Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley
Office Hours: Tuesdays 9:15-10:15, Wednesdays 4:00-5:00, Thursdays 4:30-5:30; additional availability by appointment

Textbook: APEX Calculus 2 by Gregory Hartman, ISBN: 1719263388;
available as a free pdf or interactive html site.

## Announcements

Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.

## Syllabus

Check the syllabus for all the important class policies (grades, attendance, etc.).

## Homework

There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.

• General information. A selection of problems will be assigned to be written up individually and turned in each week.
• These problems will be due Fridays at 5 pm.
• You may work with others but the writing should be done on your own.
• You should be enrolled automatically if you have an account with your MHC email address. Please let me know if you have any issues logging in.
• Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Also, please only write what you understand so that I know where to help, and please avoid using online forums like Math StackExchange, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
• Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
• You will be allowed to submit revisions of most problems for full credit each week.
• Your revisions will be due on Fridays at 5 pm.
Assignment Due
Homework 0 Sep 9
Homework 1 Sep 16
Homework 2 Sep 23
Homework 3 Sep 30
Homework 4 Oct 14
Homework 5 Oct 21
Exam 1 revisions Oct 24
Homework 6 Oct 28
Homework 7 Nov 4
Homework 8 Nov 18
Homework 9 Dec 2
Exam 2 revisions Dec 5
Homework 10 Dec 9

## Quizzes

There will be quizzes most weeks that will be given on Wednesdays. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Problems will always be related to the previous homework and class topics.

Quiz Date Material
Quiz 1 Sep 14 Homework 0, Week 1 definitions
Quiz 2 Sep 21 Homework 1, Week 2 theorems
Quiz 3 Sep 28 Homework 2, Week 3 theorems
Quiz 4 Oct 19 Homework 4
Quiz 5 Oct 26 Homework 5
Quiz 6 Nov 2 Homework 6
Quiz 7 Nov 30 Homework 8
Quiz 8 Dec 7 Homework 9

## Exams

There will be two midterms and a final exam. The dates for the exams are subject to change slightly.

Exam Date Format Material
Exam 1 Oct 7 in-class Homework 0 to 3
Exam 2 Nov 11 in-class Homework 4 to 7
Exam 3 Dec 16 – 20 self-scheduled cumulative

## Course plan

Our plan is to cover parts of chapters 6, 7, and 8 in the textbook. Below is a rough outline of what is to be covered week by week through the semester. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.

#### Chapter 8

##### Wednesday
• Topic: Section 8.1: Sequences. We discuss the definition of a sequence (an infinite list of numbers) and get some intuition for what a sequence is and what it means for it to converge or diverge through examples. We conclude with an example that involves computing limits of certain sequences using a special algebraic technique.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 0.
##### Friday
• Topic: Section 8.2: Infinite Series. We introduce the notion of infinite series (a sum of an infinite list of numbers) and focus our attention on a particular type of infinite series, geometric series, where the ratio of successive terms is constant.
• Class materials: Lecture notes, worksheet
• After class: Try today’s worksheet and work on Homework 1.

#### Chapter 8

##### Monday
• Topic: Section 8.2: Infinite Series, continued. We wrap up our discussion on geometric series and introduce the $$n$$th term test for divergence.
• Class materials: Lecture notes
• After class: Read Definition 8.2.3, Theorem 8.2.2, and Example 8.2.3 in Section 8.2.
##### Wednesday
• Topic: Section 8.3: Comparison Tests. We introduce $$p$$-series and begin formulating the comparison test, which allows us to make conclusions about convergence of series with complicated terms by comparing them with simpler versions whose behavior we understand.
• Class materials: Lecture notes, worksheet
• After class: Read the statements of the Comparison Test and Limit Comparison Test in Section 8.3.
##### Friday
• Topic: Section 8.3: Comparison Tests, continued. We give the statements of the Comparison Test and Limit Comparison Test and work on examples where we try to prove a given series converges or diverges using these two theorems and prior knowledge about $$p$$-series and geometric series. We put an emphasis on how to structure our solutions in order to communicate clearly.
• Class materials: Lecture notes, worksheet
• After class: Try the worksheet problems above and work on Homework 2.

