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Chapter 1 - Calculus
Calculus rules from A level

The product rule
The quotient rule
Integration by parts

Differentiation and integration of trigonometric functions




Trigonometric identities




Differentiation and integration of trigonometric functions

When given a basic integral (One containing tan, sin, cos, cosec, sec, cot), use
identities to substitute unwanted…

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arcsine: arccosine: arctangent:




Domain: All real numbers
Range:

Differentiating and integrating these functions is easy since their general solutions are in the
formula booklet.

To integrate you must first get the integral into the correct form. This may require
taking out numbers or methods such as completing the square.

To…

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Chapter 2 ­ Polar co-ordinates
In this chapter polar co-ordinates are used instead of Cartesian. In this system, r represents
the distance from a fixed point and the angle between the initial line and a line in a fixed
direction.




Polar co-ordinates are written as (r, ) where the angle…

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Cardioid


a 0, b 0

| |




Rose curves


a 0, n 0

When n is odd, there are n
petals.
When n is even there are 2n
petals.
Lemniscate


a0




Spiral

n0



The area of a sector

The area of a sector can be found very simply by using…

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Chapter 3 ­ Complex numbers
Argand diagrams

A complex number can be represented on an argand diagram with modulus (length) r and
an argument (angle) of measured anticlockwise from the positive real axis. The principle
argument of z is that where .




When given a complex number in the form…

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De Moivre's theorem

De moivre's theorem states that

Example 1: Express cos 5 in terms of cos
We need to expand the RHS binomaly

(abbreviating sin and cos as s and c makes this expansion easier to manage)

Since we only need to find cos 5, we now need to…

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Roots of unity

Roots of unity are solutions to zn = 1
They are for k = 0,1,2,... n-1
They are equally spaced around a argand diagram

If we call the first complex number then the other solutions are 2,3,...n-1 and the sum
of the roots is 0.

1++2+...+n-1 =…

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Chapter 4 ­ Power series
Maclaurin expansions are using to find an approximation of a function. The general formula
for these expansions is:


The Maclaurin expansions can be calculated for many different functions. Some of which are
in the formula book.

Taylor approximations

Maclaurin expansions are centred on x=0 but…

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Chapter 5 - Matrices
Recapping from FP1

The order of a matrix is rows x columns e.g. ( ) is a 3 x 2

Matrices represent transformations

( ) ( ) are the identity matrices for 2 x 2 and 3 x 3 matrices

respectively.
The determinant (DetM) is the…

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Properties of the determinant

There are several important properties of the determinant that you need to know:

Swapping two columns of a determinant reverses its sign
| | | |
If two columns or rows of the determinant are the same then the matrix is singular
Cyclic changes of the…

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