IB Diploma Program Mathematics Course Companion Standard Level (SL) [1 ed.] 0198390114, 9780198390114, 9780199129355

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v

Contents

Chapter



Functions

1.1

Introducing

1.2

The

functions

domain

Cartesian

and

Function

1.4

Composite

1.5

Inverse

1.6

Transforming



range

4

of

a

relation

on

plane

1.3

Chapter



notation

functions

functions

functions

Quadratic

functions

a

Solving

quadratic

2.2

The

2.3

Roots

2.4

Graphs

2.5

Applications

quadratic

of

equations

Patter ns

6.2

Arithmetic

13

6.3

Geometric

14

6.4

Sigma

16

6.5

Arithmetic

21

6.6

Geometric

6.7

Convergent

6.8

Applications

6.9

Pascal’s

of

quadratic

of

and

(Σ)

and

sequences

162

sequences

164

sequences

notation

arithmetic

167

and

series

170

series

172

series

175

series

of

and

sums

geometric

to

infinity

178

and

patter ns

triangle

181

and

the

binomial

expansion

38

equations

sequences



6.1

34

formula

quadratic

Patterns,

series



2.1



8

and

equations

Chapter

184

41

functions

quadratics

43

Chapter

53

7.1



Limi ts

Limits

and

and

derivatives



convergence

196

n

Chapter



Probabi li ty



3.1

Definitions

64

3.2

Venn

68

3.3

Sample

diagrams

product

space

3.4

Conditional

3.5

Probability

Chapter



diagrams

and

the

r ule

probability

tree

diagrams

Exponential

and

Exponents

Solving

4.3

Exponential

4.4

Proper ties

4.5

Logarithmic

4.6

Laws

4.7

Exponential

4.8

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of

exponential



logarithms

functions

logarithms

logarithmic

Chapter

equations

and

of

functions

Rational

5.1

Reciprocals

5.2

The

5.3

Rational

reciprocal

functions

function

functions

7.4

The

chain

line

for

and

derivative

of

x

200

derivatives

r ule

and

higher

208

order

derivatives

Rates

7.6

The

85

7.7

More

of

215

change

derivative

on

and

and

extrema

motion

in

a

line

221

graphing

and

230

optimization

problems

Chapter



240

Descriptive

8.1

Univariate

analysis

103

8.2

Presenting

data

107

8.3

Measures

of

109

8.4

Measures

of

115

8.5

Cumulative

118

8.6

Variance

equations 127

and

Chapter

statistics



256

257

central

tendency

260

dispersion

267

frequency

and

standard

271

deviation

276

Integration

9.1

Antiderivatives

9.2

More

131







and

the

indefinite

integral

291

on

and

indefinite

9.3

Area

142

9.4

Fundamental

143

9.5

Area

147

9.6

Volume

of

9.7

Definite

integrals

other

vi

r ules

122

logarithmic

exponential

tangent

7.5

logari thmic

functions

of

More

77



4.2

The

7.3

89

functions

4.1

7.2

definite

between

integrals

297

integrals

Theorem

two

of

302

Calculus

cur ves

313

revolution

problems

with

309

318

linear

motion

and

321

Chapter



Bivariate

10.1

Scatter

10.2

The

10.3

Least

10.4

Measuring

analysis



diagrams

line

of

best

squares

fit

regression

correlation

Chapter



334

15.1

Random

15.2

The

binomial

345

15.3

The

normal

Right-angled

triangle

11.2

Applications

of



trigonometr y

right-angled

Using

the

363

triangle

trigonometr y

11.3

369

coordinate

variables

axes

in



16.1

About

16.2

Inter nal

the

16.3

How

16.4

Academic

the

Record

16.6

Choosing

11.4

The

sine

380

16.7

Getting

11.5

The

cosine

11.6

Area

11.7

Radians,

Chapter



r ule

r ule

Vectors:

triangle

12.2

Addition

and

12.3

Scalar

12.4

Vector

12.5

Application

Chapter

and

basic

391



of

of

the

13.3

Trigonometric

13.4

Graphing

13.5

Translations

of

a

line

unit

using

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circle

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Chapter



Using

a

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

1

Functions

2

Differential

3

Integral

4

Vectors

5

Statistics

572

calculus

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598

606

608

and

probability

612

stretches

sine

with

Prior

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

Number

633

2

Algebra

657

448

3

Geometr y

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454

4

Statistics

699

Chapter



Practice

papers



Practice

paper

1

708

469

Practice

paper

2

712

478

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

483

Index



sine

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Calculus



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of

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

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

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456

functions

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unit

identities

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564

420

437

circle

circular

cosine

563

topic

430

functions

equations

trigonometric

cosine

vectors

vectors

Circular

Solving

keeping

a

562

562

426

equation

Using

marked

Honesty

557

407

subtraction

13.2

13.7

sectors

concepts

13.1

13.6

389

product



556

criteria

is



386

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12.1

538

exploration

exploration

16.5

arcs

527

Exploration

assessment

373

a



520

distribution

distribution

The

trigonometr y

of

distri butions

349

Trigonometry

11.1

Probabi li ty

339

Chapter

Chapter



wi th

trigonometric

functions



14.1

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14.2

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14.3

Integral

14.4

Revisiting

of

practice

of

trigonometric

functions

withderivatives

sine

and

linear

cosine

motion

496

500

505

510

vii

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paper

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Simultaneous and quadratic equations

1.5

Tolerance

Asian

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simultaneous

linear

equations

European

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30

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50

40

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20

80

Practice paper 2

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

Functions



CHAPTER

OBJECTIVES:

2.1

Functions:

2.2

Graphs

of

domain,

range,

functions,

by

composite,

hand

and

identity

using

GDC,

and

inverse

their

functions

maxima

and

minima,

−1

asymptotes,

the

T ransformations

2.3

graph

of

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f

(x)

graphs,

translations,

reections,

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start

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coordinates.

Plot

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1

y

a

2

the

check

Plot

these

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points

B(5, −3),

on

a

coordinate

C(4, 4),

plane.

D(−3, 2),

D

C 1

points

A(4, 0),

B(0, −3),

E(2, −3),

F(0, 3).

y

A

C(−1, 1)

and

0

D(2, 1)

–2

x 1

–1

3

2

b

4

Write

down

2

the

A

–1

on

a

coordinate

plane.

coordinates

–2

1.5

of

E

H

B –3

points

A

to

1

H

–4 0.5

2

Substitute

e.g.

values

into

an

expression.

