MATH 157
Calculus II  spring 2016
Department of Mathematics
HOWARD UNIVERSITY
Richard E. Bayne
Table of Contents:
[ Text and Required Materials
 Course Objectives
 Course Content
 Pace Sheet
 Prerequisites
 Cooperative Learning Groups
 Requirements
 Administrative Policies
 Office Hours
]
Text and Required Materials
 Calculus: Early Transcendentals, by James Stewart (8th
edition)
Back to TOC
Course Content
This course deals with calculus and its
applications, and is aimed primarily at students whose majors are
science, engineering or mathematics. Topics to be discussed will include:
integration of algebraic and transcendentalfunctions; applications of
integration to geometry, physics, and engineering; sequences and infinite series;
analytic geometry including parametric and polar curves.
Approximate pacing of topics
Back to TOC
Course Objectives
This is the second course in a threesemester sequence. The primary aim
of the sequence is to help students learn, understand, explain, and use
calculus. In addition, it is desired that students will improve their
mathematical skills, further their understanding of mathematics and its
applications to the sciences, as well as increase both their intellectual
curiosity and their desire to learn more about the value of mathematics in
general and calculus in particular. This second course concentrates on
functions of integration, sequences, series and analytic geometry. At
the completion of this course, the student should be able to:
Applications of the definite integral
 interpret the definite integral geometrically as the area under a curve.
 Find the area between two curves.
 Compute the average value of a function on an interval.
 Find volume using the disk method and shell method.
 Find the volume of a solid using cross sectional areas.
 Set up and use definite integrals to calculate volumes of solids of revolution.
 Calculate the moments and centers of mass of planar regions.
 Set up and evaluate definite integrals to calculate work.
 Use a definite integral to find the arc length of a smooth curve.
 Use a definite integral to find the area of a surface of revolution.
Techniques of integration
 Evaluate integrals using the method of Integration by Parts.
 Evaluate integrals using the method of Partial Fraction Decomposition.
 Evaluate integrals using the method of Trigonometric Substitution.
 Evaluate trigonometric function integrals using identities and substitutions.
 Be able to simplify and manipulate an integrand and choose an elective
technique or combination of techniques based on the form of the integrand.
 Determine which integration method is most appropriate for a given integration problem and then apply that
method to calculate an antiderivative.
 Recognize and describe an improper integral.
 Evaluate improper integrals as limits of ordinary integrals.
Sequences and infinite series
 Conjecture a formula or a recurrence relation to define all the terms of a sequence.
 Determine if a sequence is increasing, decreasing or bounded.
 Evaluate the limit of a convergent sequence.
 Be able to recognize a harmonic series.
 Be able to recognize telescoping sums.
 Given a bound for the difference between the terms of a convergent sequence and its limit, determine the
minimal index that achieves that bound.
 Carefully define the meaning of the sum of an infinite series of numbers in terms of its partial sums.
 Use the comparison test to determine if a series is convergent or divergent.
 Use the integral test to determine if a series is convergent or divergent.
 Use the alternating series test to determine whether a series is divergent.
 Accurately approximate the sum of a convergent alternating series.
 Evaluate the limit of a convergent sequence
 Evaluate the sum of a convergent series.
 Use partial sums to estimate the sum of a convergent series, and find error bounds where appropriate.
 Use power series to represent functions.
 Create Taylor (or Maclaurin) series to find a power series representation of a function.
 Perform algebra and calculus operations on power series.
 Recognize a geometric series and correctly apply the convergence theorem.
 Distinguish between absolute convergence and conditional convergence of a series.
 Use the ratio test to determine if a series is absolutely convergent or divergent.
 Use the ratio test to determine the intervals on which a power series converge.
 Determine the radius of convergence and the interval of convergence of a power series.
 Prove that a function is equal to its power series expansion.
 Use power series to evaluate limits.
 Be able to recognize functions that are represented as a power series.
 Derive the leading terms in the Taylor Polynomial for a function of one variable.
 Illustrate convergence and divergence of sequences graphically.
Analytic Geometry
 Sketch graphs of polar equations.
 Convert rectangular equations to polar equations.
 Convert polar equations to rectangular equations.
 Use derivatives to find slopes and rates of change in polar coordinates.
 Use integrals to find areas in polar coordinates.
 Sketch graphs of parametric equations in twodimensions.
 Classify conic sections by their eccentricity.
 Describe conics in terms of polar coordinates.
 Given a general seconddegree equation in two variables, determine which
type of conic it is.
 Sketch the graph of a conic, given its equation.
 Given the equation of a parabola, determine its vertex and axis.
 Given the equation of a ellipse, determine its center, foci, and axis.
 Given the equation of a hyperbola, determine its center, foci, and axis.
Back to TOC
Prerequisites
To be successful in this course, you should have mastery of college
algebra. You should have received a satisfactory grade in MATH
156 (Calculus I). Please see me if you have any
questions about your preparation for this course.
Back to TOC
Cooperative Learning Groups
A part of this course may be run using a cooperative learning approach.
Classes can be highly interactive and vigorous class participation is
expected. Early in the semester, each student will be assigned to
a group of four to five students.
Working well in a group is an important skill. Some of you may enjoy
the group work more than others, but all of you will benefit from further
