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\leftline{Mathematics 136, section 4 -- Exam 1 -- September 30, 2003}
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{\it Directions:} Answer questions I - V below in the blue exam booklet. 
There are 100 ordinary and 5 Extra Credit points, distributed as indicated.
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I. 
\item{A.} (10) Find a possible formula for the function $y(t)$ plotted
below:
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\item{B.} (10) Suppose you ``zoomed in'' on the graph of your function
near $t = 0$.  Would the graph look locally linear?  If so,
what would the equation of the line be?  If not, say why not.
(Use your formula from A, but don't do any actual ``zooming''.)
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II. A population of wildebeest grows in a habitat that can support 
no more than $P_1$ animals.  At time $t = 0$, the number of 
wildebeest present is  $P_0 < P_1$.  The population grows more and
more rapidly at first, but then the rate of growth decreases and
the population approaches $P_1$, without exceeding it.  
\item{A.} (10) Sketch a graph
of the population of wildebeest as a function of time
that fits this description.  
\item{B.} (10)  Give a qualitative graph of the derivative of the 
population function.
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III.  
\item{A.}  (10) What is the exact mathematical definition of the 
{\it derivative\/} of a function $f(x)$ at $x = a$?  
\item{B.}  (10) Use the definition (not the ``shortcut rules'') 
to compute the derivative of 
$f(x) = 4x^2 - 2x + 2$ at a general $x$.
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IV.  Find the derivative functions of each of the following functions
defined by formulas:
\item{A.}  (5) $f(x) = 2x^3 + 6\sqrt{x} - {3\over \root 3\of x} + e^\pi$
\item{B.}  (7.5) $f(x) = {\sin(x) \over 1 + \cos(x)}$
\item{C.}  (7.5) $g(x) = \arctan(e^x - 2^x)$
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V. (20)  Sketch the parametric curve whose coordinate functions
$x = f(t)$ and $y = g(t)$ are given in the following plots:
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(Over for extra credit problem.)
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{\bf Extra Credit} (10)  The graph below shows $y = f'(x)$, the derivative
of a function $f(x)$.  Sketch
the graph $y = f(x)$ using the information given.  Indicate on your
sketch any {\it local maximum, local minimum, or inflection points\/}
that $f(x)$ has.
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