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\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Exam II -- November 4, 2005}
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{\it Directions:}  Do all work in the blue exam booklet.  You
may use the supplied calculator on any of these problems.
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I.  The following graph shows $y = f(x)$. 
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\item{A)}  (5)  Identify all points where $f$ appears
not to be {\it differentiable}.
\item{B)}  (15)  Using the information in the graph,
construct a ``qualitative'' plot of $y = f'(x)$ for 
$x\ne $ your points from part A.
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II.  (20) Using the limit definition, 
find $f'(x)$ for $f(x) = x^2 - x + 4$.
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III.  Find the derivatives of each of the following functions by applying
the appropriate ``shortcut'' derivative rules:
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\item{A)} (10) $$f(x) = 2^x (x^4 + {3\over {\root 3\of x}})$$
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\item{B)} (10) $$g(x) = (e^{\pi x} + 2)^3$$
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\item{C)} (10) $$h(x) = {x^2 - 3\over 2x^2 + 3x}$$
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IV.  (15) The price of an antique rocking chair at time $t$ years after
1950 is approximated by $P(t) = 100(1.05)^t$.  What was the 
price in 2000, and how fast was the price changing then?  Was it
increasing or decreasing?
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V.  (15) It costs $C(q) = 50 q^{1/4} + 1000$ dollars for an publishing
company to manufacture $q$ copies of its star author's
latest novel.   Compute $C''(q)$.  Is the graph of the cost function 
concave up or down?  What does your answer mean in everyday terms?
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