Modular Mathematics: Pure Mathematics 2 Module B

Functions: polynomial functions; rational functions; exponential functions; periodic/continuous functions; the modulus of a function. Curve sketching: curve sketching using transformations; the graph of the reciprocal of a function. Algebra: partial fractions; the cover-up method; logarithms; the laws of logarithms; exponential and logarithmic equations. Straight lines and circles: the angle between two lines; the distance of a point from a straight line; loci; the equation of a circle; tangents to circles; touching circles. Trigonometric identities and equations: the trigonometric functions; general solutions of trigonometric equations; compound angle identities; double angle identities; half-angle identities. Inequalities: use of graphical methods to solve inequalities; solving rational inequalities; problems involving the range of a function. Consolidation A. Differentiation of products and quotients: standard derivatives and stationary points; the second derivative and its use in distinguishing between different types of stationary points; differentiating a product of functions; differentiating a quotient of functions. Further trigonometric identities: the factor formulae. The exponential and logarithmic functions: the definition of e and the exponential function; natural logarithms; the logarithmic function and its derivative. Series: finding the sum of a number series using the method of differences; the natural number series; the sum to infinity of a number series; the binomial theorem for any real value of n; the factorial notation; approximations. Reduction of a relationship to a linear law. consolidation B. Trigonometric functions: expressing a cos 0 + b sin 0 as r sin (0+-a); solving the equation a cos 0 + b sin 0 = c; solving the equation cos A = cos B; inverse trigonometric functions; approximation for sin 0 cos 0 and tan 0 when 0 is small. Differentiation of trigonometric functions: deriavatives of the trig functions and the inverse trig functions. Differentiating implicit and parametric functions: differentiating implicit functions; logarithmic differentiation; differentiating a x; sketching and finding the gradient function of a curve with parametric equations. Co-ordinate geometry and curves: intersection of curves and the significance of coincident points of intersection; the cartesian and parametric equations of the parabola; the ellipse and the rectangular hyperbola. Consolidation C: integration: standard results; integrating by recognition; integration of products by recognition and by change of variable; integration by parts. Further integration: integrating fractions by recognition; by substitution and by using partial fractions; the use of partial fractions in differentiation; integrating compound trigonometric functions.