# Math 131: Calculus II

```   Department of Mathematics and Computer Science
Hobart and William Smith Colleges

Spring 2017.

Instructor:  David J. Eck  (eck@hws.edu)

Monday, Wednesday, Friday, 12:20–1:15 PM
Room Eaton 111.

Lab: Tuesday, 1:30–2:55 PM
Room Gulick 206A.
```

The textbook for this course is Single Variable Calculus: Early Transcendentals (2nd Edition), by Briggs, Cochran, and Gillett (ISBN 0321954237). This is the same textbook that was used in Math 130. We will cover much of Chapters 5 through 9, omitting some sections. In particular, we will omit many of the applications of integration and integration techniques in Chapters 6 and 7.

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 with 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.

### Labs

This course has a required lab component. 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 mostly 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!)

The lab period for the course is on Tuesday. Occasionally, I might 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.

### Homework

Reading the textbook is an important 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.

There will be weekly assignments using WebWork, the web site for on-line math homework. You might be familiar with WebWork from Calculus I.

There will also be weekly written homework assignments. Each week, I will assign a fairly small number of exercises from the textbook to be handed in. 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. Written homework is very different from WebWork. The goal is not just to find an answer, 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 assignments will be announced in class and will be posted on the course web page.

### Quizzes

Quizzes will be given at the beginning of some classes and labs, maybe once a week. Some quizzes will be announced in advance, but most will not. Quizzes might sometimes include material from assigned readings before that material is covered in class, so it is important to keep up with the reading! Quizzes will cover old as well as recent material; there is no restriction on how far back into the course the questions can go. It is not generally possible to make up quizzes that you miss. However, your two lowest quiz grades will be dropped.

### Tests and Final Exam

There will be three in-class tests in addition to a final exam. The tests will be given on February 15, March 23, and April 19 (which are all Wednesdays). The final exam will take place during the officially scheduled exam time for the course, which is Tuesday, May 9, at 1:30 PM.

The final exam will be comprehensive, covering material from the entire term, with some emphasis on material covered during the last part of the course.

Your numerical grade for the course will be determined as follows (except that there is some possibility that we will cut back on or abandon Webwork and redistribute some of those points to other items in the list):

```             First Test:         12%
Second Test:        12%
Third Test:         12%
Final Exam:         18%
Quizzes:            10%
Webwork:            10%
Written Homework:   12%
Labs:               14%
```

Letter grades are assigned as follows: 90-100: A; 80-89: B; 65-79: C; 55-64: D; 0-54: F. Grades near a cutoff get a plus or minus.

### No Technology During Lecture!

Use of a laptop, cell phone, or other device is not allowed during lecture. (The only exception is if you have a verified medical reason to take class notes on a computer.) For note taking, you should use paper.

There is substantial research showing that taking notes on paper can improve retention of the material, compared to note-taking on computer. My real advice is to take notes in outline form, noting down important ideas and examples, and to make a more formal copy of the notes after class, filling in any missing details. There is also research showing that the multitasking that you are likely to engage in if you have a computer open in front of you is detrimental to learning.

### Attendance Policy

I assume that you understand the importance of attending class. While I do not take attendance in every class, I expect you to be present unless circumstances make that impossible. Participation in lab is particularly important, and I do take attendance at lab. If you miss a lab without a good reason, you can still turn in a lab report for that lab, but your grade on the lab report might be reduced.

If you miss a quiz, test, or final exam without an extremely good excuse, you will receive a grade of zero. If you think you have an excuse for missing a test, please discuss it with me, in advance if possible. If I judge that your excuse is reasonable, I will — depending on the circumstances — either give you a make-up test, or I will average your other grades so that the missing grade does not count against you. Since your two lowest quiz grades will be dropped, make-up quizzes will be extremely rare.

Also, I ask that in the absence of real necessity, you do not walk in and out of class during lecture.

### Disability Statement from the CTL

Disability Accommodations: If you are a student with a disability for which you may need academic accommodations in this course, you should self-identify, provide appropriate documentation of your disability, and register for services with Disability Services at the Center for Teaching and Learning (CTL). 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/academics/ctl/disability_services.aspx

### 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 Chris Wilson (Hobart '16). Math intern hours will be announced in class; there should be hours afternoon and evening, Sunday through Thursday.

The math intern can offer help on homework and lab problems, and can help you to understand the course material. Note, however, that the math intern is not a substitute for your professor! Use my office hours, or make an appointment, when you need help.

### Office Hours, E-mail, WWW

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. I will announce regular office hours and post them on my office door and on the course web page as soon as my schedule is determined, but note that your office visits are certainly not restricted to my regular office hours!

The home page for this course on the World Wide Web is located at http://math.hws.edu/eck/math131/index.html. That page will contain a weekly guide to the course and links to lab worksheets. You will want to bookmark the page! This courses does not use Canvas.

### Tentative Schedule

Here is a tentative schedule of readings for this course. We will try to keep approximately to this schedule, and we should cover all the major topics listed. However the actual reading assignments will be posted weekly on the course web page.

Dates Topics
Jan. 17 and 19 Section 5.1: Area under curves; Riemann sums.
Jan. 22, 24 and 26 Section 5.2: Definite integrals.
Section 5.3: The Fundamental Theorem of Calculus.
Jan. 29; Feb. 1 and 3 Section 5.4: Some properties of integrals.
Section 5.5: Integration by substitution.
Feb. 6, 8, and 10 Section 6.1: Applications to motion and change.
Section 6.2: Area between curves.
Feb. 13, 15, and 17 Section 6.3: Volumes by slicing.
Test on Wednesday, February 15.
Feb. 20, 22, and 24 Section 6.4: Volumes by shells.
Section 6.5: Arc length.
Feb. 27; Mar. 1 and 3 Section 6.8: Applications to logs and exponentials.
Section 7.1: Some basic integration techniques.
Mar. 6, 8, and 10 Section 7.2: Integration by parts.
Section 7.4: Trig substitution.
Spring Break, March 11–19
Mar. 20, 22, and 24 Section 7.8: Improper Integrals.
Test on Wednesday, March 22.
Mar. 27, 29, and 31 Section 7.9: Introduction to differential equations.
Section 8.1: Introduction to sequences and series.
Apr. 3, 5, and 7 Section 8.2: Infinite sequences.
Section 8.3: Infinite series.
Apr. 10, 12, and 14 Section 8.4: Convergence tests for series.
Section 8.5: More convergence tests.
Apr. 17, 19, and 21 Section 8.6: Alternating series.
Section 9.1: Taylor polynomials.
Test on Wednesday, April 19.
Apr. 24, 27, and 29 Section 9.2: Power series.
Section 9.3: Taylor and Maclaurin series.
May 1 Wrapping up the course!
May 9 Final Exam: Tuesday, May 9, 2017, 1:30 PM