# Core 4 Notes Edexcel

Notes for Core 4 maths, done for the edexcel syllabus but should be helpful with other exam boards.

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- Created on: 07-06-11 19:40

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## Pages in this set

### Page 1

Par

ti

al

Fract

ions

(x-3)(2x-1)

3x+5

An

algebr

aic

fr

act

ion

such

as

can

be

broken

down

int

o si

mpl

er

par

ts

A B

cal

led

par

ti

al

fr

act

i

o ns

so

that

i

t

is

in

the

for

m (x-3) + (2x-1)…

ti

al

Fract

ions

(x-3)(2x-1)

3x+5

An

algebr

aic

fr

act

ion

such

as

can

be

broken

down

int

o si

mpl

er

par

ts

A B

cal

led

par

ti

al

fr

act

i

o ns

so

that

i

t

is

in

the

for

m (x-3) + (2x-1)…

### Page 2

must be divided first

to

obt

ain

a number

and

a pr

oper

f

ract

ion

bef

ore

it

can

be

exp r

ess ed as

p a r

ti

al

fr

act

ions

Coor

dinat

e Geomet

ry

A

pa r

ame t

ri

c equation

of a

curve…

to

obt

ain

a number

and

a pr

oper

f

ract

ion

bef

ore

it

can

be

exp r

ess ed as

p a r

ti

al

fr

act

ions

Coor

dinat

e Geomet

ry

A

pa r

ame t

ri

c equation

of a

curve…

### Page 3

Bi

nomi

nal

Expansi

on

The

previ

ous ver

sion

of

the

binomial

t

heorem onl

y wor

ks

when

n i

s a

posi

ti

ve

i

nteger

.

If

n

is

any f

ract

ion,

t

he bi

nomial

theor

em becomes:

2 3

n(n-1)x

(1 + x)n…

nomi

nal

Expansi

on

The

previ

ous ver

sion

of

the

binomial

t

heorem onl

y wor

ks

when

n i

s a

posi

ti

ve

i

nteger

.

If

n

is

any f

ract

ion,

t

he bi

nomial

theor

em becomes:

2 3

n(n-1)x

(1 + x)n…

### Page 4

I

mpl

ici

t

D i

ff

erent

iat

ion

N orma l

ly,

w he n

dif

ferent

iat

ing,

it

i

s deal

ing

w i

th

`expl

ici

t

funct

ions'

of

x,

where

a

valu e

of y

is

d efi

ned only

in

terms of…

mpl

ici

t

D i

ff

erent

iat

ion

N orma l

ly,

w he n

dif

ferent

iat

ing,

it

i

s deal

ing

w i

th

`expl

ici

t

funct

ions'

of

x,

where

a

valu e

of y

is

d efi

ned only

in

terms of…

### Page 5

x

Di

ff

erent

iat

inga

Thi

s f

unct

ion

descr

ibes

growth

and decay,

and

it

s der

ivat

ive

giv

e s

a

measur

e of

t

he r

ate

of

change

of

thi

s gr

ow t

h/decay.

Si

ncey = ax

,

taki

ng

logs…

Di

ff

erent

iat

inga

Thi

s f

unct

ion

descr

ibes

growth

and decay,

and

it

s der

ivat

ive

giv

e s

a

measur

e of

t

he r

ate

of

change

of

thi

s gr

ow t

h/decay.

Si

ncey = ax

,

taki

ng

logs…

### Page 6

The

key

to

d oi

ng t

he se

probl

ems i

s t

o i

denti

fy

thr

ee comp onent

s and

w r

it

e t

hem

down

ma t

hema ti

call

y:

Wh at

you

a r

e gi

ven

Wh at

i

s r…

key

to

d oi

ng t

he se

probl

ems i

s t

o i

denti

fy

thr

ee comp onent

s and

w r

it

e t

hem

down

ma t

hema ti

call

y:

Wh at

you

a r

e gi

ven

Wh at

i

s r…

### Page 7

f (ax + b)dx = 1

a f (ax + b) + C

Usi

ng

the

rever

se

of

the

chai

n r

ule,

t

he

fol

lowi

ng

gener

ali

sat

ions

can

be

found:

n

(ax+b)

(ax + b)n dx = 1

a n+1 +C

eax+b dx =…

### Page 8

I

f

necessar

y,

use

u=f(

x)

to

change

the

val

ues f

or

the

li

mit

s of

i

ntegr

ati

on

Put

your

x'

s

back

in

agai

n at

t

he end

and

fi

nish

up.

Exampl

e

3

Suppose

we

wis

h

to

fi

nd

(9…

f

necessar

y,

use

u=f(

x)

to

change

the

val

ues f

or

the

li

mit

s of

i

ntegr

ati

on

Put

your

x'

s

back

in

agai

n at

t

he end

and

fi

nish

up.

Exampl

e

3

Suppose

we

wis

h

to

fi

nd

(9…

### Page 9

dv

Car

e must

be

taken

over

t

he

choi

ce

of

uand

dx .

The

aim

is

to

ensur

e t

hat

i

t

is

si

mpl

er

to

int

egr

ate

thanv du

dx .

So

choose

u t

o be

easy

to

dif

fer

ent

iat

e…

Car

e must

be

taken

over

t

he

choi

ce

of

uand

dx .

The

aim

is

to

ensur

e t

hat

i

t

is

si

mpl

er

to

int

egr

ate

thanv du

dx .

So

choose

u t

o be

easy

to

dif

fer

ent

iat

e…

### Page 10

Di

ff

erent

ial

Equat

ions

I

n gener

al

a di

ff

erent

ial

equat

io n

ma y ha ve x an d

y t

erms

on bot

h si

des,

but

i

f

the

dy

equat

ion

is

of

a

cert

ain

form

f (x)g(y) dx =…

ff

erent

ial

Equat

ions

I

n gener

al

a di

ff

erent

ial

equat

io n

ma y ha ve x an d

y t

erms

on bot

h si

des,

but

i

f

the

dy

equat

ion

is

of

a

cert

ain

form

f (x)g(y) dx =…

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