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\centerline{Mathematics 131 -- Calculus for Physical and Life Sciences 1}
\centerline{Sample Exam 3}
\centerline{November 30, 2004}
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\noindent
I.  Sketch the following parametric curves:
\item{A)}  $x = 2 + t, y = 3 - 2t$, $0 \le t \le 1$.
\item{B)}  $x = \cos(2t), y = \sin(2t)$, $0 \le t \le \pi/2$.
\item{C)}  $x = 3\sin(t), y = 4\cos(t)$, $0 \le t \le 2\pi$.
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II.  
\item{A)}  Differentiate implicitly, and find the equation 
of the tangent line to the curve 
$$-x^3 + 2y^3 - xy = 0$$
at $(1,1)$.
\item{B)}  Boyle's Law states that the pressure $P$ and volume $V$
of an ideal gas are related by an equation $PV = c$, where
$c$ is constant.  If a sample of gas has $P = 1$ atm, $V = 3$ liters and 
$P$ is changing at a rate of $.2$ atm/min, how is $V$ changing?
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\noindent
III.  Compute the following limits:
\item{A)}  
$$\lim_{x\to 0} {\sin(4x)\over \tan(3x)}$$
\item{B)}  
$$\lim_{x\to \infty} {3\ln(x) + 2 \over x}$$
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\noindent
IV.  All parts of this question refer to the function 
$$f(x) = x^2 e^{-2x}$$
\item{A)}  Find all critical points of $f(x)$ and classify
them as local maxima, local minima, or neither.
\item{B)}  Find all inflection points of $f(x)$.
\item{C)}  Sketch the graph $y = f(x)$ accurately, using
the information from parts A and B.
\item{D)}  What are the (global) maximum and minimum values
of $f(x)$ for $-1 \le x \le 2$?
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\noindent
V.  10000 feet of fencing will be used to enclose a
large rectangular area and subdivide it into 
six smaller rectangular
playing fields in two rows of three fields each.
What dimensions for the overall rectangle will maximize
the total area enclosed?
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