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{\bf Review Questions, Calc I and App. E, 5.1-5.2}
\end{center}
Here are some selected topics from Calculus I that you might want 
to review if its been a while since you've seen them:


\bigskip


\begin{tabular}{|l|l|}
Topic: & Section\\
\hline
Definition of continuity&Section 2.5, p. 122\\
Where is $f(x)$ continuous?&Theorem 7, p. 127\\
What is the domain of $f(x)$?&Section 1.1, p. 12\\
 & Appendix A\\
$\displaystyle \lim_{x\rightarrow \infty}f(x)$ & Section 2.6, p. 137\\
(horizontal asympt)&Examples 3, 4,5 \\
Some antiderivatives&Section 4.10\\
\hline
\end{tabular}

\bigskip

Test your understanding by answering the following questions
from this material!
\begin{enumerate}
\item If $f(x)=\sqrt{\frac{1-x^2}{x^2-4}}$, where is $f$ continuous?
\item  Compute the limit, if it exists:
\begin{enumerate}
\item $\displaystyle \lim_{n\rightarrow \infty}\frac{3n^3+5n^2+2n}{6n^3+2n+1}$
\item $\displaystyle \lim_{n\rightarrow \infty}\frac{1-\sqrt{n}}{1+\sqrt{n}}$
\item $\displaystyle \lim_{n\rightarrow \infty} 6+\frac{18}{5n^2}\cdot
n(n+2)$
\item $ \displaystyle \lim_{x\rightarrow \infty}\sqrt{x^2+1}-\sqrt{x^2-1}$
\end{enumerate}
\item The following questions give a Riemann Sum.  What definite integral
does each represent?
\begin{enumerate}
\item $\displaystyle \lim_{n\rightarrow \infty}\sum_{i=1}^n \frac{1}{n}
\left[ \left(\frac{i}{n}\right)^3+1 \right]$
\item $\displaystyle \lim_{n\rightarrow \infty}\sum_{i=1}^n \frac{3}{n}
\sqrt{1+\frac{3i}{n}}$
\item $\displaystyle \lim_{n\rightarrow \infty}\frac{1}{n}
\sum_{i=1}^n \frac{1}{1+(i/n)^2} $
\end{enumerate}
\item  Give the general antiderivative, $F(x)$, where $f(x)$ is
given as:
\begin{enumerate}
\item $ f(x)=2x+5(1-x^2)^{-1/2}$
\item $ f(x)=\frac{3}{x^2}-\frac{5}{x^4}+\sqrt[3]{x^2}$
\item $ f(x)=4-3(1+x^2)^{-1}, F(1)=0$ (Give the specific antiderivative)
\end{enumerate}
\item Set up, but do not evaluate, the integral of the given function
on the given interval {\it using the definition}:
\begin{enumerate}
\item $f(x)=1+x, 2\leq x \leq 3$
\item $f(x)=\cos(x), \pi \leq x \leq 2\pi$
\item $f(x)=2+x+3x^2, -1 \leq x \leq 3$
\end{enumerate}
\item Evaluate each limit.  You may use the formulas:
$$
\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}, \quad 
\sum_{i=1}^n i^3=\left( \frac{n(n+1)}{2} \right)^2
$$
\begin{enumerate}
\item $\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^n \frac{1}{n}
\left( \frac{i}{n} \right)^2$
\item $\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^n 
\frac{1}{n} \left( \frac{i^3}{n^3}+1 \right)$
\item $\displaystyle \lim_{n\rightarrow \infty} \sum_{i=1}^n 
\frac{2}{n} \left[ \left( \frac{2i}{n} \right)^3 +  5
\left( \frac{2i}{n} \right)\right]$
\end{enumerate}
\item What definite integral did each of those limits represent?
\end{enumerate}
\end{document}
