Sequences (3 weeks): Review of background material. Definitions, the sum, product "rules", squeeze and monotone, subsequences, Bolzano-Weierstrass, Cauchy. Numerical applications including finding roots.
Series (3 weeks): Definitions, convergence tests, applications to Riemann sums and quadrature. Power series, Taylor polynomials, Taylor series and and tail estimation, Power series in practice including Newton's method and root finding.
Several variable calculus – differentiation (3 weeks): Limits and continuity, partial derivatives, chain rule, Taylor's Theorem. Extrema, Newton's method for systems, tangent planes and directional derivatives, constrained optimisation.
Several variable calculus – integrations (3 weeks): Definitions, double integrals, polar coordinates, change of variables and triple integrals. Applications of multiple integrals