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Use Newton's method to find the first 5 approximations of the solution to the equation \(\displaystyle e^{- x}= \frac{9 x^{3}}{125} - 6\) using \(\displaystyle x_0=5\).
Using the formula for Newton's method gives \begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{125} + 6 + e^{- x_{n}}}{- \frac{27 x_{n}^{2}}{125} - e^{- x_{n}}} \end{equation*} Using \(\displaystyle x_0 = 5\) and \(\displaystyle n = 0,1,2,3,\) and \(\displaystyle 4\) gives: \begin{equation*}x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{125} + 6 + e^{- (5.0000000000)}}{- \frac{27 (5.0000000000)^{2}}{125} - e^{- (5.0000000000)}} = 4.4463829980\end{equation*} \begin{equation*}x_{2} = (4.4463829980) - \frac{- \frac{9 (4.4463829980)^{3}}{125} + 6 + e^{- (4.4463829980)}}{- \frac{27 (4.4463829980)^{2}}{125} - e^{- (4.4463829980)}} = 4.3722276141\end{equation*} \begin{equation*}x_{3} = (4.3722276141) - \frac{- \frac{9 (4.3722276141)^{3}}{125} + 6 + e^{- (4.3722276141)}}{- \frac{27 (4.3722276141)^{2}}{125} - e^{- (4.3722276141)}} = 4.3709675304\end{equation*} \begin{equation*}x_{4} = (4.3709675304) - \frac{- \frac{9 (4.3709675304)^{3}}{125} + 6 + e^{- (4.3709675304)}}{- \frac{27 (4.3709675304)^{2}}{125} - e^{- (4.3709675304)}} = 4.3709671706\end{equation*} \begin{equation*}x_{5} = (4.3709671706) - \frac{- \frac{9 (4.3709671706)^{3}}{125} + 6 + e^{- (4.3709671706)}}{- \frac{27 (4.3709671706)^{2}}{125} - e^{- (4.3709671706)}} = 4.3709671706\end{equation*}
\begin{question}Use Newton's method to find the first 5 approximations of the solution to the equation $e^{- x}= \frac{9 x^{3}}{125} - 6$ using $x_0=5$.
\soln{9cm}{Using the formula for Newton's method gives
\begin{equation*}x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{125} + 6 + e^{- x_{n}}}{- \frac{27 x_{n}^{2}}{125} - e^{- x_{n}}} \end{equation*}
Using $x_0 = 5$ and $n = 0,1,2,3,$ and $4$ gives:
\begin{equation*}x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{125} + 6 + e^{- (5.0000000000)}}{- \frac{27 (5.0000000000)^{2}}{125} - e^{- (5.0000000000)}} = 4.4463829980\end{equation*}
\begin{equation*}x_{2} = (4.4463829980) - \frac{- \frac{9 (4.4463829980)^{3}}{125} + 6 + e^{- (4.4463829980)}}{- \frac{27 (4.4463829980)^{2}}{125} - e^{- (4.4463829980)}} = 4.3722276141\end{equation*}
\begin{equation*}x_{3} = (4.3722276141) - \frac{- \frac{9 (4.3722276141)^{3}}{125} + 6 + e^{- (4.3722276141)}}{- \frac{27 (4.3722276141)^{2}}{125} - e^{- (4.3722276141)}} = 4.3709675304\end{equation*}
\begin{equation*}x_{4} = (4.3709675304) - \frac{- \frac{9 (4.3709675304)^{3}}{125} + 6 + e^{- (4.3709675304)}}{- \frac{27 (4.3709675304)^{2}}{125} - e^{- (4.3709675304)}} = 4.3709671706\end{equation*}
