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\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Sample Exam II -- October 28, 2005}
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I. The following graph shows {\it the derivative} $y = f'(x)$
for some function $f(x)$.
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\item{A)} (15) Using the information here,
construct a ``qualitative'' plot of $y = f''(x)$. 
\item{B)} (10) Over which intervals is $f$ increasing?  
\item{C)} (5) Is $f'$ continuous at $x = 1$?  Why or why not?
What happens on the graph $y = f(x)$ at $x = 1$?
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II. 
\item{A)} (10) The function $H(t)$ gives the number of hours of daylight
$t$ days after the start of the year in Worcester.  At $t = 304$ days
(October 31 in a non-leap year), $H'(304) = -0.083$.  Give the meaning of
this equation as a sentence, using appropriate units.
\item{B)} (10) The table below gives the position $s$ (in miles) of a 
freight train moving
along a straight line track as a function of time $t$ (in hours).  
$$\matrix{t & .5 & 1 & 1.5 & 2 & 2.5 \cr
          s & 10 & 25 & 42 & 50 & 55 \cr}$$
Estimate the train's instantaneous velocity at $t = 1.5$ hours
as closely as you can from this information.   
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III.  (15) Using the limit definition, find $f'(x)$ for $f(x) = 1/x$.
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IV.  Find derivatives of each of the following functions by applying
the appropriate ``shortcut'' derivative rules:
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\item{A)} (10) $f(x) = 5x^7 - {3\over \sqrt{x}} - 4^{2x}$
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\item{B)} (10) $g(x) = (x^2+1)^{12} 2^x$
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\item{C)} (10) $h(x) = {x^2\over e^x - 1}$
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V. (5)  Say whether the following statement is true or false,
and explain your reasoning:  If the time interval is short enough,
then we expect the average velocity of a car over the interval will be
close to its instantaneous velocity at any time in the interval.
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