Math 104: Introduction to Analysis
UC Berkeley | Summer 2017
LEC 001: MTuWTh 10 AM - 12 PM in 3107 Etcheverry Hall
Instructor: Theodore Zhu
Email:
Office: 816 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 13, 10 AM - 12 PM (Exam) (Solutions)
- Final Exam: Thursday, August 10, 10 AM - 12 PM (Information)
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 30. The last day to change your grading option is Friday, July 28.
- 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
- 06/19: Introduction, rational numbers, ordered fields, real numbers, notion of distance. Worksheet 1. (Sections 2-3)
- 06/20: Notion of infinity, infimum and supremum, least upper bound property, completeness axiom, Archimedean Principle. Worksheet 2. (Sections 4-5)
- 06/21: Denseness of Q in R, sequences, convergence and divergence of sequences, limit theorems for sequences. Worksheet 3. (Sections 4, 7-9)
- 06/22: More on limits of sequences, divergence to infinity. Worksheet 4. (Section 9)
- 06/26: Monotone sequences, monotone convergence theorem, liminf and limsup. Worksheet 5. (Section 10)
- 06/27: Cauchy sequences, subsequences, Bolzano-Weierstrass theorem. (Sections 10-11)
- 06/28: More on liminf and limsup, subsequential limits. Worksheet 6. (Sections 11-12)
- 06/29: Metric spaces, sequences in metric spaces, some topological concepts. Worksheet 7. (Section 13, Notes)
- 07/03: Completeness of R^k, Bolzano-Weierstrass theorem in R^k, more on metric spaces. (Section 13)
- 07/04: NO CLASS
- 07/05: Compactness. Worksheet 8. (Section 13, Notes)
- 07/06: Finite intersection property, Cantor set, Heine-Borel theorem. (Section 13, Notes)
- 07/10: Infinite series, convergence and divergence of series, Cauchy criterion, tests for convergence and divergence of series. (Section 14)
- 07/11: p-series, alternating series test, continuity of functions. Worksheet 9. (Sections 15, 17)
- 07/12: Review
- 07/13: Midterm Exam
- 07/17: Real-valued functions, continuity, intermediate value theorem. Worksheet 10. (Sections 17-18)
- 07/18: Uniform continuity, continuous extension theorem (Section 19)
- 07/19: Functions between metric spaces, continuity, uniform continuity, properties of continuous functions on compact sets. Worksheet 11. (Sections 18, 21)
- 07/20: Continuous extension theorem, intermediate value theorem and related results, strictly increasing/decreasing functions (Sections 18, 21)
- 07/24: Power series, radius of convergence, pointwise and uniform convergence of functions (Sections 23-24)
- 07/25: Uniform limit theorem, uniformly Cauchy sequences of functions, series of functions, Weierstrass M-test, convergence of power series (Sections 24-26)
- 07/26: Abel's theorem, differentiation, basic properties of the derivative (Sections 26, 28)
- 07/27: Chain rule, Rolle's theorem, mean value theorem (Sections 28-29)
- 07/31: Dini's theorem, connection between uniform continuity and differentiability, intermediate value theorem for derivatives, generalized mean value theorem. Worksheet 12. (Sections 19, 29-30)
- 08/01: L'Hospital's rule, Taylor series. (Sections 30-31)
- 08/02: Taylor's theorem, Riemann integration. (Sections 31-32)
- 08/03: Properties of the Riemann integral. Worksheet 13. (Section 33)
- 08/07: More properties of the Riemann integral, intermediate value theorem for integrals, convergence theorems for integrals. Worksheet 14. (Section 33)
- 08/08: Fundamental theorem of calculus. Integration and differentiation of power series. (Sections 26, 34)
- 08/09: Review
- 08/10: Final Exam
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