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Integration, Differential Equations and Approximations

A graphing calculator will be a useful thing to have. Most of the
calculator handouts will be based on the TI-83/TI-83 Plus/TI-
84/TI-84 Plus family of calculators. Many students like the TI-86
and TI-89 calculators because of their advanced capabilities. If
you select a more advanced calculator, make sure you are very
familiar with it and keep the user manual handy. You will be
allowed to use your calculator on quizzes and exams although
some quiz and test questions will explicitly forbid the use of

Important Dates:

Add/Drop deadline (Tuition) Monday, September 8.
Last day to drop (no W) Monday, November 3.
Last day to drop (W) Friday, December 5.
Unit exam 1 Friday, September 19, in class.
Unit exam 2 Friday, October 24, in class.
Unit exam 3 Friday, November 21, in class.
Final exam (cumulative) Scheduled by registrar.

These exam dates are firm. Non-university related travel plans are not a sufficient reason
to reschedule an exam.

Class Attendance:

We will cover the material at a rapid pace – probably considerably faster than you are
used to if your last calculus course was in high school. I have been teaching college
classes for more than 15 years and one of the strongest factors that I have seen in
determining who passes and who fails each class is class attendance patterns. The

material is covered so quickly that missing a week of class is almost certainly going to
influence your learning of the material and performance in the course.

I don’t have a formal attendance policy, nor do I have a formal component of the course
grade devoted to class attendance. On the other hand, I am not afraid to assign final
grades of “C,” “D” or “R” either. The overwhelming majority of students who stopped
coming to classes that I taught in the past ended up with a final grade that was a lot lower
than they were hoping to get (and a lot lower than they had the potential to get).


The main forms of assessment used in the course will be as follows:
Quizzes 10%
Homework 10%
Unit exam 1 20%
Unit exam 2 20%
Unit exam 3 20%
Final Exam (cumulative) 20%
The curve that I will use to determine your final grade is as follows:

Score Range Letter Grade

90-100 A
80-89 B
70-79 C
60-69 D
0-59 R

The only circumstances under which I will deviate from this curve are to ensure that: (a)
at least 20% of the class gets a final grade in the course of “A,“ and (b) at least 50% of
the class gets a final grade in the course of “B“ or higher. That is, the grade cut-offs
listed above may be lowered at the end of the semester. They will not be raised, so for
example, if you achieve a weighted average of 90 for the semester, you are guaranteed to
get an A no matter what.

In the case of a non-integer, weighted average, your average will be rounded up to the
nearest whole number.

Unit Exams:

Approximately once every four weeks we will have a major exam. The material covered
on each of these exams will correspond roughly to one or two of the major mathematical
areas that we will have covered.

Unit Exam 1: Friday, September 19, 2008.
Held during class (lecture) time.
This test will concentrate on techniques of integration,
including u-substitution, integration by parts, partial
fractions and long division, completing the square, and
trigonometric substitutions and identities. The test will also
include questions on approximating integrals using sums
(e.g. Riemann sums, midpoint, trapezoid and Simpson’s
rules) and convergence/divergence of improper integrals.

Unit Exam 2: Friday, October 24, 2008.
Held during class (lecture) time.

This test will concentrate on applications of integration,
including volumes, arc lengths, area, work and hydrostatic
pressure (applications to physics). The test will also feature
a significant number of questions on setting up and solving
differential equations, including separation of variables,
variation of parameters, integrating factors, Euler’s method
and slope fields.

Unit Exam 3: Friday, November 21, 2008.
Held during class (lecture) time.

This test will concentrate on issues of convergence and
divergence of infinite series using definitions and
convergence/divergence tests. Convergence and
divergence tests will include the n term, integral, ratio,
comparison and alternating series test. You may also be
asked to use series and summation formulas (including
those for geometric series) to solve real-world problems.
The test will also include questions about power series
(including Taylor series) and their radius or interval of

Approximately one week before each unit exam, a set of review problems will be posted
on the course web site. The lecture and recitation section immediately before the exam
will be devoted to working through review problems in preparation for the exam.

Unit Exam Do-Overs:

When I write each of the unit exams, I will write two versions of the test. These won’t be
exact clones of each other but will be over the same unit of material and be of
approximately the same level of difficulty.

