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I.  Given a graph $y = f(x)$ for some function:
\item{A)}  Sketch the graph $y = 2f(x-2) + 1$.  
\item{B)}  Sketch the graph $y = - f(-x)$  
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({\it Note: Be prepared for any problem of this type -- i.e. 
for any graph I might give you.})
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II.  Given a graph $y = f(x)$ ({\it again, be prepared for any
graph I might give you}): 
\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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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 (both parts).
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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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V.   Compute the following integrals.  
\item{A)} $\int_0^1 \cos(\pi x) - x^2e^{x^3}\, dx$
\item{B)} $\int x^2 \sin(3x)\, dx$  (Use parts; check work with the table entry for
the appropriate reduction formula.)
\item{C)} $\int {x \over (x + 3)(x^2 + 1) }\, dx$   (Use partial fractions.)
\item{D)} $\int {dx \over \sqrt{x^2 - 16x + 65} }$  (Complete the square under
the radical, then do an appropriate trigonometric substitution.)
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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 $x = 3$.
\item{C)} Find the coordinates of the centroid of $R$.  
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VII.  All parts of this question refer to the differential equation
$${dy\over dx} = x y. $$
\item{A)}  Sketch the slope field for this 
equation, showing the slopes at points with $ x y = -2, -1, 0, 1, 2$.  
\item{B)}  Solve the differential equation to obtain a formula
for $y(x)$ satisfying ${dy \over dx} = x y$ and $y(0) = 4$.  
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VIII.  Newton's Law of Cooling (and Heating) states that if an
object is placed in a surrounding medium of constant temperature
$A$, then the rate of change of the object's temperature $T$ with respect
to time $t$ is proportional to the difference $T - A$.  
\item{A)}  State Newton's Law of Cooling as a differential 
equation.  
\item{B)}  Your differential equation in part A should be separable.
Solve it to get an explicit formula for $T$ as a function of $t$.
\item{C)}  When a batch of Christmas cookies is taken out of the
oven it has temperature $T(0) = 200$ degrees Fahrenheit.  The sheet
is placed in a room heated to a constant $70$ degrees Fahrenheit. 
5 minutes later the cookies have cooled to $170$ degrees F.
When will the cookies reach $120$ degrees F and be cool enough to
eat with burning your mouth? 
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IX.  A large punchbowl is a hemisphere with radius $.5$ meters.  If
the punchbowl is full of Christmas punch with density $950$ kg per
cubic meter, how much work will be done in emptying the punchbowl by
``pumping'' the contents out the top?
(The acceleration of gravity is $g = 9.8$ meters per second squared.)
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X. 
\item{A)}  Does the geometric series 
$$1 + {e\over \pi} + \left({e\over \pi}\right)^2 + \cdots $$
converge?  If so, what is its sum?
\item{B)}  For which $p$ does the improper integral $\int {1\over x^p}\ dx$ converge?
\item{C)}  Find the Taylor series of $g(x) = e^{2x}$ 
at $a = 0$.
\item{D)}  Compute the Taylor polynomial $p_5(x)$ of $f(x) = \sin(x)$ at
$a = 0$, and use it to approximate $f(.6)$.  How close is the approximation
to the value of $\sin(.6)$ given by your calculator?  (Note:  Your calculator
must be set to compute the sine of an angle {\it in radians} for 
this problem to make any sense!)


          
 
