Noteful
Mathematics

Partial Fractions

• Decompose rational functions into partial fractions
• Denominator will be up to squared terms and maximum three terms
• Numerator will be numerial or linear
• Use of factor theorem and polynomial division to form rational functions
• For Integration see later

Parametric Equations

• Understand and use parametric equations of curves
• Convert between cartesian and parametric forms
• Sketch simple parametric curves
• Use parametric curves in modelling in a variety of contexts

Calculus with Parametric Equations

• Be able to differentiate simple functions defined parametrically
• Find the gradient of a point on a curve and use to find the equations of tangents and normals, and solve associated problems

Binomial Expansion

• Understand and be able to use the binomial expansion for positive integers of n
• Coefficient notations
• Calculate binomial coefficents
• Relationship to Pascal's traingle
• Know that 0! = 1
• Understand and know the link to binomial probabilities (see probability section)

• Extend the binomal expansion for any rational n
• Write (a+bx)ⁿ as aⁿ(1 + ^bx⁄_a)ⁿ
• Know when the expansion is valid

Trigonometry

Inverse Functions

• Understand and use definitions of arcsin x, arccos x and arctan x and their relationship to sin x, cos x and tan x respectively
• Know their graphs and relate them to the graphs of sin x, cos x and tan x

Modelling

• Be able to use trigonometric functions to solve problems in context

Compound & Double Angle Formulae

• Understand and be able to use the double angle formulae and compound angle formulae
• Understand the geometric proofs of these formulae
• Be able to prove the double angle formulae
• Be able to construct proofs involving trigonometric functions and identities
• Use the formulae to solve trigonometric equations, prove identities or evaluate integrals

Reciprocal Functions

• Understand and be able to use the definitions of secant (sec x), cosecant (cosec x) and contangent (cot x)
• Understand and be able to use their relationship with sin x, cos x and tan x
• Understand the graphs of the functions, their ranges and domains
• Understand and be able to use the identities (see notes)
• Use the identities to solve trigonometric equations, prove identities or evaluate integrals

a cos x + b sin x

• Understand and be able to use expressions for acosx + bsinx in the equivalent form of Rcos(x±a) or Rsin(x±a).
• Sketch graphs of acosx + bsinx.
• Determine features of the graphs including minimum or maximum points.
• Solve equations of the form acosx + bsinx = c.

Connected Rates of Change

• Be able to differentiate using the chain rule, including problems involving connected rates of change and inverse functions.
• Be able to construct simple differential equations in pure mathemiatcs and in context e.g. kinematics, population growth and modelling the relationship between price and demand.

Inverse Functions

• Be able to differentiate using the chain rule, including problems involving connected rates of change and inverse functions.

Definite Integrals and Areas

• Be able to use a definite integral to find the area between two curves.
• This may include finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between two curves or between a line and a curve.
• This includes curves defined parametrically.

Partial Fractions and Integration

• Be able to integrate functions using partial fractions that have linear terms in the denominator.

Discrete Probability Distributions

• Understand and be able to use simple, finite, discrete probability distributions, defined in the form of a table or formula

Binomial Probability Distributions

Summary

Overview and Combining Forces

• Understand the concept and vector nature of a force.
• Identify the forces acting on a system and represent them in a force diagram.
• Understand and be able to use Newton's first law.
• Understand and be able to use Newton's second law for motion on a straight line of bodies of constant mass moving under the action of constant forces.
• Understand and be able to use Newton's second law in simple cases of forces given as 2D vectors.

Types of Forces and Equilibrium

• Be able to complete a diagram with the force(s) required for a given body to remain in equilibrium.
• Understand and be able to use the weight of a body to model the motion in a staight line under gravity.
• Understand the gravitational acceleration, g, and its value (may be assumed to take a constant value of 9.8ms_-2 but should be aware it is not a universal constant).
• Understand and be able to use Newton's third law.
• Use the concept that a system in which none of its components have any related motion may be modelled as a single particle.
• Understand and be able to use the concept of a normal reaction force.
• Be able to use the model of a smooth contact and understand the limitations of the model.
• Be able to use the concept of equilibrium together with 1D motion in a straight line to solve problems that involve connected particles and smooth pulleys.
• Be able to solve problems involving simple cases of equilibrium of forces on a particle in 2D using vectors, including connected particles and smooth pulleys.

Resolving Forces

• Be able to extend use of Newton's second law to situations where forces need to be resolved.
• Be able to extend use of Newton's third law to situations where forces need to be resolved (restricted to 2D).
• Understand the term resultant.
• Be able to use vector addition in solving problems involving resultants and components of forces.
• Be able to find and use perpendicular components of a force, for example to find the resultant of a system of forces or calculate the magnitude and direction of a force.
• Be able to solve problems involving the dynamics of motion for a particle moving in a plane under the action of a force or forces.

Friction, Inclined Planes, Contact Forces

• Be able to resolve forces for more advanced problems involving connected particles and smooth pulleys.
• Understand the concept of a frictional force and be able to apply it in contexts where force is given in vector or component form, or the magnitude and direction of force are given.
• Be able to represent contact forces between two rough surfaces by two components (the normal and frictional contact forces).
• Understand and be able to use the coefficient of friction and model of friction in 1D and 2D.
• Understand the concept of limiting friction.
• Understand and be able to solve problems regarding the static equilibrium of a body on a rough surface and solve problems regarding limiting equilibrium.
• Understand and be able to solve problems regarding the motion of a body on a rough surface.

Further Equilibrium

• Be able to use the principle that a particle is in equilibrium if and only if the sum of resolved parts in a given direction is zero.
• Problems may involve resolving forces including cases where it is sensible to resolve horizontally and vertically, resolve parallel and perpendicular to an inclined plane, resolve in directions to be chosen by the learner, or use a polygon of forces.