Mathematics 21-122
Fall 2003

Course Outline: In his book Il Saggiatore (The Analyzer) Galileo wrote, "The book of the universe is written in the language of mathematics... without the help of which it is impossible to conceive a single word of it, and without which one wanders in vain through a dark labyrinth." In Math 21-122 we will continue to gain fluency in the language of mathematics by studying the following three topics in single variable calculus:

• Methods and Applications of Integration
• Differential Equations
• Infinite Polynomials

You should already be familiar with the definite integral, its interpretation as the limit of Riemann sums, its calculation via antiderivatives (and u-substitution), and its application to the calculation of areas and volumes. We will begin the course by learning other sophisticated techniques of integration which we will apply to the calculation of work, surface area, probability and other useful quantities.

We will then turn our attention to equations which relate a function to its rate of change and possibly higher order derivatives. Such differential equations are common in physics, chemistry and the life sciences. For example, under ideal conditions it is reasonable to assume that the rate of change of a population with respect to time is proportional to the size of the population itself. We will ponder similar models and develop techniques to recover the underlying function from the differential equation. Techniques to analyze the behavior of the underlying function without actually finding a formula for it will also be presented.

Lastly, we will show how one can replace complicated functions such as exponentials or sines with large polynomials. While the resulting expression will not be exact, it can frequently be made to be as exact as necessary by choosing a large enough polynomial. For example, sin(x) is very close to x whenever x is small. Even better approximations are x - x³/6 and x - x³/6 + x⁵/125. We will see how to continue getting better and better such approximations.

Classes and Recitation Sessions: There are two large sections of math 21-122, one meeting mwf from 8:30 to 9:20 in Baker A51 and the other meeting mwf from 9:30 to 10:20 in Wean 7500. Your Teaching Assistant (TA) will also hold two weekly recitation sessions. I strongly encourage you to attend these recitation sessions as they are an integral part of the course and will be devoted primarily to working problems and amplifying the material. Click here for more information about your TA and the recitation sessions.

Help: In addition to class, recitation sessions, and office hours, the University operates a walk-in Peer Tutoring Center in the Mudge Library and the Donner Reading Room on Sunday-Thursday evenings from 8:00 to 11:00pm. Individualized tutoring and other help options are also available through Academic Development.

Homework: Homework exercises are an integral part of the course. It is difficult to understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Please think about the problems posed, your strategies, the meaning of your computations, and the answers you get.

Homework is due at the beginning of the class period following the one in which it has been assigned. Although discussion of the homework with your peers is encouraged, copying any part of another person's homework is not permitted. As a courtesy to the TAs late homework will generally not be accepted. If extreme circumstances cause an assignment to be late, the TA can determine whether to accept the homework. Homework solutions will be posted on the website each Friday afternoon.

Text: Calculus, Early Transcendentals; 5th edition, by James Stewart. It is published by Brooks/Cole (2003) and is available at the bookstore.

Exams: There will be three in-class midterms and a cumulative final exam. The dates of the midterm exams are as follows:

Exam 1: Tuesday, March 6

Exam 2: Tuesday, April 17

Exam 3: Tuesday, April 17

Grading: Your course grade will be determined as follows:

Each of the two high midterm scores: 25%
The low midterm score: 10%
Homework: 15%
Final Exam: 25%

Calculators: We encourage you to not rely too heavily on a graphing calculator as you work through your homework problems. Use the calculator to check your graphs if you must. That said, the use of a quality calculator can prove very helpful in understanding a good number of topics in the course from integration and successive approximation to differential equations. Calculators will not be allowed during exams unless we explicitly state otherwise.

Week by week schedule (tentative):

Wednesday, January 31: Course Orientation, 8 a.m., Sc C

Week 1 (Feb 5-9):

• Section 6.1 Areas between Curves
• Section 6.2 Volumes
• Section 6.3 Volumes by Cylindrical Shells

Week 2 (Feb 12-16):

• Section 7.1 Integration by Parts
• Section 7.2 Trigonometric Integrals
• Section 7.3 Trigonometric Substitution

Week 3 (Feb 20-23):

• Section 7.4 Integration of Rational Functions by Partial Fractions
• Section 7.7 Approximate Integration

Week 4 (Feb 26-Mar 2):

• Section 7.8 Improper Integrals
• Section 8.1 Arc Length
• Section 8.5 Continuous Probability

First Mid-Term March 6

Week 5 (Mar 5-9):

• Section 9.1 Modeling with Differential Equations
• Section 9.2 Direction Fields and Euler's Method

Week 6 (Mar 12-16):

• Section 9.3 Separable Equations
• Section 9.4 Exponential Growth and Decay
• Section 9.5 The Logistic Equation

Week 7 (Mar 19-23):

• Section 9.7 Predator-Prey Systems
• Section 11.1 Sequences
• Section 11.2 Series

Week 8 (April 2-6):

• Section 11.3 The Integral Test and Estimates of Sums
• Section 11.4 The Comparison Tests
• Section 11.5 Alternating Series

Week 9 (April 9-13):

• Section 11.6 Absolute Convergence and the Ratio and Root Tests
• Section 11.8 Power Series
• Section 11.9 Representations of Functions as Power Series

Second Mid-Term April 17

Week 10 (April 16-20):

• Section 11.10 Taylor and Maclaurin Series
• Section 11.11 The Binomial Series

Week 11 (April 23-27):

• Section 11.12 Applications of Taylor Polynomials
• Section 17.1 Second-Order Linear Equations
• Section 17.2 Nonhomogeneous Linear Equations

Week 12 (April 30 - May 4):

• Section 17.3 Applications of Second-Order Differential Equations
• Section 17.4 Series Solutions

Reading Period (May 5 - 16)

Final Examination, Scheduled by the Registrar

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