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--- calculus:resources:calculus_flipped_resources:applications:3.9_antiderivatives_tex [2015/08/29 02:19] – Page moved from people:jbrennan:calculus_flipped_resources:applications:3.9_antiderivatives_tex to calculus:resources:calculus_flipped_resources:applications:3.9_antiderivatives_tex nye
+++ calculus:resources:calculus_flipped_resources:applications:3.9_antiderivatives_tex [2015/08/29 02:35] (current) – external edit 127.0.0.1
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+TeX code compiled with \documentclass{beamer} using the Amsterdam theme.\\
+There is one png image needed to compile slides:\\
+[[http://www2.math.binghamton.edu/lib/exe/fetch.php/calculus/resources/calculus_flipped_resources/antiderivative.png|antiderivative.png]]\\
+<nowiki>
+\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}
+</nowiki>