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\centerline{\textbf{\large Exam \#1 $\cdot$ Wednesday, September 17, 2008}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

% ----------------------------------------------------------------
\textbf{Problem 1.}
Let
$$
	f(y) = 3y+7
$$
and
$$
	g(y) = y^2 e^{-2y}.
$$

\textbf{Part (a).}
Find $g(f(y))$.
\vspace*{1.5in}

\textbf{Part (b).}
Find $g(y)f(y)$.

\vspace*{1.5in}

% ----------------------------------------------------------------
\textbf{Problem 2.}
Solve for $z$ exactly:
$$
	6 \cdot 3^z = 7 \cdot 2^z.
$$

\vspace*{2.0in}

% ----------------------------------------------------------------
\textbf{Problem 3.}
An unknown brownish substance weighing 1.5 pounds was placed in your roommate's
refrigerator.  It was then observed to increase in weight by 7 percent per day.
Find the number of days (to the nearest tenth of a day) it will take for the
substance to weigh 6 pounds.

\vspace*{0.1in}

% ----------------------------------------------------------------
\newpage
% ----------------------------------------------------------------
\textbf{Problem 4.}
The following data represent profit $P$ (in dollars) versus quantity $q$ of
wristwatches sold at a nearby store.
$$
\begin{tabular}{|r|r|}
\hline $q$   & $P$ \\
\hline
\hline $140$ & $45$ \\
\hline $180$ & $75$ \\
\hline $220$ & $105$ \\
\hline $260$ & $135$ \\
\hline
\end{tabular}
$$

\textbf{Part (a).}
Write down a linear equation for $P$ as a function of $q$.  Simplify your work
as much as possible.  Then use the equation to find $P(90)$.
% P = 0.75 q - 60
\vspace*{1.5in}

\textbf{Part (b).}
Verbally describe the significance of the horizontal intercept, from a business
point of view.
\vspace*{1.3in}

\textbf{Part (c).}
Find the horizontal intercept.

% q = 80

\vspace*{1.3in}

\textbf{Part (d).}
Verbally describe the significance of the vertical intercept, from a business
point of view.

\vspace*{1.3in}

% ----------------------------------------------------------------
\textbf{Problem 5.}
Let
$$
	f(x) = \frac{x^2 - kx + 3}{x-3}.
$$
Find $k$ such that $\lim_{x \to 3} f(x)$ exists.

\vspace*{0.1in}

% ----------------------------------------------------------------
\newpage
% ----------------------------------------------------------------
\textbf{Problem 6.}
On December 9, 1999 the Mars Polar Lander entered the Martian atmosphere.  The
following is a reconstruction of its altitude $A(t)$ versus time $t$ in
seconds, measured in seconds from the start of the final landing phase.  At an
altitude of 40 meters, the landing rockets switched off prematurely, letting
the lander free-fall to the Martian surface --- after which time the lander was
never heard from again.

\vspace*{0.2in}
\includegraphics{figures/marsfig.eps}
\vspace*{0.2in}

\textbf{Part (a).}
Using the graph, estimate $A(24)$.  Describe verbally what this quantity means.
\vspace*{1.3in}

\textbf{Part (b).}
Using the graph, estimate $A^{-1}(20)$.  Describe verbally what this quantity
means.
\vspace*{1.3in}

\textbf{Part (c).}
Using the graph, estimate the time of impact.
\vspace*{1.3in}

\textbf{Part (d).}
Using the graph, estimate the lander's speed at time of impact.

% 40 m:  2.4 m/s
% Then free fall at 3.7 m/s^2

% A(t) = at^2 + bt + c on both pieces: before and after 26.

% Before 26:
% A(20) = 100
% A(26) =  40
% A'(26) = -2.5  (rocket-assisted descent)

% After 26:
% A(26) = 40     (height continuity)
% A'(26) = -2.5  (velocity continuity)
% A''(26) = -3.7 (Martian free-fall)

% Solve for a,b,c on both pieces.  Results are in figures/marsfig.m.

% ----------------------------------------------------------------
\newpage

% ----------------------------------------------------------------
\textbf{Problem 7.}
Invert the following function:
$$
	f(x) = C a^x.
$$
(You may assume $C>0$ and $a>0$.)

\vspace*{1.25in}

% ----------------------------------------------------------------
\textbf{Problem 8.}
Evaluate the following limit, using algebraic properties of limits:
$$
	\lim_{h \to 0} \frac{(x+h)^2 + 2(x+h) - x^2 - 2x}{h}.
$$

\vspace*{4.0in}

% ----------------------------------------------------------------
\textbf{Problem 9.}
Numerically estimate the following limit to 3 decimal places:
$$
	\lim_{\theta \to 0} \frac{2\theta}{\sin(3\theta)}.
$$

\vspace*{0.1in}

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\end{document}
