# Pure

## Trigonometric Functions

Graphs Inverse Trigonometric Functions ● sin, cos and tan are all many-to-one functions ● Therefore the inverse functions would be one-to-many, and so therefore would not by definition be functions ● To solve this we restrict the domains of the trigonometric...

## Sequences and Series

Types of Sequence ● Progression → sequence or series ● Increasing sequence → each term greater than the previous term ● Decreasing sequence → each term less than the previous term ● Arithmetic sequence → difference between one term and the next is always the same ●...

## Parametric Equations

Definition of a Parametric Equation ● Cartesian equation - y = f(x) ● Parametric equation - x = f(t), y = g(t) ● Both x and y are defined in terms of a third variable (usually t or θ ) ● Parametric equations can be used for a complicated curve which doesn’t have a...

## Numerical Methods

Change of Sign Methods ● Sometimes there is no easy way of finding the roots of an equation by factorising (e.g x3 - 7x + 3 = 0 ) ● Alternative is to look at graph to find the interval where the roots lie (e.g between 2 and 3) ● Then try plugging in values of x...

## Functions

Language ● Mapping = rule which associates two sets of items ● Input or Object is something which is to be mapped to something else (the Output or Image) ● Set of possible inputs of mapping is called domain and the range is the set of outputs for a...

## Integration

Finding Areas ● Can find the area under a curve by integrating ● The same idea can be used to find the area between a curve and the y-axis ● Instead of integrating with respect to x, we are integrating with respect to y ● Therefore we use y-values as the limits and...

## Further Differentiation

Differentiating Functions of ‘y’ with Respect to ‘x’ ● Sometimes you want to differentiate ‘y2’ or sin y instead of a function of x ● To do this, follow the golden rule: Differentiating Implicit Functions ● Sometimes x and y are given as a jumbled equation rather...

## Differentiation

Concave Upwards and Concave Downwards Curves ● Relates to the second derivative ● Concave Upwards: Stationary Points of Inflection Non-Stationary Points of Inflection ● Some points of inflection are NOT stationary (ie dy/dx =/ 0 ) ● Either on an increasing function or...

## Differential Equations

Rates of Change Formulating Differential Equations ● You can use differential equations to model the rate of change of a variable ● If a is proportional to b then a = kb , and we can find k (the constant of proportionality) by plugging in known values of a and b ● We...

## Algebra

General Binomial Expansion If n is a positive integer there will be a finite number of terms (since eventually there will be a factor of 0 in the numerator)j ● This can be used for any values of n , although in all other cases this is an infinite series ● Can be used...