Math 104: Introduction to Analysis
UC Berkeley | Summer 2018
LEC 001: MTuWTh 10 AM - 12 PM in 3105 Etcheverry Hall
Instructor: Theodore Zhu
Email:
Office: 1085 Evans Hall
Office Hours: Mondays 12-1, Wednesdays 12:30-2:30
Course Description
The real number system. Sequences, limits, and continuous functions. Metric spaces. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.
Textbook
Elementary Analysis: The Theory of Calculus (2nd Edition)
Exams
- Midterm: Thursday, July 12, 10 AM - 12 PM. . .
- Final Exam: Thursday, August 9, 10 AM - 12 PM
Homework
There are 7 graded homework assignments. Homework is typically due on Thursdays at the beginning of class. Collaboration is welcome and encouraged, but each student must turn in his or her own assignment. At the end of the semester, the lowest homework score for each student will be dropped.
Grading
- Course components will be weighted as follows: Homework 20%, Midterm 35%, Final Exam 45%.
- Important dates: The last day to add or drop this course is Friday, June 29 (no refund after June 24). The last day to change your grading option is Friday, July 27.
- Incomplete grades will be assigned only to students with a documented medical, personal or family emergency prior to completion of the course. Students receiving such grades are required to have been doing work of passing quality up to the intervention of the emergency.
Schedule
- 06/18: Introduction, rational numbers, ordered fields, real numbers. Worksheet 1. (Sections 2-3)
- 06/19: Distance, notion of infinity, infimum and supremum, least upper bound property, completeness axiom, Archimedean Principle. Worksheet 2. (Sections 3-5)
- 06/20: Denseness of Q in R, sequences, convergence and divergence of sequences, limit theorems for sequences. Worksheet 3. (Sections 4, 7-9)
- 06/21: More on limits of sequences, monotone sequences, monotone convergence theorem. Worksheet 4. (Sections 9-10)
- 06/25: liminf and limsup Worksheet 5. (Section 10)
- 06/26: Cauchy sequences, subsequences, Bolzano-Weierstrass theorem (Sections 10-11)
- 06/27: More on liminf and limsup, subsequential limits, metric spaces. Worksheet 6. (Sections 11-13)
- 06/28: Metric spaces, sequences in metric spaces, some topological concepts. Worksheet 7 (Section 13, notes)
- 07/02: Completeness of R^k, Bolzano-Weierstrass theorem in R^k, more on metric spaces. Worksheet 8 (Section 13)
- 07/03: Compactness. Worksheet 9 (Section 13)
- 07/04: NO CLASS
- 07/05: Sequential compactness, finite intersection property, Heine-Borel theorem. Worksheet 10 (Section 13)
- 07/09: Cantor set. Worksheet 11 (Section 13)
- 07/10: Infinite series, convergence and divergence of series, Cauchy criterion, tests for convergence and divergence of series, p-series (Sections 14-15)
- 07/11: REVIEW
- 07/12: MIDTERM EXAM
- 07/16: Alternating series test, real-valued functions, continuity. Worksheet 12 (Sections 15,17-18)
- 07/17: More on continuity, intermediate value theorem, uniform continuity. Worksheet 13 (Sections 17-19)
- 07/18: Continuous extension theorem, functions between metric spaces, continuity, preimages. Worksheet 14 (Sections 18-19,21)
- 07/19: Uniform continuity, properties of continuous functions on compact sets (Sections 18,21)
- 07/23: Generalized continuous extension theorem, limits of functions, power series. Worksheet 15 (Sections 20,23)
- 07/24: power series, radius of convergence, pointwise and uniform convergence of functions, uniform limit theorem (Sections 23-24)
- 07/25: uniformly Cauchy sequences of functions, series of functions, Weierstrass M-test, convergence and continuity of power series. Worksheet 16 (Sections 24-26)
- 07/26: Dini's theorem, Abel's theorem, differentiation. Worksheet 17 (Sections 26,28)
- 07/30: properties of derivatives, chain rule, Rolle's theorem, mean value theorem. Worksheet 18 (Sections 28-29)
- 07/31: connection between uniform continuity and differentiability, intermediate value theorem for derivatives, generalized mean value theorem. Worksheet 19 (Sections 19, 29-30)
- 08/01: L'Hospital's rule, Taylor series, Taylor's theorem (Sections 30-31)
- 08/02: Riemann integration. Worksheet 20 (Section 32)
- 08/06: properties of Riemann integration (Section 32)
Links
bCourses
Past Exams
Student Learning Center