Calculus II - Mathematics 142, Honors Section 2

Frank Thorne - Fall 2014
University of South Carolina

Welcome to Math 142! Calculus is a beautiful, important, and fascinating subject.

It is also challenging. Please plan on a lot of hard work; we are here to help you succeed.

Instructional Staff :

• Instructor: Frank Thorne, LeConte 400G, thorne [at] math.sc.edu.
Office Hours: Wednesdays 3:30-5:00 and Fridays 10:30-12:00.

(Note: These might occasionally be rescheduled when we have meetings.)

• Supplemental Instructor: TBA.
The supplemental instructor is an undergraduate who has done very well in Math 142 in the past. She will attend all the lectures and is here to help.
In place of office hours she holds SI sessions (see below) that you can go to.
(Note: I am not sure yet whether or not the honors section will have an SI.)

Learning outcomes:

"Education is what survives when what has been learned has been forgotten." -- B.F. Skinner

Successful students will:

• Have ample opportunities to practice and use algebra, trigonometry, and first semester calculus, increasing their skills in these areas. These topics are absolute prerequisities for Calculus II, but at the same time the course will be directed towards students whose background skills may be slightly rusty and in need of practice.
• Understand quantitative data and its presentation. In calculus, this usually takes the shape of a function. Functions will be presented in terms of equations, or in terms of graphs, or in English text, and students will be required to seamlessly translate between these.
• Develop the ability to explain their work clearly. As with the previous bullet point, this will involve a combination of equations, graphs, and English text. Students are expected to learn to write well.
• Understand what definitions and theorems are. The student will be able to give precise definitions as well as informal explanations and will be able to explain the correspondence.
• Tackle problems that require more than one step to solve, or whose solution is not obvious. Typically this means that you try something, and if it doesn't work you try something else.
The above is true of any math class. Typically, all of this is best learned in some specific context. Therefore, successful students will also:
• Master techniques for evaluating integrals and determining the convergence and divergence of series.
• Learn a variety of applications of integration, including areas, volumes, and arc lengths.
• Further study the relationship between calculus, algebra, and geometry, through the study of parametric curves, polar coordinates, and various area and volume problems. A key point is understanding alternative ways in which functions may be represented.
This Honors section has the additional component that you will study Thompson's Calculus Made Easy and write short essays on it. By doing this, successful students will:
• Compare and contrast two approaches to calculus. This will encourage you to think critically about each book: what are its strengths and shortcomings? What are its goals in its presentation of calculus, and what shortcuts is the author willing to take?
• By writing short essays, develop the ability to explain mathematical topics in plain English.

Warning. There is a ton of homework. It will be collected and graded. That is because this is the most effective way for you to learn. But:

• The grading scale for the class is: A=90+, B=78+, C=64+, D=50+. There is extra credit on each of the homeworks, and there will also be one extra credit question on the final exam, so scores above 100% are possible.
• All of the assignments for this course are posted below, subject to only minor changes. Some exam questions will come verbatim from these assignments (including the `additional problems'). So everything is predictable, and you can look before you leap.
• If the numerical average on any midterm exam is at least 80%, then we will have a pizza lunch (in class) the day I hand back the exams.

This class doesn't cover exactly the same material as most sections of Math 142. Compared to a recent version of the course:

• The treatment of Calculus Made Easy is unique to this section of the course.
• We will de-emphasize some of the more tedious aspects of the course. For example the book treats partial fractions as a long algorithm, broken up into many cases, and it doesn't explain why any of them actually work. We will do fewer of the cases, and focus both on how to do them and why they work.
• We will add the sections on arc length (8.1) and parametric curves (10.1-10.2). The purpose is to give you a greater understanding of the flexibility of calculus and its relation to geometry, and to introduce you to some subjects which you are likely to encounter again later.
• We will emphasize explicit approximation for convergence tests. We will strongly keep in mind that ''converges'' means ''equals a number''.
• We will de-emphasize the problem-solving aspect of convergence tests: whenever you are asked to determine whether or not a series converges, a suitable test will be suggested to you. Also, a list of convergence tests will be provided for you on the exams, which means that you won't have to memorize it.

Texts : James Stewart, Calculus, Early Transcendentals, 6th edition. We will use a custom edition of this book, available in the campus bookstore. However, the Calculus II portion of the book is identical to the non-custom version. You might consider buying your book used on Amazon or elsewhere online. You might also buy a used version from an upperclassman.

We will also read Calculus Made Easy by Silvanus Thompson, as revised by Martin Gardner. To my knowledge this book is not available at the campus bookstore -- you can buy it here among many other places.

Meeting schedule :

• Lectures: T-Th, 11:40-12:55, LeConte 121.
• Maple Lab: M, 12:00-12:50, TBD.
• Discussion: W, 12:00-12:50, Gambrell 104.
Exam schedule :
• Midterm Exam 1: September 25 (all midterms in class)
• Midterm Exam 2: October 21
• Midterm Exam 3: November 20
• Final Exam: Tuesday, December 9, time TBA.
All exams will be held in the usual classroom, during class meetings (except for the final).

