Math 424 - Fundamental Concepts of Analysis I
See UW General Catalog for course description and prerequisite information.
Textbooks
- Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick
- Principles of Mathematical Analysis (Third Edition) by Walter Rudin
Suggested Syllabus
Continuity
- epsilon-delta definition and sequence definition of continuity
- epsilon-delta and sequence definitions of a limit
- pointwise convergence, uniform convergence, uniformly Cauchy sequences
- uniform limits and continuity
Differentiation
- definition of the derivative
- tangent line approximation
- algebra of derivatives and the chain rule
- Rolle’s Theorem and the Mean Value Theorem
- higher derivatives and Taylor’s Theorem
- Intermediate Value Theorem for derivatives
The Riemann-Stieltjes Integral
- definition of the Riemann-Stieltjes integral
- integrability of continuous and monotone functions
- properties of the integral
- change of variables
- Fundamental Theorem of Calculus
- integration by parts
- Mean Value Theorem for Riemann integrals
Uniform Convergence and Power Series
- uniform convergence and integration
- uniform convergence and differentiation
- Weierstrass M-test
- power series and radius of convergence
- differentiation of power series
- Weierstrass Approximation Theorem