Core 3 Revision Sheet.doc

7/22/2019 Core 3 Revision Sheet.doc

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Core 3 Revision Sheet

Functions

The domainis the inputof the function

The rangeis the outputof the function

Modulus functions

x means the magnitude of x and is always positive.

When sketching modulus functions, the graph will be reflected in the x axis.

When solving modulus equations, remember to find the solutions when it is positive and

negative.

Example

olve 3x ! "

x # $ ! " %& x # $ ! '"

x ! () x ! '*

Composite Functions

When finding composite functions, apply the functions one at a time, starting with the

function closest to x.

Example

f+x ! x- $ g+x ! x /

0ind fg+x fg+x ! f+x /

! +x /- $

! x- ()x 1/ $

! x- ()x 12

To find the range of composite functions, find the range of the first function being applied

then use that as the domain of the next function being applied. This output will give the

range of the overall composite function.

Inverse Functions


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Remember that an inverse function can only be found if the original function is one-to-one.

This can be achieved by restricting the domain.

Example

f+x ! 1

32

+xx

find f 1 +x

let y !1

32

+

x

xy+x # ( ! 1x $

yx # y ! 1x $

yx # 1x ! $ y

x+y # 1 ! $ y

x !2

3

+

yy

Therefore f 1 +x !2

3

+

x

x

Drawing graphs of inverse functions

&eflect the original graph in the line y ! x

3ou may be asked to draw inverse graphs of exponential, natural logarithms and

trigonometric graphs. 4ake sure you are familiar with them.

Differentiation

5hain &ule dx

du

du

dy

dx

dy

=


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6roduct &uledx

duv

dx

dvu

dx

uvd+=

)(

7uotient &ule

2v

dx

dvu

dx

dud

v

u

dy

d =

' this is in your formula book

x as a function of ydx

dy !

Integration

Integration b substitution

determine your value for u

0ind and rearrange it into the form dx ! du

ubstitute du in for dx and simplify where necessary

8f necessary, substitute any x values in terms of u

9ow integrate your new function and substitute x back in at the end if required

8f it is a definite integral, you must either rearrange the limits for u or put the answer

in terms of x before substituting in the limits of x.

Example

dxx

x +23

22

u ! $x- 1dx

du!:x dx !

x6

1du substitute this into the integral

and cancel

u

x2x

x6

1du

u3

1du ! 2

1

3

1

u du ! 32

u 21

c %& u3

2 c

in terms of x 3

223 2 +x c

Integration b parts

dxdx

duvuvdx

dx

dvu = this is in your formula book


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;etermine your value for u and your value fordx

dv

HINT the value for u should get simplerwhen differentiated

0inddxdu and v

ubstitute the values into the formula which should leave you with a simpler integral

at the end to work out. 8f not, integrate by parts again.

Example


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Exponentials and %ogarithms

4ake sure you recognise and could sketch the exponential and natural logarithm graphs,

including transformations of them.

;ifferentiating exponential functions= y ! e kxdx

dy! ke kx

8ntegrating exponential functions= ! c

;ifferentiating natural logarithms= y ! ln xdx

dy!x

1

8ntegrating natural logarithms= dx ! ln x c

pecial 0unctions dx ! ln +f+x c, if f+x > )

&rigonometr

Reciprocal Functions

cosec

sin

1= sec

cos

1= cot

tan

1=

3ou must be able to sketch the graphs of these functions

&rigonometric Identities

tan !

cossin cot !

sincos


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sin +2 cos 2 ! ( +? cos sec 2 ! ( tan 2

sin +2 cos 2 ! ( +? sin cosec 2 ! ( cot 2

3ou must learn all these identities.

Inverse &rigonometric Functions

3ou must be able to recognise and sketch the following inverse trigonometric graphs=

y ! sin 1 x for '(@ x @ ( y ! cos 1

x for '(@ x @ ( y ! tan 1 x for

'A @ x @ A

&emember, the domain and range are the opposite for the inverse functions.

The domain must be restricted in to make it a one'to'one function in order to be able to find

its inverse.

Differentiating &rigonometric Functions

xxdx

dcos)(sin = xx

dx

d 2sec)(tan =

xxdx

dsin)(cos =

Integrating &rigonometric Functions

ckxk

kxdx += sin1

cos cxk

kx += tan1

sec2

ckxk

kxdx += cos1

sin

'ou must learn all of the differentiation and integration trigonometric functions

(umerical methods and solutions

Change of sign method

The equation must be e)ual to *erobefore the change of sign method can be used.

Example

how that the equations y ! x- ( and y !x

1have a solution in the interval ).:2 and ).:B.

0irst make the equations equal to each other.

x- ( !x

1


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4ake it equal to Cero

x- ( 'x

1! )

&earrange if necessary

multiply by x

xD x # ( ! )

f+).:2 ! ).:2D ).:2 # ( ! ').))//:2 f+).:B ! ).:BD ).:B # ( ! ).)(2/)B

change of sign, therefore root in interval ).:2 E x E ).:B

Iterative Methods

This involves programming your calculating so that the process is repeated to find a

sequence of approximations to the required solution +converges to the solution.

3ou need to be able to decide if the convergence produces a cobweb or staircase diagram,

and be able to sketch it.

Mid!ordinate rule

++++=

b

a nnyyyyhydx

2

1

2

3

2

3

2

1 ..... where h !n

ab

Simpson+s rule

odds evens

[ ])...(2)...(43

2421310 +++++++++= nnnb

ayyyyyyyy

hydx

These formulae are both in your formula book.

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Core 3 Revision Sheet.doc