D

C

B

0

Given x = 2, y = 3 and z = −5,

–2

x

–1

1

2

3

0.5 2

find

the

value

of

a

4x

+

2y

y

b

−

3z –1

4x

a

+

2y

=

4(2)

+

2(3)

=

8

+

6

=

G

14 –0.5

2

y

b

2

−

3z

=

(3)

−3(−5)

=

9

+

15

=

24 F –2

3

Solve

linear

equations. 2

e.g.

Solve

6

4x

6

−

4x

=

Given

that

x

=

4,

y

=

6

and

z

=

−10,

find

0 

 

2

a

−

=

0

⇒

6

=

4x

+

3y

z

b

−

3y

y

c

−

z

d

4x  3

1.5

=

x

⇒

x

=

1.5

Solve

y

 6

4

Use

your

GDC

to

a

3x

−

6

=

6

5x

b

+

7

=

−3

c

   

graph  4

a

function.

4

Graph

these

functions

on

your

GDC

2

e.g.

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f (x)

=

within 0 –6

2x

−

1,

–3

≤

x

≤

–4

–2

the

given

domain.

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x 2

4

6

3

sketch

the

functions

on

paper.

–4

a

y

=

2x

b

y

=

10

−

3,

−4

≤

x

≤

7

–6

−

2x,

−2

≤

x

≤

5

–8

2

5

Expand

e.g.

linear

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(x

binomials.

+

3)

(x

−

2)

c

5

y

=

x

–

3,

–3

≤

x

≤

3.

Expand

2

=



x

+

x

Functions

−

6

a

(x

+

4)

(x

+

5)

c

(x

+

5)

(x

−

4)

b

(x

−

1)

(x

−

3)

The

Inter national

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Station

(ISS)

has

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the

Ear th

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International

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over

15

have

times

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philosopher

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is

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mathematical

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called

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and

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how

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a He

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Chapter



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handshakes

complete

Number

people

are

this

at

business

meeting.

If

there

there

there

for

are

4

meetings

are

3

2

to

people

shake

there

handshakes

and

is

so

on.

people?

table.

Number

of Y ou

of

people

might

nd

it

helps

handshakes to

tr y

this

out

with

a

2 group

3

in

of

your

friends

class.

4

5

6

7

8

Do

not

join

the

points

9

in

this

case

as

we

10 are

dealing

with c

the

Plot

points

on

a

Car tesian

coordinate

plane

with

the

whole

number

numbers.

of

people

Write

d

a

the

formula

number

of

Relations

Distance

on

x-axis

for

people,

and

(m)

and

the

the

number

number

of

T ime

The

(s)

200

34

300

60

400

88

way

➔

A

There

of

and

is

showing

data

is

nothing

provided

In

other

that

Functions

in

a

a

table

time

within

set

of

it

shows

takes

words,

order.

brackets

all

any

numbers

cer tain

is

(400, 88).

ordered

at

r un

information

and

specific

special

these

this

(300, 60)

enclosed

relation

relation.



of

(200, 34),

pieces

comma

H,

in

the

terms

y-axis.

of

the

the

for

a

amount

student

15

to

two

handshakes,

on

n

of

(100, 15),

handshakes

functions

100

Another

of

as ordered pairs:

Each

They

in

distances.

the

are

ordered

separated

form

pair

by

a

(x, y).

pairs.

about

group

come

the

of

in

numbers

numbers

pairs.

that

is

a

are

in

relation

a

has

only

(discrete)

➔

The

is

domain

ordered

pairs.

domain

of

the

set

of

all

the

first

numbers

(x-values)

of

the

The

The

the

ordered

pairs

above

is

{100,

200,

300,

curly

mean

➔

The

range

is

the

set

of

the

second

numbers

brackets,

{

},

400}.

( y-values)

in

'the

set

of'.

each

pair.

The

range

of

Example

Find

the

the

ordered

pairs

above

is

{15,

34,

60,

88}.



domain

a

{(1, 4),

(2, 7),

b

{(−2, 4),

and

range

(3, 10),

(−1, 1),

of

these

relations.

(4, 13)}

(0, 0),

(1, 1),

(2, 4)}

Answers

a

b

The

domain

is

The

range

{4,

The

The

is

domain

range

is

is

{1,

2,

7,

10,

{−2,

{0,

1,

3,

−1,

4}

First

13}

0,

1,

2}

4}

elements

Second

elements

Do

repeat

not

there

are

ordered

➔

A

of

function

the

element

be

a

is

a

domain

of

the

function

mathematical

of

the

range

no

two

of

the

is

pairs

in

ordered

the

values

4s

such

and

that

associated

function.

ordered

two

the

pairs

ordered

even

two

pairs

though

1s

in

the

pairs.

relation

function

in

In

may

each

with

order

have

for

the

element

exactly

a

one

relation

same

to

first

element.

Example

Which

of



these

sets

of

ordered

pairs

a

{(1, 4),

(2, 6),

(3, 8),

(3, 9),

(4, 10)}

b

{(1, 3),

(2, 5),

(3, 7),

(4, 9),

(5, 11)}

c

{(−2, 1),

(−1, 1),

(0, 2),

(1, 4),

are

functions?

(2, 6)}

Answers

a

Not

a

function

number

3

because

occurs

twice

the

in

the

domain.

b

A

function;

elements

c

A

function;

elements

all

are

all

are

of

the

first

different.

of

the

first

different.

Note

some

that

of

it

the

doesn’t

matter

y-values

are

that

the

same.

Chapter





Exercise

1

2

Which

A

of

these

sets

a

{(5, 5),

b

{(−3, 4),

c

{(4, 1),

d

{(−1, 1),

(0, 3),

e

{(−4, 4),

(−4, 5),

f

{(1, 2),

For

each

whether

(4, 4),

(3, 3),

(−1, 6),

(4, 2),

(2, 2),

ordered

(2, 2),

(0, 5),

(4, 3),

(2, −1),

relation

is

a

(2, 8)}

(−2, 8)}

(5, 2)}

the

domain

and

range

and

say

function.

y

a

functions?

(3, −1)}

(−3, 7),

(4, 2),

identify

are

(4, 5)}

(1, 7),

(−3, 6),

(3, 2),

pairs

(1, 1)}

(4, 4),

(1, 6),

diagram,

the

of

y

b

2 2

Write

down

the

1 1

coordinates

x

0 1

2

3

as

x

0 1

–1

2

ordered

3

pairs.

