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