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\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Midterm Exam 3 -- December 2, 2005}
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{\it Directions:}
Do all work in the blue exam booklet.  There are 100 possible points.
You may use any of the following formulas without comment:
\item{1)}  Cylinders:  $V = \pi r^2 h$
\item{2)}  Circular cones:  $V = {1\over 3}\pi r^2 h$
\item{3)}  Spheres: $V = {4\over 3} \pi r^3$, $A = 4\pi r^2$.    
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I. All parts of this question refer to the function $f(x) = xe^{-x^2}$.
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\item{A)} (10) Compute $f'(x)$. 
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\item{B)} (10)  What are the (global) maximum and minimum values
of $f(x)$ on the interval $-2\le x \le 0$?
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\item{C)} (5) Does your minimum from part B occur at a {\it local 
minimum} of $f(x)$?  Explain.
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II.  Compute the derivatives of the following:
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\item{A)} (10) 
$$f(x) = {\sin(x)\over \cos(x) + \sin(x)}$$
\item{B)} (10) 
$$g(x) = \ln\left(3x + {1\over x}\right)$$
\item{C)} (10)
$$h(x) = \arcsin(4x)$$
\item{D)} (10) Find ${dy\over dx}$ if $xy^2 + x^2y = 3$.
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III.  (20)  The Holy Cross Buildings and Grounds department wants
to build a new secure parking lot for fans of visiting football teams.  To 
save costs they decide to make the lot by fencing in a rectangular area 
and using the existing fence at the non-enclosed end of Fitton Field
(the end behind the scoreboard) as one of the four sides.  The area
of the lot must be 500 square yards.  What should the dimensions
of the lot be to minimize the length of new fencing needed?
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IV.  (15)  The radius of a spherical ice ball is increasing 
at the rate of $3$ mm/min.  At what rate is the volume increasing
when the radius is 6 mm?
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