4 –1

3

Look

it

back

takes

for

between

The

a

at

a

can

Y ou

can

relation

student

line

represent

use

is

a

table

distance

vertical

Y ou

the

the

on

to

r un

traveled

4

that

cer tain

and

shows

the

distances.

time

taken

a

Is

amount

the

of

time

relationship

function?

test

relations

ver tical

function

page

or

and

line

not,

test

by

functions

to

on

a

determine

drawing

Car tesian

whether

ver tical

lines

a

plane.

par ticular

across

the

Car tesian

graph.

and

the

plane

➔

A

relation

intersect

line

is

the

a

function

graph

if

more

any

than

ver tical

once.

line

This

drawn

is

called

will

not

after

the vertical

René

test

Example

Which

of

a

coordinates

Car tesian

are

named

Frenchman

Descar tes

(1596 – 1650).



these

relations

y

are

functions?

b

y

y

c

y

=

|x|

0 0

x

0

{



Functions

x

x

Continued

on

next

page

Answers

a

y

b

c

y

y

Crosses

0

a

A

function

Exercise

1

0

x

Which

A

b

x

x

0

function

Not

c

a

twice

function

B

of

these

relations

a

are

functions?

b

y

c

y

y

Draw,

or

3

imagine, 2

ver tical 1

lines

0

x

x

0

x

on

the

–1

graph.

d

e

y

f y

If

the

a

‘solid

function

has

y 2

indicates

1

value

x

0

dot’

0

x

0

is

•,

that

this

the

included

in

x

2

the

function.

–1

If

the

function

has

a

–2

‘hollow

dot’

,

this

°

indicates

value

the

g

y

h

is

that

not

the

included

in

function.

i y

y 3

2

1

0

x 1

–1

–2

2

3

4

5

2

2

1

1

0 –4

–3

–2

x

0

–1

x 1

–1

–1

–2

–2

Chapter





Use

2

your

GDC

to

sketch

these

straight

line

graphs. Indicate

a

y

=

x

e

Are

f

Will

y

b

=

x

+

2

y

c

=

2x

−

3

y

d

=

x-

they

all

all

functions?

straight

lines

Explain

be

your

functions?

and /or

the

region

y




6

} Inter val

often

notation

considered

efcient

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set

of

x-values

such

that

x

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less

than

Set

Description

builder

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notation

notation

are

+∞)

x

is

greater

than

–2

{x : x

>

the

4]

x

is

less

x

lies

than

or

equal

to

4

{x : x

≤

3)

between

including

(–∞,

5)

∪

[6,

x

+∞)

is

less

than

or

−3

−3

but

than

5

equal

to

and

not

or

3

{x : −3

≤

x


0,

Logarithmic

x

exponents

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x

x

of

log

x



e

functions

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x

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

lnx ,

x

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a

Relationship

x

between

x ln a

these

log

x

functions

x a

a

=

e

,

log

a

=

x;

a

=

x,

x

> 0

a

x

2.7

Solving

2.8

Applications

Before

Y ou

1

equations

you

should

Evaluate

of

the

graphing

form

a

skills

x

=

and

b,

a

y

=

b

solving

equations

to

real-life

situations

start

know

simple

of

how

positive

to:

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exponents

1

check

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4

e.g.

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⎛



⎛

⎞

a

4

3





3

=

3

×

3

×

3

×

3

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⎜

81

 ⎞

⎝



⎟

⎜

⎠

⎝

⎟ 

⎠

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⎛

e.g.

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3

⎜

⎟ 

⎝

c

0.001

⎠

 

 







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



2





 



  





Conver t

numbers



to

exponential

form

2

State

e.g.

Find

n

given

=

2

the

value

of

n

in

n

n

128

a

7

these

equations.

n

=

343

=

625

b

3

=

243

7

128

=

2

so

n

=

n

7

c

5

2

3

Transform

graphs

3

Transform

the

graph

2

2

e.g.

Given

the

graph

of

y

=

x

sketch

2

graph

of

y

=

x

+

3

y 2

y

=

x

+

3

8

6

4

2 2

y

=

x

x –3



–2

–1

0

Exponential

1

2

3

and logarithmic functions

the

graph

of

y

=

(x

− 2)

of

y

=

x

to

give

the

Facebook,

the

social

y

media

Facebook

users

600

giant,

celebrated

with

more

up

million

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Febr uar y

than

from

in

450

huge

200

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from

0

when

90-ceD

80-ceD

70-ceD

million

60-ceD

one

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there

400

40-ceD

2004

100

2008

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2010

stinU

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http://www .facebook.com/press/info.

users

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growth

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the

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A

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could

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number

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millions,

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to

predictions

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could

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the

equation

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future

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estimate

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to

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the

date

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‘extrapolation’.

are

and

as

will

its

you

come

across

opposite,

move

many

other

exponential

along

the

examples

decay

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of

exponential

the

gradient

growth

decreases

a

model

to

cur ve).

Malcolm

Imagine

again,

How

Gladwell

taking

until

high

1

Fold

2

For

a

you

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the

have

sheet

number

can

of

this

piece

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think

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large

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each

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the

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size)

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to

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many

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times

the

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the

paper

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is

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rst

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few

Number

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Number

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of

have

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done

Thickness

(km)

As

thick

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0

1

1

×

10

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of

paper

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1

2

2

×

10

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4

4

×

10

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8

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6

7

8

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How

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than

would

you

height

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a

height

paper

to

make

the

table?

of

be

and

a

man?

after

and logarithmic functions

50

folds?

paper

of

possible.

folds,

formed.

km.

entries

as

mm

−7

that

Point

again

be?

show

thickness

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book

folding

would

in

table

the

his

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times.

stack

this

and

that

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paper ,

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nal

complete

paper

problem

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it

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this

estimate

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with

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the

reached. problems

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of

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called

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before

you

can’t Does

fold

the

paper

any

more.

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seven

folds

the

paper

is

already

about

how

thick

the

man.

a

50

thick.

of

a

table

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folds

This

Paper

17

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of

fact

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folds

after

after

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about

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the

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paper

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only

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it

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much

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than

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two-storey

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would

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a

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the

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a

function

of

Ear th

the

The

113

to

the

g rowth.

ter ms

number

million

of

km

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‘number

in

folds,

n,

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n

f

(n)

f

(n)

In

=

2

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this

their

exponential

chapter

inverses,

.

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growth

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lear n

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more

called

about

exponential

functions

and

functions .

logari thmic

Exponents

Exponents

are

multiplication

a

shor thand

of

a

number

way

by

of

representing

the

repeated

itself.

