Real numbers, topology of Euclidian space, integration, continuity, differentiability, sequences, and series of functions. (Offered spring semester only.)

MATH 211 (Calculus III) or equivalent

1. Understand and use the fundamental theorem of calculus.
   
2. Familiarity with power series and regions of convergence.
   
3. Ability to use a computer algebra system or graphics calculator.
   
4. Ability to analyze functions and their graphs.
   
5. Familiarity with sequences and limits.
   
6. Familiarity with vector spaces.

Herbert S. Gaskill and P.P. Narayanaswami, Element of Real Analysis, Prentice Hall 1998

Douglas C. Phillippy, Ph.D., Associate Professor of Mathematics

Students majoring in mathematics.

1. To develop a rigorous understanding of the real number system and its properties.
   
2. To develop a rigorous understanding of real-valued functions with an emphasis on the
   properties of continuity, differentiability, and integrability.
   
3. To develop a rigorous understanding of sequences and series, including sequences and
   series of real-valued functions.
   
4. To introduce students to graduate-level mathematics.
   
5. To develop an ability to state and prove theorems.
   
6. To enhance the learning process by using technology as a tool to reinforce, expedite, and
   display mathematical concepts.

1. An introduction to set theory: a) the algebra of sets, b) functions, and c) infinite sets.
   
2. The real number system: a) algebraic properties, b) order properties, c) completeness,
   d) intervals and cluster points, and e) open and closed sets.
   
3. Sequences: a) the limit of a sequence, b) Monotone sequences, c) the Bolzano-
   Weierstrass Theorem, d) the Cauchy criterion, e) Sequences of functions, and f) pointwise
   and uniform convergence.
   
4. Limits and continuity: a) the limit of a function, b) continuous functions, c) continuity
   on intervals, d) uniform continuity, and e) inverse functions.
   
5. Differentiation: a) the Mean Value Theorem, b) L’Hospital’s rule, and c) Taylor’s Theorem.
   
6. Integration: a) the Riemann integral, b) the Fundamental Theorem of Calculus, and c) The
   integral as a limit.
   
7. Infinite Series: a) convergence, b) tests for convergence, and c) Series of functions.