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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \sin{\left(x \right)}= \frac{663 x^{3}}{1000} - 4\) using \(\displaystyle x_0=1\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{663 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 4}{- \frac{1989 x_{n}^{2}}{1000} + \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{663 (1.0000000000)^{3}}{1000} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{1989 (1.0000000000)^{2}}{1000} + \cos{\left((1.0000000000) \right)}} = 3.8842946335\end{equation*} \begin{equation*}x_{2} = (3.8842946335) - \frac{- \frac{663 (3.8842946335)^{3}}{1000} + \sin{\left((3.8842946335) \right)} + 4}{- \frac{1989 (3.8842946335)^{2}}{1000} + \cos{\left((3.8842946335) \right)}} = 2.7286527805\end{equation*} \begin{equation*}x_{3} = (2.7286527805) - \frac{- \frac{663 (2.7286527805)^{3}}{1000} + \sin{\left((2.7286527805) \right)} + 4}{- \frac{1989 (2.7286527805)^{2}}{1000} + \cos{\left((2.7286527805) \right)}} = 2.1519703547\end{equation*} \begin{equation*}x_{4} = (2.1519703547) - \frac{- \frac{663 (2.1519703547)^{3}}{1000} + \sin{\left((2.1519703547) \right)} + 4}{- \frac{1989 (2.1519703547)^{2}}{1000} + \cos{\left((2.1519703547) \right)}} = 1.9704690136\end{equation*} \begin{equation*}x_{5} = (1.9704690136) - \frac{- \frac{663 (1.9704690136)^{3}}{1000} + \sin{\left((1.9704690136) \right)} + 4}{- \frac{1989 (1.9704690136)^{2}}{1000} + \cos{\left((1.9704690136) \right)}} = 1.9518155454\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{663 x^{3}}{1000} - 4$ using $x_0=1$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{663 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 4}{- \frac{1989 x_{n}^{2}}{1000} + \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{663 (1.0000000000)^{3}}{1000} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{1989 (1.0000000000)^{2}}{1000} + \cos{\left((1.0000000000) \right)}} = 3.8842946335\end{equation*}
\begin{equation*}x_{2} = (3.8842946335) - \frac{- \frac{663 (3.8842946335)^{3}}{1000} + \sin{\left((3.8842946335) \right)} + 4}{- \frac{1989 (3.8842946335)^{2}}{1000} + \cos{\left((3.8842946335) \right)}} = 2.7286527805\end{equation*}
\begin{equation*}x_{3} = (2.7286527805) - \frac{- \frac{663 (2.7286527805)^{3}}{1000} + \sin{\left((2.7286527805) \right)} + 4}{- \frac{1989 (2.7286527805)^{2}}{1000} + \cos{\left((2.7286527805) \right)}} = 2.1519703547\end{equation*}
\begin{equation*}x_{4} = (2.1519703547) - \frac{- \frac{663 (2.1519703547)^{3}}{1000} + \sin{\left((2.1519703547) \right)} + 4}{- \frac{1989 (2.1519703547)^{2}}{1000} + \cos{\left((2.1519703547) \right)}} = 1.9704690136\end{equation*}
\begin{equation*}x_{5} = (1.9704690136) - \frac{- \frac{663 (1.9704690136)^{3}}{1000} + \sin{\left((1.9704690136) \right)} + 4}{- \frac{1989 (1.9704690136)^{2}}{1000} + \cos{\left((1.9704690136) \right)}} = 1.9518155454\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{663 x^{3}}{1000} - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B663%20x%5E%7B3%7D%7D%7B1000%7D%20-%204%20" alt="LaTeX: \displaystyle \sin{\left(x \right)}= \frac{663 x^{3}}{1000} - 4 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{663 x^{3}}{1000} - 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><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{663 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 4}{- \frac{1989 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%7B663%20x_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%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{663 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 4}{- \frac{1989 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{663 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 4}{- \frac{1989 x_{n}^{2}}{1000} + \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{663 (1.0000000000)^{3}}{1000} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{1989 (1.0000000000)^{2}}{1000} + \cos{\left((1.0000000000) \right)}} = 3.8842946335 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%281.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B663%20%281.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%20%281.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%7D%20%3D%203.8842946335%20" alt="LaTeX: x_{1} = (1.0000000000) - \frac{- \frac{663 (1.0000000000)^{3}}{1000} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{1989 (1.0000000000)^{2}}{1000} + \cos{\left((1.0000000000) \right)}} = 3.8842946335 " data-equation-content=" x_{1} = (1.0000000000) - \frac{- \frac{663 (1.0000000000)^{3}}{1000} + \sin{\left((1.0000000000) \right)} + 4}{- \frac{1989 (1.0000000000)^{2}}{1000} + \cos{\left((1.0000000000) \right)}} = 3.8842946335 " />
