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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.
Intended learning outcomes
Students completing this subject should be able to:
- determine limits of functions of one or several variables
- apply theorems about continuity to investigate functions
- determine convergence and divergence of sequences and series
- sketch and manipulate hyperbolic and inverse hyperbolic functions
- evaluate integrals using integration by parts and substitution, in particular using trigonometric and hyperbolic substitutions, and partial fractions
- find solutions of first and second order ordinary differential equations, and use these equations to model some physical and biological systems
- compute Taylor polynomials of functions of one variable
- calculate partial derivatives and gradients for functions of two or more variables, and use these to find maxima and minima
- write a simple mathematical proof
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis
- collaborative skills: the ability to work in a team
- time-management skills: the ability to meet regular deadlines while balancing competing commitments
Last updated: 8 August 2023