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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \sin{\left(x \right)}= \frac{19 x^{3}}{500} - 4\) using \(\displaystyle x_0=5\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{19 x_{n}^{3}}{500} + \sin{\left(x_{n} \right)} + 4}{- \frac{57 x_{n}^{2}}{500} + \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{19 (5.0000000000)^{3}}{500} + \sin{\left((5.0000000000) \right)} + 4}{- \frac{57 (5.0000000000)^{2}}{500} + \cos{\left((5.0000000000) \right)}} = 4.3341000296\end{equation*} \begin{equation*}x_{2} = (4.3341000296) - \frac{- \frac{19 (4.3341000296)^{3}}{500} + \sin{\left((4.3341000296) \right)} + 4}{- \frac{57 (4.3341000296)^{2}}{500} + \cos{\left((4.3341000296) \right)}} = 4.3249340444\end{equation*} \begin{equation*}x_{3} = (4.3249340444) - \frac{- \frac{19 (4.3249340444)^{3}}{500} + \sin{\left((4.3249340444) \right)} + 4}{- \frac{57 (4.3249340444)^{2}}{500} + \cos{\left((4.3249340444) \right)}} = 4.3249330518\end{equation*} \begin{equation*}x_{4} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518\end{equation*} \begin{equation*}x_{5} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{19 x^{3}}{500} - 4$ using $x_0=5$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{19 x_{n}^{3}}{500} + \sin{\left(x_{n} \right)} + 4}{- \frac{57 x_{n}^{2}}{500} + \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{19 (5.0000000000)^{3}}{500} + \sin{\left((5.0000000000) \right)} + 4}{- \frac{57 (5.0000000000)^{2}}{500} + \cos{\left((5.0000000000) \right)}} = 4.3341000296\end{equation*}
\begin{equation*}x_{2} = (4.3341000296) - \frac{- \frac{19 (4.3341000296)^{3}}{500} + \sin{\left((4.3341000296) \right)} + 4}{- \frac{57 (4.3341000296)^{2}}{500} + \cos{\left((4.3341000296) \right)}} = 4.3249340444\end{equation*}
\begin{equation*}x_{3} = (4.3249340444) - \frac{- \frac{19 (4.3249340444)^{3}}{500} + \sin{\left((4.3249340444) \right)} + 4}{- \frac{57 (4.3249340444)^{2}}{500} + \cos{\left((4.3249340444) \right)}} = 4.3249330518\end{equation*}
\begin{equation*}x_{4} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518\end{equation*}
\begin{equation*}x_{5} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518\end{equation*}
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\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}<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{19 x^{3}}{500} - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B19%20x%5E%7B3%7D%7D%7B500%7D%20-%204%20" alt="LaTeX: \displaystyle \sin{\left(x \right)}= \frac{19 x^{3}}{500} - 4 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{19 x^{3}}{500} - 4 " /> 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{19 x_{n}^{3}}{500} + \sin{\left(x_{n} \right)} + 4}{- \frac{57 x_{n}^{2}}{500} + \cos{\left(x_{n} \right)}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20x_%7Bn%7D%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20x_%7Bn%7D%5E%7B2%7D%7D%7B500%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{19 x_{n}^{3}}{500} + \sin{\left(x_{n} \right)} + 4}{- \frac{57 x_{n}^{2}}{500} + \cos{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{19 x_{n}^{3}}{500} + \sin{\left(x_{n} \right)} + 4}{- \frac{57 x_{n}^{2}}{500} + \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{19 (5.0000000000)^{3}}{500} + \sin{\left((5.0000000000) \right)} + 4}{- \frac{57 (5.0000000000)^{2}}{500} + \cos{\left((5.0000000000) \right)}} = 4.3341000296 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%285.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20%285.0000000000%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20%285.0000000000%29%5E%7B2%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%285.0000000000%29%20%5Cright%29%7D%7D%20%3D%204.3341000296%20" alt="LaTeX: x_{1} = (5.0000000000) - \frac{- \frac{19 (5.0000000000)^{3}}{500} + \sin{\left((5.0000000000) \right)} + 4}{- \frac{57 (5.0000000000)^{2}}{500} + \cos{\left((5.0000000000) \right)}} = 4.3341000296 " data-equation-content=" x_{1} = (5.0000000000) - \frac{- \frac{19 (5.0000000000)^{3}}{500} + \sin{\left((5.0000000000) \right)} + 4}{- \frac{57 (5.0000000000)^{2}}{500} + \cos{\left((5.0000000000) \right)}} = 4.3341000296 " />
