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\centerline{Mathematics 136, section 4 -- AP Calculus}
\centerline{Review Sheet for Exam 2}
\centerline{October 24, 2003}
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{\it General Information}
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The second exam for the course will be given next Friday, October 31, 
as announced in the course syllabus. It will cover the material
from Chapters 4 - 7 in the text that we have discussed starting 
from immediately before the first exam through and including the material 
from Friday October 24. There will be 4 or 5 problems, some possibly with 
several parts. The format will be similar to that of the first exam. 
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{\it Topics to be Covered}
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\item{1.} Critical points, local maxima and minima, global maxima and 
minima on an interval.  In this connection, I might ask you 
something concerning critical points of a family of curves (as in Lab 3).
\item{2.} Applied optimization problems.
\item{3.} Riemann sums, the definite integral of a function over an interval 
$[a,b]$. Be prepared to compute a Riemann sum for a particular function and 
a given number of subdivisions (small, of course!) Also know how to derive the 
exact value of a definite integral of a function cx + d by taking the limit of 
the Riemann sums.
\item{4.}  The Fundamental Theorem of Calculus (know the statement of this 
theorem -- both parts)
\item{5.} First applications of integrals:  Total change of a function, 
areas, average values.
\item{6.}  Methods of integration, including substitution, integration by 
parts, use of the table of integrals.  (Note:  In order to give you 
time to prepare on other topics for the exam, there will not be a problem
set covering this last point before the exam.)  The review problems
from Chapter 7 of the text are an extremely good source of practice
problems, though!
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{\it 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.  Unfortunately, my 
evening and late afternoon schedule is going to be very full next week -- I 
am basically only available on Tuesday (October 28).  If that time is
not convenient, I would be willing to consider doing an in-class review 
Wednesday, October 29.
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{\it Suggested Practice Problems}
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{\it From the text:}
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\item{$\bullet$} From Section 4.5: 8, 11, 12, 14.
\item{$\bullet$} From Review Problems for Chapter 4: 5,6,17 and 19 (ignore the 
part of the instructions on graphing),22,23,27,31,35
\item{$\bullet$} From Review Problems for Chapter 5: 3,4,5,9-13,23,31
\item{$\bullet$} From Review Problems for Chapter 6: be able to do ``simple''
integrals like 1-29 using basic rules, 37, 41, 43, 48, 49
\item{$\bullet$} From Practice Integration problems at end of Chapter 7: look at a good sample of these and see if you can tell which method will apply. 
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{\it Sample Exam} -- Note the real exam will contain different questions,
even different types of questions.  But the topics covered here are a representative
sample, and the actual exam will be about this length.
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I.  Section 4.5 (page 202)/16
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II. Terminology. 
\item{A.}  What is the definition of the definite integral of a function $f(x)$
over the interval $[a,b]$? 
\item{B.} What does the Fundamental Theorem of Calculus say about definite integrals? 
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III.  Review Problems for Chapter 5 (page 252)/1
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IV.  Review Problems for Chapter 6 (page 285)/47
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V. Compute each of the following integrals (the table of integrals from the text will 
be provided, and you may use it for parts B,C,D if necessary). 
\item{A.}  Show that entry II.17 in the table of integrals is correct, using 
integration by parts. 
\item{B.} $$\int x^5 e^{3x^2}\ dx$$
\item{C.} $$\int \cos(2x) (\sin(2x) + 4)^{1/3}\ dx$$ 
\item{D.} $$\int_2^4 {x + 3 \over 4x^2 + 4x - 3}\ dx$$
\bye 


