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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 6}{- \frac{3 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}} \end{equation*} Using \(\displaystyle x_0 = 11\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (11.0000000000) - \frac{- \frac{(11.0000000000)^{3}}{1000} + \cos{\left((11.0000000000) \right)} + 6}{- \frac{3 (11.0000000000)^{2}}{1000} - \sin{\left((11.0000000000) \right)}} = 3.6632691524\end{equation*} \begin{equation*}x_{2} = (3.6632691524) - \frac{- \frac{(3.6632691524)^{3}}{1000} + \cos{\left((3.6632691524) \right)} + 6}{- \frac{3 (3.6632691524)^{2}}{1000} - \sin{\left((3.6632691524) \right)}} = -7.4350176849\end{equation*} \begin{equation*}x_{3} = (-7.4350176849) - \frac{- \frac{(-7.4350176849)^{3}}{1000} + \cos{\left((-7.4350176849) \right)} + 6}{- \frac{3 (-7.4350176849)^{2}}{1000} - \sin{\left((-7.4350176849) \right)}} = -16.5537410336\end{equation*} \begin{equation*}x_{4} = (-16.5537410336) - \frac{- \frac{(-16.5537410336)^{3}}{1000} + \cos{\left((-16.5537410336) \right)} + 6}{- \frac{3 (-16.5537410336)^{2}}{1000} - \sin{\left((-16.5537410336) \right)}} = -10.2674653278\end{equation*} \begin{equation*}x_{5} = (-10.2674653278) - \frac{- \frac{(-10.2674653278)^{3}}{1000} + \cos{\left((-10.2674653278) \right)} + 6}{- \frac{3 (-10.2674653278)^{2}}{1000} - \sin{\left((-10.2674653278) \right)}} = -4.2291049005\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\cos{\left(x \right)}= \frac{x^{3}}{1000} - 6$ using $x_0=11$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 6}{- \frac{3 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}  \end{equation*}
Using $x_0 = 11$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} =  (11.0000000000) - \frac{- \frac{(11.0000000000)^{3}}{1000} + \cos{\left((11.0000000000) \right)} + 6}{- \frac{3 (11.0000000000)^{2}}{1000} - \sin{\left((11.0000000000) \right)}} = 3.6632691524\end{equation*}
\begin{equation*}x_{2} =  (3.6632691524) - \frac{- \frac{(3.6632691524)^{3}}{1000} + \cos{\left((3.6632691524) \right)} + 6}{- \frac{3 (3.6632691524)^{2}}{1000} - \sin{\left((3.6632691524) \right)}} = -7.4350176849\end{equation*}
\begin{equation*}x_{3} =  (-7.4350176849) - \frac{- \frac{(-7.4350176849)^{3}}{1000} + \cos{\left((-7.4350176849) \right)} + 6}{- \frac{3 (-7.4350176849)^{2}}{1000} - \sin{\left((-7.4350176849) \right)}} = -16.5537410336\end{equation*}
\begin{equation*}x_{4} =  (-16.5537410336) - \frac{- \frac{(-16.5537410336)^{3}}{1000} + \cos{\left((-16.5537410336) \right)} + 6}{- \frac{3 (-16.5537410336)^{2}}{1000} - \sin{\left((-16.5537410336) \right)}} = -10.2674653278\end{equation*}
\begin{equation*}x_{5} =  (-10.2674653278) - \frac{- \frac{(-10.2674653278)^{3}}{1000} + \cos{\left((-10.2674653278) \right)} + 6}{- \frac{3 (-10.2674653278)^{2}}{1000} - \sin{\left((-10.2674653278) \right)}} = -4.2291049005\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}

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{x^{3}}{1000} - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7Bx%5E%7B3%7D%7D%7B1000%7D%20-%206%20" alt="LaTeX:  \displaystyle \cos{\left(x \right)}= \frac{x^{3}}{1000} - 6 " data-equation-content=" \displaystyle \cos{\left(x \right)}= \frac{x^{3}}{1000} - 6 " />  using  <img class="equation_image" title=" \displaystyle x_0=11 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D11%20" alt="LaTeX:  \displaystyle x_0=11 " data-equation-content=" \displaystyle x_0=11 " /> . </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{x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 6}{- \frac{3 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%7Bx_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%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{x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 6}{- \frac{3 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 6}{- \frac{3 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}   " /> 
Using  <img class="equation_image" title=" \displaystyle x_0 = 11 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%2011%20" alt="LaTeX:  \displaystyle x_0 = 11 " data-equation-content=" \displaystyle x_0 = 11 " />  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} =  (11.0000000000) - \frac{- \frac{(11.0000000000)^{3}}{1000} + \cos{\left((11.0000000000) \right)} + 6}{- \frac{3 (11.0000000000)^{2}}{1000} - \sin{\left((11.0000000000) \right)}} = 3.6632691524 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%2811.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%2811.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%2811.0000000000%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%20%2811.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%2811.0000000000%29%20%5Cright%29%7D%7D%20%3D%203.6632691524%20" alt="LaTeX:  x_{1} =  (11.0000000000) - \frac{- \frac{(11.0000000000)^{3}}{1000} + \cos{\left((11.0000000000) \right)} + 6}{- \frac{3 (11.0000000000)^{2}}{1000} - \sin{\left((11.0000000000) \right)}} = 3.6632691524 " data-equation-content=" x_{1} =  (11.0000000000) - \frac{- \frac{(11.0000000000)^{3}}{1000} + \cos{\left((11.0000000000) \right)} + 6}{- \frac{3 (11.0000000000)^{2}}{1000} - \sin{\left((11.0000000000) \right)}} = 3.6632691524 " /> 
