Math 131: Calculus II

   Department of Mathematics and Computer Science
   Hobart and William Smith Colleges

   Spring, 2011.

   Instructor:  David J. Eck.

   Monday, Wednesday, Friday, 9:05--10:00 AM, Room Eaton 111.
   Lab:  Thursday, 8:45--10:10 AM, Room Gulick 206A

About This Course

The textbook for Math 131 is Single Variable Calculus: Early Transcendentals, by William Briggs and Lyle Cochran (ISBN 978-0-321-66414-3). This is the same textbook that was used in Math 130 in the Fall semester. We will cover Chapters 5 through 9, although we will not have time to cover every topic in those chapters.

This is a second course in calculus. Calculus I covered differential calculus and was primarily concerned with derivatives, their definition in terms of limits, their properties, and their applications. At the very end of the semester, antiderivatives and indefinite integrals were introduced.

For most of Calculus II, we will cover integral calculus, which is primarily concerned with integrals and their definition, properties, and applications. In addition to the indefinite integrals, or antiderivatives, that you have already seen, there is the another type of integral, the definite integral. The definition of the definite integral has nothing to do with derivatives. One of the most important facts in calculus is that there is nevertheless a close relationship between definite integrals and derivatives. That relationship is called the Fundamental Theorem of Calculus. The first part of the course (most of Chapter 5) defines the definite integral and proves the Fundamental Theorem. The second part (the end of Chapter 5 and Chapter 6) covers various applications of integration. The third part (Chapter 7) covers some techniques for computing definite integrals and antiderivatives.

The last part of the course (Chapters 8 and 9) turns to a completely different topic: infinite sequences and infinite series. Although it is part of calculus, this material is not closely related to derivatives and integrals. It is, instead, another application of limits. The term infinite sequence refers to an unending list of numbers, such as 1, 1/2, 1/4, 1/8, 1/16, 1/32, ... An infinite sequence can approach a limit. It should be clear that for the example, the limit is zero. An infinite series is what you get when you add up all the terms in an infinite sequence, such as 1 +1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... This infinite series adds up to two (which might not be completely clear). Not every infinite sequence has a limit, and not every infinite series has a sum. We will spend the last part of the course looking at the definition, properties, and applications of infinite sequences and series.

Homework

Reading the textbook is a major part of the homework for this course. I also suggest going over the notes that you take in class, preferably on the same day as the class. Ideally, you should make a neat copy of your notes, filling in details and making sure that you understand everything. Anything that you don't understand should be brought up in class, in my office hours, or with the math intern.

Each week, I will assign a fairly small number of exercises from the textbook to be handed in as homework. You are allowed to work with other people on the homework, but you should write up the solutions on your own, in your own words. Your solutions should not be identical to those of the people that you work with. The goal is not just to solve the problem, but to present the solution in an organized, clear, mathematically correct way. Your solutions will be graded for presentation and style, as well as for correctness. In no case will you get any points simply for stating a correct final answer.

In addition to the homework that I collect, I will suggest additional problems for you to work on. You should do all these problems and as many more as are necessary for you to master the material. Note that answers to odd-numbered problems are in the back of the book.

Homework will generally be collected on Wednesday of the week following the week when the homework is assigned. Assignments will be announced in class and will also be posted on the course web site. After collecting the homework, I will post my solutions to the collected homework on the web site.

Labs

There is a required lab component for this course. In the lab, you will work with in a group of three students. Your group will work on some problems and will turn in one set of solutions to be graded. Everyone in the group will get the same grade for the lab. Unless you have a very good excuse, you must be present at the lab to get a grade for it.

Each lab will consist of a few challenging problems. (There might also be a few warm-up problems, which won't be turned in.) The problems are not meant to be straightforward exercises. In some cases, they will be open-ended problems without a single solution. For each problem, your group should turn in an essay explaining your solution and how you found it. In cases where you don't find a solution, you should explain what you did to try to solve the problem, and you should present any partial results that you obtain. (There might be a few problems that don't have solutions, just to keep you on your toes!)

Problems from one lab will ordinarily be collected at the following lab. I will post my solutions to the collected lab problems on the course web site.

Occasionally, I will ask someone in the class to present a solution to one of the lab problems. Everyone in your group should be prepared to present the group's solution to any lab problem.

