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\noindent
I.  
\item{A)} 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)} The following graph shows $y = f(x)$ for some function.
Sketch the graph $y = 2f(x-2) + 1$.  ({\it Be prepared for any 
problem of this type})
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\noindent
II.  The following graph shows $y = f(x)$.  ({\it Be prepared for 
any question of this type}) 
\item{A)}  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)}   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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\noindent
III. 
\item{A)}  What is the precise definition of the {\it derivative\/}
of a function $f$ at $x = a$?   
\item{B)}  What is the precise definition of the {\it definite integral\/} 
of a function $f$ over an interval $[a,b]$?
\item{C)}  Give a precise statement of the Fundamental Theorem
of Calculus.
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\noindent
IV.   A rectangular metal plate of uniform thickness extends from 
$x = 0$ to $x = 6$ and $y = 0$ to $y = 4$.
The density of the metal decreases with $x$ according to the following
table:
$$\matrix{ x {\rm\ in\ inches}             &0  &1  &2  &3  &4  &5 &6\cr
     {\rm density\ in\ grams/square\ inch} &200&190&170&140&100&90&80\cr}$$
(The density is independent of $y$.)
\item{A)}  Approximate the total mass of the plate using a suitable
Riemann sum.
\item{B)}  Do you think your estimate is larger or smaller than the
true mass?  Explain, including any assumptions you are making.    
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\noindent
V.   Compute the following integrals.  
\item{A)} $\int_0^1 \cos(\pi x) - x^2e^{x^3}\, dx$
\item{B)} $\int x^3 \sin(3x)\, dx$
\item{C)} $\int {x - 2 \over x^2 + 6x + 10}\, dx$
\item{D)} $\int {dx \over \sqrt{x^2 - 16x + 63} }$
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\noindent
VI.  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)} Find the volume of the solid obtained by revolving $R$
around the $x$-axis.
\item{B)} Same, but revolving around the line $x = 3$.
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\noindent
VII.  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)}  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)}  Suppose $y(0) = 5$.  What happens to the amount of the
compound as $t \to \infty$?
\item{C)}  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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\noindent
VIII. 
\item{A)}  Compute the Taylor polynomial $p_5(x)$ of $f(x) = \sin(x)$ at
$a = 0$, and use it to approximate $f(.6)$.
\item{B)}  Find the Taylor series of $g(x) = e^{2x^3}$ 
at $a = 0$ using any applicable method.
\item{C)}  For which $x$ in the real numbers does the power series
$$\sum_{k=1}^\infty {5^k \over 2^k k!} x^k$$
converge?



          
 