5

The

expression

The

3

in

this

,

3

for

example,

expression

is

the

represents

base

3

number

×

3

and

×

3

the

×

5

3

is

×

3.

the

exponent.

Other

names

for

exponent

are

power

and

index

4

It

Y ou

can

also

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a

variable

as

the

base,

for

is

quicker

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x

example, than

x

×

x

×

x

×

x

4

x

=

x

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x

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of

x

×

x

exponents

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5

5

x

3

x

×

x

=

(x

×

x

=

x

=

x

Simplify

3

×

x

×

x

×

×

x

x

×

×

x

x

×

×

x)

x

×

×

x

(x

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x

x

×

×

x)

Remove

brackets.

x

8

5

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x

3

×

x

(5 + 3)

=

m

➔

a

x

n

×

a

8

=

x

m+n

=

a 5

Notice

that

the

two

are

the

Y ou

3

×

x

variables

simplify

3

×

y

using

5

x

base

x

same.

cannot

5

x

in

,

for

this

3

×

y

example,

law.

5

=

x

3

y

Chapter





Division Cancel

5

Simplify

÷

x

   5

factors.

 

   





 

 

3

x

÷

2

x

=

=   

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common

3

x

÷

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 



x

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x

=

x



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x

=

x Notice

that

you

2

=

x 5

can’t

m

➔

a

n

÷

m

a

=

because

n

to

a

5

Simplify

5

(x

(x

the

x

3

÷

y

bases

a

are

Raising

simplify

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the

same.

power

3

)

3

)

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(x

=

x

=

x

×

×

x

x

×

×

x

x

×

×

x

x

×

×

x)

x

×

×

x

(x

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×

x

x

×

×

x

x

×

×

x

x

×

×

x

x)

×

×

x

(x

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x

×

×

x

x

×

×

x

x

×

×

x

×

x)

x

15

5

So

=

(x

m

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3

5×3

)

=

15

x

n

=

x

mn

)

=

a

Example



2

Expand

(2xy

3

)

Answer

2

(2xy

Don’t

3

)

2

=

(2xy

2

)

×

(2xy

)

(2xy

)

You

don’t

need

to

show

this

line

of

to

working.

3

=

3

2

×

x

2

×

(y

3

)

3

=

8x

in

raise

the

you

have

numbers

Apply

the

power

of

3

to

ever y

ter m

in

bracket

to

bracket.

power

as

x-

y-terms.

and

well

as

the

the

A

Simplify

1



 

3

2

x

a

×

2

x

b

4

3p

×

2p

2

q

 

c





×







3

d

(x

2

y

4

)(xy

)

Remember

to

multiply



the

constants

Simplify

2



 5

2

x

a

÷

7

x

b

2a

3

÷

2a

7

c

2a



(2a)

well

Simplify

3

(x

a

The

4

2

)

b

power

3

)

3

c

3(x

zero

2

Simplify

(3t

2

x

÷

x







=







=











= 

But 



0

Therefore

x

=

Exponential

1

and logarithmic functions

together

as



d 

3

(numbers)

3

÷

 



the

6

y

the

Exercise

forget

2

×

2

y

2

)

2

d

(−y

3

)

as

the

variables.

0

➔

a

=

‘Anything

1

Any

base

raised

to

the

power

of

zero

is

equal

to

zero

1.

is

‘Zero

Fractional

is

to

1. ’

power

0. ’

0

So

× 

what

How 











 



 

 

decide



Who



we

?









should

0





what



about



+ 

Law



any







But

equal

to

power





Using

the

exponents



Simplify

to

this

is

equal

should

to?

decide?







so



=









Similarly











 

 





and



















 

 









and

so







 Y ou

can

assume

always

that

a









➔

=



is

positive

when

considering

Roots

of





2

=

2

x





a.



6

x

roots



Simplify

Since

even

×



2

x

×



=



x





 

 

2

=

x

6

3

=

x



 





➔

=

Example

(

 







)

=

(

)





=





‘Evaluate’

Without

using

a

calculator,

means

evaluate:

4

‘work

out

the

value

1

⎛

1

⎞

3

2

a

36

b

⎜ ⎝

of ’.

⎟ 27

⎠

Answers

1 1

2

a

36

=

36

=

n

6

Since

n

a

=

a

4 4

1

⎛ ⎛ b

1

⎞

3

⎜

⎞ ⎛

1

⎞

3

⎝

⎟ 27

⎠

⎜ ⎜ ⎝ ⎝

27

n m

⎟

= ⎜

Since

⎟

⎟

⎠

⎠

(a

mn

)

=

a

4

1

⎛

⎜

⎟

3

⎝

⎞

27

⎠

4

⎛ 1 ⎞ ⎜

⎟

⎝ 3 ⎠

1

81

Chapter





Negative

exponents

3

Simplify

5

x

÷

x

 3

x









5

÷

x

= 









   



=  × 



= 



3

Also

5

x

÷

x

3−5

=

−2

x

=

x

 

And



therefore

= 





Y ou



➔



must

learn

the

= 

laws



as

for

they

exponents

are

Formula

Example

Without



using

a

calculator

evaluate

2

⎛

−2

a

6

3

⎞

b

⎜ ⎝

⎟ 4

⎠

Answers

1

1

1

n

2

a

6

=

=

Use

a

= n

2

a

36

6

2

⎛ b

3

1

1

⎞

=

= ⎜ ⎝

⎟ 4

2

⎠

⎛

3

⎜ ⎝

⎞

⎟ 4

⎠

⎛

9

⎜ ⎝

⎞

⎟ 16

⎠

16 =

9

Exercise

✗ 1

B

Evaluate

2

1

1

3

3

2

a

9

b

125

c

64

2

2

⎛

8

⎞

3

3

d

2

8

e

⎜

⎟

⎝

27 ⎠

Evaluate 

−3

a





2

b





c





⎛



d

(



)



⎞





e

⎜

⎟

⎝  ⎠



Exponential

and logarithmic functions





not

in

booklet.

the

Example



Here

Simplify

these

expressions.

‘simplify’

1

2

0

−3

5d

a

2

6x

b

÷

(2x

3

3

)

⎛

6

27 a

c

d

⎞

9v

⎜

⎝

2

means

write

these

⎟

4

expressions

16w

using

only

⎠

positive

exponents.

Answers

0

0

5d

a

=

5

×

1

−3

2

6x

b

=

÷

(2 x

5

Use

3

−3

)

=

a

=

m

6

6x

÷

8x

6

1.