developing this skill. After graduation, most of you will be working in
jobs which will require you to function as a member of a project team.
One objective of group work in this course is to help you to develop skills
in working effectively as part of a team. Another is to encourage discussion
about the concepts.
One of the primary objectives of this course is to help you learn
to think about problems mathematically and to solve the problems on your
own. Working with your colleagues in this class
and talking about problems with your group members are strategies to help
you better understand a problem situation from several points of view.
Experience has shown that those students that actually do work with
their groups not only do better in the course, they also learn more. Those
who for one reason or another refuse to fully participate in their
cooperative group invariably do worse.
Back to TOC
Requirements
 Regular attendance, participation, homework:
Regular attendance is expected. Some material will be presented
in class from a different perspective than that given in the text. "Getting
someone's notes" is a poor substitute for being present and involved
in class discussion. However, if you must miss a class, it is your
responsibility to find out what you missed. Make a friend!
Each student will be expected to do the following:
1. Attend every class.
2. Devote a minimum of 12 hours of study per week to the course.
3. Come to each class on time and ready to participate.
4. Be willing to help your classmates.
5. Be able to explain concepts to the instructor or to other students.
6. Meet with group members at least twice each week to
review and discuss course material.
7. Do all class activities and homework assignments.
Please note that only under the most unusual circumstances will
class activities or homework assignments be accepted after the due date.
 EXAMS:
 There will be four hour exams each worth 100 points. The material
to be covered on each test will be announced in advance of the scheduled
test date. Tests are tentatively scheduled for the following dates, but please
note that these dates may change depending on class progress and unforseen
circumstances.
Exam #1 week 4 Exam #2 week 8
Exam #3 week 12 Exam #4 week 15
Final Exam April 26
Please note: ALL EXAMS COUNT;  NO SCORES WILL BE DISCARDED.
Please also note that these dates may change depending on class progress and unforseen
circumstances.
Ordinarily, there are no makeup tests; exceptions to this policy will
be considered on a casebycase basis. You must determine BEFORE
the exam date whether your excuse will be acceptable.
Generally, incomplete grades will not be given. If there is an emergency
which causes a student to be unable to finish course requirements, the
emergency must be documented by the student's advisor or by the advisory
center.
If you have concerns about your progress or ability to keep up with
course assignments, please discuss these with me as soon as possible. DO NOT
WAIT until late in the semester.
 Final Exam: 200 points
 The final exam is cumulative and is listed in the
University's class schedule.
WeBWorK:
In addition to homework problems that will be assigned from the text,
there will be continuing assignments of problems on line using WeBWorK.
WeBWorK is an online system that allows you to work homework problems on
the web. You will have the opportunity to work the problems more than
once and generally will be able to work them until you get the correct
answer. You should read through the Student Introduction to WeBWorK (accessible on Blackboard)
before the end of the first week of classes.
Homework
Homework will count for 100 points. These points will be determined
by work done from the textbook and work done on WeBWorK, as described
above. The percentage from each is generally weighted in the student's
favor, usually 55% of the greater and 45% of the lesser.
Back to TOC
Administrative Policies
GRADES Grades will be determined as follows:
A 85%  100% of total points available
B 75%  84% of total points available
C 62%  74% of total points available
D 50%  61% of total points available
F 0%  49% of total points available
Academic Integrity Policy
Students who cheat violate their own integrity and the integrity of
the university by claiming credit for work they have not done and knowledge
they do not possess. All students are expected to recognize and to abide by
the policy on academic integrity found in the Student Handbook. Because you
will be asked to do a lot of work in collaboration with your group members,
I will ask you to sign all homework assignments attesting to the fact that
you have actively participated in the work.
Back to TOC
AMERICAN DISABILITIES ACT
Howard University is committed to providing an educational environment that is
accessible to all students. In accordance with this policy, students in need of
accommodations due to a disability should contact the Office of the Dean for Special
Student Services (2022382420, bwilliams@howard.edu) for verification and determination
of reasonable accommodations as soon as possible after admission and at the beginning of
each semester as needed.
Office Hours
Although the Mathematics Department office is located in 204 ASB, my office
is located in 236 Annex III, on the corner of 4th and College Streets, and
can be reached from either of the two southfacing doors which are
accessible from the driveway between Annex III and the C. B. Powell
building. As a rule, I am available for students on Mondays, Wednesdays,
and Thursdays.
My office hours for spring 2016 are
 Mon 1:00 pm  3:00 pm
 Wed 1:00 pm  3:00 pm
If you are unable to meet at these times, it is possible to make an
appointment at a different time that will be convenient for both of us.
If you need to reach me between classes:
 you may send an email message
to rbayne@howard.edu
 you may leave a voice message at 2028067673, or
 you may drop by my office and see if I am available.
I regularly check email several times a day both from home and at school,
but I check voice messages only when I am on campus.
The easiest way to contact me is to
send an email message to Richard Bayne.
Back to TOC
I. Intro & Review of Previous Material 



