### Sidebar

calculus:resources:calculus_flipped_resources:limits:1.4-2.1_rates_of_change_tex

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} The cost (in dollars) of producing $x$ units of a certain commodity is $C(x) = 5000 + 6x + 0.05x^2$. \vskip 40pt Find the average rate of change of $C$ with respect to $x$ when the production level is changed from $x = 100$ to the given value: (Round your answers to the nearest cent.) \begin{enumerate} \item $x = 103$ \item $x = 101$ \end{enumerate} \end{frame} \begin{frame} \Large Each limit below represents the derivative of some function $f$ at some number $a$, find them. \vskip 15pt \LARGE \begin{enumerate}[a)] \item $\dlim_{h\to 0}\dfrac{(16+h)^{1/4}-2}{h}$ \vskip 15pt \item $\dlim_{x\to\pi/4}\dfrac{\tan(x)-1}{x-\pi/4}$ \vskip 15pt \item $\dlim_{t\to 1}\dfrac{t^5+t-2}{t-1}$ \vskip 15pt \end{enumerate} \end{frame} \begin{frame} The number of gallons of water in a tank $t$ minutes after the tank has started to drain is $Q(t)=200(30-t)^2$. \begin{enumerate} \item \begin{enumerate} \item What is the average rate at which the water flows out during the first ten minutes? \item during the five minutes from $t=5$ to $t=10$? \item during the two minutes from $t=8$ to $t=10$? \item during the minute from $t=9$ to $t=10$? \end{enumerate} \item Estimate how fast the water is running out of the tank at the end of ten minutes. \item Draw a graph of the function $Q$ for $0\leq t\leq20$. Draw the secant lines for the four time intervals used in part a). What are their slopes? \end{enumerate} \end{frame} \begin{frame} \Large The cost (in dollars) of producing $x$ units of a certain commodity is $C(x) = x^2 -2x + 10$. \vskip 10pt \begin{enumerate}[a)] \item Find the average rate of change of $C$ with respect to $x$ when the production level is changed from $x = 5$ to $x=7$ and for the change from $x=5$ to $x=6$. \vskip 15pt \item Find the instantaneous rate of change of $C$ with respect to $x$ when $x = 5$. \end{enumerate} \end{frame} \end{document}