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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Review Sheet -- Midterm Exam 3}
\centerline{November 22, 2005}
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\noindent
{\it General Information}
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As you know from the course syllabus, the third and final 
midterm exam for the course will be given in class on 
Friday, December 3.  This be an individual closed book exam
similar in format to the other midterms we have done this
semester.  I will be happy 
to hold a late afternoon or evening review session to help you 
prepare.  Late afternoon times are possible Wednesday, December 1,
but I will have to leave campus no later than 5:30pm to get to 
a rehearsal, so Wednesday evening will not work for me that week.  
Tuesday or Thursday evening would be OK too.  
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\noindent
{\it Topics to be Covered}
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The exam will cover the material we have covered since the last 
exam, starting with the material on limits of functions, and 
going through the discussion of the Fundamental Theorem of 
Calculus from class on Monday, November 22.
This is sections 20, 21, 22, 25, 26, 27, 29, 30, 31 in the text, but 
as before, not all the topics in those sections were discussed in 
class. You are responsible for only what we did talk about:
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\item{1)}  The definition of the statement
$\lim_{x\to c} f(x) = L$, consequences, limit theorems, 
techniques for computing limits.
\item{2)}  The definition of continuity and its consequences,
key properties of continuous functions on a closed
interval:  The Intermediate Value and Extreme Value Theorems.
\item{3)}  The definition of differentiability and examples,
the Mean Value Theorem and its consequences.
\item{4)}  L'Hopital's Rule for limits.
\item{5)}  The definition of integrability, computations
of definite integrals from the definition, the Fundamental
Theorem of Calculus.
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\noindent
{\it What to Expect}
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The exam will have four or five questions, each possibly with several
parts.  Some questions will ask for a precise statement of a 
definition or a theorem we have discussed.  Be prepared to give
careful statements of the definitions noted above and know 
how to use them (for instance how to show that a limit
exists using the $\varepsilon,\delta$ definition).
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Also know and be able to give these proofs:
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\item{1)}  The Mean Value Theorem. 
\item{2)}  A monotone increasing function on an interval $[a,b]$
is integrable.
\item{3)}  The Fundamental Theorem of Calculus, ``part 1''
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\noindent
The other questions will be similar to questions from the problem sets
and discussions. 
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\noindent
{\it (over for review problems)}
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\noindent
{\it Suggested Review Problems}
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 Some good review problems to look at are
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\item{} Section 20/3bdfh,4b,5,15
\item{} Section 21/5,8
\item{} Section 22/questions like 4-9
\item{} Section 25/3bcd,6,8
\item{} Section 26/10 (Hint: use the MVT),11,15
\item{} Section 27/4
\item{} Section 29/3,11
\item{} Section 30/12,13
\bye