(a

Use

n

)

mn

= a

.

3 9

m

=

x

=

Use

a

n

÷

a

m

=

–

n

a

9

8

4 x

1

1

1 1

3

6

6

=

(

27 a

= 27

)

n

6

3

3

27 a

c

(

m

m n

3

a

a

Use

)

= (a

)

.

2

= 3a

1

1

2

⎛ d

⎜

⎝

4 2

⎞

9v

4

16w

=

⎟

⎠

⎛ 16w

⎞

⎜

⎟

⎝

2

9v

2

1 n

a

Use

= n

a

⎠

1

4

(16w

2

2

)

=

4w =

1

3v 2

(9v

Exercise

C

Simplify

1

2

)

these

exponential

expressions.



In

this

exercise,



















(  

a

)

 

b

c

 











d







make

sure

your

e

 

 





answers







 

have

positive



exponents.

Simplify

2

these

expressions.











a



b

÷ 







c







.









 







Solving

Exponential



exponential

equations

are

equations

equations

involving

‘unknowns’

x

as

exponents,

for

example,

5

=

25. y

x

Y ou

can

write

Example

3

exponential

equation

in

the

form a

=

b



x –1

Solve

an

5x

=

3

Answer

x

1

3

x

5x

=

− 1 =

3

Both

5x

sides

powers

of

of

3

the

so

equation

the

two

are

exponents

are

equal. −1 =

4 x

1

x

=

− 4

Chapter





Example

3x

Solve



+

1

For

3

=

this

many

of

the

questions

Answer

3 x +1

3

example

=

to

81

learn

3 x +1

2

4

=

Write

3

81

as

a

power

of

1

=

1

=

3

=

9

1

=

2

3

=

4

3

exponents. 2

=

3

2

3

2

x

powers.

4 2

3x

3

3.

2

Equate + 1 =

need

0

=

1

3x

following

you

these

0

3

and

81

3

=

8

3

=

16

3

=

32

3

=

64

=

128

=

1

7

=

5

7

4

= 1

2

5

2

=

27

=

81

=

243

=

1

=

7

4

5

6

2

Exercise

D

7

2

Solve

1

these

equations

for

x

0

5

x

✗

1−2x

2

a

=

32

3

b

1

=

243

5

1

2

2

x

2 x

5

2x−1

3

c

0

=

27

5

d

−

25

=

2

=

25

7

=

125

7

=

625

3

5

x 4

=

5

49

Solve

2

these

x−3

3

a

equations

for

2−x

=

x

3x

3

b

5

d

2

x−2

=

25

   +



c

2−3x



=

x−1

=

4





EXAM-STYLE

QUESTION

 +

 

Solve

3

Example





=







3

5

Solve

3x

= 24

Answer

3

Divide

both

sides

by

3.

5

3x

= 24

3

Multiply

5

x

the

exponent

a

reciprocal

5

since

−

5

x

)

3

= 8

5

3

Replace

3

x

=

x

=

(

)

2

3

−5

2

1 x =

32



its

Exponential

and logarithmic functions

8

with

2

b ×

b

5 3

(

by

=8

3

49

343

3

1 1

7

e

=

=

0

= 1

−

a

Exercise

Solve

1

E

these

equations

for

x

4

5

2x

a

=

162

x

b

−

−2

c

x

=

e

27x

16

d

8x

Solve

=

0

f

27x

3

=

−2

2

32

−3

(8x)

−3

=

81x

these

equations

for

=

64

x











a

=





b







c

= 





=





d

= 





 





e

=



f

=





Solve

3

these

equations

for

x

3



2



x

a

=

125



b

=









c

.

=

192

and



d

Exponential

Graphs

➔

216



=

16

functions

properties

of

An exponential function

is

a

exponential

function

of

the

functions

We

form

could

also

write

x

f

:

x

→

a

x

f

(x)

where

=

a

a

is

a

positive

Investigation

Using

a

GDC,

sketch

–

real

number

graphs

the

graphs

of

(that

of

is,

a

>

0)

and

exponential

these

exponential

a

≠

1.

functions

1

functions. Think

about

the

x

a

y

=

3

b

y

=

5

domain,

range,

x

intercepts

on

the

x

y

c

=

axes,

10

asymptotes,

shape

Look

at

your

three

each

What

can

you

and

behavior

of

graphs.

deduce

about

the

exponential

graph

as

x

tends

function, to

innity.

x

f (x)

=

a

,

Whatever

when

a

>

positive

1?

value

a

has

in

the

equation

y

x

f

(x)

=

x

, the graph will

a

always

have

the

same

f(x)

=

a

shape.

x

f

(x)

=

a

is

an

exponential

growth

function

1 (0, 1)

0

x

Chapter





x

➔

The

domain

The

range

The

cur ve

The

graph

value

of

x

of

is

f

the

does

(x)

=

set

a

of

not

is

all

the

of

positive

intercept

approaches

set

the

closer

all

real

real

numbers.

numbers.

x-axis.

and

closer

to

the x-axis

The

y-intercept

The

graph

is

of

f

passes

through

the

points

 ⎞

,

−  ⎜



⎝

(1,

The

Now

between

at

0

increases

the

and

graphs

a

1)

⎠

GDC

exponential

continually .

of

exponential

functions

when

the

base a

is

1.

Investigation

Using

(0,

⎟

a).

graph

look

the

1.

⎛

and

as

decreases.

sketch

-

the

graphs

graphs

of

of

exponential

functions



these

functions.

–x

y

a

=

−x

3

y

=

3

is

the

–x

b

y

=

5

c

y

=

10

1

equivalent

of

y

or

= x

–x

y

–x

What

can

you

deduce

about

the

exponential

function,

f (x)

=

a

a

>

1,

from

these

three

graphs?

−x

Whatever

will

positive

always

have

–x

f(x)

=

value

this

a

has,

the

graph

of

shape.

y

a

(0, 1) 1

0

x

– x

f



(x)

=

a

is

an

Exponential

exponential

decay

and logarithmic functions

function

f

(x)

=

a

=

⎛ 1

⎞

⎜

⎟

⎝ 3

⎠

so

the

base

, is

when

3

x

between

0

and

1.

The

The

natural

base

e

is

exponential

exponential

one

that

you

you

come

across

often

in

functions.