week 1 

A. Goals for course 




B. Working in Groups 




C. Review of Ideas from Calc I 









II. Areas & Volumes 



weeks 2, 3 & 4 

A. Area between curves 




B. Volumes 




C. Average Value 









III. Techniques of Integration 



weeks 4, 5 & 6 

A. Integration by Parts 




B. Trigonometric Integrals 




C. Trigonometric Substitutions 








D. Partial Fractions 








E. Other Techniques 









IV. Additional Applications of Definite Integrals 



weeks 7 & 8 

A.Arc Length 




B.Physics & Engineering Applications 




C. Social Sciences Applications 




V. Sequences & Series 



weeks 9, 10 & 11 

A. Sequences 




B. Series 




C. Convergence Tests 




D. Power Series 









VI. Polar Coordinates & Curves 



weeks 12 & 13 

A. Parametric Equations 




B. Polar Coordinates & Polar Curves 




VII. Conic Sections 



weeks 13 & 14 

A. Parabolas 




B. Ellipses 




C.Hyperbolas 




D. Rotation of Axes 








Dates to remember<>
1/11 Classes begin
1/18 Martin L. King Holiday
2/4 EXAM #1
2/15 Presidents' Day
2/25 EXAM #2
3/4 Charter Day
3/12 Spring Break begins
3/20 Spring Break ends
3/24 EXAM #3
4/1 Last Day to Withdraw
4/19 Senior Finals begin
4/20 EXAM #4
4/22 Last Day of Classes
4/26 Final Exam
This page was updated December 2015.