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\leftline{Mathematics 133 -- Intensive Calculus for Science 1}
\leftline{Sample Exam II -- October 26, 2001}
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\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.
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\noindent
{\it Solution:}  The numerator must have zeroes at $x = \pm 2$, so 
it looks like $ax^2 + b = a(x+2)(x-2) = ax^2 - 4a$.  The denominator
must have zeroes at $x = 1,3$, so it looks like 
$x^2 + dx + e = (x-1)(x-3) = x^2 - 4x + 3$.  The $y$-level of the
horizontal asymptote is the limit of $r(x)$ as $x \to \pm\infty$, which
is 
$$\eqalign{
\lim_{x\to\pm\infty} {ax^2 - 4a\over x^2 - 4x + 3} &=
\lim_{x\to\pm\infty} {ax^2 - 4a\over x^2 - 4x + 3}\cdot{{1\over x^2}\over{1\over x^2}}\cr
&=\lim_{x\to\pm\infty} {a - 4{a\over x^2}\over 1 - {4\over x} + {3\over x^2}}\cr
&=a\cr}$$
Hence $a = 6$, and the full equation is
$$r(x)= {6x^2 - 24\over x^2 - 4x + 3}$$
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\noindent
II.  The following graph shows $y = f(x)$.  Using the information here,
construct a ``qualitative'' plot of $y = f'(x)$.
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III.  Using the limit definition, find $f'(x)$ for $f(x) = x^2 - x + 4$.
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\noindent
{\it Solution:}  
$$\eqalign{f'(x) &= \lim_{h\to 0} {f(x+h) - f(x)\over h}\cr
&=\lim_{h\to 0} {(x+h)^2 - (x+h) + 4 - x^2 + x - 4\over h}\cr
&=\lim_{h\to 0} {x^2 + 2xh + h^2 - x - h + 4 - x^2 + x - 4\over h}\cr
&=\lim_{h\to 0} {2xh - h + h^2 \over h}\cr
&=\lim_{h\to 0} 2x - 1 + h\cr
&=2x - 1\cr}$$
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\noindent
IV.  Find derivatives of each of the following functions by applying
the appropriate ``shortcut'' derivative rules:
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\item{A)} (10) $f(x) = 3^x + x^6 - {3\over {\root 3\of x}}$
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{\it Solution:}  $f(x) = 3^x + x^6 - 3 x^{-1/3}$, so applying the 
sum, power, and exponential rules:
$$f'(x) = 3^x\ln(3) + 6x^5 + x^{-4/3}$$
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\item{B)} (10) $g(x) = x^{e^2} + {(e^x)}^2$
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{\it Solution:}  The first term is a power; the second can be rewritten
as the exponential $e^{2x} = (e^2)^x$.  So 
$$g'(x) = e^2 x^{e^2-1} + (e^2)^x \ln(e^2) = e^2 x^{e^2-1} + 2 e^{2x}$$
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\item{C)} (10) $h(x) = {4x^2 - 3\over x^5}$
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{\it Solution:}  This is one where you need ``preliminary algebra''
(if you don't use the quotient rule!):
$$\eqalign{h(x) &= 4x^{-3} - 3x^{-5}\qquad {\rm so}\cr
           h'(x) &= -12 x^{-4} + 15 x^{-6}\cr
                 &={-12x^2 + 15\over x^6}\cr}$$

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\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?
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{\it Solution:}  1990 is 15 years
after 1975, so $P(15) = 100(1.25)^{15} \doteq 2842.17$ and 
$P'(t) = 100(1.25)^t\ln(1.25)$, so $P'(15) = 100(1.25)^{15}\ln(1.25) \doteq
624.21$.  The price was increasing ($P'(t) > 0$) at a rate of 624.12 \$/year. 
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\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?
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{\it Solution:}  $C'(q) = {25\over 2} q^{-3/4}$, so $C''(q) = {-75\over 8}q^{-7/4}$.
This shows $C''(q)$ is always negative, so the graph is {\it concave down}.
In everyday terms, this says that the more CD's are produced, the 
lower the cost per CD (in economics, this is called an 
``economy of scale'').
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