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 (excluding the limit comparison test), 10.14, the
criterion for summability of the *p*th powers, 10.15, 10.16

**Absolute convergence:** 10.18 (the definition and Theorem 10.18 **only**)

**The limit of a function:** The sequential definition of the limit

**Basic theorems on limits:** Uniqueness of limits, the limit of a scaling of a function, the limits of sums and
products of functions

**The limit of a function:** The "ε-δ" definition: 3.1, 3.2

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

**Continuity:** 3.3, 3.6–3.8

**Bolzano's Theorem, the intermediate-value theorem, and applications:** 3.9–3.11

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

**The extreme-value theorem for continuous functions:** 3.16

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

**Basic differential calculus:** 4.4–4.6, 4.10, 4.13 (Section 4.12 assigned for **self-study**
excluding the discussion on "implicit differentiation")

**Points of absolute/global and relative/local extremum:** 4.13–4.15

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

**Inverse functions and their derivatives:** 3.12, 3.13, 6.20–6.22

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

**Integration:** 1.16, 1.17, 1.24

**Uniform continuity**

**Integrability of continuous functions:** 3.17, 3.18

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

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

**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, 6.14, 6.16. **Note:** The material
in Sections 6.14 and 6.16 was developed through homework problems.

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

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

**Linear transformations:** 16.1, 16.4

**Matrix representations of linear transformations:** 16.10

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

**The null space and range of a linear transformation, injective linear transformations:** 16.2, 16.7