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Questions: Algebra BusinessCalculus
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Find the value of \(\displaystyle c\) for \(\displaystyle f(x)=- x^{2} + 3 x - 5\) that satisfies the Mean Value Theorem on \(\displaystyle [-7, 8]\).
The slope of the secant line is \(\displaystyle m = \frac{f(8) -f(-7)}{8-(-7)}=2\). Setting the derivative equal to the slope of the secant line gives the equation \(\displaystyle 2 = 3 - 2 x\). The solution is \(\displaystyle c = \frac{1}{2}\).
\begin{question}Find the value of $c$ for $f(x)=- x^{2} + 3 x - 5$ that satisfies the Mean Value Theorem on $[-7, 8]$.
\soln{9cm}{The slope of the secant line is $m = \frac{f(8) -f(-7)}{8-(-7)}=2$. Setting the derivative equal to the slope of the secant line gives the equation $2 = 3 - 2 x$. The solution is $c = \frac{1}{2}$. }
\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>Find the value of <img class="equation_image" title=" \displaystyle c " src="/equation_images/%20%5Cdisplaystyle%20c%20" alt="LaTeX: \displaystyle c " data-equation-content=" \displaystyle c " /> for <img class="equation_image" title=" \displaystyle f(x)=- x^{2} + 3 x - 5 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D-%20x%5E%7B2%7D%20%2B%203%20x%20-%205%20" alt="LaTeX: \displaystyle f(x)=- x^{2} + 3 x - 5 " data-equation-content=" \displaystyle f(x)=- x^{2} + 3 x - 5 " /> that satisfies the Mean Value Theorem on <img class="equation_image" title=" \displaystyle [-7, 8] " src="/equation_images/%20%5Cdisplaystyle%20%5B-7%2C%208%5D%20" alt="LaTeX: \displaystyle [-7, 8] " data-equation-content=" \displaystyle [-7, 8] " /> . </p> </p><p> <p>The slope of the secant line is <img class="equation_image" title=" \displaystyle m = \frac{f(8) -f(-7)}{8-(-7)}=2 " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7Bf%288%29%20-f%28-7%29%7D%7B8-%28-7%29%7D%3D2%20" alt="LaTeX: \displaystyle m = \frac{f(8) -f(-7)}{8-(-7)}=2 " data-equation-content=" \displaystyle m = \frac{f(8) -f(-7)}{8-(-7)}=2 " /> . Setting the derivative equal to the slope of the secant line gives the equation <img class="equation_image" title=" \displaystyle 2 = 3 - 2 x " src="/equation_images/%20%5Cdisplaystyle%202%20%3D%203%20-%202%20x%20" alt="LaTeX: \displaystyle 2 = 3 - 2 x " data-equation-content=" \displaystyle 2 = 3 - 2 x " /> . The solution is <img class="equation_image" title=" \displaystyle c = \frac{1}{2} " src="/equation_images/%20%5Cdisplaystyle%20c%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20" alt="LaTeX: \displaystyle c = \frac{1}{2} " data-equation-content=" \displaystyle c = \frac{1}{2} " /> . </p> </p>