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 *p*th 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

**Continuity:** 3.3, 3.6

The topics above comprised the syllabus of the mid-term examination. **They will
also be a part of the syllabus of the final 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

**Uniform continuity**

**Integrability of continuous functions:** 3.18

**The first and second Fundamental Theorems of Calculus:** 5.1, 5.3, 5.5

**The substitution rule:** 5.7, 5.8

**Integration by parts:** 5.9, **excluding** Theorem 5.5. **Note:** Section 5.10 is for
self-study, as you have seen most of the problems in them in high school. **For help and
solved examples, see Section 5.9**

**The logarithm and the exponential functions:** 6.3, 6.7, 6.12, 6.15, 6.16

**Vector spaces and subspaces:** 15.2, 15.3, 15.5, 15.6

**Linear independence and bases:** 15.7–15.9

**Linear transformations:** 16.1, 16.2, 16.4

**The Rank-Nullity Theorem:** 16.3, 16.4

**Matrix representations of linear transformations:** 16.10

**Composition of linear transformations, matrices, invertibility of matrices:** 16.5, 16. 15, 16.16

**One-to-one linear transformations:** 16.7