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Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{269 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{807 x_{n}^{2}}{1000} + \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{269 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{807 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.8640638541\end{equation*} \begin{equation*}x_{2} = (2.8640638541) - \frac{- \frac{269 (2.8640638541)^{3}}{1000} + \sin{\left((2.8640638541) \right)} + 6}{- \frac{807 (2.8640638541)^{2}}{1000} + \cos{\left((2.8640638541) \right)}} = 2.8580258232\end{equation*} \begin{equation*}x_{3} = (2.8580258232) - \frac{- \frac{269 (2.8580258232)^{3}}{1000} + \sin{\left((2.8580258232) \right)} + 6}{- \frac{807 (2.8580258232)^{2}}{1000} + \cos{\left((2.8580258232) \right)}} = 2.8580140070\end{equation*} \begin{equation*}x_{4} = (2.8580140070) - \frac{- \frac{269 (2.8580140070)^{3}}{1000} + \sin{\left((2.8580140070) \right)} + 6}{- \frac{807 (2.8580140070)^{2}}{1000} + \cos{\left((2.8580140070) \right)}} = 2.8580140069\end{equation*} \begin{equation*}x_{5} = (2.8580140069) - \frac{- \frac{269 (2.8580140069)^{3}}{1000} + \sin{\left((2.8580140069) \right)} + 6}{- \frac{807 (2.8580140069)^{2}}{1000} + \cos{\left((2.8580140069) \right)}} = 2.8580140069\end{equation*}

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\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $\sin{\left(x \right)}= \frac{269 x^{3}}{1000} - 6$ using $x_0=3$. 
    \soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} =  x_{n} - \frac{- \frac{269 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{807 x_{n}^{2}}{1000} + \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{269 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{807 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.8640638541\end{equation*}
\begin{equation*}x_{2} =  (2.8640638541) - \frac{- \frac{269 (2.8640638541)^{3}}{1000} + \sin{\left((2.8640638541) \right)} + 6}{- \frac{807 (2.8640638541)^{2}}{1000} + \cos{\left((2.8640638541) \right)}} = 2.8580258232\end{equation*}
\begin{equation*}x_{3} =  (2.8580258232) - \frac{- \frac{269 (2.8580258232)^{3}}{1000} + \sin{\left((2.8580258232) \right)} + 6}{- \frac{807 (2.8580258232)^{2}}{1000} + \cos{\left((2.8580258232) \right)}} = 2.8580140070\end{equation*}
\begin{equation*}x_{4} =  (2.8580140070) - \frac{- \frac{269 (2.8580140070)^{3}}{1000} + \sin{\left((2.8580140070) \right)} + 6}{- \frac{807 (2.8580140070)^{2}}{1000} + \cos{\left((2.8580140070) \right)}} = 2.8580140069\end{equation*}
\begin{equation*}x_{5} =  (2.8580140069) - \frac{- \frac{269 (2.8580140069)^{3}}{1000} + \sin{\left((2.8580140069) \right)} + 6}{- \frac{807 (2.8580140069)^{2}}{1000} + \cos{\left((2.8580140069) \right)}} = 2.8580140069\end{equation*}
}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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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 \sin{\left(x \right)}= \frac{269 x^{3}}{1000} - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%3D%20%5Cfrac%7B269%20x%5E%7B3%7D%7D%7B1000%7D%20-%206%20" alt="LaTeX:  \displaystyle \sin{\left(x \right)}= \frac{269 x^{3}}{1000} - 6 " data-equation-content=" \displaystyle \sin{\left(x \right)}= \frac{269 x^{3}}{1000} - 6 " />  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>

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{269 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{807 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%7B269%20x_%7Bn%7D%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28x_%7Bn%7D%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%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{269 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{807 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}   " data-equation-content=" x_{n+1} =  x_{n} - \frac{- \frac{269 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{807 x_{n}^{2}}{1000} + \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{269 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{807 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.8640638541 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%283.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B269%20%283.0000000000%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%20%283.0000000000%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%283.0000000000%29%20%5Cright%29%7D%7D%20%3D%202.8640638541%20" alt="LaTeX:  x_{1} =  (3.0000000000) - \frac{- \frac{269 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{807 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.8640638541 " data-equation-content=" x_{1} =  (3.0000000000) - \frac{- \frac{269 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{807 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.8640638541 " /> 