#### Chapter 8

##### Monday
• Topic: Section 8.4: Ratio Test. We introduce a new test for convergence that says when the terms of a series have an asymptotic common ratio, the series converges under criteria similar to that for a geometric series.
• Class materials: Lecture notes, worksheet
• After class: Work on the worksheet above.
##### Wednesday
• Topic: Section 8.4: Ratio Test, continued. We learn how to use the ratio test and work on examples that get us to practice identifying which test is appropriate to use.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and read Section 8.5 up through Example 8.5.1.
##### Friday
• Topic: Section 8.5: Absolute Convergence and Alternating Series Tests. We introduce the notions of alternating series, absolute convergence, and conditional convergence. We also introduced our last convergence tests for the moment, the Absolute Convergence Test and the Alternating Series Test, giving a picture based proof of the latter.
• Class materials: Lecture notes, worksheet
• After class: Finish today’s worksheet and start on Homework 3

#### Chapters 5, 8

##### Monday
• Topic: Section 8.5: Absolute Convergence and Alternating Series Tests, continued. We get some more practice identifying absolute and conditional convergence and discuss the Riemann Rearrangement Theorem.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 5.1.1, 5.1.2, and 5.2.2.
##### Wednesday
• Topic: Chapter 5: Basics of Integration review. We discuss antiderivatives, definite integrals, the Fundamental Theorem of Calculus, and some basic algebraic properties of the definite integral.
• Class materials: Lecture notes, worksheet
• After class: Read Theorem 5.4.3 and Example 5.4.7.
##### Friday
• Topic: Chapter 5: Basics of Integration review. We discuss finding areas between two curves using the definite integral.
• Class materials: Lecture notes, worksheet
• After class: Begin studying for Exam 1 by working on old homework and worksheet problems, and similar problems from the textbook.

#### Chapter 5

##### Monday
• Topic: Section 6.1: Substitution. We introduce a new technique for finding antiderivatives. It comes from the chain rule for derivatives and is called $$u$$-substitution.
• Class materials: Lecture notes, worksheet
• After class: Study for Exam 1.
##### Wednesday
• Topic: Exam review.
• Class materials: worksheet, solutions
• After class: Study for Exam 1.
##### Friday
• Topic: Exam 1.
• After class: Enjoy your break!

#### Chapter 6

##### Monday
• Topic: Fall break, no class.
##### Wednesday
• Topic: Section 6.1: Substitution, continued. We study more examples using the method of substitution, now focusing on definite integrals and changing the limits of integration.
• Class materials: Lecture notes, worksheet
• After class: Read Section 6.2 up through Example 6.2.3.
##### Friday
• Topic: Section 6.2: Integration by parts. We learn a method for integration that is derived from the product rule for differentiation. The main difficulty in this method is making a choice for which part of the integrand should be $$u$$ and we use the mnemonic LIATE as a general rule of thumb.
• Class materials: Lecture notes, worksheet
• After class: Read Section 6.5 up through Example 6.5.1.

#### Chapter 6

##### Monday
• Topic: Section 6.5: Partial fraction decomposition. We discuss integrating rational functions using an algebraic technique that is akin to the reverse of adding fractions through finding common denominators.
• Class materials: Lecture notes, worksheet
• After class: Read Example 6.1.12 and Section 6.8 up through Example 6.8.1.
##### Wednesday
• Topic: Section 6.8: Improper integration. We wrap up our discussion of partial fractions, and how to handle terms that require the use of arctangent. We begin to discuss definite integrals over infinitely long integrals.
• Class materials: Lecture notes, worksheet
• After class: Read Definition 6.8.2 and Example 6.8.2.
##### Friday
• Topic: Section 6.8: Improper integration, continued. We conclude our discussion on improper integrals by introducing a new type: integrals where the integrand has a vertical asymptote.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.1.1, 7.1.2, and 7.1.3.

#### Chapter 7

##### Monday
• Topic: Section 7.1: Area Between Curves. We continue a past discussion on using integrals to find areas between curves. One focus will be on the idea that it’s often possible to set up these integrals with respect to either the $$x$$ variable or the $$y$$ variable.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.2.2 and 7.2.3.
##### Wednesday
• Topic: Section 7.2: Disk Method. We discuss volumes of three-dimensional solids formed by revolving a region in the $$xy$$-plane around an axis. Our approach will be to find the volumes of such solids by slicing them into thin disks and summing the volumes of these disks.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.2.2 and 7.2.3.
##### Friday
• Topic: Section 7.2: Washer Method. We expand our discussion of the disk method to include solids of revolution which have a hole in them. Instead of slicing the solids into thin disks, the solids will be sliced into washers, ie. disks with a hole in the middle.
• Class materials: Lecture notes, worksheet
• After class: Read Section 7.3 up through Example 7.3.1.

#### Chapter 7

##### Monday
• Topic: Section 7.3: Shell Method. We introduce a final method for finding volumes of revolution. This method involves slicing our region so that cylindrical shells are formed when a slice is revolved around the axis of revolution.
• Class materials: Lecture notes, worksheet
• After class: Read Section 7.5 up through Example 7.5.2.
##### Wednesday
• Topic: Section 7.5: Work. We discuss another application of integration through the physical notion of work. Our initial focus will be on problems involving lifting a mass over a distance and compressing springs.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 7.5.5 and 7.5.6.
##### Friday
• Topic: Section 7.5: Work, continued. We do some more examples of computing work that involving pumping fluids from containers of various shapes.
• Class materials: Lecture notes, worksheet
• After class: Read Section 8.3 up through Example 8.3.1.

