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\centerline{Mathematics 136 -- AP Calculus}
\centerline{Makeup Exam 2 -- November 9, 2003}
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{\it Directions.}  Do all work in the blue exam booklet.
There are 100 points.  
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I. (20)  In an alternate universe, the ``super-attractive'' force
exerted by two objects of mass $m_1,m_2$ 
is {\it directly} proportional to the square of the distance between
them: $F = S m_1m_2 r^2$, where $S$ is a constant and $r$ is the 
distance between the masses.  Four 1kg masses are fixed 
at points $x = $ respectively along a straight line
in this alternate universe.  At what location $x$ along that line should
another 2kg mass be placed to {\it minimize} the sum of the super-attractive
forces exerted on it by the four unit masses?
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II.  In a car moving at 90 ft/sec, the driver 
suddenly saw an obstacle 400 feet ahead and braked
to a stop in 10 seconds.  The car's velocity was recorded
by a sensor in its onboard computer every two seconds:
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$$\matrix{t\ (sec) & 0 & 2 & 4 & 6 & 8 & 10\cr
     v(t)\ (ft/sec) & 90 & 75 & 50 & 25 & 5 & 0 \cr}$$
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\item{A)} (10)  Can you say for sure from the information
here whether the car hit the obstacle or not?  Explain, using 
the left- and right-hand Riemann sums for $v(t)$.
\item{B)} (10)  From the given information, what is your
best estimate about whether the car hit the obstacle?
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III. 
\item{A)} (10) State the first and second parts of the 
Fundamental Theorem of Calculus.
\item{B)} (10)  The following graph shows $y = f(x)$ for 
some function.  Sketch the graph of the antiderivative
$F(x)$ of $g(x)$ with $F(x) = 0$.  What is $F(4)$?
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IV.  Methods of integration. In B,C,D you may use the table of integrals.
\item{A)} (10) Using integration by parts, show that
$$\int x^n \cos(ax)\ dx = {x^n\over a} \sin(ax) - {n\over a}\int x^{n-1} \sin(ax)\ dx.$$ 
\item{B)} (10)  $\int \tan(\sqrt{x})/\sqrt{x}\ dx$
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\item{C)} (10)  $\int (3x + 2)/(x^2 + 8x + 7)\ dx$
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\item{D)} (10) $\int (4 - x^2)^{-3/2}\ dx$
\bye
