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.
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.
Determine whether or not a given function is differentiable.
The Riemann Integral
Use the Trapezoidal Rule and Simpson's Rule to find approximations to integrals.
Find integrals and derivatives, using the Fundamental Theorems of Calculus.
Determine whether or not a function is integrable.
Compute the value of a Riemann sum.
Prove statements about monotone and inverse functions.
Determine whether or not a function is uniformly continuous.
Prove statements about continuous functions on intervals.
Prove continuity using rules for combinations of continuous functions.
Prove statements about continuous functions.
Sequences, Series, and Limits
Determine limits of functions.
Prove statements about infinite series.
Prove statements about sequences and subsequences.
Determine convergence and sums of series of real numbers.
Determine convergence and limits of sequences of real numbers.
Preliminaries and Real Numbers
Prove statements about intervals.
Prove statements using the Completeness and Supremum Properties of real numbers.
Find subsets of R defined by specified conditions.
Prove statements about the sets R of real numbers and Q of rational numbers.
Prove statements about finite and infinite sets.
Use the Principle of Mathematical Induction to prove statements about the set of natural numbers.
Prove statements about sets and functions.
Find the results of operations on finite sets.
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