Course Description:
This advanced course is an introduction to the theory of calculus of functions of a single real variable. In the usual introductory calculus sequence (Calculus I, II and III), students learn how to perform routine calculations with little attention to the theory that makes these calculations possible and meaningful. In this course, students will learn how to establish calculus on a rigorous foundation: familiar concepts will be revisited in the context of mathematical proof. This advanced course is an in-depth, behind-the-scenes look into the theory of calculus of functions of a single real variable.
Topics covered in this course include: the real numbers, limits of sequences, completeness, the Monotone Convergence Theorem, the Bolzano-Weierstrass Theorem, the Cauchy Criterion, the Cantor Set, open and closed sets, sequential compactness, limits of functions, continuity, the Intermediate Value Theorem, the derivative, the Mean Value Theorem, the Riemann Integral, and the Fundamental Theorem of Calculus.
The primary audience for this course is students who wish to concentrate in either mathematics or applied mathematics. Students concentrating in a tangentially related field, such as physics, may also be interested in this course.
Prior to enrolling in this course, students should be fluent in differential and integral Calculus, as would be acquired in a three or four course sequence of Calculus. Also, students should be comfortable with mathematical proof (both reading and writing of proofs). Such knowledge may be obtained from courses such as Logic & Proof, Discrete Mathematics, or Linear Algebra, for example.
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