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\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Sample Exam III -- November 22, 2005}
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\noindent
I.   Find derivatives of each of the following functions.  Any correct 
form of the derivative is OK here.
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\item{A)} $f(x) = \ln(x) + 2^{4x}$
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\item{B)} $g(x) =  {\sin(x)\cos(x^2)}$
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\item{C)} $h(x) = \arctan(x^2 + e^{2x} + 1)$
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\item{D)} Find ${dy\over dx}$ if $x^2y^2 - 2xy + x^3 = 1$.
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\noindent
II.  All parts of this question refer to the function 
$$f(x) = {x\over x^2 + 4}$$
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\item{A)} Compute $f'(x)$.  Simplify and factor the numerator.
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\item{B)} Using your $f'(x)$, 
find all critical points of $f$ and classify each
as a local maximum, local minimum, or neither using the 
{\it First Derivative Test}.
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\item{C)} Compute $f''(x)$.  Simplify and factor the numerator 
again.
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\item{D)} Using your $f''(x)$, at which $x$ are the inflection points 
on the graph $y = f(x)$ located?
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\noindent
III.  The hypotenuse of a right triangle has one end at $(0,0)$
and the other at a point on $y = x^2 e^{-3x}$ with $x \ge 0$.
One of the other two sides lies along the $x$-axis and 
the other is parallel to the $y$-axis.  Find the maximum 
area of such a triangle.   
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\noindent
IV.  A chemical storage tank is an ``upside-down'' cone.  The 
depth is 12 meters and the top radius is 5 meters.  When 
the depth of the chemical in the tank is 1 meter the 
level is falling at the rate of 1 meter per minute.  How
fast is the volume of chemical changing then?
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