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Calculus with Theory I >> Content Detail

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References are to the textbook: Apostol, Calculus, Vol. I, Second Edition (1967); Readings are from the course notes by Prof. J. Munkres.


SES #	TOPICS	REFERENCES	READINGS


1	Real Numbers	I 3.1 - 3.4, 3.7	Chapter A (PDF)


2	Integers, Rationals; Mathematical Induction	I 3.6, 4.1 - 4.6


3	Completeness Axiom; Triangle Inequality	I 3.8 - 3.13, 4.8	Chapter B (PDF)


4	Functions; Step Functions	1.2 - 1.10


5	Definition of the Integral	1.12 - 1.17	Chapter C.1 - C.2 (PDF)


6	Piecewise-Monotonic Functions	1.20 - 1.22	Chapter E (PDF)


7	Integral of a Power of X, Properties of the Integral	1.23 - 1.25	Chapter C.3 - C.11 (PDF)


8	Proof of the Properties; Some Applications	1.27, pp. 88-90	Chapter D (PDF)


9	Limits	3.1 - 3.4


10	Basic Limit Theorems	3.5 - 3.7	Chapter F (PDF)


11	Intermediate Value Theorem	3.9 - 3.12


12	Inverse Functions	3.12 - 3.14	Chapter G (PDF)


13	Quiz


14	Extreme-value Theorem; Uniform Continuity	3.16 - 3.17	Chapter H (PDF)


15	Derivatives	4.3 - 4.8


16	Composite and Inverse Functions	4.10, 6.20	Chapter I (PDF)


17	Mean Value Theorem	4.13 - 4.18	Chapter I (PDF)


18	The Fundamental Theorem of Calculus	5.1 - 5.3, 5.6	Chapter K (PDF) Chapter L (PDF)


19	Integration by Substitution; Primitives of Elementary Functions	5.7 - 5.8	Chapter N (PDF)


20	Integration by Parts	5.9


21	Logarithms and Exponentials	6.3 - 6.7, 6.12 - 6.16	Chapter M (PDF)


22	Inverse Trigonometric Functions, Trig Substitutions	6.21


23	Integration by Partial Fractions	6.23


24	Taylor's Formula	7.1 - 7.2, 7.5	Chapter O (PDF)


25	Applications of Taylor's Formula	7.2, 7.4


26	L'Hopital's Rule, Infinite Limits	7.12 - 7.15	Chapter P (PDF)


27	Behaviour of Log and Exp	7.16


28	Sequences and Series, Geometric Series	10.2 - 10.6, 10.8


29	Tests for Convergence I	10.11 - 10.16


30	Quiz


31	Alternating Series, Absolute Convergence	10.17 - 10.18


32	Tests for Convergence II	10.19	Chapter Q (PDF)


33	Improper Integrals	10.23


34	Series of Functions	11.1


35	Power Series	11.6 - 11.7	Chapter R (PDF)


36	Taylor's Series	11.9


37	Convergence of Taylor Series	11.10 - 11.11	Chapter S (PDF)


38	Fourier Series	P. 575


39	Review		Chapter T (PDF - 1.2 MB)

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