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\centerline{Mathematics 136, section 4 -- AP Calculus}
\centerline{Final Examination}
\centerline{December 19, 2003}
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{\it Directions}
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Answer questions I - VIII.  There are 200 regular points and 10 Extra
Credit points.  Don't panic and remember to have some fun!
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I. Consider the functions:
$$y = f_1(x) = {2 \sin(3x) + 1\over 1+x^2}, \qquad y = f_2(x) = x + 
3 x^{5/3} + 1, 
\qquad y = f_3(x) = x e^{2x} + e^{x^2}$$  
\item{A)} (15) Find the derivatives of $f_1,f_2,f_3$.
\item{B)} (5) What line would you see if you ``zoomed in'' 
on the point $(0,2)$ on the graph of $y = f_1(x) + f_3(x)$?
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II.  The graph below is the graph $y = f(x)$ for some function $f(x)$:
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\item{A)} (15) Using the graph above, 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.
\item{B)} (15) Sketch the graph of the {\it antiderivative\/} $F$
of $f(x)$ with $F(0) = 0$.
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III.  A rectangular poster is to have an area no larger than 400 square
inches.  It will consist of a central printed
area, surrounded by margins of 1 inch along the top and sides and 
a 2 inch margin along the bottom. 
\item{A)}  (15) What overall dimensions for the poster will yield the 
maximum printed area?
\item{B)} (5) How can you tell that your answer from A is really 
a maximum for the printed area?
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IV.  
\item{A)}  (15) The hull of a boat is 15 feet long.  At a distance
$s$ feet from the bow (the front), the cross section of the hull
perpendicular to its long axis has the shape of the region in the 
$xy$-plane below $y = 0$ ($y = 0$ is the water line) and above 
the parabola $y = ax^2 - b$, where $a,b$ are given in the following table:
$$\matrix{s & 5 & 10 & 15 \cr
          a & 2 & 3 & 4 \cr
          b & 2 & 3 & 4 \cr}$$
Estimate the {\it volume\/} enclosed by the hull below the water line
using a suitable Riemann sum.
\item{B)} (10) The region $R$ in the 
$xy$-plane is bounded by $y = x^3 + 1$, $x = 1$, $x = 2$, and the 
$x$-axis.  Find the volume of the solid obtained by rotating $R$
about the $x$-axis.
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V.  Compute each of the following integrals.  If you use an entry 
from the table, identify it by number.
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\item{A)} (5) $\int x \sin(5x^2) \cos(5x^2)\ dx$
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\item{B)} (10) $\int x^2 \ln(4x)\ dx$
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\item{C)} (10) $\int_0^1 {x - 2\over x^2 - 4x + 5}\ dx$
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\item{D)} (10) $\int {1\over \sqrt{x^2+2x}}\ dx$
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VI. {\it When an object (with no heat
source or refrigeration of its own) is placed
into a surrounding medium of constant temperature $A$, the rate of
change of the temperature $T$ of the object is proportional to 
the difference between $T$ and $A$ at all times\/}.  
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\item{A)} (5)  Restate the above as
a differential equation for the temperature function $T$.
\item{B)} (5)  Find the general solution of your equation from A.
\item{C)} (10)  When a frozen  chicken pot pie is removed
from the freezer its temperature is $0^\circ$ F.  At that time, it is
placed into a preheated $350^\circ$ F oven (maintained at that
temperature).  
After 10 minutes, the temperature of the pot pie has reached $40^\circ$ F.
How much longer will it take for the pot pie to reach $160^\circ$ F and
be ready to serve?
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VII.  
\item{A)} (10) Sketch the slope field of the differential 
equation 
${dy\over dx} = y^2 - y$,
showing the slopes at points on the lines $y = -1/2, 0, 1/2, 1, 3/2$.
\item{B)} (10)  Find the solution of this differential equation
with $y(0) = 1/2$.  
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VIII.    
\item{A)} (10) Consider the function $f(x) = \sqrt{1 + x}$.
Compute the Taylor polynomial of degree 4 for $f$ at
$a = 0$, and use it to approximate the value $\sqrt{1.2}$.
\item{B)} (10) Find the Taylor series of $g(x) = e^{-x/2}$
at $a = 0$, using any convenient method.
\item{C)} (10) Use the Ratio Test to determine the radius of 
convergence of the power series
$$\sum_{n=1}^{\infty} (-1)^n {x^n\over n^2}.$$

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{\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
$$A(y){dy \over dt} = -\sqrt{2gy}$$
(where $g$ is the acceleration of gravity at the surface of the 
Earth).  
\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 Greeks actually constructed tanks of this
shape, and called them ``water-clocks.''  They were used
to measure the time allotted to speakers in legal 
proceedings, for instance.)
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\centerline{\it Have a joyous and peaceful holiday season!}
\bye
