Math 334 syllabus

Advanced Calculus I

Course Description: This is the first of a two course sequence designed to provide students with the theoretical context of concepts encountered in MA 125 through MA 227. Topics covered include Completeness Axiom, sequences of real numbers, suprema and infima, Cauchy sequences, open sets and accumulation points in Euclidean space, completeness of Euclidean space, series of real numbers and vectors, compactness, Heine- Borel Theorem, connectedness, continuity, Extremum Theorem, Intermediate Value Theorem, differentiation of functions of one variable.
Prerequisites:  MA 227 Minimum Grade of C and MA 237 Minimum Grade of C and MA 320 Minimum Grade of C

Suggested Text:   Edward D. Gaughan, Introduction to Analysis (5th ed.), American Mathematical Society 2009

Coverage: Chapters 0 through 4

(Note : Division of material between MA 334 and 335 may vary.) 


Learning outcomes: Upon the successful completion of the course a student will:

Work with axioms and formal definitions,
Discover and write proofs and construct counterexamples in the context of real analysis of a single variable.
Understand the reals as an ordered field with least upper bound property;
Be familiar with fundamental concepts of metric topology of the reals, such as open closed sets, compactness and connectedness; 
Know the definitions of convergence of sequences and upper and lower limits, and methods of determining limits and convergence;
Know and be able to apply rigorous definitions of limit, continuity, uniform continuity and derivative of functions of a real variable;
Know the statements and understand the proofs of fundamental results such as the Intermediate Value Theorem, Mean Value Theorem, and Inverse Function Theorem, and their consequences;
Construct “epsilon-delta” proofs and other types of proof that arise routinely in analysis.