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calculus:resources:calculus_flipped_resources:applications:5.5_average_tex

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\begin{document} \begin{frame} Find the average value of each function on the given interval. \vskip 10pt \begin{enumerate}[a)] \item $f(x) = 10x - x^2$ on the interval $[0, 2]$ \vskip 15pt \item $f(\theta) = 11 \sec^2(\theta/4)$ on the interval $[0,\pi]$ \vskip 15pt \item $h(x) = 7 \cos^4(x)\sin(x)$ on the interval $[0,\pi]$ \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = 3\sqrt{x}$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[0, 16]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = (x-5)^2$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[4,7]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Consider the function $$f(x) = 9 sin(4x)$$ \begin{enumerate}[a)] \item Find the average value $f_{\mbox{ave}}$ of $f$ on the interval $[-\pi, \pi]$. \item Find all values $c$ such that $f_{\mbox{avg}}= f(c)$. \item Sketch the graph of $f$ and, in the same picture, a rectangle whose area is the same as the area under the graph of $f$. \end{enumerate} \end{frame} \begin{frame} Find all numbers $b$ such that the average value of $$f(x) = 7 + 10x - 9x^2$$ on the interval $[0, b]$ is equal to 8. \vskip 65pt The velocity $v$ of blood that flows in a blood vessel with radius $R$ and length $L$ at a distance $r$ from the central axis is $$v(r) =\frac{ P}{4\eta L}(R^2 - r^2)$$ where $P$ is the pressure difference between the ends of the vessel and $\eta$ is the viscosity of the blood. Find the average velocity (with respect to $r$) over the interval $0 \leq r \leq R$. \end{frame} \end{document}