Real Analysis – mth440 (3 credits)
This course focuses on the functions of real variables. Students investigate the topology of the real line and plane, and use these concepts to prove limit and convergence theorems about sequences, series, and functions. Students also examine continuity, differentiation, and the Riemann integral.
Sequences, Series, and Limits
- Prove statements about sequences and subsequences.
- Determine limits of functions.
- Determine convergence and sums of series of real numbers.
- Determine convergence and limits of sequences of real numbers.
- Prove statements about infinite series.
- Prove statements about continuous functions.
- Prove continuity using rules for combinations of continuous functions.
- Prove statements about continuous functions on intervals.
- Determine whether or not a function is uniformly continuous.
- Prove statements about monotone and inverse functions.
- Determine whether or not a given function is differentiable.
- Use L'Hospital's Rules to find limits of functions.
- Use Taylor's Theorem to develop polynomial approximations to functions.
- Use the Mean Value Theorem to prove the existence of relative extrema.
The Riemann Integral
- Compute the value of a Riemann sum.
- Determine whether or not a function is integrable.
- Find integrals and derivatives, using the Fundamental Theorems of Calculus.
- Use the Trapezoidal Rule and Simpson's Rule to find approximations to integrals.
Preliminaries and Real Numbers
- Prove statements about intervals.
- Find the results of operations on finite sets.
- Prove statements about sets and functions.
- Use the Principle of Mathematical Induction to prove statements about the set of natural numbers.
- Prove statements about finite and infinite sets.
- Find subsets of R defined by specified conditions.
- Prove statements using the Completeness and Supremum Properties of real numbers.
- Prove statements about the sets R of real numbers and Q of rational numbers.
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