Description

A rigorous analysis of functions of one variable.

Prerequisites

A C- or better in (MATH 240, MATH 241, and MATH 310) or (MATH 340, MATH 341, and MATH 310)

Topics

The Real Numbers

The Completeness Axiom: The Natural, Rational,
and Irrational Numbers

Sequences of Real Numbers

The Convergence of Sequences
The Monotone Convergence Theorem, the Bolzano-Weierstrass
Theorem, and the Nested
Interval Theorem

Continuous Functions and Limits

Continuity
The Extreme Value Theorem
The Intermediate Value Theorem
Uniform Continuity
Limits

Differentiation

The Algebra of Derivatives
Differentiating Inverses and Compositions
The Lagrange Mean Value Theorem and
Its Geometric Consequences
The Cauchy Mean Value Theorem and Its Analytic Consequences

The Elementary Functions as Solutions of Differential Equations

The Natural Logarithm and the Exponential Functions
The Trigonometric Functions
The Inverse Trigonometric Functions

Integration

The Definition of the Integral and Criteria for Integrability
The First Fundamental Theorem of Calculus
The Convergence of Darboux Sums and Riemann Sums
Linearity, Monotonicity, and Additivity over Intervals

The Fundamental Theorems of Calculus and Their Consequences

The Second Fundamental Theorem of Calculus
The Existence of Solutions of Differential Equations
The Approximation of Integrals

Approximation by Taylor Polynomials

Taylor Polynomials and Order of Contact
The Lagrange Remainder Theorem
The Convergence of Taylor Polynomials
The Cauchy Integral Remainder Formula and
the Binomial Expansion
The Weierstrass Approximation Theorem

The Convergence of Sequences and Series of Functions

Sequences and Series of Numbers
Pointwise Convergences and Uniform Convergence
of Sequences of Functions
The Uniform Limit of Continuous Functions,
of Integrable Functions, and of
Differentiable Functions
Power Series