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\noindent
\textbf{Math 105 Exam III.}
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\hbox to 3.3in{Name:\enspace\hrulefill}

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\noindent
April 13, 1999
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\hbox to 3.3in{Instructor's name:\enspace\hrulefill}

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\noindent
\hspace{3.15in}
\hbox to 3.3in{Time of MWF class:\enspace\hrulefill}

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Show all work and put your answers in the spaces provided.  You will
receive no credit if the answer is not in the space provided and no
partial credit for a wrong answer if you do not show your work.
Answers may be approximated by decimals if desired, but full points
will also be awarded for correct answers left in exact form in
terms of fractions, logarithms, or exponentials.

\vspace{0.2in}

\noindent 
Sections of the course: 
\begin{tabbing}
Section 01 \hspace{0.2in}  \= Brown  \hspace{0.2in} \= MWF
10:40-11:30 am \\
Section 02                 \> Palmieri \> MWF 1:55-2:45 pm
\end{tabbing}

\newpage

\begin{questions}

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\question A company determines that when producing $x$ units, its
profit is given by this function: $\displaystyle P(x) = -\frac{1}{3}
x^{2} + 200x - 1800$.

\begin{parts}

\part[4] What is the marginal profit function?

\vspace{1in}

\part[2] What is the marginal profit when the production level $x$ is
150?

\vspace{1in}

\part[2] What does that tell you about the change in profit as the
production level increases from 150 to 151?

\vspace{1in}

\end{parts}

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\question[8] Use linear approximation to estimate $(1.03)^{2/3}$.

\vspace{1.8in}

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\question[6] A piece of petrified wood is found at an archaeological site
and contains $87\%$ of the original carbon-14.  How old is the piece
of wood?  (The decay constant for carbon-14 is $k=0.000121$.)

\newpage

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\question[15] Find the derivatives of these functions:

\begin{parts}

\part $\ln (4x^{2/5})$

\vspace{1.6in}

\part $(x^{3}-3x+2)^{17}$

\vspace{1.6in}

\part $\sqrt{6^{x}}$

\vspace{1.6in}

\end{parts}

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\question[8] A square, with edge of length $x$, is expanding.  When the
edge is 5 ft., the edge is growing at 3 ft./min.  How fast is the area 
growing at this point?

\vspace{1.5in}

\newpage

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\question[10] You get a cup of coffee from the drive-through window of a
fast food restaurant.  The coffee starts at $190^{\circ}$---too hot to
drink---and the temperature in your car is a constant $70^{\circ}$.
After 1 minute, the coffee cools to $175^{\circ}$.  You like your
coffee at $150^{\circ}$; how long until it reaches this temperature?

\vspace{4.5in}

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\question[7] Let $\displaystyle y=\frac{x-2}{x+1}$.  Find the equation of
the line tangent to this curve at the point $(0,-2)$.

\vspace{2in}

\newpage

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\question[8] Let $f(x)=xe^{-3x}$.

\begin{parts}

\part Where is $f(x)$ increasing?

\vspace{2.5in}

\part Where is $f(x)$ decreasing? 
 
\vspace{1.5in}

\end{parts}

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\question[8] Where does the function $f(x)=x^{3}-3x^{2}$ have critical
points?  For each of those critical points, is it a relative maximum,
a relative minimum, or neither?

\vspace{1.5in}

\newpage

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\question[8] A circle of radius $r$ has area $A=\pi r^{2}$.  The radius is 
changing: at time $t$, the radius is given by $r=f(t)=3t+2$.  What is
$\displaystyle \frac{dA}{dt}$, the rate of change of the area with
respect to time?  (Write your answer in terms of $t$, not $r$.)

\vspace{2.5in}

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\question[14] Find the derivatives of these functions:

\begin{parts}

\part $\displaystyle \frac{e^{x^{4}+7}}{x+2}$ 

\vspace{2in}

\part $x^{2} \ln (x^{2} +7x - 2)$

\end{parts}

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\end{questions}

\end{document}

