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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Final Examination -- December 18, 2009}
\vskip 10pt
\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 200 regular points in
problems I through IX, and 10 Extra Credit points.  {\it If a problem
includes specific directions about how to solve it, full credit will be
given only for a solution by that method.} Take a deep breath before
plunging in, don't panic, and have some fun!
\vskip 10pt
\noindent
I. Consider the functions:
$$y = f_1(x) = {2 \sin(x) + 1\over 1+x^2}, \qquad y = f_2(x) = x + x^2 + 1, 
\qquad y = f_3(x) = x e^{3x} + e^{x^3}$$  
\begin{itemize}
\item[A)] (15) Find the derivatives of $f_1,f_2,f_3$.
\item[B)] (5) Find the equation of the tangent line to 
the graph $y = f_1(x) + f_3(x)$  at $(0,2)$.
\end{itemize}
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\noindent
II.  Figure 1 (at the top of the next page) 
shows the graph $y = f(x)$ for some function $f(x)$.
\begin{itemize}
\item[A)] (10) Sketch the graph $y = -f(x + 1)$.  
\item[B)] (10) Sketch the graph $y = f'(x)$, 
and explain the features of your graph, including its $x$-axis intercepts, 
its local maxima/minima, points where $f'(x)$ is undefined, and so forth.
\end{itemize}
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\noindent
III.  
\begin{itemize}
\item[A)]  (10) State the definition of the definite
integral of a function $f(x)$ from $a$ to $b$.
\item[B)]  (10) State both parts of the Fundamental
Theorem of Calculus.
\end{itemize}  
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\noindent
IV.  The region $R$ in the 
$xy$-plane is bounded by $y = x^3 - 1$, $x = 1$, $x = 2$, and the 
$x$-axis. 
\begin{itemize}
\item[A)]  (10)  Find the volume of the solid obtained by rotating $R$
about the $x$-axis.
\item[B)]  (5) Set up the integral to compute the volume of 
the solid obtained by rotating $R$ about the line $x = 1$.  You
{\it do not} need to evaluate.  
\item[C)]  (10)  Find the coordinates of the centroid of $R$.  
\end{itemize}
\begin{figure}
\begin{center}
\includegraphics[width=3in]{FI.ps}
\caption{Figure for Problem II}
\end{center}
\end{figure}
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\noindent
V.  Compute each of the following integrals.  If you use an entry 
from the table, identify it by number.
\begin{itemize}
\item[A)] (5) $\displaystyle{\int x^2 \sin(3x^3) \cos(3x^3)\ dx}$
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\item[B)] (5) $\displaystyle{\int_0^\infty {dx \over x^2 + 9}}$
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\item[C)] (10) $\displaystyle{\int x^2 \ln(2x)\ dx}$  (use parts)
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\item[D)] (15) $\displaystyle{\int { x^2 \over \sqrt{25 - x^2}}\ dx}$ (use a trigonometric substitution)
\end{itemize}
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\noindent
VI. (15)  Find the arclength of the catenary curve 
$\displaystyle{y = \frac{e^x + e^{-x}}{2}}$
from $x = 0$ to $x = 2$.   
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\noindent
VII.  All parts of this question refer to the differential equation
$${dy\over dx} = y^2 - y.$$
\begin{itemize}
\item[A)] (10) Sketch the slope field 
showing the slopes at points on the lines $y = -1/2, 0, 1/2, 1, 3/2$.
\item[B)] (10)  Find the solution with $y(0) = 1/2$.  
\end{itemize}
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\noindent
VIII.  A hive of honey bees is attacked by a fungal infection
at $t = 0$, and the population declines at 
a rate proportional to the {\it square root\/} of the population.
\begin{itemize}
\item[A)] (5) Express this statement about the rate of growth of the 
population $P$ as a differential equation.
\item[B)]  (10) Use separation of variables to find the general solution
of your differential equation.
\item[C)]  (5) At $t = 0$ there were 900 bees; 441 were
left after 6 weeks.  When did the bee population disappear entirely?
\end{itemize}
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\noindent
IX.  Series.
\begin{itemize}
\item[A)] (5)  Find the sum of the infinite series 
$$4 - {8\over 3} + {16\over 9} - {32\over 27} + \cdots.$$
\item[B)] (10) Find the Taylor series of $g(x) = e^{-x}$
at $a = 0$.
\item[C)] (10) Compute the Taylor polynomial of degree 2 
for $f(x) = \sqrt{x}$ at $a = 1$ and use it to approximate
$\sqrt{1.3}$.  How close is the approximation?
\end{itemize}
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\noindent
{\it Extra Credit.}  For a water tank with cross-section area $A(y)$ at
height $y$ above the bottom and an opening
of area $1$ at $y = 0$, Torricelli's Law implies that as 
the water drains out through the opening,
the height $y$ of the water in the tank changes according to the equation
$$A(y){dy \over dt} = -\sqrt{2gy}$$
(where $g$ is the acceleration of gravity at the surface of the 
Earth).  
\begin{itemize}
\item[A)] (5) Suppose that the tank has the shape of the 
solid of revolution formed by rotating the area between a curve
$x = f(y)$ and the $y$-axis, $0 \le y \le h$, about the $y$-axis.
How should $f(y)$ be chosen in order for the 
water level to decrease at a {\it constant rate\/}?
\item[B)]  (5)  Suppose you had such a tank. Explain
how you could use it as a {\it time-measuring device\/}.  (Note:  
The ancient Egyptians actually constructed tanks of this
shape and used them to measure time 
as early as the 16th century BC (i.e. long before calculus!).  
The Greeks took over the idea and their
name for this sort of device was a {\it clepsydra}.)  
\end{itemize}
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\centerline{\it Have a joyous and peaceful holiday season!}

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