%13 October 1998, Russell Brown

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\course \hfill Russell Brown \\
Review 2  \hfill \today

The second hour exam is on Tuesday, 20 October from
7:30pm--9:30pm in CB 114. You may use a graphing calculator on the
exam. You may 
not use notes, textbooks, a computer or a calculator with symbolic
manipulation capabilities such as a TI-92.  

Below are a selection of problems to help you prepare for the
exam. You should also review material covered in lecture and the
problems assigned for recitation. 


\begin{enumerate}
\item
Estimate the following limits. Your answers should be correct to 2
decimal places.
$$
\lim _{x\rightarrow 0} \sin (e^x-1)/ x , \qquad \lim _{x\rightarrow 0}
\frac {2^x-3^x} x .
$$

\item Find the following limits using the limit laws, if they exist. Explain each
step of your solution. If a limit does not exist, determine if it
$\infty$ or $-\infty$.
$$
\lim_{x\rightarrow 2} \frac {x-2} {x^2-4}, \qquad \lim _{x\rightarrow
2} \frac {x+2} {x^2+4}, \qquad \lim_{x\rightarrow 2} \frac {x+3}{x-2}.
$$

\item Find the following limits, if they exist. If the limit does not
exist, explain why.
$$
\lim _{x\rightarrow 0} \frac {|x|} x, \qquad \lim _{x\rightarrow
\infty} \frac { x^2 + 3} { 3x^2+4x}.
$$

\item Use the squeeze theorem to evaluate the following limits.
$
\lim_{x\rightarrow \infty} \frac {\sin x } x$,  $ \lim _{x\rightarrow
0} x \sin (1/x)$.



\item Let $f(x)= 2^x$. Using the definition, estimate the derivative
at $x=2$.

\item Let $f(x) = \sqrt  {2x} $. Find the derivative, $f'(x)$, using the
definition. Give the domain of the derivative.



\item State the definition of the derivative of $f$ at $a$.  

\item If $f$ is differentiable at $a$, is $f$ continuous at $a$? If
your answer is yes, provide a proof. If your answer is no, provide an
example.

\item If $f$ is continuous at $a$, is $f$ differentiable at $a$?  If
your answer is yes, provide a proof. If your answer is no, provide an
example.

\item On which intervals are the following functions continuous? On
which intervals are the functions differentiable.
$$
f(x) = |x|, \qquad g(x) = \frac 1 {x^2-4}.
$$


\item  Show that the equation $\cos x = x$ has a solution.

\item Let $f(x) = e^{2x}$. a) Estimate $f'(0)$. Explain how you
arrived at your answer. b)  Use the tangent line at $0$ to approximate
$e^{1.02} $  and $ e^{0.99}$.  

\item If the prevailing price for milk is  $p$ dollars per gallon,
then a store will sell $Q(p)$ gallons of milk. 
a) What are the units for $Q'(p)$?
b) Do you expect $Q'(p) >0$ or $Q'(p) < 0 $? Why?

\item State the intermediate value theorem.

\item Draw the graph of a function which is defined, but not continuous, at $x=2$ and
which is continuous, but not differentiable, at $x=4$.

\item Below are the graphs of $f$, $f'$ and $f''$. Which is which?
Explain your answer.

\vspace{2 in}

\item  Below is the graph of a function $f$. Sketch the graph of
$f'$. 

\vspace{2 in}



\item Below is the graph of a function $f'$. Sketch two possibilities
for the graph of $f$.

\vspace{2 in}
\end{enumerate}
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