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This subject extends students' knowledge of functions and calculus and introduces them to the topics of vectors and complex numbers. Students will be introduced to new functions such as the inverse trigonometric functions and learn how to extend the techniques of differentiation to these. Integration techniques will be applied to solving first order differential equations.
Differential calculus: graphs of functions of one variable, trigonometric functions and their inverses, derivatives of inverse trigonometric functions, implicit differentiation and parametric curves. Integral calculus: properties of the integral, integration by trigonometric and algebraic substitutions and partial fractions with a variety of applications. Ordinary differential equations: solution of simple first order differential equations arising from applications such as population modelling. Vectors: dot product, scalar and vector projections, plane curves specified by vector equations. Complex numbers: arithmetic of complex numbers, sketching regions in the complex plane, De Moivre's Theorem, roots of polynomials, the Fundamental Theorem of Algebra.
Intended learning outcomes
Students completing this subject should:
- be familiar with functions of a single, real variable including injective and bijective functions, composition of functions and conditions under which inverse functions can be defined;
- be able to graphically represent and analyse key features of polynomial, circular, inverse circular and reciprocal functions;
- be able to manipulate simple trigonometric identities and compound and double angle formulas for sine, cosine and tangent;
- understand the arithmetic of vectors in two and three dimensions, scalar products and application to vector projections and resolutes, plane curves specified parametrically by a vector equation and determination of corresponding cartesian equations;
- extend differentiation techniques to implicit differentiation, derivatives of inverse circular functions, second and higher order derivatives and be able to apply these to problems including curve sketching;
- be able to evaluate integrals using algebraic and trigonometric substitutions, and partial fractions;
- be able to apply integration techniques to problems including the area between curves and the solution of simple ordinary differential equations;
- understand the extension of the real numbers to the set of complex numbers and their arithmetic, including Cartesian representation and polar form.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- 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; and
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 7 September 2023