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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{741 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{2223 x_{n}^{2}}{1000} - \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{741 (1.0000000000)^{3}}{1000} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{2223 (1.0000000000)^{2}}{1000} - \sin{\left((1.0000000000) \right)}} = 1.5871494019\end{equation*} \begin{equation*}x_{2} = (1.5871494019) - \frac{- \frac{741 (1.5871494019)^{3}}{1000} + \cos{\left((1.5871494019) \right)} + 2}{- \frac{2223 (1.5871494019)^{2}}{1000} - \sin{\left((1.5871494019) \right)}} = 1.4388179601\end{equation*} \begin{equation*}x_{3} = (1.4388179601) - \frac{- \frac{741 (1.4388179601)^{3}}{1000} + \cos{\left((1.4388179601) \right)} + 2}{- \frac{2223 (1.4388179601)^{2}}{1000} - \sin{\left((1.4388179601) \right)}} = 1.4253065071\end{equation*} \begin{equation*}x_{4} = (1.4253065071) - \frac{- \frac{741 (1.4253065071)^{3}}{1000} + \cos{\left((1.4253065071) \right)} + 2}{- \frac{2223 (1.4253065071)^{2}}{1000} - \sin{\left((1.4253065071) \right)}} = 1.4251985222\end{equation*} \begin{equation*}x_{5} = (1.4251985222) - \frac{- \frac{741 (1.4251985222)^{3}}{1000} + \cos{\left((1.4251985222) \right)} + 2}{- \frac{2223 (1.4251985222)^{2}}{1000} - \sin{\left((1.4251985222) \right)}} = 1.4251985153\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\cos{\left(x \right)}= \frac{741 x^{3}}{1000} - 2$ using $x_0=1$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{741 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{2223 x_{n}^{2}}{1000} - \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{741 (1.0000000000)^{3}}{1000} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{2223 (1.0000000000)^{2}}{1000} - \sin{\left((1.0000000000) \right)}} = 1.5871494019\end{equation*}
\begin{equation*}x_{2} =  (1.5871494019) - \frac{- \frac{741 (1.5871494019)^{3}}{1000} + \cos{\left((1.5871494019) \right)} + 2}{- \frac{2223 (1.5871494019)^{2}}{1000} - \sin{\left((1.5871494019) \right)}} = 1.4388179601\end{equation*}
\begin{equation*}x_{3} =  (1.4388179601) - \frac{- \frac{741 (1.4388179601)^{3}}{1000} + \cos{\left((1.4388179601) \right)} + 2}{- \frac{2223 (1.4388179601)^{2}}{1000} - \sin{\left((1.4388179601) \right)}} = 1.4253065071\end{equation*}
\begin{equation*}x_{4} =  (1.4253065071) - \frac{- \frac{741 (1.4253065071)^{3}}{1000} + \cos{\left((1.4253065071) \right)} + 2}{- \frac{2223 (1.4253065071)^{2}}{1000} - \sin{\left((1.4253065071) \right)}} = 1.4251985222\end{equation*}
\begin{equation*}x_{5} =  (1.4251985222) - \frac{- \frac{741 (1.4251985222)^{3}}{1000} + \cos{\left((1.4251985222) \right)} + 2}{- \frac{2223 (1.4251985222)^{2}}{1000} - \sin{\left((1.4251985222) \right)}} = 1.4251985153\end{equation*}
}

\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}

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 \cos{\left(x \right)}= \frac{741 x^{3}}{1000} - 2 " src="/equation_images/%20%5Cdisplaystyle%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B741%20x%5E%7B3%7D%7D%7B1000%7D%20-%202%20" alt="LaTeX:  \displaystyle \cos{\left(x \right)}= \frac{741 x^{3}}{1000} - 2 " data-equation-content=" \displaystyle \cos{\left(x \right)}= \frac{741 x^{3}}{1000} - 2 " />  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>

