In this course we give a rigorous formalization of "limits" of sequences and functions by means of the "epsilon-delta" definitions.
We also learn the differential calculus rigorously, and in particular the Taylor expansion as an excellent polynomial approximation.
Each lecture will be followed by a recitation (a problem-solving session). This course will be succeeded by "Introduction to Analysis II" in the second quarter.
The students will learn how to write the mathematical analysis logically. The students will become familiar with the "epsilon-delta" definitions and proofs, and be able to describe limits of real numbers rigorously.
At the end of this course, students are expected to:
-- Understand the construction of irrational numbers
-- Be familiar with limits superior and limits inferior
-- Be able to state and prove propositions about limits of sequences and functions by using "epsilons and deltas".
-- Understand some important properties of continuous functions, such as the intermediate value theorem, and the existence of the maximum/minimum.
-- Be able to calculate polynomial approximations of a given function by using the Taylor expansion or asymptotic expansion.
continuity of real numbers, infimum, supremum, limit superior, limit inferior, Cauchy sequence, continuous function, derivative, Taylor expansion
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course with recitation sessions.
Homework will be assigned every week. There will be occasional quizzes.
Course schedule | Required learning | |
---|---|---|
Class 1 | Continuity of real numbers, supremum and infimum | Details will be provided in class. |
Class 2 | Recitation session | Details will be provided in class. |
Class 3 | Limit of series, subsequence, and accumulation point | Details will be provided in class. |
Class 4 | Recitation session | Details will be provided in class. |
Class 5 | limit superior, limit inferior, Cauchy sequence | Details will be provided in class. |
Class 6 | Recitation session | Details will be provided in class. |
Class 7 | Convergence of series, interchanging the order of summation | Details will be provided in class. |
Class 8 | Recitation session | Details will be provided in class. |
Class 9 | Limits and continuity of functions | Details will be provided in class. |
Class 10 | Recitation session | Details will be provided in class. |
Class 11 | Continuous functions | Details will be provided in class. |
Class 12 | Recitation session | Details will be provided in class. |
Class 13 | Differentiability, Rolle's theorem, mean-value theorem | Details will be provided in class. |
Class 14 | Recitation session | Details will be provided in class. |
Class 15 | higher order derivatives and Taylor expansion, quiz | Details will be provided in class. |
None required
None required
Final exam 50%, assignments and quizzes 50%.
Students are expected to have passed Calculus I/II, Linear Algebra I/II, and their reciｔations.