<img class="equation_image" title=" x_{2} = (4.3341000296) - \frac{- \frac{19 (4.3341000296)^{3}}{500} + \sin{\left((4.3341000296) \right)} + 4}{- \frac{57 (4.3341000296)^{2}}{500} + \cos{\left((4.3341000296) \right)}} = 4.3249340444 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%284.3341000296%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20%284.3341000296%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28%284.3341000296%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20%284.3341000296%29%5E%7B2%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.3341000296%29%20%5Cright%29%7D%7D%20%3D%204.3249340444%20" alt="LaTeX: x_{2} = (4.3341000296) - \frac{- \frac{19 (4.3341000296)^{3}}{500} + \sin{\left((4.3341000296) \right)} + 4}{- \frac{57 (4.3341000296)^{2}}{500} + \cos{\left((4.3341000296) \right)}} = 4.3249340444 " data-equation-content=" x_{2} = (4.3341000296) - \frac{- \frac{19 (4.3341000296)^{3}}{500} + \sin{\left((4.3341000296) \right)} + 4}{- \frac{57 (4.3341000296)^{2}}{500} + \cos{\left((4.3341000296) \right)}} = 4.3249340444 " />
<img class="equation_image" title=" x_{3} = (4.3249340444) - \frac{- \frac{19 (4.3249340444)^{3}}{500} + \sin{\left((4.3249340444) \right)} + 4}{- \frac{57 (4.3249340444)^{2}}{500} + \cos{\left((4.3249340444) \right)}} = 4.3249330518 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%284.3249340444%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20%284.3249340444%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28%284.3249340444%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20%284.3249340444%29%5E%7B2%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.3249340444%29%20%5Cright%29%7D%7D%20%3D%204.3249330518%20" alt="LaTeX: x_{3} = (4.3249340444) - \frac{- \frac{19 (4.3249340444)^{3}}{500} + \sin{\left((4.3249340444) \right)} + 4}{- \frac{57 (4.3249340444)^{2}}{500} + \cos{\left((4.3249340444) \right)}} = 4.3249330518 " data-equation-content=" x_{3} = (4.3249340444) - \frac{- \frac{19 (4.3249340444)^{3}}{500} + \sin{\left((4.3249340444) \right)} + 4}{- \frac{57 (4.3249340444)^{2}}{500} + \cos{\left((4.3249340444) \right)}} = 4.3249330518 " />
<img class="equation_image" title=" x_{4} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%284.3249330518%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20%284.3249330518%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28%284.3249330518%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20%284.3249330518%29%5E%7B2%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.3249330518%29%20%5Cright%29%7D%7D%20%3D%204.3249330518%20" alt="LaTeX: x_{4} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " data-equation-content=" x_{4} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " />
<img class="equation_image" title=" x_{5} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%284.3249330518%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B19%20%284.3249330518%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Csin%7B%5Cleft%28%284.3249330518%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B57%20%284.3249330518%29%5E%7B2%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%284.3249330518%29%20%5Cright%29%7D%7D%20%3D%204.3249330518%20" alt="LaTeX: x_{5} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " data-equation-content=" x_{5} = (4.3249330518) - \frac{- \frac{19 (4.3249330518)^{3}}{500} + \sin{\left((4.3249330518) \right)} + 4}{- \frac{57 (4.3249330518)^{2}}{500} + \cos{\left((4.3249330518) \right)}} = 4.3249330518 " />
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