### 2021　Introduction to Analysis I

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Instructor(s)
Kagei Yoshiyuki  Miura Tatsuya
Course component(s)
Lecture / Exercise    (対面型/ZOOM)
Day/Period(Room No.)
Mon3-4(H103)  Mon5-8(H103)
Group
-
Course number
MTH.C201
Credits
2
2021
Offered quarter
1Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

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.

### Student learning outcomes

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.

### Keywords

continuity of real numbers, infimum, supremum, limit superior, limit inferior, Cauchy sequence, continuous function, derivative, Taylor expansion

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

This is a standard lecture course with recitation sessions.
Homework will be assigned every week. There will be occasional quizzes.

### Course schedule/Required learning

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, Cauchy sequence Details will be provided in class.
Class 4 Recitation session Details will be provided in class.
Class 5 Convergence of series, interchanging the order of summation Details will be provided in class.
Class 6 Recitation session Details will be provided in class.
Class 7 Limits and continuity of functions Details will be provided in class.
Class 8 Recitation session Details will be provided in class.
Class 9 Continuous functions Details will be provided in class.
Class 10 Recitation session Details will be provided in class.
Class 11 Differentiability, Rolle's theorem, mean-value theorem Details will be provided in class.
Class 12 Recitation session Details will be provided in class.
Class 13 higher order derivatives and Taylor expansion Details will be provided in class.
Class 14 Recitation session Details will be provided in class.

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

None required

None required

### Assessment criteria and methods

Based on the attendance, quizzes, reports, and the problem solving situation in the recitation sessions. Details will be provided in the class.

### Related courses

• MTH.C202 ： Introduction to Analysis II
• MTH.C203 ： Introduction to Analysis III
• MTH.C204 ： Introduction to Analysis IV

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Calculus I/II, Linear Algebra I/II, and their reciｔations.