[ List of Lectures | Math 1100 Index ]
1. Questions (a) through (e) refer to the graph of the function $f$ given below.
\input{pieces}
\begin{enumerate}
\item $\displaystyle \lim_{x\rightarrow 1} f(x)=
$
\ansmt{0}{1}{2}{4}{\mbox{does not exist}}
\vfill
\item $\displaystyle \lim_{x\rightarrow 2^{+}} f(x)=
$
\ansmt{0}{1}{2}{4}{\mbox{does not exist}}
\vfill
\item A good estimate of $f'(-2)$ is
\ansmt{-1}{0}{1}{2}{\mbox{there is no good estimate}}
\item A good estimate of $f'(-1)$ is
\ansmt{-1}{0}{1}{2}{\mbox{there is no good estimate}}
\item A good estimate of $f'(2)$ is
\ansmt{-1}{0}{1}{2}{\mbox{there is no good estimate}}
\end{enumerate}
2. The line tangent to the graph of a function $f$ at the point $(2,3)$
on the graph also goes through the point $(0,7)$. What is $f'(2)$?
\ansmt{-2}{-1}{0}{1}{2}
3. What is the slope of the tangent line to the graph of
$f(x)=(2x)^{-2}$ at the point (8,1/256)?
\ansmt{-2}{-1}{-1/2}{-1/4}{-1/8}
On all the following questions, {\bf show your work.}
4. (20 points) The total weekly cost in dollars incurred by the Lincoln Record Company
in pressing x playing records is given by $C(x)=2000+3x-0.001x^2$ for $x$ in the range
0 to 6000.
\begin{enumerate}
\item Find the marginal cost function $C'(x)$.
\vspace{1in}
\item Find the average cost function $\overline{C}(x)$.
\vspace{1in}
\item Find the marginal average cost function $\overline{C}'(x)$.
\vspace{1in}
\item Interpret your results.
\vspace{1in}
\end{enumerate}
5. (15 points) Find the following derivatives.
\begin{enumerate}
\item $\frac{d}{dx} \sqrt{x^3+1}$
\vspace{1.8in}
\item $\frac{d}{dx} ((2x+1)^4\cdot 3x^2)$
\vspace{1.8in}
\item $\frac{d}{dx} \frac{2x-1}{3x+2}$
\vspace{1.8in}
\end{enumerate}
6. (15 points) Let $f(x)=1/(2x)$.
\begin{enumerate}
\item Construct $\frac{f(3+h)-f(3)}{h}$
\vspace{1.8in}
\item Simplify and take the limit of the expression in (a) as $h$ approaches 0
to find $f'(3)$.
\vspace{1.8in}
\item Use the information found in (b) to find an equation for the line
tangent to the graph of $f$ at the point $(3,1/6)$.