\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}

\begin{document}
\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day on Consequences of FTC, ``Part II''}
\centerline{January 31/February 1, 2018}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
Recall that the ``FTC, part II'' says:  If $f$ is continuous on $[a,b]$, then
$$\frac{d}{dx} \int_a^x f(t)\ dt = f(x)$$
for all $x$ in $[a,b]$.  (Note: at the endpoints $x = a$ and $x = b$, we may
be talking only about ``one-sided derivatives'' if $f$ is not defined for $x < a$
or $x > b$.)  

\vskip 10pt
\noindent
{\it Questions}

\vskip 10pt
\begin{enumerate}
\item[(0)]  Look at the function $f(t)$ plotted on the back.  What is 
$\frac{d}{dx} \int_0^x f(t)\ dt$ at $x = 3$? 
\item[(1)]  What is 
$$\frac{d}{dx} \int_3^x \sqrt[3]{t^5 + 3t + 1}\ dt?$$
\item[(2)]  Same question for 
$$\frac{d}{dx} \int_0^x \frac{\tan(t)}{t^3 + 2}\ dt$$
\item[(3)]  What is 
$$\frac{d}{dx} \int_x^4 e^{4t^2 + 7}\ dt?$$
\item[(4)]  What about
$$\frac{d}{dx} \int_0^{x^2} \sqrt{t^2 + 1}\ dt?$$
(Use the chain rule.)
\item[(5)]  What is 
$$\frac{d}{dx} \int_{-x^3}^{x^4} \frac{t}{t + 1}$$
(You'll need to use the interval union property, the interchange of limits
property and the chain rule here!)
\item[(6)]  Finally, suppose someone asked you for a function whose derivative
was $f(x) = \displaystyle{\frac{x^2}{x^4 + x + 1}}$.  How could you ``cook one up'' using
what we have talked about?  (In case you are worried about possible 
discontinuities from vertical asymptotes, you don't need to: this function $f$ is
continuous for all real $x$ -- the denominator is never zero.)
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=.8\linewidth]{132FIS.png}
\end{center}
\caption{The region}
\end{figure}
\end{document}



