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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle \cos{\left(x \right)}= \frac{489 x^{3}}{500} - 8\) using \(\displaystyle x_0=1\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{489 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 8}{- \frac{1467 x_{n}^{2}}{500} - \sin{\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{489 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 8}{- \frac{1467 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 3.0030089852\end{equation*} \begin{equation*}x_{2} = (3.0030089852) - \frac{- \frac{489 (3.0030089852)^{3}}{500} + \cos{\left((3.0030089852) \right)} + 8}{- \frac{1467 (3.0030089852)^{2}}{500} - \sin{\left((3.0030089852) \right)}} = 2.2707516957\end{equation*} \begin{equation*}x_{3} = (2.2707516957) - \frac{- \frac{489 (2.2707516957)^{3}}{500} + \cos{\left((2.2707516957) \right)} + 8}{- \frac{1467 (2.2707516957)^{2}}{500} - \sin{\left((2.2707516957) \right)}} = 2.0130802820\end{equation*} \begin{equation*}x_{4} = (2.0130802820) - \frac{- \frac{489 (2.0130802820)^{3}}{500} + \cos{\left((2.0130802820) \right)} + 8}{- \frac{1467 (2.0130802820)^{2}}{500} - \sin{\left((2.0130802820) \right)}} = 1.9813054003\end{equation*} \begin{equation*}x_{5} = (1.9813054003) - \frac{- \frac{489 (1.9813054003)^{3}}{500} + \cos{\left((1.9813054003) \right)} + 8}{- \frac{1467 (1.9813054003)^{2}}{500} - \sin{\left((1.9813054003) \right)}} = 1.9808453318\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\cos{\left(x \right)}= \frac{489 x^{3}}{500} - 8$ using $x_0=1$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{489 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 8}{- \frac{1467 x_{n}^{2}}{500} - \sin{\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{489 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 8}{- \frac{1467 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 3.0030089852\end{equation*}
\begin{equation*}x_{2} = (3.0030089852) - \frac{- \frac{489 (3.0030089852)^{3}}{500} + \cos{\left((3.0030089852) \right)} + 8}{- \frac{1467 (3.0030089852)^{2}}{500} - \sin{\left((3.0030089852) \right)}} = 2.2707516957\end{equation*}
\begin{equation*}x_{3} = (2.2707516957) - \frac{- \frac{489 (2.2707516957)^{3}}{500} + \cos{\left((2.2707516957) \right)} + 8}{- \frac{1467 (2.2707516957)^{2}}{500} - \sin{\left((2.2707516957) \right)}} = 2.0130802820\end{equation*}
\begin{equation*}x_{4} = (2.0130802820) - \frac{- \frac{489 (2.0130802820)^{3}}{500} + \cos{\left((2.0130802820) \right)} + 8}{- \frac{1467 (2.0130802820)^{2}}{500} - \sin{\left((2.0130802820) \right)}} = 1.9813054003\end{equation*}
\begin{equation*}x_{5} = (1.9813054003) - \frac{- \frac{489 (1.9813054003)^{3}}{500} + \cos{\left((1.9813054003) \right)} + 8}{- \frac{1467 (1.9813054003)^{2}}{500} - \sin{\left((1.9813054003) \right)}} = 1.9808453318\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 \cos{\left(x \right)}= \frac{489 x^{3}}{500} - 8 " src="/equation_images/%20%5Cdisplaystyle%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B489%20x%5E%7B3%7D%7D%7B500%7D%20-%208%20" alt="LaTeX: \displaystyle \cos{\left(x \right)}= \frac{489 x^{3}}{500} - 8 " data-equation-content=" \displaystyle \cos{\left(x \right)}= \frac{489 x^{3}}{500} - 8 " /> 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{489 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 8}{- \frac{1467 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20x_%7Bn%7D%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20x_%7Bn%7D%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%7D%20%20%20" alt="LaTeX: x_{n+1} = x_{n} - \frac{- \frac{489 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 8}{- \frac{1467 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{489 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 8}{- \frac{1467 x_{n}^{2}}{500} - \sin{\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{489 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 8}{- \frac{1467 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 3.0030089852 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%281.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20%281.0000000000%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20%281.0000000000%29%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%7D%20%3D%203.0030089852%20" alt="LaTeX: x_{1} = (1.0000000000) - \frac{- \frac{489 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 8}{- \frac{1467 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 3.0030089852 " data-equation-content=" x_{1} = (1.0000000000) - \frac{- \frac{489 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 8}{- \frac{1467 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 3.0030089852 " />
