Aims & Learning Objectives:
Aims: To give a thorough grounding, through rigorous theory and exercises, in the method and theory of modern calculus. To define the definite integral of certain bounded functions, and to explain why some functions do not have integrals.
Students should be able to quote, verbatim, and prove, without recourse to notes, the main theorems in the syllabus. They should also be capable, on their own initiative, of applying the analytical methodology to problems in other disciplines, as they arise. They should have a thorough understanding of the abstract notion of an integral, and a facility in the manipulation of integrals.
Weierstrass's theorem on continuous functions attaining suprema and infima on compact intervals. Intermediate Value Theorem.
Functions and Derivatives. Algebra of derivatives. Leibniz Rule and compositions. Derivatives of inverse functions. Rolle's Theorem and Mean Value Theorem. Cauchy's Mean Value Theorem. L'Hopital's Rule. Monotonic functions. Maxima/Minima.
Uniform Convergence. Cauchy's Criterion for Uniform Convergence. Weierstrass M-test for series. Power series. Differentiation of power series.
Reimann integration up to the Fundamental Theorem of Calculus for the integral of a Riemann-integrable derivative of a function. Integration of power series. Interchanging integrals and limits. Improper integrals.