Lecture 20: Review of Chapter 3

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The lecture

Test 2 took place today.

Test two

What follows is the TEX code used to produce test 2. When the software to convert this code to a more readable format, I will convert it.

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 $(3,1/6)$.


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