Spring 2006 - Math 16B - Section 1 Webpage

LECTURER: Don Sarason

OFFICE: 779 Evans Hall

OFFICE PHONE: 642-3521

HOURS: M 10:30-11:30am & 4:30-5:30pm, Tu 4:00-5:00pm, F 9:30-11:30am, or by appointment

E-MAIL: In a large lecture course such as Math 16B, it is not possible for the lecturer to respond to email inquires from individual students. If you need to communicate with Mr. Sarason about the course, see him after lecture, come to his office (preferably during office hours or by appointment), or use the phone. Except in dire emergencies, email messages will not be answered.

TEXTBOOK: Calculus & Its Applications, Tenth Edition, Goldstein-Lay-Schneider, Pearson Education, Inc., 2004. The book is available both as a single volume, which contains the material for both Math 16A and Math 16B, or as two half-volumes, one with the 16A material (Chapters 0-6), one with the Math 16B material (Chapters 7-12).

PREREQUISITES: The only formal prerequisite is Math 16A or the equivalent. In practice, however, you will be at a disadvantage if you lack a reasonably good mastery of basic algebra. If your algebra is rusty, or if your earlier calculus course was not taken recently, you had best review the relevant material early in the semester. If you are not confident about your mastery of the 16A material, you may want to use the long version of the textbook.

EXAMS: There will be midterm exams in lecture on Wednesday, February 15, on Chapter 7, and on Wednesday, April 5, on Chapters 8-10. The final exam is scheduled for Tuesday, May 16, 12:30-3:30 p.m. (Exam Group 11). Do not enroll in the course if you have a conflict; there will be no make-up exams.

GRADING SYSTEM: There are two grading options. The one yielding the higher grade will be applied on an individual basis.

Option 1: Section 20%, each midterm 20%, final 40%
Option 2: Section 20%, higher midterm 20%, final 60%

The section grade will be based 50% on homework and 50% on quizzes.

Grade points will be assigned for each exam and for section. Their weighted average will be converted to a letter grade according to the following table.

[4.15, infinity)A+
[3.75, 4.15)A
[3.50, 3.75)A-
[3.25, 3.50)B+
[2.75, 3.25)B
[2.50, 2.75)B-
[2.25, 2.50)C+
[1.75, 2.25)C
[1.50, 1.75)C-
[1.25, 1.50)D+
[0.75, 1.25)D
[0.50, 0.75)D-
[0.00, 0.50)F

INCOMPLETE GRADES: According to University regulations, you may receive an I grade only if "your work in a course has been of passing quality but is incomplete for reasons beyond your control."

HOMEWORK: Homework will be due in section each Tuesday, except for the first week of the semester and the weeks of the midterm exams. Late papers will not be accepted. (Your GSI will define what "late" means.)

Unfortunately, the Math Department, for budgetary reasons, is unable to hire readers to grade homework. Your GSI, however, will look over your homework carefully enough to determine whether you have made a serious effort at doing the exercises.

LECTURES: The lectures will follow the textbook but not parrot it. Their purpose is to introduce and illustrate the main ideas. There is too much material in the course for the lectures to cover everything, so, even if you attend the lectures, expect to depend on the textbook for much of what you learn. You will follow the lectures better if you go over the relevant parts of the textbook beforehand.

SYMBOL CRUNCHING: Calculus, as you have experienced, involves a good dose of symbolism. It is important that you acquire a facility with the various algebraic manipulations that come up (adding fractions, differentiating products and quotients, applying the chain rule, and the like). That is done through practice. It is important, also, to remember that reliance on symbols is designed to aid thought, not to eliminate it. When symbol crunching, do not lose sight of what the symbols mean. This will help you to avoid silly errors leading to absurd conclusions.

A PLEA: I hope you will not approach Math 16B as a cookbook course where the goal is to memorize recipes for problem solving. Math may be hard to understand, but, with effort, you will be able to gain understanding. If you can grasp the basic ideas behind the jungle of symbolism, the need for mindless memorization will be greatly reduced; you will have internalized a context that will enhance retention.

