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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) 
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II.  The velocity of a drag racing car is measured yielding the following data.
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$$\matrix{t\ (sec) & 0 & 1 & 2 & 3 & 4 & 5\cr
     v(t)\ (ft/sec) & 0 & 10 & 25 & 40 & 45 & 50 \cr}$$
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\item{A)} (10) What is your best estimate of the total
distance the car travels between $t = 0$ and $t = 5$ seconds?
\item{B)} (10) How often would you need to measure the velocity
in order to get an estimate of the distance traveled
accurate to within 1 foot?
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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)  $$\int {\tan(\sqrt{x})\over \sqrt{x}} dx$$
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\item{C)} (10)  $$\int {3x + 2 \over x^2 + 8x + 7}\ dx$$
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\item{D)} (10) $$\int {dx \over (4 - x^2)^{3/2}}$$
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