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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \sin{\left(x \right)}= \frac{81 x^{3}}{125} - 7\) using \(\displaystyle x_0=3\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{81 x_{n}^{3}}{125} + \sin{\left(x_{n} \right)} + 7}{- \frac{243 x_{n}^{2}}{125} + \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{- \frac{81 (3.0000000000)^{3}}{125} + \sin{\left((3.0000000000) \right)} + 7}{- \frac{243 (3.0000000000)^{2}}{125} + \cos{\left((3.0000000000) \right)}} = 2.4398526347\end{equation*} \begin{equation*}x_{2} = (2.4398526347) - \frac{- \frac{81 (2.4398526347)^{3}}{125} + \sin{\left((2.4398526347) \right)} + 7}{- \frac{243 (2.4398526347)^{2}}{125} + \cos{\left((2.4398526347) \right)}} = 2.2966874470\end{equation*} \begin{equation*}x_{3} = (2.2966874470) - \frac{- \frac{81 (2.2966874470)^{3}}{125} + \sin{\left((2.2966874470) \right)} + 7}{- \frac{243 (2.2966874470)^{2}}{125} + \cos{\left((2.2966874470) \right)}} = 2.2873183584\end{equation*} \begin{equation*}x_{4} = (2.2873183584) - \frac{- \frac{81 (2.2873183584)^{3}}{125} + \sin{\left((2.2873183584) \right)} + 7}{- \frac{243 (2.2873183584)^{2}}{125} + \cos{\left((2.2873183584) \right)}} = 2.2872791709\end{equation*} \begin{equation*}x_{5} = (2.2872791709) - \frac{- \frac{81 (2.2872791709)^{3}}{125} + \sin{\left((2.2872791709) \right)} + 7}{- \frac{243 (2.2872791709)^{2}}{125} + \cos{\left((2.2872791709) \right)}} = 2.2872791703\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{81 x^{3}}{125} - 7$ using $x_0=3$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{81 x_{n}^{3}}{125} + \sin{\left(x_{n} \right)} + 7}{- \frac{243 x_{n}^{2}}{125} + \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{- \frac{81 (3.0000000000)^{3}}{125} + \sin{\left((3.0000000000) \right)} + 7}{- \frac{243 (3.0000000000)^{2}}{125} + \cos{\left((3.0000000000) \right)}} = 2.4398526347\end{equation*}
\begin{equation*}x_{2} = (2.4398526347) - \frac{- \frac{81 (2.4398526347)^{3}}{125} + \sin{\left((2.4398526347) \right)} + 7}{- \frac{243 (2.4398526347)^{2}}{125} + \cos{\left((2.4398526347) \right)}} = 2.2966874470\end{equation*}
\begin{equation*}x_{3} = (2.2966874470) - \frac{- \frac{81 (2.2966874470)^{3}}{125} + \sin{\left((2.2966874470) \right)} + 7}{- \frac{243 (2.2966874470)^{2}}{125} + \cos{\left((2.2966874470) \right)}} = 2.2873183584\end{equation*}
\begin{equation*}x_{4} = (2.2873183584) - \frac{- \frac{81 (2.2873183584)^{3}}{125} + \sin{\left((2.2873183584) \right)} + 7}{- \frac{243 (2.2873183584)^{2}}{125} + \cos{\left((2.2873183584) \right)}} = 2.2872791709\end{equation*}
\begin{equation*}x_{5} = (2.2872791709) - \frac{- \frac{81 (2.2872791709)^{3}}{125} + \sin{\left((2.2872791709) \right)} + 7}{- \frac{243 (2.2872791709)^{2}}{125} + \cos{\left((2.2872791709) \right)}} = 2.2872791703\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{81 x^{3}}{125} - 7 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B81%20x%5E%7B3%7D%7D%7B125%7D%20-%207%20" alt="LaTeX: \displaystyle \sin{\left(x \right)}= \frac{81 x^{3}}{125} - 7 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{81 x^{3}}{125} - 7 " /> 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><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{81 x_{n}^{3}}{125} + \sin{\left(x_{n} \right)} + 7}{- \frac{243 x_{n}^{2}}{125} + \cos{\left(x_{n} \right)}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20x_%7Bn%7D%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20x_%7Bn%7D%5E%7B2%7D%7D%7B125%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{81 x_{n}^{3}}{125} + \sin{\left(x_{n} \right)} + 7}{- \frac{243 x_{n}^{2}}{125} + \cos{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{81 x_{n}^{3}}{125} + \sin{\left(x_{n} \right)} + 7}{- \frac{243 x_{n}^{2}}{125} + \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{- \frac{81 (3.0000000000)^{3}}{125} + \sin{\left((3.0000000000) \right)} + 7}{- \frac{243 (3.0000000000)^{2}}{125} + \cos{\left((3.0000000000) \right)}} = 2.4398526347 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%283.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20%283.0000000000%29%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20%283.0000000000%29%5E%7B2%7D%7D%7B125%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%7D%20%3D%202.4398526347%20" alt="LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{81 (3.0000000000)^{3}}{125} + \sin{\left((3.0000000000) \right)} + 7}{- \frac{243 (3.0000000000)^{2}}{125} + \cos{\left((3.0000000000) \right)}} = 2.4398526347 " data-equation-content=" x_{1} = (3.0000000000) - \frac{- \frac{81 (3.0000000000)^{3}}{125} + \sin{\left((3.0000000000) \right)} + 7}{- \frac{243 (3.0000000000)^{2}}{125} + \cos{\left((3.0000000000) \right)}} = 2.4398526347 " />
