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\centerline{Mathematics 136, section 1 -- AP Calculus}
\centerline{Information for Exam 3}
\centerline{November 25, 2003}
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\noindent
{\it 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 5.  The 
format will be similar to that of the other exams this semester.
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\noindent
{\it Topics to be Covered}
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This exam will cover the material we have discussed since 
Exam 2, up to the material on geometric and power series and
convergence covered in class on Monday and Tuesday, November 24, 25.  
This includes the applications of integrals and
the material on differential equations and population 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 (Cavalieri's Principle), 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.   
\item{6)}  Geometric series, power series (Sections 9.1, 9.4)
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\noindent
{\it 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 Tuesday, December
2.  I have commitments Wednesday and Thursday evenings (orchestra rehearsals 
in Boston that I cannot miss), so those times are not possible for me.  
We can discuss this in class on Monday, December 1 and set it up.
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\noindent
{\it Some good review problems}
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\noindent
From the text: 
\item{$\bullet$} From the review problems for Chapter 8 (p. 397): 
7,8 (also be able to set up the integrals for the total mass and 
center of mass of a wire in a shape
like one of these if the density function is given), 9-13,25,31
\item{$\bullet$} From the review problems for Chapter 11 (p. 552): 
8-23, 31, 34,35, 36, 42.  ``Check Your Understanding'' problems
8,9,10.
\item{$\bullet$} From Chapter 9/Section 1: 1 - 14, Chapter 9/Section 4: 11-14,
From the review problems for Chapter 9: 16, 17.
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{\it Sample Exam}
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Note:  This is slighly longer than the actual exam will be.  Nevertheless,
it may help, as a part of your preparation, to set aside an hour 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.  Both parts of this problem refer to the region $R$ bounded
by $y = 1 + x^2$, $x = 0$, $x = 1$, and the $x$-axis.
\item{A)}  Find the volume of the solid obtained by 
rotating $R$ about the $x$-axis.
\item{B)}  Find the volume of the solid obtained by rotating 
$R$ about the line $y = -3$.  
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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 general 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?
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\noindent
IV.  
\item{A)}  Find the sum of the infinite geometric series
$$ 3 + {3\over 7} + {3\over 49} + {3\over 343} + \cdots $$
\item{B)}  Use the Ratio Test to determine the interval 
of convergence of the power series
$$\sum_{k=0}^\infty {2^n x^n\over n!}$$ 
\bye
  
