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\centerline{\textbf{\large Exam \#4 $\cdot$ Thursday, December 4, 2008}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.}
Let
\begin{align*}
	x(t) &= 2t^3 - 15t^2 + 24t + 7 \\
	y(t) &= t^2 + t + 1.
\end{align*}
Find all values of $t$ such that this curve has a vertical tangent line.

\vspace*{2.5in}

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\textbf{Problem 2.}

\textbf{Part (a).}
Find the average value of
$$
	G(y) = \frac{A}{y} + B y
$$
over the interval $-4 \le y \le -1$, where $A$ and $B$ are positive constants.
\vspace*{3.0in}

\textbf{Part (b).}
If the units of $y$ and $G(y)$ are inches and gallons, respectively,
what are the units of the average value found in part (a)?

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\newpage
\textbf{Problem 3.}
Suppose the rate of change of the price of a stock is $R(t)$ where $t$ is
measured in days since the start of the year and $R(t)$ is dollars per day.

\vspace*{0.1in}

\textbf{Part (a).}  What are the units of $\int_{31}^{59} R(t) dt$?
\vspace*{1.2in}

\textbf{Part (b).}  Give a practical interpretation of $\int_{31}^{59} R(t) dt$.
\vspace*{1.2in}

\textbf{Part (c).} If $R(t) < 0$ for $30 \le t \le 60$, can you tell what the
sign of $\int_{31}^{59} R(t) dt$ is?  If so, what is the sign?  If not, why
not?
\vspace*{1.2in}

\textbf{Part (d).} If $R(t) < 0$ for $30 \le t < 40$ and $R(t) > 0$ for $40 < t
\le 60$, can you tell what the sign of $\int_{31}^{59} R(t) dt$ is?  If so,
what is the sign?  If not, why not?
\vspace*{1.5in}

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\textbf{Problem 4.}
Find the exact area of the region between $f(x) = x^{2/3}$ and $g(x)=\sin(x)$
over the interval $1/8 \le x \le \pi$.

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\newpage
\textbf{Problem 5.}
Label the following statements as correct (C) or incorrect (I):

\textbf{Part (a).}
If $f'(x) < 0$ on an interval then $f(x)$ is decreasing on that interval.
\vspace*{0.7in}

\textbf{Part (b).}
If $f'(x) = 0$ at a point then $f(x)$ has an inflection point at that point.
\vspace*{0.7in}

\textbf{Part (c).}
If $f'(x)$ has a maximum or minimum at a point then $f(x)$ has a critical
point there.
\vspace*{0.7in}

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\textbf{Problem 6.}
Let
$$
	\frac{dP}{dt} = t(1-t).
$$
\vspace*{0.1in}

\textbf{Part (a).}  Find a general solution for $P$.
\vspace*{3.0in}

\textbf{Part (b).}  Find a specific solution for $P$ if $P(3)=2$.

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\newpage
\textbf{Problem 7.}
Let $F(x)= \int_{10}^x f(t) dt$, with the following known values for $f(t)$:
$$
\begin{tabular}{|c||c|c|c|c|c|}
\hline $t$    & $0$     & $10$ & $20$ & $30$ & $40$ \\
\hline $f(t)$ & $-1.25$ & $-1.2$ & $-1.1$ & $-0.9$ & $-0.7$ \\
\hline
\end{tabular}
$$
Estimate $F(20)$ and $F(30)$.
\vspace*{3.0in}

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\textbf{Problem 8.}

\textbf{Part (a).}
Find
$$
	\frac{d}{d\theta} \;
	\int_{\pi/2}^\theta \frac{\sin(x)}{x} dx.
$$
\vspace*{1.5in}

\textbf{Part (b).}
Find
$$
	\frac{d}{d\theta} \;
	\int_{\pi/2}^{\pi-e^{-\theta}} \frac{\sin(x)}{x} dx.
$$

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\end{document}
