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 pth 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