Investigation

When

will

function

invest

–

money

compound

it

earns

interest

interest.

n t

r

⎛

We

use

this

A = C

formula

⎜

r

is

of

the

is

the

nal

interest

happens

£1

a

is

if

in

you

invested

How

much

(capital

expressed

a

at

will

year ,

star t

an

and

as

t

+

a

is

interest),

decimal,

the

total

compounding

interest

you

calculate

the

interest,

n ⎠

amount

rate

compoundings

What

1

A

to

⎟

⎝

where

⎞

1 +

have

rate

if

this

of

is

is

the

the

number

more

100%

is

n

C

and

for

capital,

number

of

years.

more

1

frequently?

year .

compounded

year ly?

100

P

=

1,

r

=

100%

=

=

1,

n

=

1,

t

=

1

and

n

=

100

1

1 ⎞

⎛

A

= C

⎜

1

+

b

How

C

=

⎟ 1

⎝

much

1,

r

=

=

2

(since

r

=

1

1)

⎠

will

you

100%

=

have

1,

n

if

=

this

4,

t

is

=

compounded

quar terly?

1

4

1

⎛

A

=

⎜

1

+

2

Copy

and

⎟ 4

⎝

⎞

=

2

44 140 625

⎠

complete

Compounding

the

table.

Calculation

F inal

all

amount

gures

on

(write

calculator)

1

1 ⎞

⎛

Y early

⎜

1

+

2

⎟ 1

⎝

⎠

2

1

⎛

Half-Y early

⎜

1

+

⎟ 2

⎝

⎞

2.25

⎠

4

1

⎛

Quar terly

⎜ ⎝

1

+

⎞

⎟ 4

2.44 140 625

⎠

Monthly

Weekly

Daily

Hourly

Ever y

minute

Ever y

second

Chapter





The

final

amount

compoundings

smaller

value

The

and

is

value

of

impor tant

subject

e

is

an

the

called

increases

decreases

final

as

but

amount

the

inter val

each

between

separate

converges

on

increase

a

value.

is

This

‘e’.

e

is

approximately

number

in

2.71828

mathematics

and

which

it

has

is

an

exceptionally

applications

in

many

areas.

number.

irrational

Jacob

Bernoulli

(1654–1705)

Mathematics

beautiful

sometimes

throws

out

some

surprising

one

and

is

one

results.

such

20

decimal

Swiss

is

no

places

e

=

2.718

281

828

459

045

235

obvious

pattern

to

this

chain

of

look

at

this

series,

which

gives

a

1

1

+

value

1

+

of

2 × 1

3 × 2 × 1

the

tried

to

nd

4 × 3 × 2 × 1

1 ⎞

⎛

+ ... of

⎜

5 × 4 × 3 × 2 × 1

tends might

[See

and

the

wonder

Theor y

discussion

about

of

on

the

connection

Knowledge

beauty

in

page

at

between

the

end

this

of

series

this

and

chapter

the

for

value

of

1 +

to

thoughts

used

of

the

exponential

function f

(x)

=

e

is

a

graph

growth

is

the

exponential

binomial

and

the

graph

of

f

(x)

=

is

e

limit

to

a

show

had

2

to

and

considered

that

lie

3.

to

This

be

graph the

of

He

of

x

exponential

innity.

between

x

graph

n

⎠

mathematics.]

the

The

as

⎟ n

e.

theorem

➔

the n

+

⎝

Y ou

problem

interest,

e:

limit 1

at

compound

1

+

was

numbers.

he 1

e = 1 +

he

36…

of

However

of

Bernoulli

When

looking

There

great

result. family.

T o

the

mathematicians

the

Here

of

was

rst

approximation

decay . found

y

for

e.

y

x

f(x)

=

e –x

y

=

e

An

(0, 1)

(0, 1)

x

0

Transformations

Now

you

function,

Chapter



know

you

1

to

the

can

help

Exponential

of

exponential

general

use

you

the

x

0

shape

r ules

sketch

of

for

graph

of

an

transformations

graphs

and logarithmic functions

the

functions

of

other

exponential

of

graphs

exponential

from

functions.

number

cannot

be

expressed

exactly

as

a

a

1

irrational

decimal.

fraction

or

➔

f

(x)

±

units

k

translates

ver tically

up

f

(x)

or

through

k

y

down

y

f

(x

±

units

or

k)

translates

horizontally

f

(x)

to

through

the

y

k

=

=

f(x

(x)

(−x)

pf

+

2

f(x)

+

2)

right

reflects

f

(x)

in

the

y

x-axis y

f

f(x)

left

y

−f

=

(x)

reflects

f

stretches

scale

factor

f

stretches

(qx)

(x)

f

in

(x)

the

f(x)

f(x)

–f(x)

y

y-axis

ver tically

=

=

=

=

f(–x)

y

with

=

y

y

(x)

f(x)

2f(x)

p

f

=

=

f(x)

horizontally y

=

f(2x)



with

scale

y

factor

=

f(x)



Example



x

The

diagram

shows

the

sketch

of

f

(x)

=

y

2

x−2

On

the

same

axes

sketch

the

graph

of

g (x)

=

8

2

6

4

2

x

0 –3

–1

1

3

Answer

y

8

You

find

f

through

(x)

g (x)

by

2

translating

units

to

the

right.

6

4

The

graph

of

2

the

(0, 1)

point

⎜

0,

⎝

–1

will

pass

through

⎞ ⎟

4

⎠

x

0 –3

g (x)

1

⎛

1

3

4

5

Both

graphs

get

closer

and

closer

to

1

4

the

x-axis

as

the

value

of

x

decreases.

Chapter





Exercise

1

Given

the

F

the

graph

graph

of

intercepts

of

g (x)

on

the

on

f (x)

=

(x),

and

the

axes

x

a

f

without

same

and

set

any

of

using

axes

a

calculator,

showing

g (x)

=

x

2

+

3

b

f (x)

=

g (x)

8

8

6

6

4

4

2

2

x

–1

1

1

–4

–4

–6

–6

–8

–8

–10

–10

  ⎜ ⎝



 ⎞ x

=

⎟

⎜

⎠

⎝

d

⎟ 

f (x)

=

x+1

e

g (x)

=

e

⎠

y

y

8

8

6

6

4

4

2

2

x

0 –3

3



⎛

 ⎞

=

x

–1

–2



 

3

0 –3

3

–2

⎛



=

y

0

c

any

–x

3

y

–3

clearly

asymptotes.

x

2

sketch

–1

x

0

3

–3

–1

1

–2

–2

–4

–4

–6

–6

–8

–8

–10

–10

3

2x

x 



⎛  ⎞ e



 

= ⎜ ⎝

⎟ 

 

=



f

⎜

⎠

⎝

f

⎟ 

⎛ 1 ⎞

⎛ 1 ⎞

⎛  ⎞

(

x

)

=

⎜ ⎝

⎠

y

g

⎟ e

(

x

)

=

⎜ ⎝

⎠

⎟ e

y

8

8

6

6

4

4

2

x

0 –3

2



State

–1

the

–3

–1

–2

–4

–4

–6

–6

–8

–8

–10

–10

domain

Exponential

and

range

x

0

3

–2

of

each g(x)

and logarithmic functions

function

1

in

3

question

1.

