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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 3 -- Things To Know}
\centerline{April 23, 2001}
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{\it General Information}
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As announced in the course syllabus, the third full-period
midterm exam of the semester will be given in class on Friday, 
April 27.  The format will be similar to that of the midterm exams
last semester and Exams 1 and 2 this time.  The exam will cover the material
since the last exam (Quizzes 7,8,9), including the following material from 
Chapters 9 and 10 of the text:
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\item{1)}  Taylor polynomials and Taylor approximations
\item{2)}  Taylor series and interval of convergence
\item{3)}  ``Shortcuts'' (substitution, algebraic
manipulations, term-by-term integration and differentiation)
for computing Taylor series from 
the known series for 
$$e^x,\ \sin(x),\ \cos(x),\ {1\over (1-x)},\ (1+x)^p.$$
\item{4)}  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)}  Approximate values of solutions via Euler's Method
\itemitem{d)}  Formulas for solutions via separation of variables
\itemitem{e)}  Applications to growth/decay problems
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A copy of the table of integrals from our text will be provided.
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There will be a review for the exam in class on Thursday, April 26.  
I will also try to set up an evening session with Jodi Marani on Thursday 
for last minute questions.
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\noindent
{\it Sample Exam}
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Caution:  The following problems indicate {\it only} the kinds of topics
that will be used in,
the approximate amount of computation required for, and the level of difficulty
of the questions that will appear on this exam.  The actual exam questions 
{\it will look different from these\/} and the exam itself
will be slightly shorter.  It is important to 
understand the methods involved in the solutions to these problems
as well as the answers.  Do other 
practice problems besides these ones if you want to be well-prepared.
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\noindent
I.  
\item{A)}  Compute the 4th degree Taylor polynomial for $f(x) = \ln(1+x)$
at $a = 0$, {\it using the definition} of Taylor polynomials.
\item{B)}  Use your polynomial to approximate the value of $\ln(1.1)$.
What is the error in your approximation?
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II.  
\item{A)}  Using our ``shortcut'' rules, determine the first
three nonzero terms in the Taylor series at $a = 0$ of the functions
$cos(x)$ and $(1 - x^2)^{1\over 2}$.
\item{B)}  Using the Taylor series, determine which of the two functions 
in part A is {\it larger} for small values of $x$ (say $x < .1$).  
Explain how you can tell.
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III.  All parts of this question refer to the differential equation
$${dy\over dx} = 4 - y$$
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\item{A)}  Sketch the slope field of this equation, showing 
the slopes at points on the lines $y = 0,1,2,3,4,5,6$
\item{B)}  On your slope field, sketch the graph of the 
solution of the equation with $y(0) = 1$.
\item{C)}  Using separation of variables, solve the differential 
equation with the initial condition $y(0) = 1$.
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IV. 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)}  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{C)}  Use separation of variables to find the formula for the 
general solution of your differential equation.
\item{D)}  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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Also see Problem Sets 6,7,8, the last three quizzes, and 
the Review Problems from the ends of Chapters 9,10 in
the text for other practice problems.
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