<img class="equation_image" title=" x_{2} = (3.8842946335) - \frac{- \frac{663 (3.8842946335)^{3}}{1000} + \sin{\left((3.8842946335) \right)} + 4}{- \frac{1989 (3.8842946335)^{2}}{1000} + \cos{\left((3.8842946335) \right)}} = 2.7286527805 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%283.8842946335%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B663%20%283.8842946335%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.8842946335%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%20%283.8842946335%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.8842946335%29%20%5Cright%29%7D%7D%20%3D%202.7286527805%20" alt="LaTeX: x_{2} = (3.8842946335) - \frac{- \frac{663 (3.8842946335)^{3}}{1000} + \sin{\left((3.8842946335) \right)} + 4}{- \frac{1989 (3.8842946335)^{2}}{1000} + \cos{\left((3.8842946335) \right)}} = 2.7286527805 " data-equation-content=" x_{2} = (3.8842946335) - \frac{- \frac{663 (3.8842946335)^{3}}{1000} + \sin{\left((3.8842946335) \right)} + 4}{- \frac{1989 (3.8842946335)^{2}}{1000} + \cos{\left((3.8842946335) \right)}} = 2.7286527805 " />
<img class="equation_image" title=" x_{3} = (2.7286527805) - \frac{- \frac{663 (2.7286527805)^{3}}{1000} + \sin{\left((2.7286527805) \right)} + 4}{- \frac{1989 (2.7286527805)^{2}}{1000} + \cos{\left((2.7286527805) \right)}} = 2.1519703547 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.7286527805%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B663%20%282.7286527805%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.7286527805%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%20%282.7286527805%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.7286527805%29%20%5Cright%29%7D%7D%20%3D%202.1519703547%20" alt="LaTeX: x_{3} = (2.7286527805) - \frac{- \frac{663 (2.7286527805)^{3}}{1000} + \sin{\left((2.7286527805) \right)} + 4}{- \frac{1989 (2.7286527805)^{2}}{1000} + \cos{\left((2.7286527805) \right)}} = 2.1519703547 " data-equation-content=" x_{3} = (2.7286527805) - \frac{- \frac{663 (2.7286527805)^{3}}{1000} + \sin{\left((2.7286527805) \right)} + 4}{- \frac{1989 (2.7286527805)^{2}}{1000} + \cos{\left((2.7286527805) \right)}} = 2.1519703547 " />
<img class="equation_image" title=" x_{4} = (2.1519703547) - \frac{- \frac{663 (2.1519703547)^{3}}{1000} + \sin{\left((2.1519703547) \right)} + 4}{- \frac{1989 (2.1519703547)^{2}}{1000} + \cos{\left((2.1519703547) \right)}} = 1.9704690136 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.1519703547%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B663%20%282.1519703547%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.1519703547%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%20%282.1519703547%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.1519703547%29%20%5Cright%29%7D%7D%20%3D%201.9704690136%20" alt="LaTeX: x_{4} = (2.1519703547) - \frac{- \frac{663 (2.1519703547)^{3}}{1000} + \sin{\left((2.1519703547) \right)} + 4}{- \frac{1989 (2.1519703547)^{2}}{1000} + \cos{\left((2.1519703547) \right)}} = 1.9704690136 " data-equation-content=" x_{4} = (2.1519703547) - \frac{- \frac{663 (2.1519703547)^{3}}{1000} + \sin{\left((2.1519703547) \right)} + 4}{- \frac{1989 (2.1519703547)^{2}}{1000} + \cos{\left((2.1519703547) \right)}} = 1.9704690136 " />
<img class="equation_image" title=" x_{5} = (1.9704690136) - \frac{- \frac{663 (1.9704690136)^{3}}{1000} + \sin{\left((1.9704690136) \right)} + 4}{- \frac{1989 (1.9704690136)^{2}}{1000} + \cos{\left((1.9704690136) \right)}} = 1.9518155454 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%281.9704690136%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B663%20%281.9704690136%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%281.9704690136%29%20%5Cright%29%7D%20%2B%204%7D%7B-%20%5Cfrac%7B1989%20%281.9704690136%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.9704690136%29%20%5Cright%29%7D%7D%20%3D%201.9518155454%20" alt="LaTeX: x_{5} = (1.9704690136) - \frac{- \frac{663 (1.9704690136)^{3}}{1000} + \sin{\left((1.9704690136) \right)} + 4}{- \frac{1989 (1.9704690136)^{2}}{1000} + \cos{\left((1.9704690136) \right)}} = 1.9518155454 " data-equation-content=" x_{5} = (1.9704690136) - \frac{- \frac{663 (1.9704690136)^{3}}{1000} + \sin{\left((1.9704690136) \right)} + 4}{- \frac{1989 (1.9704690136)^{2}}{1000} + \cos{\left((1.9704690136) \right)}} = 1.9518155454 " />
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