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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle e^{- x}= \frac{69 x^{3}}{125} - 8\) using \(\displaystyle x_0=3\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{69 x_{n}^{3}}{125} + 8 + e^{- x_{n}}}{- \frac{207 x_{n}^{2}}{125} - e^{- x_{n}}} \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{69 (3.0000000000)^{3}}{125} + 8 + e^{- (3.0000000000)}}{- \frac{207 (3.0000000000)^{2}}{125} - e^{- (3.0000000000)}} = 2.5416403283\end{equation*} \begin{equation*}x_{2} = (2.5416403283) - \frac{- \frac{69 (2.5416403283)^{3}}{125} + 8 + e^{- (2.5416403283)}}{- \frac{207 (2.5416403283)^{2}}{125} - e^{- (2.5416403283)}} = 2.4502870559\end{equation*} \begin{equation*}x_{3} = (2.4502870559) - \frac{- \frac{69 (2.4502870559)^{3}}{125} + 8 + e^{- (2.4502870559)}}{- \frac{207 (2.4502870559)^{2}}{125} - e^{- (2.4502870559)}} = 2.4468603146\end{equation*} \begin{equation*}x_{4} = (2.4468603146) - \frac{- \frac{69 (2.4468603146)^{3}}{125} + 8 + e^{- (2.4468603146)}}{- \frac{207 (2.4468603146)^{2}}{125} - e^{- (2.4468603146)}} = 2.4468556034\end{equation*} \begin{equation*}x_{5} = (2.4468556034) - \frac{- \frac{69 (2.4468556034)^{3}}{125} + 8 + e^{- (2.4468556034)}}{- \frac{207 (2.4468556034)^{2}}{125} - e^{- (2.4468556034)}} = 2.4468556034\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $e^{- x}= \frac{69 x^{3}}{125} - 8$ using $x_0=3$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{69 x_{n}^{3}}{125} + 8 + e^{- x_{n}}}{- \frac{207 x_{n}^{2}}{125} - e^{- x_{n}}} \end{equation*}
Using $x_0 = 3$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} = (3.0000000000) - \frac{- \frac{69 (3.0000000000)^{3}}{125} + 8 + e^{- (3.0000000000)}}{- \frac{207 (3.0000000000)^{2}}{125} - e^{- (3.0000000000)}} = 2.5416403283\end{equation*}
\begin{equation*}x_{2} = (2.5416403283) - \frac{- \frac{69 (2.5416403283)^{3}}{125} + 8 + e^{- (2.5416403283)}}{- \frac{207 (2.5416403283)^{2}}{125} - e^{- (2.5416403283)}} = 2.4502870559\end{equation*}
\begin{equation*}x_{3} = (2.4502870559) - \frac{- \frac{69 (2.4502870559)^{3}}{125} + 8 + e^{- (2.4502870559)}}{- \frac{207 (2.4502870559)^{2}}{125} - e^{- (2.4502870559)}} = 2.4468603146\end{equation*}
\begin{equation*}x_{4} = (2.4468603146) - \frac{- \frac{69 (2.4468603146)^{3}}{125} + 8 + e^{- (2.4468603146)}}{- \frac{207 (2.4468603146)^{2}}{125} - e^{- (2.4468603146)}} = 2.4468556034\end{equation*}
\begin{equation*}x_{5} = (2.4468556034) - \frac{- \frac{69 (2.4468556034)^{3}}{125} + 8 + e^{- (2.4468556034)}}{- \frac{207 (2.4468556034)^{2}}{125} - e^{- (2.4468556034)}} = 2.4468556034\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}<p> <p>Use Newton's method to find the first 5 approximations of the solution to the equation <img class="equation_image" title=" \displaystyle e^{- x}= \frac{69 x^{3}}{125} - 8 " src="/equation_images/%20%5Cdisplaystyle%20e%5E%7B-%20x%7D%3D%20%5Cfrac%7B69%20x%5E%7B3%7D%7D%7B125%7D%20-%208%20" alt="LaTeX: \displaystyle e^{- x}= \frac{69 x^{3}}{125} - 8 " data-equation-content=" \displaystyle e^{- x}= \frac{69 x^{3}}{125} - 8 " /> 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><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{69 x_{n}^{3}}{125} + 8 + e^{- x_{n}}}{- \frac{207 x_{n}^{2}}{125} - e^{- x_{n}}} " src="/equation_images/%20x_%7Bn%2B1%7D%20%3D%20%20x_%7Bn%7D%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20x_%7Bn%7D%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20x_%7Bn%7D%7D%7D%7B-%20%5Cfrac%7B207%20x_%7Bn%7D%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20x_%7Bn%7D%7D%7D%20%20%20" alt="LaTeX: x_{n+1} = x_{n} - \frac{- \frac{69 x_{n}^{3}}{125} + 8 + e^{- x_{n}}}{- \frac{207 x_{n}^{2}}{125} - e^{- x_{n}}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{69 x_{n}^{3}}{125} + 8 + e^{- x_{n}}}{- \frac{207 x_{n}^{2}}{125} - e^{- x_{n}}} " />