One of these versions of the test will be given as the unit exam during class on the day
announced in this document.

The second version of the test will be given approximately ten days to two weeks later
(when everyone has received their graded unit exam back). This test will be given
outside of regular class time and will be open to anyone who wants to take it.

The score that we will use when computing your final grade in the course will be the
higher of these two scores. If you take the one of the tests (the version given in class or
the version given later), you will receive the score you get on that test. If you take both
tests (the version given in class and the version given later) you will receive the higher of
the two scores. If you take neither test, you will receive a score of zero.

The purpose of this policy is to recognize that during your first semesters in college, you
are learning a lot about the expectations of college-level courses, what you do and do not
know (and how to recognize this), what is expected of you when you take an exam in
college, how long it will take to prepare yourself for a test, and what you should do to get
yourself prepared. All of this can be quite different from the courses you are accustomed
to, and you will need time to adapt. The idea of giving you the opportunity to take a
similar test at a later date is to allow you to start to figure these things out for yourself
without doing severe damage to your grade in the class.

Because of the obvious logistical problem of the end of the semester, there is no “do-
over” opportunity for the cumulative final exam.

Note Card Policy:

You will be allowed to make and use note cards for the each of the unit exams and on the
final exam. On each unit exam, you will be allowed to use one (1) note card measuring
not more than three (3) inches by five (5) inches. I will bring a pair of scissors to the
exams and will be happy to trim larger cards down to size for you before the test begins.
On the cumulative final exam, you will be allowed to use up to two (2) note cards, each
measuring not more than three (3) inches by five (5) inches.

The reason for two (rather than 3) note cards on the final exam is so that you have a
powerful incentive to carefully examine the state of your mathematical knowledge at the
end of the semester and review what you do and do not know well. This will help you
prepare for the cumulative final exam. If you were allowed three note cards on the final
exam, some people would be tempted to cut corners and just use the cards they had made
for the unit exams.


During recitation on Thursdays, you will have the opportunity to complete a quiz on the
material covered in the preceding three lessons. Some of the best things that you can do
to prepare for each quiz are:
(a) Attend lecture and recitation regularly,

(b) Participate in class and ask questions when you are confused,
(c) Complete as many suggested problems from the textbook as you can (you will be
able to find these posted on the course web site), and,
(d) Seek help when/if you need it.

Each Thursday, your TA will begin the recitation by giving you the opportunity to ask
about the review problems you have been assigned. The TA will do his or her best to
answer as many questions and work as many problems as possible. When the questions
have been answered, or when there is only a limited amount of time left in the recitation,
the quiz will begin.

During the semester, there will be a total of ten (10) quizzes given. When computing
your final grade in the course, we will drop the lowest two (2) quiz scores. If you are
absent from recitation when a quiz is given and you don’t have a good excuse, then the
missed quiz will count as one of the scores that will be dropped.

Your TA will not give make-up quizzes except in cases of a serious, documented
emergency (e.g. serious illness that prevents you from attending recitation and/or


Once per week you will be required to complete a homework assignment consisting of
problems from the course textbook. Each homework assignment will consist of
approximately ten (10) problems from the textbook. Of these, five (5) problems will be
graded. As there is no way of knowing which problems will be graded, you should do
your best to complete all of the problems on each homework assignment.

I explicitly encourage you to feel free to work with other students in the course and seek
out help either from me or from your recitation instructor when working on the
homework problems. The whole point of doing the homework is for you to develop your
understanding of math. The points that the homework contributes to your final grade at
the end of the semester are nice but not the main point of the exercise. To this end,
simply copying another person’s homework is not permitted because it deprives you
of the opportunity to figure out what is going on. If the grader detects copying (e.g.
exactly the same nonsensical work on two papers) then all students involved will receive
zero credit for the assignment and the matter will be referred to the head of the Math
Department, the Dean and the Dean of Student Affairs.

There will be ten (10) homework assignments due during the course of the semester.
Homework assignments can be found on the course web site. Completed homework is
due in at the start of recitation on Tuesdays, starting with Tuesday, September 2.

If you cannot hand in your homework at the start of the recitation section when it is due,
you can still hand in the homework by taking the completed assignment to the
Mathematical Sciences office (6113 Wean Hall), putting the assignment in your TA’s
mail box, and sending your TA an e-mail to let them know it is there. If you do all this,
then your homework will be graded although you will receive a 50% penalty for
handing it in late.

