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 pth 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 Rn 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