Introduction to Calculus

Course Head: John Mackey, Science Center 331, 496-5211, jfm@math.harvard.edu

Introduction and Goals: If you draw the graph of a function and then pick a point on the graph, you should be able to draw the line tangent to the graph at that point. You can then estimate the slope of this tangent line by taking the quotient of the rise over the run. The beautiful notion is that for most functions given by a formula one can find another formula (called the derivative) which will enable you to find the exact value of the slope at any given point on the function. This may not seem to be such a big deal at first, but consider the fact that at points where a smooth function reaches a maximum or minimum value the slope of the tangent line must be 0. Thus, one can locate the exact maximum and minimum values achieved by a smooth function by using its derivative to locate the places where the function has a flat tangent line. One can probably imagine that this is an important idea. For example, an economist may wish to determine the number of units to produce in order to maximize profit.

In Math 1a we will begin by introducing tangent lines and their possible uses. We will then learn about the tools needed to calculate the derivative. Once we have a thorough understanding of the derivative and how to calculate it, we will move on to explore its applications. Finally, we will talk about a method for calculating the exact area between a given function and the x-axis between any two fixed points. This method (called integration) is surprisingly linked to the derivative. When you leave Math 1a, you will carry with you concepts and ideas from calculus that can be applied later, both in mathematics and in other fields.

Necessary Background: Some of you have had calculus before. However, those of you who haven't need not be alarmed: in the past students without a calculus background have done as well as those with it. Doing well in Math 1a requires a solid background in precalculus, as demonstrated by a score of 18 or more on part 1 of the Harvard Math Placement Test.

Classes and Problem Sessions: Math 1a is taught in sections which meet three hours per week. Each section has a Section Leader who will teach the class and help with the design of exams. Each section also has a Course Assistant (CA) who will grade homework, work in the Math Question Center and hold a weekly problem session. I strongly encourage you to attend these problem sessions as they are an integral part of the course and will be devoted primarily to working problems and amplifying the material presented in class. A schedule of problem sessions will be posted on the course website.

As the Course Head, I will work with the Section Leaders and CAs to ensure uniformity and excellence among the sections. If you have any comments or input regarding Math 1a you can email the calculus ombudsman (tenured faculty member not involved with Math 1a) at this address.

Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates a Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise. It is intended to supplement the office hours held by your Section Leader.

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 CAs late homework will generally not be accepted. If circumstances cause an assignment to be late, please contact me and I will determine whether to accept the homework. Homework solutions will be posted on the website each Tuesday afternoon.

Text:  Single Variable Calculus: Concepts and Contexts, 2nd edition by James Stewart. Available at the COOP.

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

Exam 1: Monday, March 4

Exam 2: Monday, April 15

Exam 1: 25%
Exam 2: 25%
Homework: 20%
Final Exam: 30%

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 limits and successive approximation to graphing. Calculators will not be allowed during exams.

Week by week schedule (tentative):

Week 1 (February 4-8):

• Section 2.1 The Tangent and Velocity Problems
• Section 2.2 The Limit of a Function
• Section 2.3 Calculating Limits Using the Limit Laws

Week 2 (February 11-15):

• Section 2.4 Continuity
• Section 2.5 Limits Involving Infinity
• Section 2.6 Tangents, Velocities, and Other Rates of Change

Week 3 (February 20-22):

• Section 2.7 Derivatives
• Section 2.8 The Derivative as a Function

Week 4 (February 25 - March 1):

• Section 2.9 Linear Approximations
• Section 2.10 What Does f' Say about f?
• Section 3.1 Derivatives of Polynomials and Exponential Functions

First Mid-Term
Monday, March 4 from 5 to 7pm in SC D

Week 5 (March 4-8):

• In-class Review for Midterm 1
• Section 3.2 The Product and Quotient Rules
• Section 3.3 Rates of Change in the Natural and Social Sciences

Week 6 (March 11-15):

• Section 3.4 Derivatives of Trigonometric Functions
• Section 3.5 The Chain Rule
• Section 3.6 Implicit Differentiation

Week 7 (March 18-22):

• Section 3.7 Derivatives of Logarithmic Functions
• Section 3.8 Linear Approximations and Differentials
• Section 4.1 Related Rates

Week 8 (April 1-5):

• Section 4.2 Maximum and Minimum Values
• Section 4.3 Derivatives and the Shapes of Curves
• Section 4.5 Indeterminate Forms and l'Hospital's Rule

Week 9 (April 8-12):

• Section 4.6 Optimization Problems
• More Optimization Problems
• Section 4.7 Applications to Economics

Second Mid-Term
Monday, April 15 from 5 to 7pm in SC D

Week 10 (April 15-19):

• In-class Review for Second Midterm
• Section 4.8 Newton's Method
• Section 4.9 Antiderivatives

Week 11 (April 22-26):

• Section 5.1 Areas and Distances
• Section 5.2 The Definite Integral
• Section 5.3 Evaluating Definite Integrals

Week 12 (April 29 - May 3):

• Section 5.4 The Fundamental Theorem of Calculus
• Section 5.5 The Substitution Rule
• In-class review for final