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I.  
\item{A)} (10) One of the following tables gives a function that 
is approximately {\it exponential\/}: given by $ca^x$.  Identify which one it 
is and give values for $c$ and $a$:
$$\matrix{x&0  &.2 &.4 &.6 &.8\cr
       g(x)&3.10&3.40&3.70&4.00&4.30\cr}\qquad
  \matrix{x&0  &.2 &.4 &.6 &.8\cr
       h(x)&3.10&3.56&4.09&4.70&5.40\cr}$$
\item{B)} (10) The following graph shows $y = f(x)$ for some function.
Sketch the graph $y = 2f(x-2) + 1$.
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II.  The following graph shows $y = f(x)$.
\item{A)} (10) Sketch the graph $y = f'(x)$ qualitatively.  Show any points 
where $f'(x)$ does not exist by hollow circle(s) at those $x$'s.
\item{B)} (10)  Let $g(x)$ be the {\it antiderivative\/} of $f(x)$
with $g(0) = 0$.  Sketch the graph $y = g(x)$ qualitatively.
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III. 
\item{A)}  (5)  What is the precise definition of the {\it derivative\/}
of a function $f$ at $x = a$?   
\item{B)}  (5) What is the precise definition of the {\it definite integral\/} 
of a function $f$ over an interval $[a,b]$?
\item{C)} (10)  Give a precise statement of the Fundamental Theorem
of Calculus.
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IV.  All parts of this question refer to the function $f(x) = \sin(x)$.
\item{A)}  (10)  Compute the Taylor polynomial $p_5(x)$ of $f(x)$ at
$a = 0$, and use it to approximate $f(.6)$.
\item{B)}  (10)  Using the Error Bound for Taylor polynomial approximation,
determine the smallest degree polynomial you would need to approximate 
$f(.6)$ to 7-decimal place accuracy.  (You do not need to compute the 
approximation.)
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V.   A circular metal plate of uniform thickness has total radius $6$ inches.
The density of the metal decreases with radius according to the following
table:
$$\matrix{ r {\rm\ in\ inches}             &0  &1  &2  &3  &4  &5 &6\cr
     {\rm density\ in\ grams/square\ inch} &200&190&170&140&100&90&80\cr}$$
\item{A)} (15)  Approximate the total mass of the plate using a suitable
Riemann sum.
\item{B)} (5)  Do you think your estimate is larger or smaller than the
true mass?  Explain, including any assumptions you are making.    
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VI.   Compute the following integrals.  
\item{A)} (10)  $\int_0^1 \cos(\pi x) - x^2e^{x^3}\, dx$
\item{B)} (10)  $\int x^3 \sin(3x)\, dx$
\item{C)} (10)  $\int {x - 2 \over \sqrt{4 - x^2}}\, dx$
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VII.  Both parts of this question refer to the region $R$ in the
first quadrant bounded
by the graph $y = 4 - x^2$, the $x$-axis, and the $y$-axis.
\item{A)} (10) Find the volume of the solid obtained by revolving $R$
around the $x$-axis.
\item{B)} (10) Same, but revolving around the line $x = 3$.
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VIII.  In a chemical reaction, the amount $y = y(t)$ of a certain compound
that is present changes over time and satisfies the differential equation
$${dy\over dt} = k(y-2)(y-6)$$
for some $k > 0$.
\item{A)} (10)  Taking $k = 1$, sketch the slope field for this 
equation, showing the slopes at points with $y = 1,2,3,4,5,6,7$.  
\item{B)} (5)  Suppose $y(0) = 5$.  What happens to the amount of the
compound as $t \to \infty$?
\item{C)} (15)  Solve the differential equation to obtain a formula
for $y(t)$.  (Hint: you will need to use an entry from the table of 
integrals for this.  State the one you use for partial credit.)
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IX.  Consider the parametric curve $(x(t),y(t)) = (\cos(2t),\sin(t))$.
\item{A)} (10) Sketch the curve for $t \in [-\pi,\pi]$.  
\item{B)} (5) Derive an $xy$-equation for this curve by eliminating $t$.  
(Hint: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$.)
\item{C)} (5) Is every point $(x,y)$ that satisfies the equation 
from B on the curve?  Why or why not?



          
 