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\centerline{Mathematics 136, section 4 -- AP Calculus}
\centerline{Exam 3, December 5, 2003}
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\noindent
{\it Directions:}  Answer questions I - III in the blue exam booklet. 
There are 100 ordinary and 10 Extra Credit points. Please {\it
show all work\/} you do to derive your solutions.   
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\noindent
I.  All parts of this question refer to the region
$R$ under the graph $y = (16 - x^2)^{1\over 4}$, above
the $x$-axis, between $x = 0$ and $x = 4$.  
\item{A)} (10)  Set up an integral computing the volume of the solid
obtained by rotating $R$ about the $x$-axis.
\item{B)} (15)  Using the table of integrals and the Fundamental Theorem
of Calculus, compute the volume.  (Partial credit for identifying the 
correct table entry that applies.)
\item{C)} (10)  Set up the integral to compute the volume
of the solid obtained if the region $R$ is rotated about the line
$y = 3$.  {\it Note: Do not try to compute this integral.}
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\noindent
II.  The population of a prolific breed of hamsters increases at 
a rate proportional to the {\it square\/} of the population at all times.
\item{A)} (10)  Express the above statement as a differential equation
for the hamster population as a function of time.
\item{B)} (10)  There should be a constant of proportionality, say $k$,
in your equation from A.  Set $k = 1$ for this part of the question {\it only}.  
Sketch the slope field for the equation, showing slopes at the points
with integer coordinates, $0\le t \le 3, 0 \le P \le 3$.
\item{C)} (10)  Find the general solution of your 
differential equation from part A by separation of variables.
\item{D)}  (5) There were 2 hamsters at time $t = 0$ weeks and 4 hamsters
at $t = 3$ months.  What happens between $t = 3$ and $t = 6$?  Is
this a realistic population model?
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\noindent
III.  
\item{A)} (15) Does the infinite geometric series
$$8 + {16\over 3} + {32\over 9} + {64\over 27} + {128\over 81} + \cdots $$
converge?  If so, find the sum.  If not, say why not.
\item{B)}  (15) Use the Ratio Test to find the interval of convergence
of the power series:
$$ \sum_{k=1}^\infty {k x^k\over 2^k}= {x\over 2} + {x^2\over 2} + {3 x^3\over 8} + \cdots $$
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\noindent
{\it Extra Credit} (10) Find the {\it arclength} of the curve
$y = x^{3/2}$ from $x = 1$ to $x = 4$.
\bye