frac{111 (4.1849852842)^{3}}{1000} + \sin{\left((4.1849852842) \right)} + 9}{- \frac{333 (4.1849852842)^{2}}{1000} + \cos{\left((4.1849852842) \right)}} = 4.1849850689 " data-equation-content=" x_{4} =  (4.1849852842) - \frac{- \frac{111 (4.1849852842)^{3}}{1000} + \sin{\left((4.1849852842) \right)} + 9}{- \frac{333 (4.1849852842)^{2}}{1000} + \cos{\left((4.1849852842) \right)}} = 4.1849850689 " /> 
 <img class="equation_image" title=" x_{5} =  (4.1849850689) - \frac{- \frac{111 (4.1849850689)^{3}}{1000} + \sin{\left((4.1849850689) \right)} + 9}{- \frac{333 (4.1849850689)^{2}}{1000} + \cos{\left((4.1849850689) \right)}} = 4.1849850689 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%284.1849850689%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B111%20%284.1849850689%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%284.1849850689%29%20%5Cright%29%7D%20%2B%209%7D%7B-%20%5Cfrac%7B333%20%284.1849850689%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.1849850689%29%20%5Cright%29%7D%7D%20%3D%204.1849850689%20" alt="LaTeX:  x_{5} =  (4.1849850689) - \frac{- \frac{111 (4.1849850689)^{3}}{1000} + \sin{\left((4.1849850689) \right)} + 9}{- \frac{333 (4.1849850689)^{2}}{1000} + \cos{\left((4.1849850689) \right)}} = 4.1849850689 " data-equation-content=" x_{5} =  (4.1849850689) - \frac{- \frac{111 (4.1849850689)^{3}}{1000} + \sin{\left((4.1849850689) \right)} + 9}{- \frac{333 (4.1849850689)^{2}}{1000} + \cos{\left((4.1849850689) \right)}} = 4.1849850689 " /> 
</p> </p>