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{69 (3.0000000000)^{3}}{125} + 8 + e^{- (3.0000000000)}}{- \frac{207 (3.0000000000)^{2}}{125} - e^{- (3.0000000000)}} = 2.5416403283 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%283.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20%283.0000000000%29%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20%283.0000000000%29%7D%7D%7B-%20%5Cfrac%7B207%20%283.0000000000%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%283.0000000000%29%7D%7D%20%3D%202.5416403283%20" alt="LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{69 (3.0000000000)^{3}}{125} + 8 + e^{- (3.0000000000)}}{- \frac{207 (3.0000000000)^{2}}{125} - e^{- (3.0000000000)}} = 2.5416403283 " data-equation-content=" x_{1} = (3.0000000000) - \frac{- \frac{69 (3.0000000000)^{3}}{125} + 8 + e^{- (3.0000000000)}}{- \frac{207 (3.0000000000)^{2}}{125} - e^{- (3.0000000000)}} = 2.5416403283 " />
<img class="equation_image" title=" x_{2} = (2.5416403283) - \frac{- \frac{69 (2.5416403283)^{3}}{125} + 8 + e^{- (2.5416403283)}}{- \frac{207 (2.5416403283)^{2}}{125} - e^{- (2.5416403283)}} = 2.4502870559 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%282.5416403283%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20%282.5416403283%29%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20%282.5416403283%29%7D%7D%7B-%20%5Cfrac%7B207%20%282.5416403283%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%282.5416403283%29%7D%7D%20%3D%202.4502870559%20" alt="LaTeX: x_{2} = (2.5416403283) - \frac{- \frac{69 (2.5416403283)^{3}}{125} + 8 + e^{- (2.5416403283)}}{- \frac{207 (2.5416403283)^{2}}{125} - e^{- (2.5416403283)}} = 2.4502870559 " data-equation-content=" x_{2} = (2.5416403283) - \frac{- \frac{69 (2.5416403283)^{3}}{125} + 8 + e^{- (2.5416403283)}}{- \frac{207 (2.5416403283)^{2}}{125} - e^{- (2.5416403283)}} = 2.4502870559 " />
<img class="equation_image" title=" x_{3} = (2.4502870559) - \frac{- \frac{69 (2.4502870559)^{3}}{125} + 8 + e^{- (2.4502870559)}}{- \frac{207 (2.4502870559)^{2}}{125} - e^{- (2.4502870559)}} = 2.4468603146 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%282.4502870559%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20%282.4502870559%29%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20%282.4502870559%29%7D%7D%7B-%20%5Cfrac%7B207%20%282.4502870559%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%282.4502870559%29%7D%7D%20%3D%202.4468603146%20" alt="LaTeX: x_{3} = (2.4502870559) - \frac{- \frac{69 (2.4502870559)^{3}}{125} + 8 + e^{- (2.4502870559)}}{- \frac{207 (2.4502870559)^{2}}{125} - e^{- (2.4502870559)}} = 2.4468603146 " data-equation-content=" x_{3} = (2.4502870559) - \frac{- \frac{69 (2.4502870559)^{3}}{125} + 8 + e^{- (2.4502870559)}}{- \frac{207 (2.4502870559)^{2}}{125} - e^{- (2.4502870559)}} = 2.4468603146 " />
<img class="equation_image" title=" x_{4} = (2.4468603146) - \frac{- \frac{69 (2.4468603146)^{3}}{125} + 8 + e^{- (2.4468603146)}}{- \frac{207 (2.4468603146)^{2}}{125} - e^{- (2.4468603146)}} = 2.4468556034 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%282.4468603146%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20%282.4468603146%29%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20%282.4468603146%29%7D%7D%7B-%20%5Cfrac%7B207%20%282.4468603146%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%282.4468603146%29%7D%7D%20%3D%202.4468556034%20" alt="LaTeX: x_{4} = (2.4468603146) - \frac{- \frac{69 (2.4468603146)^{3}}{125} + 8 + e^{- (2.4468603146)}}{- \frac{207 (2.4468603146)^{2}}{125} - e^{- (2.4468603146)}} = 2.4468556034 " data-equation-content=" x_{4} = (2.4468603146) - \frac{- \frac{69 (2.4468603146)^{3}}{125} + 8 + e^{- (2.4468603146)}}{- \frac{207 (2.4468603146)^{2}}{125} - e^{- (2.4468603146)}} = 2.4468556034 " />
<img class="equation_image" title=" x_{5} = (2.4468556034) - \frac{- \frac{69 (2.4468556034)^{3}}{125} + 8 + e^{- (2.4468556034)}}{- \frac{207 (2.4468556034)^{2}}{125} - e^{- (2.4468556034)}} = 2.4468556034 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%282.4468556034%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B69%20%282.4468556034%29%5E%7B3%7D%7D%7B125%7D%20%2B%208%20%2B%20e%5E%7B-%20%282.4468556034%29%7D%7D%7B-%20%5Cfrac%7B207%20%282.4468556034%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%282.4468556034%29%7D%7D%20%3D%202.4468556034%20" alt="LaTeX: x_{5} = (2.4468556034) - \frac{- \frac{69 (2.4468556034)^{3}}{125} + 8 + e^{- (2.4468556034)}}{- \frac{207 (2.4468556034)^{2}}{125} - e^{- (2.4468556034)}} = 2.4468556034 " data-equation-content=" x_{5} = (2.4468556034) - \frac{- \frac{69 (2.4468556034)^{3}}{125} + 8 + e^{- (2.4468556034)}}{- \frac{207 (2.4468556034)^{2}}{125} - e^{- (2.4468556034)}} = 2.4468556034 " />
</p> </p>