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\centerline{Mathematics 36, section 5 -- AP Calculus}
\centerline{Information for Exam 3}
\centerline{November 23, 1999}
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\noindent
{\bf General Information}
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As per the syllabus given out at the beginning of the 
semester, it's almost time for the third midterm exam.  This exam will
be given the Friday of the week after Thanksgiving, December 3.  The 
format will be similar to that of the other exams this semester.
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\noindent
{\bf Topics to be Covered}
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This exam will cover the material we have discussed since 
Exam 2, starting from applications of integrals (also covered on 
Exam 2, but worth a second look!), through the material 
on differential equations and growth models. 
The topics are:
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\item{1)}  Applications of integration (setting up problems via
Riemann sums; in limit a definite integral is obtained) -- volumes
by slices, arclengths, physical examples like mass from
non-constant density functions, etc.
(A copy of the table of integrals
from the text will be provided with the exam; any integral you 
need to compute will be do-able by some combination of substitution, 
integration by parts, and/or consultation with the table).
\item{2)}  Differential equations -- slope fields and solutions
\item{3)}  Euler's Method for numerical solutions of differential equations
\item{4)}  Solving differential equations via separation of variables
and integration, 
\item{5)}  Exponential and logistic population growth models.   
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\noindent
{\bf Review Session}
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If there is sufficient interest, I would be happy to hold a 
review session in the afternoon or evening of Thursday, December
2.  I have a commitment Wednesday evening (an orchestra rehearsal in 
Boston), so that time is not good for me.  We can discuss this in class on 
Monday, November 29 and set it up.
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\noindent
{\bf Some good review problems}
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\noindent
From the text: 
\item{$\bullet$} From the review problems from Chapter 8 (p. 405): 
1-8,10,16,17,18
\item{$\bullet$} From the review problems from Chapter 10 (p. 584-8): 1 - 22,
30,31,32,36
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{\bf Sample Exam}
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THE EXAM PROBLEMS WILL DEAL WITH DIFFERENT GEOMETRIC AND PHYSICAL
SITUATIONS, AND DIFFERENT DIFFERENTIAL EQUATIONS MODELING BEHAVIOR.
BE SURE YOU UNDERSTAND THE METHOD USED TO DERIVE 
THE SOLUTION -- DO NOT JUST MEMORIZE THE SOLUTIONS OF THESE 
PARTICULAR PROBLEMS.  It may help, as a 
part of your preparation, to set aside 50 minutes or so
and take this as a practice test.  Solutions will be distributed
via the course homepage so that you can check your work.
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\noindent
I.  A wire in the shape of the graph $y = x^2$, $x \in [-1,2]$ 
has density $4-x$ at the point $(x,x^2)$.
\item{A)}  Set up a Riemann sum approximating the {\it total mass\/}
of the wire, and explain how you got it.
\item{B)} What definite integral computes the total mass?
\item{C)} Evaluate your integral (using the table from the text as
needed).  
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\noindent
II.  The amount of radioactive carbon-14 in a sample of living tissue
from a plant produces about 13.5 atom disintegrations per minute per
gram of carbon.  In 1977, a charcoal fragment found near Stonehenge
in England recorded 8.2 disintegrations per minute per gram of carbon.
Assuming that the rate of change of the amount of carbon-14 is 
proportional to the amount at all times, and that it takes 
5370 years for 50\% of any sample to disintegrate, how old is the 
charcoal?
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\noindent
III.  The population of fish in a lake is attacked by a microscopic water-borne
parasite at $t = 0$, and as a result the population declines at 
a rate proportional to the {\it square root\/} of the population from that
time on.
\item{A)}  Express this statement about the rate of growth of the 
population $P$ as a differential equation.
\item{B)}  There should be a constant of proportionality, say $-k$, 
in your equation.  Setting $-k = -1$, sketch
the slope field for the corresponding
differential equation for $t \in [0,4]$, $P \in [0,4]$, indicating
the slope field segment at each point with integer coordinates.  
\item{C)}  Use $n = 4$ steps of Euler's Method to approximate the 
value of the solution of your differential equation with $P(0) = 200$
at $t = 1$.
\item{D)}  Use separation of variables to find the analytic solution
of your differential equation.
\item{E)}  At $t = 0$ there were 900 fish in the lake; 441 were
left after 6 weeks.  When did the fish population disappear entirely?
\bye