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\centerline{\textbf{\large Exam \#2 $\cdot$ Thursday, March 1, 2007}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 8 $\cdot$ Spring 2007  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

% xxx conceptual ...

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\textbf{Problem 1.}  Let $f(x) = x^x$.  Numerically approximate $f'(2)$ using
difference quotients.  Use at least three successively smaller values of $h$.

\vspace*{2.0in}

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\textbf{Problem 2.}  On a mountain-climbing expedition, you find that the air
becomes cooler as you climb.  Let $y$ be your altitude above sea level,
measured in feet; let $H$ be the air temperature in degrees Fahrenheit.

\vspace*{0.2in}

\textbf{Part (a).} What are the units of $H'(y)$?

\vspace*{0.5in}

\textbf{Part (b).} What is the sign of $H'(y)$?

\vspace*{0.5in}

\textbf{Part (c).} Give a practical interpretation of $H^{-1}(35)$.

\vspace*{1.0in}

\textbf{Part (d).} Give a practical interpretation of $H'(7500)$.

\vspace*{1.0in}

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\textbf{Problem 3.}  In a lab experiment, you have microorganisms growing
in a Petri dish.  The number $m$ of microrganisms, in millions, is a
function of time $t$ in days since the start of the experiment.  This
number is given by
$$
	m(t) = 4.1 e^{0.24t}.
$$
Find the rate of change in population on day 5 of the experiment.  In your
answer, please show units.

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\newpage
\textbf{Problem 4.}  Let $G(t)=2^{-rt} \sin(at)$.  Find $G'(t)$.

\vspace*{2.0in}

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\textbf{Problem 5.}  Let $f(z)=\tan^{-1}(z)$.  Compute $f''(1)$.

\vspace*{2.0in}

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\textbf{Problem 6.}  Let $q(x)=\ln(2+2x+x^2)$.

\vspace*{0.2in}

\textbf{Part (a).} Find $q'(x)$.

\vspace*{2.0in}

\textbf{Part (b).} Find an equation for the tangent line to $q(x)$ at $x=3$.

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\newpage
\textbf{Problem 7.}  Let $F(t)$ and $G(t)$ be given by the following
graphs.

\includegraphics{e2F.eps}
\qquad
\includegraphics{e2G.eps}

\textbf{Part (a).} Find $G'(40)$.

\vspace*{1.0in}

\textbf{Part (b).} Let $H(t) = \frac{F(t)}{G(t)}$.  Find $H'(50)$.

\vspace*{1.0in}

\textbf{Part (c).} Let $H(t) = F(G(t))$.  Find $H'(30)$.
\vspace*{1.0in}

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\textbf{Problem 8.}  Let
$$
	f(x) = \begin{cases}
		\sin(x), & x \ge 0 \\
		x-x^3, & x \le 0. \\
	\end{cases}
$$
\vspace*{0.2in}

\textbf{Part (a).}  Find $f'(x)$.  Write it as a piecewise function.
\vspace*{1.0in}

\textbf{Part (b).}  Is the original function $f(x)$ differentiable at $x=0$?
Why or why not?
(Hint:  graph it.)
\vspace*{1.0in}

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\newpage
\textbf{Problem 9.}  When is $g(x)=x^3+bx^2+cx+d$ concave up?  Assume $b,c,d$
are constants.  (You will need to solve an inequality.)

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