Solutions to the homework problems will be posted on the course web site on Wednesday
afternoons. For this reason, no homework will be accepted for credit after noon on

At the end of the semester, the lowest two (2) homework scores will be dropped. If you
miss a homework assignment, the missed homework assignment will count as one of the
scores to be dropped.

Grading Homework:

When you write out your solution to each homework problem, you should clearly
indicate your final answer and show how you obtained your final answer through
appropriate mathematical calculations and manipulations. If it helps to draw a diagram or
graph, or to write a few sentences of explanation, then you should do this.

If you don’t show how you obtained a particular answer, the maximum score that you
will get for that problem is one (1) point out of a possible three (3), even if your answer is
completely correct. “Explanations” like “This is what my calculator said,” “This was
obvious to me,” etc. that do not actually show how the answer was obtained will not
impress the grader and not garner any credit.

Each problem on the homework will be graded on a 0-3 point scale, as outlined below.

Problem solution submitted Score
All sections of problem completed correctly and written up in a way that is
comprehensible to the grader.
All sections of problem completed with convincing work shown
throughout, but either some of the answers are not correct or some of the
work not comprehensible to the grader.
Correct answer given but work is either missing or incomplete.
Alternatively, some (but not all) of the problem has been attempted with
convincing work shown, but solution is incomplete.
Solution not submitted. 0

Office Hours:

I will have a number of office hours per week. I will hold my office hours on:
Office Hours 1: Monday, 1:00pm-2:00pm, 6124 Wean Hall
Office Hours 2: Tuesday, 9:00am-10:00am, 6124 Wean Hall
Office Hours 3: Wednesday, noon-1:00pm, 6124 Wean Hall
Office Hours 4: Thursday, 10:00am-11:00am, 6124 Wean Hall

These office hours are good places to get help on homework problems, help with the
course in general and help when you are preparing for a quiz or exam in the course.

If you are unable to attend these office hours, feel free to make an appointment with me
outside of these scheduled hours. I’m always happy to meet with students outside of
class time provided a mutually agreeable time can be found for the meeting.

Your TA will also hold office hours and you should check with him or her to find out
when and where these will be held.

Americans with Disabilities Act:

Any students with a disability that may entitle them to special academic consideration
need to obtain official documentation of their disability from CMU’s Disability
Resources Office. This office is located at 102 Whitfield Hall and their phone number is
(412) 268-2013. Documentation should be obtained and given to the instructor in a
timely fashion. I appreciate and require at least one week’s notice in order to make the
necessary arrangements for special accommodations in either the instructional or
assessment portions of the course.

Overview of Topics in the Course:

The overall goal of this course will be for you to gain a thorough knowledge of
fundamental topics of integration, infinite series and differential equations.

In terms of coverage, we will do our very best to cover most of the material from
Chapters 6, 7 and 8 of Stewart’s book and a number of additional topics that are
considered to be very advanced for a Calculus II course. Note that there are lots and lots
of important applications and topics in calculus and differential equations that won’t
make it into this course. An important goal that we will work together to achieve this
semester is to get you to the point where you can use your knowledge of math (and
calculus in particular) to understand the new ideas in the engineering, science and
technology courses that you will go on to when you have finished with this course.