\begin{equation*}x_{5} = (4.3709671706) - \frac{- \frac{9 (4.3709671706)^{3}}{125} + 6 + e^{- (4.3709671706)}}{- \frac{27 (4.3709671706)^{2}}{125} - e^{- (4.3709671706)}} = 4.3709671706\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{9 x^{3}}{125} - 6 " src="/equation_images/%20%5Cdisplaystyle%20e%5E%7B-%20x%7D%3D%20%5Cfrac%7B9%20x%5E%7B3%7D%7D%7B125%7D%20-%206%20" alt="LaTeX: \displaystyle e^{- x}= \frac{9 x^{3}}{125} - 6 " data-equation-content=" \displaystyle e^{- x}= \frac{9 x^{3}}{125} - 6 " /> using <img class="equation_image" title=" \displaystyle x_0=5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%3D5%20" alt="LaTeX: \displaystyle x_0=5 " data-equation-content=" \displaystyle x_0=5 " /> . </p> </p><p> <p>Using the formula for Newton's method gives
<img class="equation_image" title=" x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{125} + 6 + e^{- x_{n}}}{- \frac{27 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%7B9%20x_%7Bn%7D%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20x_%7Bn%7D%7D%7D%7B-%20%5Cfrac%7B27%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{9 x_{n}^{3}}{125} + 6 + e^{- x_{n}}}{- \frac{27 x_{n}^{2}}{125} - e^{- x_{n}}} " data-equation-content=" x_{n+1} = x_{n} - \frac{- \frac{9 x_{n}^{3}}{125} + 6 + e^{- x_{n}}}{- \frac{27 x_{n}^{2}}{125} - e^{- x_{n}}} " />
Using <img class="equation_image" title=" \displaystyle x_0 = 5 " src="/equation_images/%20%5Cdisplaystyle%20x_0%20%3D%205%20" alt="LaTeX: \displaystyle x_0 = 5 " data-equation-content=" \displaystyle x_0 = 5 " /> 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} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{125} + 6 + e^{- (5.0000000000)}}{- \frac{27 (5.0000000000)^{2}}{125} - e^{- (5.0000000000)}} = 4.4463829980 " src="/equation_images/%20x_%7B1%7D%20%3D%20%20%285.0000000000%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%285.0000000000%29%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20%285.0000000000%29%7D%7D%7B-%20%5Cfrac%7B27%20%285.0000000000%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%285.0000000000%29%7D%7D%20%3D%204.4463829980%20" alt="LaTeX: x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{125} + 6 + e^{- (5.0000000000)}}{- \frac{27 (5.0000000000)^{2}}{125} - e^{- (5.0000000000)}} = 4.4463829980 " data-equation-content=" x_{1} = (5.0000000000) - \frac{- \frac{9 (5.0000000000)^{3}}{125} + 6 + e^{- (5.0000000000)}}{- \frac{27 (5.0000000000)^{2}}{125} - e^{- (5.0000000000)}} = 4.4463829980 " />
<img class="equation_image" title=" x_{2} = (4.4463829980) - \frac{- \frac{9 (4.4463829980)^{3}}{125} + 6 + e^{- (4.4463829980)}}{- \frac{27 (4.4463829980)^{2}}{125} - e^{- (4.4463829980)}} = 4.3722276141 " src="/equation_images/%20x_%7B2%7D%20%3D%20%20%284.4463829980%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%284.4463829980%29%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20%284.4463829980%29%7D%7D%7B-%20%5Cfrac%7B27%20%284.4463829980%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%284.4463829980%29%7D%7D%20%3D%204.3722276141%20" alt="LaTeX: x_{2} = (4.4463829980) - \frac{- \frac{9 (4.4463829980)^{3}}{125} + 6 + e^{- (4.4463829980)}}{- \frac{27 (4.4463829980)^{2}}{125} - e^{- (4.4463829980)}} = 4.3722276141 " data-equation-content=" x_{2} = (4.4463829980) - \frac{- \frac{9 (4.4463829980)^{3}}{125} + 6 + e^{- (4.4463829980)}}{- \frac{27 (4.4463829980)^{2}}{125} - e^{- (4.4463829980)}} = 4.3722276141 " />