(Numbers refer to textbook sections.)

(Chap. 7)
1010.5-10.711.1No Lecture
(Chaps. 8-10)
1211.211.311.3, 11.4
Course evaluation


The purpose of the homework is to test your understanding of the course material. You should study the relevant parts of the textbook BEFORE attempting the homework exercises. Mr. Sarason and your GSI will be happy provide guidance on how to attack homework exercises.

Homework assignments should be handed in on the due date in your discussion section. Some of the assignments are quite long. You will not want to wait until the night before they are due to start them.

Besides the assigned homework exercises, some additional recommended exercises are listed below in the right column. Doing them will help you to solidify your understanding of the course material.

Assignment #Due TextbookAssigned Additional
Section ExercisesRecommended
11/247.1 2, 8, 12, 161, 7, 13, 23-26
7.22, 5, 6, 17 1
21/317.2 23, 2625,31
7.32, 8, 11, 12, 23, 31 1, 7, 16, 24, 30
32/77.4 2, 5, 6, 8, 10, 14, 211, 13, 23
7.59 10, 11
7.72, 8, 11, 12, 14 1, 7, 13
42/218.1 2, 5, 8, 181, 13
8.22, 6, 11, 21, 22, 38 1, 5, 37
8.34, 11, 12, 28, 35, 39, 40 11, 24, 36
52/288.4 3, 12, 27, 31, 32, 394, 10, 11, 18, 37
9.12, 3, 6, 29, 30, 47 1, 14, 31
9.22, 8, 15, 23, 24 1, 7, 16, 30, 32
63/79.3 2, 5, 121, 4, 19
9.45, 26, 27 6, 19, 25
9.52, 8, 12 1, 7
73/149.6 17, 29, 35, 481, 9, 10, 12, 13, 21
10.12, 11, 14, 15 1, 9, 10, 12, 13, 21
10.22, 9, 26, 35 1, 14, 25, 31
83/2110.3 11, 21, 22, 33, 34, 4012, 19, 26, 31
10.45, 6, 8, 9 7, 13
10.65, 9 1, 25
94/1111.1 5, 6, 10, 17, 238, 14, 19
104/1811.2 5, 15, 161, 24
11.33, 4, 15, 23, 27, 29 1, 8, 16, 25
11.49, 14
114/2511.5 15, 18, 23, 247, 22
12.12, 4, 5, 9 1, 6
12.26, 15, 20, 27, 28 1, 7, 19, 25
125/212.3 2, 9, 11, 141, 19
12.45, 15, 23, 27 1
12.52, 5, 8, 15 1, 7, 13





DATE:Wednesday, February 15
TIME:12:10-1:00 p.m.
ROOM:105 Northgate Hall: Students of Arun Sharma (Sections 101, 106, 113, 114)
 2050 Valley LSB: Everyone else


1. Bluebooks will not be needed. There will be space on the exam paper for answers.

2. Know your GSI's name. Bring your SID card to the exam.

3. Calculators will not be permitted.

4. The exam will be closed book except for a crib sheet of size 8-and-1/2- by-11 inches. Both sides may be used.

5. Review exercises will be posted on February 7, and hard copies will be distributed in section the same day. Answers to the review exercises will be posted on February 13.

6. Lecture on February 13, and section on February 14, will be devoted to review.

7. When you have finished the exam, you should turn in your paper to your own GSI.


Midterm 1 - Solutions


The examination was taken by 370 students. The median score was 60/65. A perfect score of 65/65 was achieved by 89 students (nearly 25% of the class).

The grading curve can be read off from the table below. To figure the number of grade points earned by the score x, locate the interval containing x in the left column, then plug x into the formula in the same row of the right column.

CONFESSIONAL. Your humble professor apologizes for the errors in the answers to the review exercises. He will try to be more accurate in the future.