The topics that we will cover are listed on the next two pages. The schedule is my best
guess at this point. I expect that we will try to stay up with this schedule but not be
rigidly bound to it as the semester goes on.
Date Topic Textbook
8/25  First day. Integration formulas. U-Substitution. 5.5
8/26  First day. U-substitution. 5.5
8/27  Integration by Parts. 6.1
8/28  Quiz #1 on class policies and integration.  
8/29  Trigonometric integrals. 6.2
9/1  Labor Day – NO CLASS.  
9/2  Homework #1 due. Trigonometric integrals.  6.2
9/3  Trigonometric substitution.  6.2
9/4  Quiz #2 on trigonometric integrals and substitution.  
9/5  Partial fractions. 6.3
9/8  More partial fractions. Integration tricks and tables. 6.3, 6.4
9/9 Homework #2 due. Approximating integrals in a calculator.  
9/10 Riemann sums. Over/under estimates. Midpoint & trapezoid rule. 6.5
9/11  Quiz #3 on integration using partial fractions and tables.  
9/12 Simpson’s rule. Error estimates. 6.5
9/15  Improper integrals. 6.6
9/16 No homework due. Comparing improper integrals to p-integrals. 6.6
9/17  Review for Unit Exam 1.  
9/18  No quiz. Review for Unit Exam 1.  
9/19 Unit Exam 1 (held during class time).  
9/22 Calculating areas and volumes (disks). 7.1, 7.2
9/21 Homework #3 due. Calculating volumes using disks. 7.2
9/24 Calculating volumes using cylindrical and spherical shells. 7.3
9/23 Quiz #4 on improper integrals and area/volume calculations.  
9/26 Calculating mass and other slicing problems. Arc length. 7.4
9/29 Calculating work and hydrostatic pressure. 7.5
9/30 Homework #4 due. Calculating the center of mass.  
10/1 Solutions of differential equations. Euler’s method. 7.6
10/2 Quiz #5 on cylindrical shells, slicing problems and physics applications.  
10/3 Numerical and graphical solutions of differential equations. 7.6
10/6 Separable equations. Separation of variables. 7.6
10/7 Homework #5 due. Modeling the spread of a disease.  
10/8 Modeling with differential equations. Solving the Logistic Equation.  
10/9 Quiz #6 on solving differential equations numerically, graphically, symbolically.  
10/10 Solving first order ODEs using integrating factors.  
10/13 Solving second order homogeneous ODEs.  
10/14 Homework #6 due today. Solving 1 and 2 order ODEs.  
10/15 Solving second order non-homogeneous ODEs using variation of parameters.  
10/17 Fall Break – NO CLASS.  
10/20 Solving second order non-homogeneous ODEs using variation of parameters.  
10/21 No homework due. Solving second order differential equations.  
10/22 Review for Unit Exam 2.  
10/23  No quiz. Review for Unit Exam 2  
10/24 Unit Exam 2 (held during class time).  
10/27  Infinite series. Definition of convergence and divergence. 8.2
10/28 Homework #7 due. Geometric series and applications.  
10/29 Convergence tests (n term, integral). Estimates of sums. 8.3
10/20 Quiz #7 on solving first and second order differential equations.  
10/31 The Ratio Test. 8.4
11/3  The Comparison Test. 8.3
11/4 Homework #8 due. Practice using convergence tests 8.3, 8.4
11/5  Alternating series. Absolute and conditional convergence. 8.4
11/6 Quiz #8 on convergence of infinite series. 8.3, 8.4
11/7  Summary of strategies for testing convergence of infinite series. 8.3, 8.4
11/10 Power series and Taylor series. 8.5, 8.7
11/11  Homework #9 due. Approximating functions with series. 8.6
11/12 Finding formulas for Taylor Series 8.6, 8.7
11/13 Quiz #9 on convergence of infinite series and Taylor series.  
11/14 Radius of convergence of a power series or Taylor series. 8.5
11/17  Applications of Taylor polynomials. 8.8
11/18 No homework due today. Radius and interval of convergence. 8.5
11/19  Review for Unit Exam 3.
11/20 No quiz. Review for Unit Exam 3.  
11/21  Unit Exam 3 (held during class time).  
11/24  Parametric equations for curves. 9.1
11/26  Thanksgiving Break – NO CLASS.  
11/27  Thanksgiving Break – NO CLASS.  
11/28  Thanksgiving Break – NO CLASS.  
12/1  Calculus of parametric curves. 9.2
12/2 Homework #10 due. Setting up parametric equations. 9.1
12/3 Polar coordinates. 9.3
12/4  Quiz #10 on Taylor series and radius of convergence.  
12/5 Review for the cumulative final exam.  

12/8-12/16 Final exam period. Final exam time and place to be set by registrar.  

Additional Help with the Course:

In addition to lecture, recitation sessions and office hours, the University operates a walk-
in peer tutoring center in the Mudge Library and the Donner Reading room.

These peer tutoring sessions are held in the evenings on Sunday-Thursday night from
8:00pm to 11:00pm.

Individualized tutoring and other help options are available through the Academic
Development Office. This is located in Cyert Hall B5 and the phone number is (412)