⎠

.

Properties

of

logarithms

3

Look

2

is

So

at

the

we

this

equation:

base

say

8

and

that

=

3

the

is

2

the

=

8

exponent

logari thm

of

8

or logari thm

to

the

base

2

is

3

and

write

this

3

as

log

In

general,

2

given

that

a

>

0:

x

➔

If

b

=

a

then

log

b

=

x

a

or,

Being

if

b

is

able

simplify

to

log

Example

Evaluate

a

to

the

change

power

x,

between

then

x

these

is

the

two

logarithm

forms

of

allows

b,

you

to

base

a

to

statements.



log

125

5

Answer

Write x

=

log

‘x

=’

the

log

statement.

125

5

x

5

Change =

125

=

5

=

3

x

5

x

Equate

3

Example

Evaluate

equation

to

exponent

for m.

exponent

for m.

exponents.



log

4 64

Answer

x

=

log

4

64

x

64

=

Change

4

equation

to

3 3

(4

x

)

3x

x

1

=

4

Write

64

=

1

Equate

1

and

as

the

solve

4

exponents

for

x.

= 3

Exercise

✗ 1

G

Evaluate

a



these



expressions.

b





2



c



Evaluate

these

log

64

d



 

2

expressions.



1

 

a

log

b



 

3

c



 

d



 

81

Chapter





Example

Evaluate



log

4

4

Answer

Write x

=

log

‘x

=’

log

statement.

4

4

Change

equation

to

exponent

x

4

=

4

=

1

for m. x

1

Equate

In

general,

➔

log

a

the

=

log

to

base

a

of

any

exponents

number

a

=

(4

=

4

).

1.

1

a

Example

Evaluate



log

1

5

Answer

x

=

log

1

5

x

5

x

=

1

=

0

Any

any

Write

number

base

➔

log

is

1

raised

to

the

power

0

is

equation

equal

to

1

in

so

exponent

the

log

of

0.

=

0

a

Exercise

✗

1

H

Evaluate

log

a

6

log

b

6

log

d

1

log

e

8

Some

you

1

log

What

log

c

find

are

solutions

1

log

f

happens

when

1

b

undefined

for

n

n

2

expressions

can’t

10

10

–

this

means

that

them.

you

tr y

to

evaluate

the

expression

(−27)?

log 3

First

write

x

=

the

log

log

equation.

(−27)

3

Then

rewrite

the

equation

in

exponent

form.

x

3

This

−27

equation

Y ou

➔

=

can

log

b

is

only

has

find

no

solution.

logarithms

undefined

for

any

of

base

a



Exponential

and logarithmic functions

posi tive

a

if

b

is

numbers.

negative.

for m.

1

in

What

2

is

the

value

of

log

0?

3

First

write

x

=

an

log

equation.

0

3

Rewrite

in

exponent

form.

x

3

This

➔

=

0

equation

log

0

is

has

no

solution.

undefined.

a

Example

13

Example

illustrates

another

proper ty

of

logarithms.



5

Evaluate

log

2

2

Answer

Write

5

x

=

log

log

equation.

2

2

Rewrite x

2

in

exponent

for m.

5

=

2

=

5

Solve. x

n

➔

log

(a

)

=

n

a

Summary

Given

that

a

of

>

properties

of

logarithms

0

b

●

If

x

=

a

then

log

x

=

b

a

●

log

a

=

1

a

●

log

●

log

1

=

0

a

b

is

undefined

if

b

is

negative

a

●

log

0

is

undefined

a

n

●

log

(a

)

=

n

a

Example

Find

the



value

of

x

if

log

x

=

5

2

Answer

log

x

=

5

=

x

Rewrite

=

32

Solve.

2

5

2

x

Exercise

1

Write

2

x

these

=

Write

a

x

equations

in

log

for m.

form.

5

2

b

these

=

exponent

I

9

a

in

log 2

8

x

=

equations

b

x

=

4

3

in

c

exponent

log 3

27

c

x

=

b

10

d

x

=

a

d

x

=

log

form.

x

=

log 10

1000

b

a

Chapter





Solve

3

these

log

a

x

=

equations.

3

log

b

4

x

=

4

log

c

3

64

=

2

x



log

d

6

=

log

e

x

x

=

−5

2



.

Logarithmic

functions

Investigation

–

What

would

kind

of

function

inverse

undo

functions

an

exponential

function

x

such

f

as

:

x



2

?

x

Copy

a

and

complete

this

table

of

values

for

the

function

y

=

2

x

x

−3

−2

1

0

1

2

f :

3

f

x

is

↦

a

2

means

function

that

under

1 which

x

is mapped

y x

8 to

2

x

The

inverse

y-values

Copy

b

and

and

function

switch

complete

of

y

=

2

will

take

all

the

x-

and

for

the

inverse

them.

this

table

of

values

x

function

of

y

=

2

.

1

x 8

y

−3

x

Using

c

and

its

What

d

Now

➔

you

find

find

and

tables

inverse

do

let’s

To

these

then

values

function

on

sketch

the

a

same

graph

set

of

of

both

y

=

2

axes.

notice?

the

an

of

equation

inverse

of

rearrange

a

to

of

the

graph

function

make

y

of

the

inverse

algebraically ,

the

function.

switch

x

and

y

subject.

x

f

:

x



2

is

another

x



−1

To

get

the

inverse

function,

f

,



of









way

of

writing

y

=

2

:

x

Write

y

=

2 y

is

the

exponent

that

y

x

log

=

2

Switch

x

=

ylog

y

=

log

2

so

2

Take

x

and

logs

to

y

the

the

base

2

of

both

sides

by

base

in

2

order

Since

log

2

=

2

Log







is

shor t



 

logarithm.