<img class="equation_image" title=" x_{3} = (4.3722276141) - \frac{- \frac{9 (4.3722276141)^{3}}{125} + 6 + e^{- (4.3722276141)}}{- \frac{27 (4.3722276141)^{2}}{125} - e^{- (4.3722276141)}} = 4.3709675304 " src="/equation_images/%20x_%7B3%7D%20%3D%20%20%284.3722276141%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%284.3722276141%29%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20%284.3722276141%29%7D%7D%7B-%20%5Cfrac%7B27%20%284.3722276141%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%284.3722276141%29%7D%7D%20%3D%204.3709675304%20" alt="LaTeX: x_{3} = (4.3722276141) - \frac{- \frac{9 (4.3722276141)^{3}}{125} + 6 + e^{- (4.3722276141)}}{- \frac{27 (4.3722276141)^{2}}{125} - e^{- (4.3722276141)}} = 4.3709675304 " data-equation-content=" x_{3} = (4.3722276141) - \frac{- \frac{9 (4.3722276141)^{3}}{125} + 6 + e^{- (4.3722276141)}}{- \frac{27 (4.3722276141)^{2}}{125} - e^{- (4.3722276141)}} = 4.3709675304 " />
<img class="equation_image" title=" x_{4} = (4.3709675304) - \frac{- \frac{9 (4.3709675304)^{3}}{125} + 6 + e^{- (4.3709675304)}}{- \frac{27 (4.3709675304)^{2}}{125} - e^{- (4.3709675304)}} = 4.3709671706 " src="/equation_images/%20x_%7B4%7D%20%3D%20%20%284.3709675304%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%284.3709675304%29%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20%284.3709675304%29%7D%7D%7B-%20%5Cfrac%7B27%20%284.3709675304%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%284.3709675304%29%7D%7D%20%3D%204.3709671706%20" alt="LaTeX: x_{4} = (4.3709675304) - \frac{- \frac{9 (4.3709675304)^{3}}{125} + 6 + e^{- (4.3709675304)}}{- \frac{27 (4.3709675304)^{2}}{125} - e^{- (4.3709675304)}} = 4.3709671706 " data-equation-content=" x_{4} = (4.3709675304) - \frac{- \frac{9 (4.3709675304)^{3}}{125} + 6 + e^{- (4.3709675304)}}{- \frac{27 (4.3709675304)^{2}}{125} - e^{- (4.3709675304)}} = 4.3709671706 " />
<img class="equation_image" title=" x_{5} = (4.3709671706) - \frac{- \frac{9 (4.3709671706)^{3}}{125} + 6 + e^{- (4.3709671706)}}{- \frac{27 (4.3709671706)^{2}}{125} - e^{- (4.3709671706)}} = 4.3709671706 " src="/equation_images/%20x_%7B5%7D%20%3D%20%20%284.3709671706%29%20-%20%5Cfrac%7B-%20%5Cfrac%7B9%20%284.3709671706%29%5E%7B3%7D%7D%7B125%7D%20%2B%206%20%2B%20e%5E%7B-%20%284.3709671706%29%7D%7D%7B-%20%5Cfrac%7B27%20%284.3709671706%29%5E%7B2%7D%7D%7B125%7D%20-%20e%5E%7B-%20%284.3709671706%29%7D%7D%20%3D%204.3709671706%20" alt="LaTeX: x_{5} = (4.3709671706) - \frac{- \frac{9 (4.3709671706)^{3}}{125} + 6 + e^{- (4.3709671706)}}{- \frac{27 (4.3709671706)^{2}}{125} - e^{- (4.3709671706)}} = 4.3709671706 " data-equation-content=" x_{5} = (4.3709671706) - \frac{- \frac{9 (4.3709671706)^{3}}{125} + 6 + e^{- (4.3709671706)}}{- \frac{27 (4.3709671706)^{2}}{125} - e^{- (4.3709671706)}} = 4.3709671706 " />
</p> </p>