advertisement

C3 Curriculum Map Taught September to January Algebra Adding and subtracting algebraic fractions Multiplying and dividing algebraic fractions Division of polynomials A simpler method of division The quotient rule of differentation Differentiation of sinx and cosx The derivatives of sin f(x) and cos f(x) Angle x in degrees The derivatives of powers of sin x and cos x Trigonometric differentiation involving the product and quotient rules Differentiation of tan x, cot x, sec x and cosec x Functions Mappings Composite functions Inverse functions Graphs of functions and their inverse functions Self-inverse functions Even and odd functions The modulus function Sketching functions involving modulus signs Equations and inequalities with modulus signs Transformation of graphs Numerical methods Introduction to numerical methods Graphical solutions Intersection points on the x-axis Intersection points of two curves Change of sign Location of roots in an interval Possible problems with location of roots Methods of locating roots Iteration Illustrating iteration graphically Trigonometry The reciprocal functions: y = sec θ,y= cosec θ and y = cot θ Identities involving the reciprocal functions Inverse trigonometrical functions The addition formulae To prove sin (A + B) = sin A cos B + cos A sin B Deriving the other addition formulae The double angle formulae The half-angle formulae Extension: The factor formulae The form a cos θ + b sin θ Choice of R cos (θ ± α) or R sin (θ ± α) The functions ex and Inx The function e x and its inverse, ln x The function ex The function lnx Graphs of ex and In x Solving equations involving exponential functions and natural logarithms 5 Differentiation The chain rule Differentiation of powers of f(x) The use of dx/dy = 1/(dy/dx) Differentiation of the exponential function and natural logarithms Differentiation of functions of the type e f(x) Differentiation of ln x Differentiation of ln kx Differentiation of In (f(x)) Simplifying a logarithmic function before differentiating The product rule of differentiation Proof Terminology Proof by deduction Using algebra in proofs Proving identities Disproof by counter-example Proof by contradiction (reduction ad absurdum) Revision using past papers C3 Mock Examination C4 Curriculum Map Taught by two teachers from January Partial fractions Partial fractions Finding partial fractions: Denominator with distinct linear factors Finding partial fractions: Denominator with a repeated linear factor Finding partial fractions for improper fractions Binomial expansion Binomial expansion for any rational index Approximations Graphical illustration of approximations Further differentiation Implicit functions Implicit differentiation Tangents and normals Second derivatives of implicit functions Stationary points Connected rates of change Exponential and logarithmic functions Exponential growth and decay Exponential growth and decay of the form Aekt The derivative of a x The derivative of log a x Coordinate geometry Parametric form Graphs of curves defined parametrically Converting between parametric form and Cartesian form Parametric differentiation Integration Integration as the limit of a sum Solids of revolution Volume of revolution Integration of e x and 1/x Limits of the definite integral ∫ 1/x dx Integration by recognition of the type ∫k(f(x))nf'(x)dx Integrals of the type ∫kf’(x)ef(x) dx Integration by substitution Definite integrals and changing the limits Integrals of the type ∫ kf’(x) dx f(x) The definite integral ∫ 1 dx ax + b Integrals of the type ∫ cx + d dx ax + b Further integration Integration of trigonometric functions Integrals of sin x, cos x and sec2 x Displacement, velocity and acceleration Integration of related trigonometric functions Recognition of other standard trigonometric integrals Integrals of tan x and cot x Integration of even powers of sin x and cos x Integrals of the type ∫cos x sinn x dx and ∫ sin x cosn x dx Integration of odd powers of sin x and cos x Integrals of the type ∫ sin ax cos bx dx, ∫cos ax cos bx dx, ∫ sin ax sin bx dx Integrals of the type ∫ secn x tan x dx and ∫ tann x sec2 x dx Integration by parts Choice of u and dy/dx Special integral ∫ lnx dx Definite integration using integration by parts Integrals of the type ∫ eax sin bx dx and ∫ eax cos bx dx Integration using parameters Area enclosed by a curve Volumes of revolution Trapezium Rule Differential equations First-order differential equations First-order differential equations of the form dy/dx = g(x) First-order differential equations of the form f(y)dy/dx=g(x) First-order differential equations of the form f(y)dy/dx = 1 Forming differential equations Exponential growth and decay Vectors Vector geometry Vector notation Scalar multiplication Addition and subtraction of vectors Position vectors and geometrical applications Geometrical applications of vectors Scalar product The vector equation of a line Pairs of lines Revision using past papers C4 Mock Examination The general case of ∫ f’(x) when f(x) is any function f(x) Using partial fractions