\magnification = \magstep1
\nopagenumbers
\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Sample Exam II -- October 26, 2001}
\bigskip
\noindent
I.  Give a formula for a rational function $r(x) = {ax^2 + b\over x^2 + dx + e}$
which has vertical asymptotes at $x = 1, 3$,  $x$-axis intercepts
at $x=-2$ and $x = 2$, and which has $y = 6$ as a horizontal asymptote.
\medskip
\noindent
II.  The following graph shows $y = f(x)$.  Using the information here,
construct a ``qualitative'' plot of $y = f'(x)$.
\vglue 3.0truein
\noindent
III.  Using the limit definition, find $f'(x)$ for $f(x) = x^2 - x + 4$.
\bigskip
\noindent
IV.  Find derivatives of each of the following functions by applying
the appropriate ``shortcut'' derivative rules:
\bigskip
\item{A)} (10) $f(x) = 3^x + x^6 - {3\over {\root 3\of x}}$
\medskip
\item{B)} (10) $g(x) = x^{e^2} + {(e^x)}^2$
\medskip
\item{C)} (10) $h(x) = {4x^2 - 3\over x^5}$
\bigskip
\noindent
V.  The price of an antique rocking chair at time $t$ years after
1975 is approximated by $P(t) = 100(1.25)^t$.  What was the 
price in 1990, and how fast was the price changing then?  Was it
increasing or decreasing?
\bigskip
\noindent
VI.  It costs $C(q) = 50 q^{1/4} + 1000$ dollars for an independent record
company to manufacture $q$ copies of its star alternative
band's latest CD.  Compute $C''(q)$. 
Is the graph of the cost function concave up or down?  What does your 
answer mean in everyday terms?
\bye