[0, 20]0
[20, 60].1(x-20)
[60, 65]4.0 + .05(x-60)


It was realized after the exams had been graded and returned that there is a serious flaw in Question 4. Fortunately, the flaw is likely not to require the revising of many scores, perhaps of none at all. However, if you wish to have your Question 4 regraded you are welcome to do so. Just return your exam paper to your GSI, who will forward it to Mr. Sarason

Question 4 asked you to maximize the length of the diagonal of a package whose sides measure x inches, y inches, z inches, under the condition that 2x+2y+z does not exceed 84. The problem is that, assuming x, y, z are all positive, there is no such maximum. The diagonal of such a package can never be as long as 84 inches, but it can be arbitrarily close to 84 (which happens when x and y are small and z is nearly equal to 84).

People (like your professor) who correctly applied Lagrange's method to optimize the length of the diagonal under the constraint 2x+2y+z = 84 actually found the dimensions of the package with the shortest diagonal under that constraint. They received full credit. Those who used the same method but had errors along the way had points deducted for the errors, a few point for minor errors, more point for more serious errors.

My apologies for the confusion. I am searching for a moral here. The best I could come up with is: Never go pheasant hunting with Mr. Sarason.


The examination is scheduled for Wednesday, April 5, 12:10-1:00 p.m. It will cover the material in Chapters 8, 9, 10. Room assignments are the same as for Midterm 1: Arun Sharma's students will take the exam in 105 Northgate Hall, everyone else in 2050 Valley LSB.

The basic ground rules are the same as for Midterm 1. In particular, bluebooks are not needed, calculators are not allowed, a crib sheet, size 8-and-1/2-by-11 inches, is permitted (both sides may be used).

Review exercises will be posted and distributed during the week before spring break. Answers will be posted on the Monday before the exam.


Midterm 2 - Solutions


The examination was taken by 364 students. The median score was 54/70. A perfect score of 70/70 was achieved by 9 students.

The grading curve can be read off from the table below. To figure the number of grade points earned by the score x, locate the interval containing x in the left column, then plug x into the formula in the same row of the right column.


[20,40].1(x - 20)
[40,50]2.0 + .05(x - 40)
[50,65]2.5 + .1(x - 50)
[65,70]4.0 + .05(x - 65)


DATE:Tuesday, May 16
TIME:12:30-3:30 p.m.
ROOMS:230 Hearst Gym - Students of Zak Mesyan and Arun Sharma
 237 Hearst Gym - Students of Tom Dorsey and Dave Penneys


1. Blue books will not be needed.

2. Scratch paper will be provided.

3. Calculators will not be permitted.

4. The exam will be closed book except for two crib sheets, size eight-and- one-half-by-eleven inches. Both sides of each sheet may be used.

5. Students should bring their SID cards to the exam.

6. Review exercises have been posted. Hard copies will be available in lecture and in sections. Answers to the review exercises will be posted on Friday, May 5. The review exercises give a good indication of what the exam will emphasize. (Some of the review exercises require a calculator. On the exam, all numerical calculations will be doable by hand.) You may also find it helpful, in preparing for the final, to look over the midterm exams and their review exercises.

You can expect many of the exam questions to involve integration. If you are shaky using the methods of substitution and integration by parts, it would be good to spend some time working examples before the exam. Try the Supplementary Exercises in Chapter 9 of the textbook.

7. The lectures on May 3, 5 and 8 will be reviews. (The May 3 session will be abbreviated to leave time for the course evaluation.)

8. Sarason's office hours after the last day of classes (May 9) will be on May 11, 12 and 15, 10:00-12-00 each day.

9. Final exam solutions will be posted within a day or two after the exam. It is hoped that the course grades will be reported by Friday, May 19. Students who wish to inspect their final exams should contact Sarason about it.

10. Guard against losing points on the exam by silly errors in arithmetic and algebra. Take your time and check your work as you go along. Answer first the questions about which you are most confident.


Final - Solutions