<img class="equation_image" title=" x_{2} = (3.0030089852) - \frac{- \frac{489 (3.0030089852)^{3}}{500} + \cos{\left((3.0030089852) \right)} + 8}{- \frac{1467 (3.0030089852)^{2}}{500} - \sin{\left((3.0030089852) \right)}} = 2.2707516957 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%283.0030089852%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20%283.0030089852%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.0030089852%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20%283.0030089852%29%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28%283.0030089852%29%20%5Cright%29%7D%7D%20%3D%202.2707516957%20" alt="LaTeX: x_{2} = (3.0030089852) - \frac{- \frac{489 (3.0030089852)^{3}}{500} + \cos{\left((3.0030089852) \right)} + 8}{- \frac{1467 (3.0030089852)^{2}}{500} - \sin{\left((3.0030089852) \right)}} = 2.2707516957 " data-equation-content=" x_{2} = (3.0030089852) - \frac{- \frac{489 (3.0030089852)^{3}}{500} + \cos{\left((3.0030089852) \right)} + 8}{- \frac{1467 (3.0030089852)^{2}}{500} - \sin{\left((3.0030089852) \right)}} = 2.2707516957 " />
<img class="equation_image" title=" x_{3} = (2.2707516957) - \frac{- \frac{489 (2.2707516957)^{3}}{500} + \cos{\left((2.2707516957) \right)} + 8}{- \frac{1467 (2.2707516957)^{2}}{500} - \sin{\left((2.2707516957) \right)}} = 2.0130802820 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.2707516957%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20%282.2707516957%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.2707516957%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20%282.2707516957%29%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28%282.2707516957%29%20%5Cright%29%7D%7D%20%3D%202.0130802820%20" alt="LaTeX: x_{3} = (2.2707516957) - \frac{- \frac{489 (2.2707516957)^{3}}{500} + \cos{\left((2.2707516957) \right)} + 8}{- \frac{1467 (2.2707516957)^{2}}{500} - \sin{\left((2.2707516957) \right)}} = 2.0130802820 " data-equation-content=" x_{3} = (2.2707516957) - \frac{- \frac{489 (2.2707516957)^{3}}{500} + \cos{\left((2.2707516957) \right)} + 8}{- \frac{1467 (2.2707516957)^{2}}{500} - \sin{\left((2.2707516957) \right)}} = 2.0130802820 " />
<img class="equation_image" title=" x_{4} = (2.0130802820) - \frac{- \frac{489 (2.0130802820)^{3}}{500} + \cos{\left((2.0130802820) \right)} + 8}{- \frac{1467 (2.0130802820)^{2}}{500} - \sin{\left((2.0130802820) \right)}} = 1.9813054003 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.0130802820%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20%282.0130802820%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.0130802820%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20%282.0130802820%29%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28%282.0130802820%29%20%5Cright%29%7D%7D%20%3D%201.9813054003%20" alt="LaTeX: x_{4} = (2.0130802820) - \frac{- \frac{489 (2.0130802820)^{3}}{500} + \cos{\left((2.0130802820) \right)} + 8}{- \frac{1467 (2.0130802820)^{2}}{500} - \sin{\left((2.0130802820) \right)}} = 1.9813054003 " data-equation-content=" x_{4} = (2.0130802820) - \frac{- \frac{489 (2.0130802820)^{3}}{500} + \cos{\left((2.0130802820) \right)} + 8}{- \frac{1467 (2.0130802820)^{2}}{500} - \sin{\left((2.0130802820) \right)}} = 1.9813054003 " />
<img class="equation_image" title=" x_{5} = (1.9813054003) - \frac{- \frac{489 (1.9813054003)^{3}}{500} + \cos{\left((1.9813054003) \right)} + 8}{- \frac{1467 (1.9813054003)^{2}}{500} - \sin{\left((1.9813054003) \right)}} = 1.9808453318 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%281.9813054003%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B489%20%281.9813054003%29%5E%7B3%7D%7D%7B500%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.9813054003%29%20%5Cright%29%7D%20%2B%208%7D%7B-%20%5Cfrac%7B1467%20%281.9813054003%29%5E%7B2%7D%7D%7B500%7D%20-%20%5Csin%7B%5Cleft%28%281.9813054003%29%20%5Cright%29%7D%7D%20%3D%201.9808453318%20" alt="LaTeX: x_{5} = (1.9813054003) - \frac{- \frac{489 (1.9813054003)^{3}}{500} + \cos{\left((1.9813054003) \right)} + 8}{- \frac{1467 (1.9813054003)^{2}}{500} - \sin{\left((1.9813054003) \right)}} = 1.9808453318 " data-equation-content=" x_{5} = (1.9813054003) - \frac{- \frac{489 (1.9813054003)^{3}}{500} + \cos{\left((1.9813054003) \right)} + 8}{- \frac{1467 (1.9813054003)^{2}}{500} - \sin{\left((1.9813054003) \right)}} = 1.9808453318 " />
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