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\centerline{Mathematics 36, section 5 -- AP Analysis}
\centerline{Information on Exam 1}
\centerline{October 1, 1999}
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\noindent
{\bf General Information}
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The first exam for the course will be given next Friday, October 8, 
as announced in the course syllabus.  It will cover the material 
discussed in class from the start of the semester
{\it through and including\/} the material from Monday, October 4.  There
will be 4 or 5 problems, some possibly with several parts.  Some
may ask for a graph or the result of a calculation; others may ask for 
a precise definition of a term or concept we have used, or a 
short description or explanation of some phenomenon (similar
to the questions from Lab Day assignments). 
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\noindent
{\bf Review Session}
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If there is interest, I will be happy to schedule a review session
outside of class time to help you get ready for the exam.  
I will be available any time after 4:00 pm on Wednesday, October
6.  {\it Important Note:  I will be leaving campus on Thursday, October 7
at about 12:00 noon to attend a conference.  The exam will be proctored
by Prof. Brevik of our department.  I will be available for 
``special office hours'' 11:00 am to 12:00 noon on Thursday for 
last minute questions, but my 
regular office hours will be cancelled that day.  Please start to 
prepare for the exam early so that you will have the opportunity
to have your questions answered at the review session or on 
Thursday morning.}
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\noindent
{\bf Suggested Practice Problems}
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\noindent
From the text:  
\item{$\bullet$} Chapter 1 Review Problems: p. 86-90/7, 8. 9,  
(use what you know about the formulas, don't just graph each of
them with your graphing calculator!), 11, 35-43, 46
\item{$\bullet$} Chapter 2 Review Problems: p. 146-7/1,2,5 (Note: 
the idea of the problem is to approximate the derivative without using
the formula.  You can gauge the accuracy of your approximation
by using the formula too!), 7 - 12
\item{$\bullet$} Be able to differentiate any functions like p. 237/1 - 30
given by formulas.
(The derivative computation might come as part of a larger problem.)
\item{$\bullet$} Chapter 5 Review Problems: p. 281/3 - 6
\item{$\bullet$} Chapter 9 Review Problems: p. 471/1,2,3,5,6,7,8,
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\noindent
Other problems:

\item{$\bullet$}  Imagine you are zooming in on the graphs of the following functions
near the origin:
\itemitem{a)} $y = \sin(x) - \tan(x)$
\itemitem{b)} $y = e^x - 1$
\itemitem{c)} $y = {x^2\over x^2 + 1}$
\itemitem{d)} $y = {x\over x^2 + 1}$
\itemitem{e)} $y = {x \over x + 1}$
\itemitem{f)} $y = |x|$ (absolute value function).
Which of them would eventually look the same?  Which would look 
locally linear?  For those that would look linear, find the equations 
of the lines you would see
{\it without\/} carrying out the zooming process.

\item{$\bullet$} State the precise definition of the derivative of a function $f$
at $x = a$.  {\it Use the definition} to compute $f'(x)$
for $f(x) = 2x^2 - 4$ at a general $x$.

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\noindent
{\bf Sample Exam}
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\noindent
I. (20) The gross domestic product (GDP) of a country has grown as in 
the following table   
$$\matrix{t = {\rm time,\ in\ years}&0&10&20&30&40&50\cr
       G(t) = {\rm GDP}&500&610&720&831&940&1051\cr}$$
Is a linear function or an exponential function a better
``fit'' for these data points?  Explain how you are deciding 
which is better and find a formula for a function that approximates the 
values in this table.
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\noindent
II.  (10) What is the exact mathematical definition of the 
{\it derivative\/} of a function $f(x)$ at $x = a$?  Explain
how this relates to the idea of the rate of change of $f$ at $a$.
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\noindent
III. A population of bacteria grows in a medium that can support no more than
$P_1$ of bacteria.  At time $t = 0$, the number of bacteria present is 
$P_0 < P_1$.  The population grows more and
more rapidly at first, but then the rate of growth decreases and
the population approaches $P_1$, without exceeding it.  
\item{A.} (15) Sketch a graph
of the population of bacteria as a function of time
that fits this description.  
\item{B.} (15)  Sketch the graph of the derivative of the population function.
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\noindent
IV.  Find the derivative functions of each of the following functions
defined by formulas:
\item{A.}  (5) $f(x) = 3x^4 + 5\sqrt{x} - {3\over \root 3\of x}$
\item{B.}  (7.5) $g(x) = {\sin(x) \over 1 + \cos(x)}$
\item{C.}  (7.5) $h(x) = \arctan(2^x)$
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\noindent
V. 
\item{A.} (15) Find the Taylor polynomial of degree 4 
for $f(x)=e^{\sin(x)}$ with $a = 0$.
\item{B.} (5)  Use your polynomial to approximate $f(.2)$.
\bye
 