#### Chapter 8

##### Monday
• Topic: Section 8.3: Integral Test. We discuss how improper integrals are used to determine whether certain series converge.
• Class materials: Lecture notes, worksheet
• After class: Study old homework problems, worksheets, and lecture examples!
##### Wednesday
• Topic: Review. We spend the day working on review problems for Exam 2.
• Class materials: worksheet, solutions
• After class: Continue your review and study by working on new problems from the textbook.
##### Friday
• Topic: Exam 2.
• After class: Review the ratio test and read Example 8.6.2.

#### Chapter 8

##### Monday
• Topic: Section 8.6: Power series. We introduce examples of series whose terms consist of powers of a variable $$x$$, as in polynomials. We focus on determining the domain of values of $$x$$ for which series converge; a notion called the interval of convergence.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 8.7.
##### Wednesday
• Topic: Section 8.7: Taylor polynomials. We introduce a generalization of linear (or tangent line) approximations of functions. We discuss how to find polynomials of arbitrary degree whose derivatives at a point match those of a given function.
• Class materials: Lecture notes
• After class: Finish Homework 8.
##### Friday
• Topic: Section 8.7: Taylor polynomials, continued. We work on a worksheet of material from last time and begin to discuss the possible error in using Taylor polynomials to approximate a function.
• Class materials: Lecture notes, worksheet
• After class: Read Examples 8.7.2 and 8.7.3.

#### Chapter 8

##### Monday
• Topic: Early break, no class.
• After class: Enjoy your break!
##### Wednesday
• Topic: November break, no class.
##### Friday
• Topic: November break, no class.

#### Chapter 8

##### Monday
• Topic: Section 8.7: Taylor polynomials, continued. We practice using Taylor’s theorem.
• Class materials: Lecture notes, worksheet
• After class: Read the first two pages of Section 8.8.
##### Wednesday
• Topic: Section 8.8: Taylor series. We discuss power series that come about by considering Taylor polynomials with infinitely many terms.
• Class materials: Lecture notes, worksheet
• After class: Read Theorem 8.8.2 and Examples 8.8.6 and 8.8.7.
##### Friday
• Topic: Section 8.8: Taylor series, continued. We discuss deriving new Taylor series using known ones, as well as how to use Taylor series to approximate integrals and compute limits that would otherwise be difficult to work with.
• Class materials: Lecture notes, worksheet
• After class: Work on Homework 10.

#### Applications

##### Monday
• Topic: Section 8.8: Taylor series, continued. We get some more practice working with Taylor series.
• Class materials: Lecture notes, worksheet
• After class: Study for Wednesday’s quiz.
##### Wednesday
• Topic: Application: Normal distribution. We discuss how to compute probabilities that involve data that is normally distributed (falls under a bell curve).
• Class materials: Lecture notes, worksheet
• After class: Finish Homework 10
##### Friday
• Topic: Application: Geometric distribution. We discuss the geometric distribution a model for understanding a random experiment that consists of repeated trials until a desired outcome occurs.
• Class materials: Lecture notes, worksheet
• After class: Study for Exam 3 by reviewing homework and quizzes since Exam 2.

#### Wrap up

##### Monday
• Topic: Review. We spend the day working on the review sheet linked below.
• Class materials: worksheet, solutions
• After class: Study for Exam 3. Enjoy your winter break! Keep in touch!

## Getting help

Here are a few ways to get help:

• Office Hours: Tuesdays 9:15-10:15, Wednesdays 4:00-5:00, Thursdays 4:30-5:30; additional availability by appointment
• TA help: We will have class TA’s who will be available for help on weeknight evenings. The schedule is as follows:
• Sundays, 7:00 - 9:00 pm, in Clapp 402
• Mondays, 7:00 - 9:00 pm, in Clapp 402
• Tuesdays, 7:00 - 9:00 pm, in Clapp 402
• Wednesdays, 7:00 - 9:00 pm, in Clapp 402 with Hannah
• Thursdays, 7:00 - 9:00 pm, in Clapp 422 with Jenny
• Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what you know on homework in order to get good feedback.
• Piazza: I’ve set up an online forum for asking and answering questions about class material, including homework problems. My plan is to check the forum every day and answer questions as they come up, but I hope everyone in the class will pitch in and answer others’ questions when possible.

## Resources

• Wolfram Alpha: a useful way to check your answers on computations. It can do algebra and calculus, among other things, and it understands a mix of English and symbols.
• Desmos: a nice website for graphing functions.