University Mathematics (MATH1010G)

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Announcement/Reminder

• Final examination. Date: 20/12 (Tuesday). Time: 0930-1130hrs. Venue: University Gymnasium.
• Please refer to the common homepage of MATH1010.

Time and Venue

• Lectures and Tutorials: Tuesdays 1030-1215hrs YIA LT5, Thursdays 1230-1415hrs YIA LT9.

For more details, please refer to course information sheet.

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Assignments

• Please refer to the common homepage of MATH1010.

Supplementary material

• 06/09
• Lecture 1: Definitions and results
• Exercise 1 (Solution)
• Exercise 2 (Solution)
• 8/09
• Lecture 2: Definitions and results
• Exercise 3 (Solution)
• Tutorial 1: Exercise 2
• 13/09
• Lecture 3: Definitions and results
• Exercise 4 (Solution)
• 15/09
• Lecture 4: Definitions and results
• Exercise 5 (Solution)
• 20/09
• Lecture 5: Definitions and results
• Exercise 6 (Solution)
• Illustration:
• Definition of one-sided limit from the right
• Definition of two-sided limit
• Standard methods for evaluating `simple' limits (Refer to Exercise 6.)
• What happens for sin(x)/x when x decreases to 0
• 22/09
• Lecture 6: Definitions and results
• Exercise 7 (Solution)
• Tutorial 2: Exercise 4, Exercise 5
• Illustration:
• Interchange of limit at 0 and limit at the infinities (Refer to Exercise 7.)
• Idea of `exponentiation drowns out powers' and examples of applications (Refer to Exercise 7.)
• Applications of `powers drowning out logarithm' (Refer to Exercise 7.)
• 27/09
• Lecture 7: Definitions and results
• Exercise 8 (Solution)
• Exercise 9 (Solution)
• Illustration:
• Example on differentiability (Refer to Exercise 8.)
• Justification of the `Product Rule' in differentiation
• Idea behind the `inductive argument' for Leibniz's Rule
• 29/09
• Lecture 8: Definitions and results
• Exercise 10 (Solution)
• Tutorial 3: Exercise 6, Exercise 7
• Illustration:
• Relation between two-sided limit and one-sided limits
• Divergence to the positive infinity, and an example (Refer to Exercise 10.)
• Divergence by oscillation, and the function sin(1/x) near 0
• Example on differentiability and continuous differentiability (Refer to Exercise 10.)
• 04/10
• Lecture 9: Definitions and results
• Exercise 11 (Solution)
• Illustration:
• Intermediate Value Theorem and examples of applications on equations (Refer to Exercise 11.)
• Another version of Intermediate Value Theorem
• Inverse function of a strictly monotonic continuous function on an interval
• Implicit Function Theorem and implicit differentiation
• 06/10
• Lecture 10: Definitions and results
• Exercise 12 (Solution)
• Tutorial 4: Exercise 8, Exercise 9
• Illustration:
• Examples on the notions of relative extrema and absolute extrema (Refer to Exercise 12.)
• Existence of absolute extrema for `nice' functions on closed and bounded intervals, and Fermat's Theorem on relative extrema
• Rolle's Theorem
• 11/10
• Lecture 11: Definitions and results
• Exercise 13 (Solution)
• Illustration:
• Cauchy's Mean Value Theorem and L'Hopital's Rule for limits of the type `0/0' at `finite points' (Refer to Exercise 13.)
• L'Hopital's Rule for other types of limits
• Miscellaneous examples on the application of L'Hopital's Rule (Refer to Exercise 13.)
• 13/10
• Lecture 12: Definitions and results
• Exercise 14 (Solution)
• Exercise 15 (Solution)
• Tutorial 5: Exercise 9, Exercise 10, and Common Assignment 1
• Illustration:
• Mean-Value Theorem
• Functions with constant zero derivatives on intervals (Refer to Exercise 14.)
• Darboux's Theorem and its applications (Refer to Exercise 14.)
• Example on obtaining inequalities from the Mean-Value Theorem (Refer to Exercise 15.)
• 18/10
• Lecture 13: Definitions and results
• Exercise 16 (Solution)
• Exercise 17 (Solution)
• Illustration:
• Notion of relative extrema (Refer to Exercise 16.)
• First Derivative Test, Second Derivative Test and their justification
• Sufficiency criteria for strict monotonicity in terms of signs of first derivatives
• Notion of convexity, concavity and point of inflexion
• 20/10
• Lecture 14: Definitions and results
• Exercise 18 (Solution)
• Tutorial 6: Exercise 13, Exercise 14, and Common Assignment 1, Common Assignment 2
• Illustration:
• From Mean-Value Theorem to Taylor's Theorem (Refer to Exercise 18.)
• 25/10
• Lecture 15: Definitions and results
• Exercise 19 (Solution)
• Illustration:
• Analyticity (Refer to Exercise 19.)
• 27/10
• Lecture 16: Definitions and results
• Exercise 20 (Solution)
• Tutorial 7: Exercise 18, Exercise 19
• Illustration:
• Examples on computation for `Taylor series expansions' (Refer to Exercise 20.)
• 01/11
• Lecture 17: Definitions and results
• Exercise 21 (Solution)
• 03/11
• Lecture 18: Definitions and results
• Exercise 22 (Solution)
• Exercise 23 (Solution)
• Tutorial 8: Exercise 19, Exercise 20, Exercise 21
• Illustration:
• Notion of primitive and indefinite integral
• How to find primitive of a continuous piecewise-defined function on an interval (Refer to Exercise 22.)
• 08/11
• Lecture 19: Definitions and results
• Exercise 24 (Solution)
• 10/11
• Lecture 20: Definitions and results
• Exercise 25 (Solution)
• Tutorial 9: Exercise 22, Exercise 23, Exercise 24
• Illustration:
• Examples of integrals of rational functions (Refer to Exercise 25.)
• Mechanism behind partial fraction decomposition (Refer to Exercise 25.)
• 15/11
• Lecture 21: Definitions and results
• Exercise 26 (Solution)
• Exercise 27 (Solution)
• 17/11
• No class.
• 22/11
• Lecture 22: Definitions and results
• Exercise 28 (Solution)
• Illustration:
• Fundamental Theorem of the Calculus for continuous functions on open intervals
• 24/11
• Lecture 23: Definitions and results
• Exercise 29 (Solution)
• Tutorial 10: Exercise 26, Exercise 28
• Taylor series expansion obtained from definite integrals
• 29/11
• Exercise 30 (Solution)
• 01/12
• Exercise 31 (Solution)
• Tutorial 11: Exercise 28, Exercise 29, Exercise 30

Test

• Please refer to the common homepage of MATH1010.

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Links

• the university library homepage
• other undergraduate courses at the Department of Mathematics