<img class="equation_image" title=" x_{2} = (2.4398526347) - \frac{- \frac{81 (2.4398526347)^{3}}{125} + \sin{\left((2.4398526347) \right)} + 7}{- \frac{243 (2.4398526347)^{2}}{125} + \cos{\left((2.4398526347) \right)}} = 2.2966874470 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%282.4398526347%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20%282.4398526347%29%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28%282.4398526347%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20%282.4398526347%29%5E%7B2%7D%7D%7B125%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.4398526347%29%20%5Cright%29%7D%7D%20%3D%202.2966874470%20" alt="LaTeX: x_{2} = (2.4398526347) - \frac{- \frac{81 (2.4398526347)^{3}}{125} + \sin{\left((2.4398526347) \right)} + 7}{- \frac{243 (2.4398526347)^{2}}{125} + \cos{\left((2.4398526347) \right)}} = 2.2966874470 " data-equation-content=" x_{2} = (2.4398526347) - \frac{- \frac{81 (2.4398526347)^{3}}{125} + \sin{\left((2.4398526347) \right)} + 7}{- \frac{243 (2.4398526347)^{2}}{125} + \cos{\left((2.4398526347) \right)}} = 2.2966874470 " />
<img class="equation_image" title=" x_{3} = (2.2966874470) - \frac{- \frac{81 (2.2966874470)^{3}}{125} + \sin{\left((2.2966874470) \right)} + 7}{- \frac{243 (2.2966874470)^{2}}{125} + \cos{\left((2.2966874470) \right)}} = 2.2873183584 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.2966874470%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20%282.2966874470%29%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28%282.2966874470%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20%282.2966874470%29%5E%7B2%7D%7D%7B125%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.2966874470%29%20%5Cright%29%7D%7D%20%3D%202.2873183584%20" alt="LaTeX: x_{3} = (2.2966874470) - \frac{- \frac{81 (2.2966874470)^{3}}{125} + \sin{\left((2.2966874470) \right)} + 7}{- \frac{243 (2.2966874470)^{2}}{125} + \cos{\left((2.2966874470) \right)}} = 2.2873183584 " data-equation-content=" x_{3} = (2.2966874470) - \frac{- \frac{81 (2.2966874470)^{3}}{125} + \sin{\left((2.2966874470) \right)} + 7}{- \frac{243 (2.2966874470)^{2}}{125} + \cos{\left((2.2966874470) \right)}} = 2.2873183584 " />
<img class="equation_image" title=" x_{4} = (2.2873183584) - \frac{- \frac{81 (2.2873183584)^{3}}{125} + \sin{\left((2.2873183584) \right)} + 7}{- \frac{243 (2.2873183584)^{2}}{125} + \cos{\left((2.2873183584) \right)}} = 2.2872791709 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.2873183584%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20%282.2873183584%29%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28%282.2873183584%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20%282.2873183584%29%5E%7B2%7D%7D%7B125%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.2873183584%29%20%5Cright%29%7D%7D%20%3D%202.2872791709%20" alt="LaTeX: x_{4} = (2.2873183584) - \frac{- \frac{81 (2.2873183584)^{3}}{125} + \sin{\left((2.2873183584) \right)} + 7}{- \frac{243 (2.2873183584)^{2}}{125} + \cos{\left((2.2873183584) \right)}} = 2.2872791709 " data-equation-content=" x_{4} = (2.2873183584) - \frac{- \frac{81 (2.2873183584)^{3}}{125} + \sin{\left((2.2873183584) \right)} + 7}{- \frac{243 (2.2873183584)^{2}}{125} + \cos{\left((2.2873183584) \right)}} = 2.2872791709 " />
<img class="equation_image" title=" x_{5} = (2.2872791709) - \frac{- \frac{81 (2.2872791709)^{3}}{125} + \sin{\left((2.2872791709) \right)} + 7}{- \frac{243 (2.2872791709)^{2}}{125} + \cos{\left((2.2872791709) \right)}} = 2.2872791703 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%282.2872791709%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B81%20%282.2872791709%29%5E%7B3%7D%7D%7B125%7D%20%2B%20%5Csin%7B%5Cleft%28%282.2872791709%29%20%5Cright%29%7D%20%2B%207%7D%7B-%20%5Cfrac%7B243%20%282.2872791709%29%5E%7B2%7D%7D%7B125%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.2872791709%29%20%5Cright%29%7D%7D%20%3D%202.2872791703%20" alt="LaTeX: x_{5} = (2.2872791709) - \frac{- \frac{81 (2.2872791709)^{3}}{125} + \sin{\left((2.2872791709) \right)} + 7}{- \frac{243 (2.2872791709)^{2}}{125} + \cos{\left((2.2872791709) \right)}} = 2.2872791703 " data-equation-content=" x_{5} = (2.2872791709) - \frac{- \frac{81 (2.2872791709)^{3}}{125} + \sin{\left((2.2872791709) \right)} + 7}{- \frac{243 (2.2872791709)^{2}}{125} + \cos{\left((2.2872791709) \right)}} = 2.2872791703 " />
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