Tests

There will be three in-class tests for this course, which will be given on Friday, February 11; Wednesday, March 9; and Monday, April 11. You should not miss a test without a very good excuse.

The final exam for this course will be given at the time scheduled by the Registrar's Office: Monday, May 9, from 8:30 to 11:30 AM. It will be in our regular classroom. Note that the exam covers the entire course, though with some emphasis on material covered in the last part of the course.

I reserve the right to adjust your grade downwards if you miss more than one or two classes or labs without a good excuse. In my grading scale, an A corresponds to 90--100%, B to 80--89%, C to 65--79%, D to 50--64%, and F to 0--49%. Grades near the endpoints of a range get a plus or minus.

Math Intern

The Colleges employ a "math intern" who is available on a regular schedule to help students in calculus and precalculus courses. This year, the intern is Emma Daley (William Smith '10). Math intern hours are:

           Sunday through Thursday
           3:00 -- 5:30 and 6:30 -- 10:30 PM
           Room Lansing 310

(Note: The Tuesday evening time slot will be covered by a student teaching assistant, Yaoxin Liu.)

Statement from the CTL

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify and register for services with the Coordinator of Disability Services at the Center for Teaching and Learning (CTL), and provide documentation of your disability.  Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: http://www.hws.edu/disabilities

Please direct questions about this process or Disability Services at HWS to David Silver, Coordinator of Disability Services, at silver@hws.edu or x3351.

Office Hours, E-mail, and Web

My office is room 313 in Lansing Hall. My office phone extension is 3398. I am on campus most days, and you are welcome to come in anytime you can find me there. My regular office hours will be:

               MWF, 10:10--11:05 AM and 1:55--2:50 PM
               Thursday, 10:30--11:30 AM

Office hours are times when I promise to try my best to be in my office. I do not generally make appointments during my office hours, since they are times when I am available to students on a first-come, first-served basis. When necessary, I am happy to make appointments for meetings outside my scheduled office hours.

My e-mail address is eck@hws.edu. E-mail is good way to communicate with me, since I usually answer messages within a day of the time I receive them.

The Web page for this course is http://math.hws.edu/eck/math131/index_s11.html. I will post weekly readings and assignments on that page.

Tentative Schedule

Here is a tentative weekly schedule of readings and topics for the course. We will probably not be able to follow this schedule exactly, but we do need to cover almost all of the material listed on the schedule.

Dates Reading Topics to be covered
Jan. 19, 21 5.1 Area under curves
Summation notation
Riemann sums
Jan. 24, 26, 28 5.2, 5.3 Definite integrals
The Fundamental Theorem of Calculus
Jan. 31; Feb. 2, 4 5.4,
5.5 (start) 		Average values of functions
Mean Value Theorem for integrals
Integration by substitution
Feb. 7, 9, 11 5.5 (finish),
6.1 (start) 		Continue with integration by substitution
TEST on Friday, February 11
Feb. 14, 16, 18 6.1 (finish)
6.2 		Velocity and net change
Area between curves
Feb. 21, 23, 25 6.3, 6.4 Volumes by slicing
Volumes by shells
Feb 28; Mar. 2, 4 6.5,
6.6 (selections)
6.8 (start) 		Arc length
Work and possibly other applications
Exponential models
Mar. 7, 9, 11 6.8 (continued)
7.1 		Integration by parts
TEST on Wednesday, March 9
SPRING BREAK
Mar. 21, 23, 25 7.2, 7.3, 7.4 Trigonometric integration
Partial Fractions
Mar. 28, 30; Apr. 1 7.5, 7.6, 7.7 Tables of integrals
Numerical integration
Improper integrals
Apr. 4, 6, 8 8.1, 8.2 Infinite sequences
Apr. 11, 13, 15 8.3, 8.4 Infinite series
Basic convergence tests
TEST on Monday, April 11
Apr. 18, 20, 22 8.5,
9.1 (start) 		More convergence tests
Taylor polynomials
Apr. 25, 27, 29 9.1, (finish)
9.2, 9.3 		Power series
Taylor series
May 2 End of term Wrap-up and review
May 9 Final Exam, Monday, May 9, 8:30 AM.