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\centerline{\textbf{\large Exam \#4 $\cdot$ Thursday, April 26, 2007}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 8 $\cdot$ Spring 2007  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.}
Let
\begin{eqnarray*}
	x(t) &=& 3t^2 - 6t \\
	y(t) &=& \frac{4}{3}\,t^3 - 4t.
\end{eqnarray*}

\textbf{Part (a).} Find the time(s) $t$, if any, when the particle comes to a stop.
\vspace*{2.2in}

\textbf{Part (b).} Find an equation for the tangent line to this curve at
$t=3$.
\vspace*{2.2in}

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\textbf{Problem 2.}  The function $H(t)$ describes the growth rate in thousands
per month of flour beetles in a jar, where $t$ is measured in months since the
start of the year.

\vspace*{0.1in}
\textbf{Part (a).}  What are the units of $\int_{4}^{7} H(t) dt$?
\vspace*{1.0in}

\textbf{Part (b).}  Give a practical interpretation of $\int_{4}^{7} H(t) dt$.
\vspace*{1.0in}

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\newpage
\textbf{Problem 3.}  The air pressure within a chamber is given by
$$
	P(t) = 2.1 + 0.4 t^{0.5}
$$
where $P$ is in units called atmospheres and $t$ is measured in hours.  Find
the average pressure over the time between $t=2$ hours and $t=5$ hours.

\vspace*{2.5in}

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\textbf{Problem 4.}  Find the exact area between $f(x) = e^x-2$ and $g(x) = -1$
on the interval $[2, 4]$.
\vspace*{2.5in}

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\textbf{Problem 5.}  Find the general antiderivative:
$$
	\int \left( \frac{y^{2.1}}{3} - \frac{7}{y} + 0.2 A e^y + B\right) \;dy.
$$

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\newpage
\textbf{Problem 6.} Let $g'(x)$ be given by the following graph,
and suppose $g(0) = 2$:
\vspace*{0.2in}
\begin{center}
\includegraphics{e4_g.eps}
\end{center}
\vspace*{0.2in}

\textbf{Part (a).}  What are the $x$-coordinates of the critical points of $g(x)$?
\vspace*{0.8in}

\textbf{Part (b).}  What are the $x$-coordinates of the inflection points of $g(x)$?
\vspace*{0.8in}

\textbf{Part (c).}  Find the values of $g(x)$ at the critical and inflection points.
\vspace*{2.2in}

\textbf{Part (d).}  Sketch a graph of $g(x)$.  Label
critical points and inflection points of $g(x)$.

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\newpage
\textbf{Problem 7.}  The quantity $A$ varies with time as specified by
$$
	\frac{dA}{dt} = 7.3\cos(t)-0.04.
$$
\vspace*{0.1in}

\textbf{Part (a).}  Write down a general solution for $A$.
\vspace*{1.5in}

\textbf{Part (b).}  Given that $A(0) = 2.1$, write down a specific solution for
$A$.
\vspace*{1.5in}

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\textbf{Problem 8.}

\textbf{Part (a).}
Let
$$
	G(x) = \int_1^x e^{t^2} dt.
$$
Determine (with justification) whether $G(x)$ is increasing,
decreasing, or constant.
\vspace*{1.5in}

\textbf{Part (b).}
Now let
$$
	G(x) = \int_1^{ax+b} e^{t^2} dt.
$$
Find $G'(x)$.

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