Course Meeting Times
Lectures: 3 sessions / week, 1 hour / session
Recitations: 2 sessions / week, 1 hour / session
Textbook
Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.
(Vol. 2 will be needed for those who wish to continue on to 18.024 Multivariable calculus with theory.)
Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.
Prerequisites
We will assume a working knowledge of Calculus I (18.01 Single Variable Calculus), but no familiarity with proofs or proof writing.
Description
We will cover the same material as 18.01 but with an emphasis on proofs and conceptual understanding rather than computation. Topics include:
- Axioms for the real numbers;
- The Riemann integral;
- Limits, theorems on continuous functions;
- Derivatives of functions of one variable;
- The fundamental theorems of calculus;
- Taylor’s theorem;
- Infinite series, power series, rigorous treatment of the elementary functions.
Recitation Assignments
At the beginning of each unit, students will receive a schedule that includes problems to complete in advance of the next recitation. Students should be prepared to present their proofs in clear detail. This will give them the opportunity to very deliberately improve their proof writing skills.
Problem Sets
Problem sets are assigned weekly, and due the following week. At the end of the course, the lowest two pset scores for each student will be replaced by his or her average score on all of the psets.
Exams
There will be three in-class, one hour exams, and one three-hour comprehensive final exam.
Grading
ACTIVITIES | PERCENTAGES |
---|---|
Problem sets | 20% |
Midterms | 40% |
Final exam | 30% |
Participation | 10% |
Calendar
LEC # | TOPICS | KEY DATES |
---|---|---|
Real numbers | ||
0 | Proof writing and set theory | |
1 | Axioms for the real numbers | |
2 | Integers, induction, sigma notation | |
3 | Least upper bound, triangle inequality | |
4 | Functions, area axioms | |
The integral | ||
5 | Definition of the integral | Pset1 due |
6 | Properties of the integral, Riemann condition | |
7 | Proofs of integral properties | |
8 | Piecewise, monotonic functions | Pset2 due |
Limits and continuity | ||
9 | Limits and continuity defined | |
10 | Proofs of limit theorems, continuity | Pset3 due |
11 | Hour exam I | Covers lectures 1-9 |
12 | Intermediate value theorem | |
13 | Inverse functions | |
14 | Extreme value theorem and uniform continuity | Pset4 due |
Derivatives | ||
15 | Definition of the derivative | |
16 | Composite and inverse functions | |
17 | Mean value theorem, curve sketching | Pset 5 due |
18 | Fundamental theorem of calculus | |
19 | Trigonometric functions | |
Elementary functions; integration techniques | ||
20 | Logs and exponentials | Pset 6 due |
21 | IBP and substitution | |
22 | Inverse trig; trig substitution | Pset 7 due |
23 | Hour exam II | Covers lectures 10-20 |
24 | Partial fractions | |
Taylor’s formula and limits | ||
25 | Taylor’s Formula | |
26 | Proof of Taylor’s formula | Pset 8 due |
27 | L’Hopital’s rule and infinite limits | |
Infinite series | ||
28 | Sequences and series; geometric series | Pset 9 due |
29 | Absolute convergence, integral test | |
30 | Tests: comparison, root, ratio | Pset 10 due |
31 | Hour exam III | Covers lectures 21-29 |
32 | Alternating series; improper integrals | |
Series of functions | ||
33 | Sequences of functions, convergence | |
34 | Power series | |
35 | Properties of power series | Pset 11 due |
36 | Taylor series | |
37 | Fourier series | |
38 | Final exam |