All numbers refer to sections in the textbook.
Your lecture notes will cover all the material
(except for a few topics assigned for self-study) in the syllabus. The chapters listed below
provide more extensive explanations, and lots of exercises for you to work on.
Basic set theory: I.2.1–I.2.5
The real line: I.3.1–I.3.4, I.3.8–I.3.10, I.3.12
Sequences and convergence: 10.2–10.4
Infinite series and their convergence: 10.5–10.9
Convergence tests for non-negative series: 10.11, 10.12, 10.14, and the
criterion for summability of the pth powers
Absolute and conditional convergence: 10.18, 10.20
The ratio and root tests: 10.15, 10.16, 10.20
Leibniz's Rule: 10.17, 10.20
The limit of a function: 3.1, 3.2, 3.4, 3.5
The sign-preservation property 3.9, 3.11
The topics above comprise the syllabus of the mid-term examination.
Bolzano's Theorem, the intermediate-value theorem, and applications 3.9–3.11
The extreme-value theorem: 3.16
Differential calculus: 4.2–4.7, 4.9
Points of absolute/global and relative/local extremum: 4.13–4.16
The chain rule and its applications: 4.10–4.12, excluding the discussion on "implicit differentiation"
The mean-value theorem and its applications: 4.14–4.17, excluding the discussion on convexity
Inverse functions and their derivatives: 3.13, 6.20–6.22
Integration, motivation, step functions: 1.8–1.13, 1.15
Integration: 1.16, 1.17, 1.21, 1.24
Integrability of continuous functions: 3.18