Homework : The homework assignments have four components:

• The required problems are, well, (you guys are Honors students...)
• The additional problems are not to be turned in, but represent good practice problems, especially for the midterm exams.
• Every week one question will ask you to address some topic in Thompson's book. These should be answered in plain English, with formulas or diagrams to match. What you read in either of your textbooks is a good model for your answer. One handwritten page should definitely be enough. Shorter answers may suffice, especially if you put some thought into your answer before you begin to write.
• Each homework also includes one or more bonus problems. These are extra credit, and are usually very interesting. I encourage you to attempt them!

You will be graded both on correctness and on quality of exposition. The standard is that someone who doesn't know the answer should be able to easily follow your work. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

The grade cutoffs are: A for 90%, B+ for 85%, B for 78%, C+ for 72%, C for 64%, and D for 50%.

 % of grade Three in-class exams: 15% x 3 Final exam: 30% Maple lab assignments: 10% Homework: 15%

The Fine Print:

Make-up policy :

If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise, makeups will only be given in case of emergency. Late homework will generally not be accepted, except in case of illness or family emergency, or unless you clear it with me in advance.

Calculators :

Calculators will not be allowed for the exams.

Here is a very, very, very powerful calculator. Do not use it to solve the homework problems, but it is a useful way to check your work.

Attendance : Skipping class is unwise, but no attendance policy will be enforced. You are responsible for all the material covered in all lectures, discussion sections, and labs.

Supplemental instruction :

TBA runs the supplemental instruction sessions. This is a valuable resource and you are strongly encouraged to take advantage of it. Please go to ask questions and meet other students. It is a particularly good place to work on your homework.

These are held [Dates and Times TBA].

There is also free drop-in tutoring available for all 100-level math courses. The Math Tutoring Center in LC 105 is open (free, drop-in) Monday-Thursday from 11am until 4pm. The Student Success Center also offers peer tutoring and online tutoring.

Other help resources : Math lab, Private tutors .

Schedule of lectures, homeworks, and exams:

Homeworks are not always due on the same days of the week. This could be a bit annoying (my apologies), but it was done to make sure no one homework is excessively long.

At this point, homeworks are mostly finished but slightly subject to change, except where noted. Also, each homework will have one question on Thompson's book and most of these haven't been added yet.

The schedule is for lecture (T/Th) only. Wednesday discussions will be dedicated to student questions about lectures and homework problems, further examples, discussion of Calculus Made Easy, and exam review.

The schedule:

• 8/21: Review I: Functions and their derivatives

• 8/26: Review II: Integrals and the fundamental theorem of calculus
• 8/28: Integration by substitution (Ch. 5.5)

Homework 1, due Friday, August 29. (Warning! It's not short.)

Sample answer for the essay question.

• 9/2: Integration by parts (7.1)
• 9/4: Trigonometric substitution (7.3)

Homework 2, due Friday, September 6.

• 9/9: Partial fractions, etc. (7.4, 7.5)
• 9/11: Improper integration (7.8)

Homework 3, due Friday, September 12.

• 9/16: Areas between curves (6.1)
• 9/18: Volumes by discs (6.2)

Homework 4, due Friday, September 19.

• 9/23: Arc length (8.1)

Homework 5, due Wednesday, September 24.

• 9/25: Midterm Exam 1.

• 9/30: Parametric curves (10.1)
• 10/2: The calculus of parametric curves (10.2)

Homework 6, due Friday, October 3.

• 10/7: Polar coordinates (10.3)

[10/9: Last day to withdraw without a grade of WF being recorded.]

• 10/9: Areas and lengths in polar coordinates (10.4)

Homework 7, due Friday, October 10.

• 10/14: Infinite sequences (11.1)
• 10/16: Infinite series (11.2)

• 10/21: Midterm Exam 2.

Homework 8, due Tuesday, October 21.

• 10/23 [Fall Break.]

• 10/28: Convergence testing: the integral test (11.3)
• 10/30: Convergence testing: the comparison test (11.4)

Homework 9, due Monday, November 3.

• 11/4: No class (Election Day)
• 11/6: Convergence testing: alternating series (11.5)

Homework 10, due Monday, November 10.

Remember that this list of convergence tests will be provided to you on the exam -- so you should also refer to it when you do the homework.

• 11/11 (*): Convergence testing: the ratio test (11.6)
• 11/13 (*): Power series (11.8)

Homework 11, due Friday, November 14.

• 11/18: Taylor and Maclaurin Series (11.10)

Homework 12, due Tuesday, November 18.

• 11/20: Midterm Exam 3.

• 11/25: Some entertaining applications of calculus.
• 11.28: No class (Happy Thanksgiving!)

• 12/3: Taylor and Maclaurin Series (cont.) (11.10)
• 12/5: Convergence and Approximation Problems for Taylor and Maclaurin Series (11.10, 11.6, 11.8)

Homework 13, due Friday, December 5.

• Monday, December 8 at 7:00 p.m.: Review Session. Review sheet.

• 12/9 (Tuesday): FINAL EXAM , 12:30 p.m.