Calculus 2: Advanced

This subject covers the same material as MAST10006 Calculus 2, but to a greater depth and with a greater emphasis on concepts, mathematical rigour, and proofs. Students are introduced to functions of several variables, the notion of a limit, and properties of continuous functions. Moreover, level curves, partial derivatives, the chain rule, directional derivatives, tangent planes and extrema for functions of several variables are discussed. For functions of one variable, the concepts of differentiation and integration are reviewed, and the Fundamental Theorem of Calculus is presented. Integration by parts, and substitution are covered as the main methods of integration. The subject also exposes students to ordinary differential equations. In particular, linear differential equations of both first and second order are discussed, as well as separable first order equations. Several special functions, such as the hyperbolic functions, and their inverses are covered in this context. Applications include the description of population models, and electrical circuits and mechanical oscillators. The subject also introduces the idea of approximating functions by polynomials. In particular, Taylor polynomials for functions of one variable are discussed, and Taylor's theorem is presented. In addition, the subject introduces infinite sequences and series, and the concepts of convergence and divergence. In addition to the intuitive understanding of convergence, students will see the mathematical definition of convergence. Finally complex functions, and complex power series are introduced. In particular, the complex exponential function, and its properties are discussed.