FP2 MEI Full Revision Notes

My notes on FP2 for MEI. All chapters included.

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  • Created by: Caraa
  • Created on: 22-05-13 13:46
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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 functions/ powers and integrate/ differentiate as
normal.
In this chapter you will also need to differentiate and integrate with inverse trigonometric
functions. It is important to know the properties of these functions in order to be able to
find correct solutions:

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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 differentiate, rules such as the chain and product may need to be used along with
the general solutions from the formula booklet. Partial fractions may sometimes be
needed.…read more

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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 is taken anticlockwise.
You will need to be able to covert between polar and cartesian co-ordinates.…read more

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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.…read more

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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 z = a + bj, you can work out r and using
pythagorus and trionometry.…read more

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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 equate only the real parts (the
ones without j):
The question wanted us to express cos 5 in terms of cos , so we must use
and subsitute this in.…read more

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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+...…read more

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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 it is possible to centre them somewhere else.
The equation you will use for this is:
The exam questions on this chapter will most likely need you to use knowledge from
chapters 1 or 6.…read more

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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 area scale factor, and can be worked out as ad ­ bc
for a 2 x 2 matrix, ( ).
If the determinant is equal to 0, the matrix is said to be singular.…read more

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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 columns leave the value of the determinant unchanged
| | | |
The determinant of a 3 x 3 matrix is the volume scale factor
Det(MN) = Det(M) x Det(N)
Interchanging rows and columns has…read more

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