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{741 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{2223 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}   " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20x_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20x_%7Bn%7D%5E%7B2%7D%7D%7B1000%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{741 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{2223 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{741 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{2223 x_{n}^{2}}{1000} - \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{741 (1.0000000000)^{3}}{1000} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{2223 (1.0000000000)^{2}}{1000} - \sin{\left((1.0000000000) \right)}} = 1.5871494019 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%281.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20%281.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20%281.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%281.0000000000%29%20%5Cright%29%7D%7D%20%3D%201.5871494019%20" alt="LaTeX:  x_{1} =  (1.0000000000) - \frac{- \frac{741 (1.0000000000)^{3}}{1000} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{2223 (1.0000000000)^{2}}{1000} - \sin{\left((1.0000000000) \right)}} = 1.5871494019 " data-equation-content=" x_{1} =  (1.0000000000) - \frac{- \frac{741 (1.0000000000)^{3}}{1000} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{2223 (1.0000000000)^{2}}{1000} - \sin{\left((1.0000000000) \right)}} = 1.5871494019 " /> 
 <img class="equation_image" title=" x_{2} =  (1.5871494019) - \frac{- \frac{741 (1.5871494019)^{3}}{1000} + \cos{\left((1.5871494019) \right)} + 2}{- \frac{2223 (1.5871494019)^{2}}{1000} - \sin{\left((1.5871494019) \right)}} = 1.4388179601 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%281.5871494019%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20%281.5871494019%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.5871494019%29%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20%281.5871494019%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%281.5871494019%29%20%5Cright%29%7D%7D%20%3D%201.4388179601%20" alt="LaTeX:  x_{2} =  (1.5871494019) - \frac{- \frac{741 (1.5871494019)^{3}}{1000} + \cos{\left((1.5871494019) \right)} + 2}{- \frac{2223 (1.5871494019)^{2}}{1000} - \sin{\left((1.5871494019) \right)}} = 1.4388179601 " data-equation-content=" x_{2} =  (1.5871494019) - \frac{- \frac{741 (1.5871494019)^{3}}{1000} + \cos{\left((1.5871494019) \right)} + 2}{- \frac{2223 (1.5871494019)^{2}}{1000} - \sin{\left((1.5871494019) \right)}} = 1.4388179601 " /> 
 <img class="equation_image" title=" x_{3} =  (1.4388179601) - \frac{- \frac{741 (1.4388179601)^{3}}{1000} + \cos{\left((1.4388179601) \right)} + 2}{- \frac{2223 (1.4388179601)^{2}}{1000} - \sin{\left((1.4388179601) \right)}} = 1.4253065071 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%281.4388179601%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20%281.4388179601%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.4388179601%29%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20%281.4388179601%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%281.4388179601%29%20%5Cright%29%7D%7D%20%3D%201.4253065071%20" alt="LaTeX:  x_{3} =  (1.4388179601) - \frac{- \frac{741 (1.4388179601)^{3}}{1000} + \cos{\left((1.4388179601) \right)} + 2}{- \frac{2223 (1.4388179601)^{2}}{1000} - \sin{\left((1.4388179601) \right)}} = 1.4253065071 " data-equation-content=" x_{3} =  (1.4388179601) - \frac{- \frac{741 (1.4388179601)^{3}}{1000} + \cos{\left((1.4388179601) \right)} + 2}{- \frac{2223 (1.4388179601)^{2}}{1000} - \sin{\left((1.4388179601) \right)}} = 1.4253065071 " /> 
 <img class="equation_image" title=" x_{4} =  (1.4253065071) - \frac{- \frac{741 (1.4253065071)^{3}}{1000} + \cos{\left((1.4253065071) \right)} + 2}{- \frac{2223 (1.4253065071)^{2}}{1000} - \sin{\left((1.4253065071) \right)}} = 1.4251985222 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%281.4253065071%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20%281.4253065071%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.4253065071%29%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20%281.4253065071%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%281.4253065071%29%20%5Cright%29%7D%7D%20%3D%201.4251985222%20" alt="LaTeX:  x_{4} =  (1.4253065071) - \frac{- \frac{741 (1.4253065071)^{3}}{1000} + \cos{\left((1.4253065071) \right)} + 2}{- \frac{2223 (1.4253065071)^{2}}{1000} - \sin{\left((1.4253065071) \right)}} = 1.4251985222 " data-equation-content=" x_{4} =  (1.4253065071) - \frac{- \frac{741 (1.4253065071)^{3}}{1000} + \cos{\left((1.4253065071) \right)} + 2}{- \frac{2223 (1.4253065071)^{2}}{1000} - \sin{\left((1.4253065071) \right)}} = 1.4251985222 " /> 
 <img class="equation_image" title=" x_{5} =  (1.4251985222) - \frac{- \frac{741 (1.4251985222)^{3}}{1000} + \cos{\left((1.4251985222) \right)} + 2}{- \frac{2223 (1.4251985222)^{2}}{1000} - \sin{\left((1.4251985222) \right)}} = 1.4251985153 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%281.4251985222%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B741%20%281.4251985222%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%281.4251985222%29%20%5Cright%29%7D%20%2B%202%7D%7B-%20%5Cfrac%7B2223%20%281.4251985222%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%281.4251985222%29%20%5Cright%29%7D%7D%20%3D%201.4251985153%20" alt="LaTeX:  x_{5} =  (1.4251985222) - \frac{- \frac{741 (1.4251985222)^{3}}{1000} + \cos{\left((1.4251985222) \right)} + 2}{- \frac{2223 (1.4251985222)^{2}}{1000} - \sin{\left((1.4251985222) \right)}} = 1.4251985153 " data-equation-content=" x_{5} =  (1.4251985222) - \frac{- \frac{741 (1.4251985222)^{3}}{1000} + \cos{\left((1.4251985222) \right)} + 2}{- \frac{2223 (1.4251985222)^{2}}{1000} - \sin{\left((1.4251985222) \right)}} = 1.4251985153 " /> 
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