Introduction to Advanced Calculus

This course "Introduction to Advanced Calculus" is a natural sequel to the course "Introduction to Calculus", also on this platform, though students who are well-prepared, with some prior calculus experience, can jump straight in. Once again, the focus and themes of this course address important foundations for applications of mathematics in science, engineering and commerce, with now a particular focus on series representations of functions and an introduction to the theory of differential equations. The course emphasises key ideas and historical motivation for calculus, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics. Students taking Introduction to Advanced Calculus will: • review key ideas of differential calculus, with further emphasis on pivotal underlying themes and results, such as the Mean Value Theorem and the Intermediate Value Theorem, and add further tools, such as L'Hopital's Rule, Newton's Method and hyperbolic functions (first week) • review key ideas of integral calculus, extending techniques of integration, including tricky substitutions, the method of integration by parts, including a proof that the number pi is irrational, the method of partial fractions, the disc and shell method for finding volumes of revolutions, formulae for arc length and surface area of revolution, application of Riemann sums to estimate work, and an introduction to improper integrals (second week) • introduce sequences and series, tests for convergence, and series representations of functions, including estimates of error terms using Taylor's theorem and a proof that Euler's number e is irrational (third week) • introduce the theory of differential equations, including discussions of separable equations, including the logistic equation and logistic function, equilibrium solutions, first order equations, solved using the integrating factor method, second order equations with constant coefficients and an introduction to solving systems of equations, modelling two interacting populations, which may be in a symbiotic or predator-prey relationship, finishing with a brief discussion of connections with linear algebra and the matrix exponential (fourth week).