Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.
Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.
SES # | TOPICS | TEXTBOOK READINGS | COURSE NOTES READINGS |
---|---|---|---|
Real numbers | |||
0 | Proof writing and set theory | I 2.1-2.4 | |
1 | Axioms for the real numbers | I 3.1-3.7 | |
2 | Integers, induction, sigma notation | I 4.1-4.6 | Course Notes A |
3 | Least upper bound, triangle inequality | I 3.8-3.10, I 4.8 | Course Notes B |
4 | Functions, area axioms | 1.2-1.10 | |
The integral | |||
5 | Definition of the integral | 1.12-1.17 | |
6 | Properties of the integral, Riemann condition | Course Notes C | |
7 | Proofs of integral properties | 88-90, 113-114 | Course Notes D |
8 | Piecewise, monotonic functions | 1.20-1.21 | Course Notes E |
Limits and continuity | |||
9 | Limits and continuity defined | 3.1-3.4 | Course Notes F |
10 | Proofs of limit theorems, continuity | 3.5-3.7 | |
11 | Hour exam I | ||
12 | Intermediate value theorem | 3.9-3.11 | |
13 | Inverse functions | 3.12-3.14 | Course Notes G |
14 | Extreme value theorem and uniform continuity | 3.16-3.18 | Course Notes H |
Derivatives | |||
15 | Definition of the derivative | 4.3-4.4, 4.7-4.8 | |
16 | Composite and inverse functions | 4.10, 6.20 | Course Notes I |
17 | Mean value theorem, curve sketching | 4.13-4.18 | |
18 | Fundamental theorem of calculus | 5.1-5.3 | Course Notes K |
19 | Trigonometric functions | Course Notes L | |
Elementary functions; integration techniques | |||
20 | Logs and exponentials | 6.3-6.7, 6.12-6.16 | Course Notes M |
21 | IBP and substitution | 5.7, 5.9 | Course Notes N |
22 | Inverse trig; trig substitution | 6.21 | |
23 | Hour exam II | ||
24 | Partial fractions | 6.23 | Course Notes N |
Taylor’s formula and limits | |||
25 | Taylor’s formula | 7.1-7.2 | |
26 | Proof of Taylor’s formula | Course Notes O | |
27 | L’Hopital’s rule and infinite limits | 7.12-7.16 | Course Notes P |
Infinite series | |||
28 | Sequences and series; geometric series | 10.1-10.6, 10.8 (first page only) | |
29 | Absolute convergence, integral test | 10.11, 10.13, 10.18 | |
30 | Tests: comparison, root, ratio | 10.12, 10.15 | Course Notes Q |
31 | Hour exam III | ||
32 | Alternating series; improper integrals | 10.17, 10.23 | |
Series of functions | |||
33 | Sequences of functions, convergence | 11.1-11.2 | |
34 | Power series | 11.3-11.4 | Course Notes R |
35 | Properties of power series | Course Notes R | |
36 | Taylor series | 11.9 | Course Notes S |
37 | Fourier series | Course Notes T |