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 natural numbers, Peano addition and multiplication**

**Fields:** Definition and examples, ordered fields

**The real line and the least upper-bound property:** I.3.1–I.3.4, I.3.8–I.3.10

**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

**The ratio test:** 10.15, 10.16 (**Note:** Many similar tests given in Apostol's book,
which are founded on the Comparison Test,
are **not** in the syllabus this year.)

**The limit of a function:** 3.1, 3.2

**Basic theorems on limits:** Theorem 3.1 from Section 3.4, 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.**

**Continuity:** 3.3, 3.6–3.8

**The extreme-value theorem:** 3.16

**Bolzano's Theorem, the intermediate-value theorem:** 3.9, 3.10 (applications deferred until Jan. 15, 2021)

**The meaning of differentiability:** 4.2, 4.3

**The meaning of differentiability:** 4.2, 4.3

**Basic differential calculus:** 4.4–4.6, 4.10, 4.13 (assigned for **self-study**)

**Rolle's Theorem, the mean-value theorem and their applications:** 4.14–4.16
(the the second-derivative test and its consequences have been **excluded** due to the lack of time)

**Inverse functions and their derivatives:** 3.12, 3.130, 6.20–6.22

**Brouwer's Fixed-point Theorem** and other applications of the intermediate-value theorem: 3.11

**The Cartesian product of sets** (the discussion on the *n*-dimensional Brouwer's Fixed-point
Theorem is **not** a part of the syllabus; the theorem was merely introduced for perspective)

**Integration, motivation, step functions:** 1.8–1.13, 1.15

**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

**Primitives, Leibnizian notation:** 5.3, 5.6, 5.8

**The substitution rule:** 5.7, 5.8

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

**Vector spaces and subspaces:** 15.2–15.6

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

**Linear transformations:** 16.1, 16.4

**The null space and range of a linear transformation** (definition and properties): 16.2, 16.4 (**relevant**
exercises only)

**Algebra of linear transformations:** 16.5, 16.8 (**relevant** exercises only)

**Matrix representations of linear transformations:** 16.10