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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Things to Know -- Final Exam}
\centerline{May 1, 2006}
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\noindent
{\it General Information}
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\item{$\bullet$} The final exam will be given in our regular
class room 8:30am - 11:30am on Wednesday, May 10.  
\item{$\bullet$} This will be a {\it comprehensive exam} -- it will
cover all of the material from Chapters 5-11 of the text
that we have studied this semester.  See the
detailed list of topics below.
\item{$\bullet$} The final will be similar in format to the 
three full period midterm exams, but roughly {\it twice as long}.
\item{$\bullet$} If you are well prepared and work steadily, it
should be possible to complete the exam in about 2 hours.  But you
will have the full-three hour period to work on the exam if 
you need that much time.
\item{$\bullet$} {\it Review Session: Monday, May 8 at 7:00pm}.
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\noindent
{\it Topics to Be Covered}
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\item{1)}  The definite integral, connections with areas, average
values, total change.
\item{2)} The Fundamental Theorem of Calculus: 1st part: If $F(x)$
is an antiderivative of a continuous function $f(x)$ on $a \le x\le b$, 
then 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
2nd part:  If $f(x)$ is continuous for $a \le x \le b$, then 
the function $F(x) = \int_a^x f(t)\ dt$ is an antiderivative of $f(x)$:
$${d \over dx} \int_a^x f(t)\ dt = f(x).$$
\item{3)}  Antiderivatives graphically and numerically
\item{4)}  The power, sum, and constant multiple rules for antiderivatives
\item{5)}  Indefinite integrals by substitution, parts, 
partial fractions, and using the table of integrals
\item{6)}  Integrals for computing volumes of solids with known 
cross-sections:  $V = \int_a^b A(x)\ dx$, where $A(x)$ is the 
cross-section area function. Volumes of solids of revolution are 
a special case of this.
\item{7)}  Integrals for computing total mass/center of mass of
wires and plates of given shapes, given the mass density function
\item{8)}  Probability density functions of distributions, cumulative 
distribution function, median, mean.
\item{9)}  Taylor polynomials and series
\item{10)}  Taylor series by ``shortcut rules'' (substitution, algebraic
manipulations, term-by-term integration and differentiation)
for computing Taylor series from 
the known series for 
$$e^x,\ \sin(x),\ \cos(x),\ (1+x)^p.$$
\item{11)}  Differential equations 
\itemitem{a)}  What it means for a function to be a solution of
a differential equation
\itemitem{b)}  Graphical meaning of solutions via slope fields
\itemitem{c)}  Euler's Method for approximate values of solutions
\itemitem{d)}  Formulas for solutions via separation of variables
\itemitem{e)}  Applications to growth/decay problems
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\noindent
{\it Suggestions on How to Study}
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\item{A)}  Reread your class notes and work through the 
examples we did in class -- everything on the final 
will be very close to something we did along the way somewhere!
\item{B)}  Look over your quizzes and exams, especially any 
questions you had difficulty with.  Read the comments and 
corrections -- they're there to help you figure out what you 
did wrong and how to solve the problem correctly.
\item{C)}  Also look at your graded problem sets and the
write-ups from the group discussion days and labs. 
\item{D)}  Look over the practice exams for the three full-period
midterms.
\item{E)}  Don't panic!  There is a lot of material here, but 
not all that many key ideas. 
{\it Everything on the final 
will be very close to something we did along the way somewhere!}
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\noindent
{\it Sample Exam Questions}
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\noindent
The following is essentially the final exam from the last time
I taught the Intensive Calculus 2 class.  As always, the real
exam may look somewhat different, but it will be roughly the same
length, of comparable difficulty, and cover most of the same 
topics.
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\noindent
I.  The following table gives the speed (in miles per hour) of 
a car, measured at 10 minute intervals.
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$$\matrix{{\rm time\ (min)} & 0  &10 &20 &30 &40 &50 &60\cr
          {\rm speed\ (mph)} & 35 &36 &71 &23 &27 &40 &35\cr}$$
\item{A)} Give your best estimate of the total distance 
traveled based on this information.
\item{B)} How often would you need to measure the speed to
get the distance accurate to within $.1$ mile?
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II.  
\item{A)} The following graph shows $y = f(x)$.  
Find $\int_0^9 f(x)\ dx$ and the average value of $f$ on 
this interval.
\vglue 2.0truein
\item{B)}  Give a precise statement of the Fundamental Theorem
of Calculus (both parts).
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\noindent
III.  Compute each of the following integrals.  If you use an entry
from the table, say which one.
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\item{A)} $\int  x^7 - {3\over \sqrt{x}} + {5\over \pi}\ dx$
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\item{B)} $\int x e^{x^2 + 1}\ dx$
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\item{C)} $\int x^2 \cos(3x)\ dx$
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\item{D)} $\int {1 \over \sqrt{1 - 4x^2}}\ dx$
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\item{E)} $\int {1\over x^3 - 9x} \ dx$
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\noindent
IV.  Let $R$ be the region bounded by $y = 4 - x^2$, the $x$-axis,
and $x = 0$, $x = 1$.
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\item{A)} Find the volume of the solid obtained by rotating
$R$ about the $x$-axis.
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\item{B)} A thin metal plate has the shape of the region $R$ ($x$ in 
cm).  The mass density of the plate at all points in $R$ is
$d(x) = x+1$ grams per square centimeter.
Find the total mass of the plate.
\item{C)} What is the $x$-coordinate of the center of mass of the 
plate from part B?  Set up, but {\it do not evaluate} the integral(s).
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\noindent
V.  The amount of snowfall $x$ (in feet) in a remote region of Alaska
in the month of January is a random variable with probability
density function $p(x) = {2\over 9}x(3-x)$ for $0 \le x \le 3$ and 
0 otherwise.  
\item{A)} What is the probability that the amount of snowfall
will be between $1$ and $2$ feet?
\item{B)} What is the mean snowfall?
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\noindent
VI.  
\item{A)}  Using the definition of Taylor polynomials, compute the
Taylor polynomial of degree $n = 3$ for $f(x) = x \cos(3x)$ at $a = 0$.  
\item{B)}  Check your answer by using our ``shortcut methods'' to 
find the first few nonzero terms in the Taylor
series for $f(x) = x\cos(3x)$ at $a = 0$.
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\noindent
VII.  
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\item{A)} Show that for any constant $c$, $y = x^2 + {c\over x^2}$
is a solution of the differential equation
$${dy\over dx} = 4x - {2\over x}y$$
\item{B)}  Newton's Law of Cooling states that the 
rate at which the temperature of an object changes is 
proportional to the difference between the object's temperature
and the surrounding temperature.  A hot cup of tea with temperature
$100^\circ$C is placed on a counter in a room maintained at constant 
temperature $20^\circ$C.  Ten minutes later the tea has cooled to 
$76^\circ$C.  How long will it take to cool off to $45^\circ$C?  
(Express Newton's Law as a differential equation, solve it for the 
temperature function, then use that to answer the question.)
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