MATH 414: Analysis I
(3-0) Cr. 3. F.S.SS.
Prereq: Minimum of C- in MATH 201
A rigorous development of calculus of functions of one real variable: real number properties and topology, limits, continuity, differentiation, integration, series.
Basic Analysis I, 5th Edition
The textbook is freely available at http://www.jirka.org/ra/
We will cover most of the material of the first 5 chapters of the book, at a pace of roughly one chapter per two and a half weeks.
- Define the set of real numbers, and state its basic properties.
- Understand the Least Upper Bound Property and the Archimedean Property.
- Define sequences of real numbers, and understand bounded and convergence sequences.
- Understand the concept of limit superior and inferior.
- Know the statement of the Bolzano-Weierstrass theorem, and its consequences.
- Understand the concept of Cauchy sequences and the completeness of R
- Define series of real numbers, and identify when they converge or diverge.
- Understand the idea of a limit of a function; in particular, the ε-δ definition.
- Define continuous functions, and give relevant examples of continuous and discontinuous functions.
- State and understand the Extreme Value Theorem and the Intermediate Value Theorem.
- Define the derivative of a function, and understand the relevant derivate rules, i.e. product, quotient, and chain.
- State and understand the Mean Value Theorem.
- Define the Riemann integral in terms of Darboux integrals.
- Understand the properties of integrals, including linearity of the integral.
- State and understand the Fundamental Theorem of Calculus.