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\centerline{\textbf{\large Exam \#3 $\cdot$ Tuesday, April 3, 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.}
Consider the following graph of $f'(x)$:

\vspace*{0.1in}

\begin{center}
\includegraphics{e3_1.eps}
\end{center}

\vspace*{0.1in}

I repeat for emphasis:  I am showing you the graph of $f'(x)$, but
I will ask you questions about $f(x)$.

\vspace*{0.1in}

\textbf{Part (a).}
Which of the labeled points are critical points of $f(x)$?

\vspace*{0.8in}

\textbf{Part (b).}
Which of those critical points are local maxima (not minima) of $f(x)$?
(In this problem, I am not interested in local extrema at boundary points.)

\vspace*{0.8in}

\textbf{Part (c).}
Which labeled points are inflection points of $f(x)$?

\vspace*{0.8in}

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\textbf{Problem 2.}
Find $dy/dx$ if
$\ln(y)\sin(y) = \cos(x)$.

\vspace*{1.75in}

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\textbf{Problem 3.}
Let $f(z) = z^4+a/z^4$.
Find $a$ such that $f(z)$ has a minimum at $z=2$.

\vspace*{0.2in}
%\vspace*{2.0in}

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\newpage
\textbf{Problem 4.}
Let $g(x)=\sin^2(x)-\cos(x)$.

\vspace*{0.1in}

\textbf{Part (a).}
Find an equation for the tangent line to $g(x)$ at $x=\pi/2$.

\vspace*{2.5in}

\textbf{Part (b).}
Write down an equation for the error function involving $g(x)$ and the linear
approximation.

\vspace*{1.0in}

%\textbf{Part (c).}
%Using your calculator, graph the error function on the interval $0 \le x \le
%3$.  Using the space provided, sketch the original function, the linear
%approximation, and the error function.  From the graph, estimate the worst
%error on that interval.
%
%\vspace*{1.5in}

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\textbf{Problem 5.}
Find $b$ and $d$ such that $f(x) = x^3+bx^2+d$ has an inflection point at
$x=3$ and $y$-intercept $-5$.

\vspace*{2.5in}

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\textbf{Problem 6.}
Does the limit
$$
	\lim_{x\to \infty} \frac{x^2}{e^{0.0024x}}
$$
exist?  If so, what is it, and why?  If not, why not?

\vspace*{0.2in}
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\newpage
\textbf{Problem 7.}
A rectangular region is to be painted inside a semicircular region
as follows:

\vspace*{0.1in}

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\includegraphics{e3_hoop.eps}
\end{center}

\vspace*{0.1in}

The semicircle has radius 7 feet.  Find $x$ that maximizes the area of the
rectangle.  (You must show all your work and use calculus.)

\vspace*{2.5in}

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\textbf{Problem 8.}
An amusement-park ride is upright and circular with 50-foot radius.  Passengers
are seated around the perimeter while the wheel turns counterclockwise at some
(as yet unknown) constant rate:

\vspace*{0.2in}

\begin{center}
\includegraphics{e3_ferris.eps}
\end{center}

\vspace*{0.1in}

The height of the labeled passenger above the ground is given by
$$
	y = 50 + 50 \sin(\theta).
$$
If the passenger's height above ground is increasing by 3.2 feet per second at
angle $\theta=\pi/3$, how fast is the wheel turning?  Include units in your
answer.

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