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\begin{document} \begin{frame} \begin{center} Which function from $\{a,b,c\}$ is an antiderivative of $f$? \end{center} \begin{center} \includegraphics[height=190pt]{antiderivative.png} \end{center} \end{frame} \begin{frame} \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 15pt An antiderivative of a sum of functions, $f+g$, is an antiderivative of $f$ plus an antiderivative of $g$. \vskip 20pt An antiderivative of a product of functions, $fg$, is an antiderivative of $f$ times an antiderivative of $g$. \end{frame} \begin{frame} Suppose you are told that the acceleration function of an object is a continuous function $a(t)$. Let's say you are given that $v(0)=1$. \vskip 20pt \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 15pt You can find the position of the object at any time $t$. \end{frame} \begin{frame} Find the most general antiderivative of each function. \vskip 5pt \begin{itemize} \item[\bf (i)] $f(x)=\dfrac{1}{2}x^2-2x+6$ \vskip 15pt \item[\bf (ii)] $g(x)=(x+5)(2x-6)$ \vskip 15pt \item[\bf (iii)] $h(x)=\dfrac{3+t+t^2}{\sqrt{t}}$ \end{itemize} \end{frame} \begin{frame} Let $f$ be a function so that $f''(x)=12x+\sin(x)$. \vskip 5pt \begin{itemize} \item[\bf (i)] If you know nothing else about $f$, give the best formula you can for $f$. \vskip 15pt \item[\bf (ii)] If you know $f'(\pi)=1$, give the best formula you can for $f$. \vskip 15pt \item[\bf (iii)] If you know $f'(\pi)=1$, and $f(\pi)=0$, give the best formula you can for $f$. \end{itemize} \end{frame} \begin{frame} Find $f$ if $f''(\theta)=\sin(\theta)+\cos(\theta)$, $f(0)=3$, and $f'(0)=3$. \vskip 100pt Find $f$ if $f'''(x)=\cos(x)$, $f(0)=5$, $f'(0)=1$, and $f''(0)=8$. \end{frame} \begin{frame} $$f(x)=\dfrac{1}{x^2}$$ If $F(x)$ is an antiderivative of $f$ with the property $F(1)=1$. \vskip 15pt \begin{block}{} \begin{center} {\LARGE {\bf True} or \bf{False}} \end{center} \end{block} \vskip 10pt $$F(-1)=3$$ \end{frame} \begin{frame} Find a function $f$ such that $f'(x)=2x^3$ and the line $2x+y$ is tangent to the graph of $f$. \vskip 100pt In each of the following, a particle is moving with the given data. Find the position function of the particle. \begin{enumerate}[a)] \item $v(t)=1.5\sqrt{t}$, $s(16)=67$. \item $a(t)=2t+5$, $s(0)=2$, $v(0)=-5$. \end{enumerate} \end{frame} \begin{frame} A stone was dropped off a cliff and hit the ground with a speed of 112 ft/s. What is the height of the cliff? (Use 32 ft/$\mbox{s}^2$ for the acceleration due to gravity.) \vskip 100pt What constant acceleration is required to increase the speed of a car from 25 mi/h to 53 mi/h in 3 s? \end{frame} \begin{frame} If a diver of mass $m$ stands at the end of a diving board with length $L$ and linear density $\rho$, then the board takes on the shape of a curve $y = f(x)$, where $$EIy'' = mg(L - x) + \frac{1}{2}\rho g(L - x)^2.$$ $E$ and $I$ are positive constants that depend on the material of the board and $g$ ($< 0$) is the acceleration due to gravity. \begin{enumerate}[a)] \item Find an expression for the shape of the curve. \item Use $f(L)$ to estimate the distance below the horizontal at the end of the board. \end{enumerate} \end{frame} \end{document}