 <img class="equation_image" title=" x_{2} =  (3.6632691524) - \frac{- \frac{(3.6632691524)^{3}}{1000} + \cos{\left((3.6632691524) \right)} + 6}{- \frac{3 (3.6632691524)^{2}}{1000} - \sin{\left((3.6632691524) \right)}} = -7.4350176849 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%283.6632691524%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%283.6632691524%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.6632691524%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%20%283.6632691524%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%283.6632691524%29%20%5Cright%29%7D%7D%20%3D%20-7.4350176849%20" alt="LaTeX:  x_{2} =  (3.6632691524) - \frac{- \frac{(3.6632691524)^{3}}{1000} + \cos{\left((3.6632691524) \right)} + 6}{- \frac{3 (3.6632691524)^{2}}{1000} - \sin{\left((3.6632691524) \right)}} = -7.4350176849 " data-equation-content=" x_{2} =  (3.6632691524) - \frac{- \frac{(3.6632691524)^{3}}{1000} + \cos{\left((3.6632691524) \right)} + 6}{- \frac{3 (3.6632691524)^{2}}{1000} - \sin{\left((3.6632691524) \right)}} = -7.4350176849 " /> 
 <img class="equation_image" title=" x_{3} =  (-7.4350176849) - \frac{- \frac{(-7.4350176849)^{3}}{1000} + \cos{\left((-7.4350176849) \right)} + 6}{- \frac{3 (-7.4350176849)^{2}}{1000} - \sin{\left((-7.4350176849) \right)}} = -16.5537410336 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%28-7.4350176849%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%28-7.4350176849%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%28-7.4350176849%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%20%28-7.4350176849%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%28-7.4350176849%29%20%5Cright%29%7D%7D%20%3D%20-16.5537410336%20" alt="LaTeX:  x_{3} =  (-7.4350176849) - \frac{- \frac{(-7.4350176849)^{3}}{1000} + \cos{\left((-7.4350176849) \right)} + 6}{- \frac{3 (-7.4350176849)^{2}}{1000} - \sin{\left((-7.4350176849) \right)}} = -16.5537410336 " data-equation-content=" x_{3} =  (-7.4350176849) - \frac{- \frac{(-7.4350176849)^{3}}{1000} + \cos{\left((-7.4350176849) \right)} + 6}{- \frac{3 (-7.4350176849)^{2}}{1000} - \sin{\left((-7.4350176849) \right)}} = -16.5537410336 " /> 
 <img class="equation_image" title=" x_{4} =  (-16.5537410336) - \frac{- \frac{(-16.5537410336)^{3}}{1000} + \cos{\left((-16.5537410336) \right)} + 6}{- \frac{3 (-16.5537410336)^{2}}{1000} - \sin{\left((-16.5537410336) \right)}} = -10.2674653278 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%28-16.5537410336%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%28-16.5537410336%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%28-16.5537410336%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%20%28-16.5537410336%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%28-16.5537410336%29%20%5Cright%29%7D%7D%20%3D%20-10.2674653278%20" alt="LaTeX:  x_{4} =  (-16.5537410336) - \frac{- \frac{(-16.5537410336)^{3}}{1000} + \cos{\left((-16.5537410336) \right)} + 6}{- \frac{3 (-16.5537410336)^{2}}{1000} - \sin{\left((-16.5537410336) \right)}} = -10.2674653278 " data-equation-content=" x_{4} =  (-16.5537410336) - \frac{- \frac{(-16.5537410336)^{3}}{1000} + \cos{\left((-16.5537410336) \right)} + 6}{- \frac{3 (-16.5537410336)^{2}}{1000} - \sin{\left((-16.5537410336) \right)}} = -10.2674653278 " /> 
 <img class="equation_image" title=" x_{5} =  (-10.2674653278) - \frac{- \frac{(-10.2674653278)^{3}}{1000} + \cos{\left((-10.2674653278) \right)} + 6}{- \frac{3 (-10.2674653278)^{2}}{1000} - \sin{\left((-10.2674653278) \right)}} = -4.2291049005 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%28-10.2674653278%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B%28-10.2674653278%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%28-10.2674653278%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B3%20%28-10.2674653278%29%5E%7B2%7D%7D%7B1000%7D%20-%20%5Csin%7B%5Cleft%28%28-10.2674653278%29%20%5Cright%29%7D%7D%20%3D%20-4.2291049005%20" alt="LaTeX:  x_{5} =  (-10.2674653278) - \frac{- \frac{(-10.2674653278)^{3}}{1000} + \cos{\left((-10.2674653278) \right)} + 6}{- \frac{3 (-10.2674653278)^{2}}{1000} - \sin{\left((-10.2674653278) \right)}} = -4.2291049005 " data-equation-content=" x_{5} =  (-10.2674653278) - \frac{- \frac{(-10.2674653278)^{3}}{1000} + \cos{\left((-10.2674653278) \right)} + 6}{- \frac{3 (-10.2674653278)^{2}}{1000} - \sin{\left((-10.2674653278) \right)}} = -4.2291049005 " /> 
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