Calculus 1

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.

本课程拓展了学生对函数和微积分的知识，并介绍了向量和复数。学生将接触到反三角函数等新函数，并学习如何将微分技术扩展到这些函数。积分技术将应用于解决一阶微分方程。微分学：单变量函数的图形，三角函数及其反函数，反三角函数的导数，隐式微分和参数曲线。积分学：积分的性质，三角和代数替换以及部分分式积分技术。

• 1recognise functions of a single, real variable including injective and bijective functions, composition of functions and conditions under which inverse functions can be defined;
• 2graphically represent and analyse key features of polynomial, circular, inverse circular and reciprocal functions;
• 3manipulate simple trigonometric identities and compound and double angle formulas for sine, cosine and tangent;
• 4calculate using the arithmetic of vectors in two and three dimensions, calculate scalar products, vector projections and resolutes, specify plane curves parametrically by a vector equation and determine the corresponding cartesian equations
• 5extend 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;
• 6evaluate integrals using algebraic and trigonometric substitutions, and partial fractions;
• 7apply integration techniques to problems including the area between curves and the solution of simple ordinary differential equations;
• 8calculate using the arithmetic of the extension of the real numbers to the set of complex numbers, including the representation of complex numbers in Cartesian and polar forms