#### 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