 <img class="equation_image" title=" x_{2} =  (2.8640638541) - \frac{- \frac{269 (2.8640638541)^{3}}{1000} + \sin{\left((2.8640638541) \right)} + 6}{- \frac{807 (2.8640638541)^{2}}{1000} + \cos{\left((2.8640638541) \right)}} = 2.8580258232 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%282.8640638541%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B269%20%282.8640638541%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.8640638541%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%20%282.8640638541%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.8640638541%29%20%5Cright%29%7D%7D%20%3D%202.8580258232%20" alt="LaTeX:  x_{2} =  (2.8640638541) - \frac{- \frac{269 (2.8640638541)^{3}}{1000} + \sin{\left((2.8640638541) \right)} + 6}{- \frac{807 (2.8640638541)^{2}}{1000} + \cos{\left((2.8640638541) \right)}} = 2.8580258232 " data-equation-content=" x_{2} =  (2.8640638541) - \frac{- \frac{269 (2.8640638541)^{3}}{1000} + \sin{\left((2.8640638541) \right)} + 6}{- \frac{807 (2.8640638541)^{2}}{1000} + \cos{\left((2.8640638541) \right)}} = 2.8580258232 " /> 
 <img class="equation_image" title=" x_{3} =  (2.8580258232) - \frac{- \frac{269 (2.8580258232)^{3}}{1000} + \sin{\left((2.8580258232) \right)} + 6}{- \frac{807 (2.8580258232)^{2}}{1000} + \cos{\left((2.8580258232) \right)}} = 2.8580140070 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.8580258232%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B269%20%282.8580258232%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.8580258232%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%20%282.8580258232%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.8580258232%29%20%5Cright%29%7D%7D%20%3D%202.8580140070%20" alt="LaTeX:  x_{3} =  (2.8580258232) - \frac{- \frac{269 (2.8580258232)^{3}}{1000} + \sin{\left((2.8580258232) \right)} + 6}{- \frac{807 (2.8580258232)^{2}}{1000} + \cos{\left((2.8580258232) \right)}} = 2.8580140070 " data-equation-content=" x_{3} =  (2.8580258232) - \frac{- \frac{269 (2.8580258232)^{3}}{1000} + \sin{\left((2.8580258232) \right)} + 6}{- \frac{807 (2.8580258232)^{2}}{1000} + \cos{\left((2.8580258232) \right)}} = 2.8580140070 " /> 
 <img class="equation_image" title=" x_{4} =  (2.8580140070) - \frac{- \frac{269 (2.8580140070)^{3}}{1000} + \sin{\left((2.8580140070) \right)} + 6}{- \frac{807 (2.8580140070)^{2}}{1000} + \cos{\left((2.8580140070) \right)}} = 2.8580140069 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.8580140070%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B269%20%282.8580140070%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.8580140070%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%20%282.8580140070%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.8580140070%29%20%5Cright%29%7D%7D%20%3D%202.8580140069%20" alt="LaTeX:  x_{4} =  (2.8580140070) - \frac{- \frac{269 (2.8580140070)^{3}}{1000} + \sin{\left((2.8580140070) \right)} + 6}{- \frac{807 (2.8580140070)^{2}}{1000} + \cos{\left((2.8580140070) \right)}} = 2.8580140069 " data-equation-content=" x_{4} =  (2.8580140070) - \frac{- \frac{269 (2.8580140070)^{3}}{1000} + \sin{\left((2.8580140070) \right)} + 6}{- \frac{807 (2.8580140070)^{2}}{1000} + \cos{\left((2.8580140070) \right)}} = 2.8580140069 " /> 
 <img class="equation_image" title=" x_{5} =  (2.8580140069) - \frac{- \frac{269 (2.8580140069)^{3}}{1000} + \sin{\left((2.8580140069) \right)} + 6}{- \frac{807 (2.8580140069)^{2}}{1000} + \cos{\left((2.8580140069) \right)}} = 2.8580140069 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%282.8580140069%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B269%20%282.8580140069%29%5E%7B3%7D%7D%7B1000%7D%20%2B%20%5Csin%7B%5Cleft%28%282.8580140069%29%20%5Cright%29%7D%20%2B%206%7D%7B-%20%5Cfrac%7B807%20%282.8580140069%29%5E%7B2%7D%7D%7B1000%7D%20%2B%20%5Ccos%7B%5Cleft%28%282.8580140069%29%20%5Cright%29%7D%7D%20%3D%202.8580140069%20" alt="LaTeX:  x_{5} =  (2.8580140069) - \frac{- \frac{269 (2.8580140069)^{3}}{1000} + \sin{\left((2.8580140069) \right)} + 6}{- \frac{807 (2.8580140069)^{2}}{1000} + \cos{\left((2.8580140069) \right)}} = 2.8580140069 " data-equation-content=" x_{5} =  (2.8580140069) - \frac{- \frac{269 (2.8580140069)^{3}}{1000} + \sin{\left((2.8580140069) \right)} + 6}{- \frac{807 (2.8580140069)^{2}}{1000} + \cos{\left((2.8580140069) \right)}} = 2.8580140069 " /> 
</p> </p>