➔

Generally

if













then











 

x

y

=

log

x

is

the

inverse

of

y

=

a

a



get

1





to

raised

2

x

2

So

is

Exponential

and logarithmic functions

for

x

x

The

graph

of

y

=

log

x

is

a

reflection

of

y

=

x

a

y

y

=

a

a

in

the

line

y

=

x y

=

x

=

log

(0,1) y

x a

x (1,0)

➔

A

logarithmic

function,

f

( x)

=

log

x,

has

these

John

proper ties:

Napier

(1550–

a

1617)

the

domain

is

the

set

of

all

positive

real

much

the

range

is

the

cur ve

does

the

set

of

all

real

intercept

of

the

y-axis

is

a

ver tical

the

x-intercept

on

logarithms.

you

graph

is

continually

Transformations

of

Again

the

once

you

logarithmic

to

consider

the

Exercise

1

Given

know

function

that

he

logarithms

or

1 discovered

the

say

asymptote invented

is

with

early

y-axis Would

the

credited

the

numbers

work

not

is

numbers

graphs

increasing.

logarithmic

you

general

can

of

them?

use

other

shape

what

functions

of

you

logarithmic

the

graph

lear nt

in

of

a

Chapter

1

functions.

J

the

function

f

( x)

=

log

x

y

describe

a

the

transformation

required

in

each

case y

=

log

x a

to

obtain

the

graph

a

g ( x)

=

log

b

g ( x)

=

log

(x)

−

of

g(x)

2

a

0

x (1, 0)

(x

−

2)

a

c

g ( x)

=

2log

x

a

EXAM-STYLE

2

Sketch

the

QUESTION

graph

of

y

=

−2log(x

−

1)

without

using

a

calculator. When

Include

on

your

graph

the

intercepts

with

the

two

the

(if

3

they

Sketch

exist).

the

of

y

=

log

(x

+

1)

+

2

clearly

base

logarithms

base

graph

no

is

given

axes

labeling

are

10.

any

2

asymptotes

4

The

sketch

on

the

shows

graph.

the

graph

of

y

=

log

x.

y

a

Find

the

value

of

(27, 3)

a

0

(1, 0)

x

−1

5

Given

that

f

(x)

=

log

x

find

f

(2)

3

Chapter





Logarithms

to

base



x

y

=

log

x

is

the

inverse

of

y

=

10

.

This

is

an

impor tant

logarithm

10

as

it

is

Base

and

one

10

of

logs

just

the

are

write

only

ones

called

log x

for

that

you

common

log

can

logs

use

and

the

you

calculator

can

omit

the

to

find.

base

x

10

There

is

a

‘log’

Example

Use

a

key

on

the

calculator.



calculator

to

evaluate

log 2

to

3 dp.

Answer

log

2

=

*Logarithms

1.1

0.301

to

3

log

dp.

0.30103

(2) 10

GDC

help

on

CD:

demonstrations

Plus

and

Casio

Alternative

for

the

TI-84

FX-9860GII

1/99

GDCs

Natural

The

are

on

the

CD.

logarithms

logari thm ,

natural

log

x

(log

to

the

base

e), is

the

other

e

impor tant

Y ou

write

logarithm.

ln x

for

log

x.

There

is

an

‘ln’

key

on

the

calculator

e

Example



Make

ln 4

Use

a

calculator

to

sure

you

close

evaluate the

ln 2

brackets

after

the

4

the

calculator

otherwise

Answer

⎛

calculate

=

2

will

*Logarithms

1.1

ln 4

In(4)

ln

4

⎞

⎜

⎟

⎝ In 2

⎠

2.

ln 2 In(2)

GDC

help

on

CD:

demonstrations

1/99

Plus

and

GDCs

Exercise

1

K

Use

a

to

significant

a

3

calculator

log 3

to

evaluate

 

4log 2

correct



c



  e

 

f

(log 3)

Exponential

2

h

log 

 

2



expressions

figures.

b

d

g

these

log 3

and logarithmic functions

Casio

are

on

Alternative

for

the

TI-84

FX-9860GII

the

CD.

➔

y

=

ln x

is

the

inverse

of

the

x

exponential

function

y

=

e

x

y

y

=

e

y

=

x

(0, 1) y

=

In x

x (1, 0)

This

relation

gives

us

x

➔

log

(a

three

log

)

=

x

and

results:

x a

a

impor tant

=

x

a

x

ln(e

lnx

)

=

x

and

e

=

x

log (10

Solve

log x

)

Example

x

=

x

and

(10

)

=

x



these

equations,

giving

your

answers

x

e

a

to

3

significant

figures.

x

=

2.3

ln x

b

=

–1.5

c

10

=

0.75

d

log x

=

3

Answers

x

e

a

=

2.3

x

ln(e

)

=

ln2.3

x

=

0.833(3 sf)

Write

in

natural

log

for m.

x

b

ln x

=

–1.5

=

e

=

0.223(3 sf)

lnx

e

x

Use

ln

–1.5

(e

)

Use

(e

)

=

Use

log(10

Use

10

x

10

c

=

x

and

evaluate.

lnx

x

and

evaluate.

x

=

0.75

)

=

log 0.75

x

=

−0.125(3 sf)

)

=

x

and

evaluate.

x

log(10

log x

log x

d

=

3

log x

10

x

=

x

and

evaluate.

3

=

10

=

1000

Example



1 2x

Given

that

f (x)

=

e

,

−1

find

f

(x).

3

Answer

1 2x

f

(x)

=

e 3

1 2x

y

=

e 3

1 2y

x

=

e

Interchange

x

and

y.

3

{

Continued

on

next

page

Chapter





2y

3x

=

e

2y

ln(3x)

=

ln e

ln(3x)

=

2y

x

Use

ln(e

)

=

x.

1

ln(3x)

=

Solve

y

for

y.

2

1 –1

So

f

(x)

=

ln(3x),

x

>

0

2

Exercise

1

Solve

L

these

equations

giving

x

a

e

d

e

answers

to

3 sf

where

x

=

1.53

x

e

b

necessar y .

=

0.003

e

c

=

1

 x

x

=

5e

e

=

0.15



2

Solve

these

equations

giving

answers

to

3 sf

where

necessar y .

 x

x

10

a

=

2.33

x

10

b

=

0.6

10

c

x

=

1

d

10

= 

3

Find

if

log x

a

4

x

=

Without

log

2

log x

b

using

a

5

−1

calculator

12

log

5

log x

c

evaluate

4



5

b

Without

using

a

0

d

evaluate

log x

=

−5.1

expressions.



ln4



c

calculator

=

these

5

5

a

=

d

these

e

expressions.

 5

ln e

a

log 100

b

ln1

c

ln e

d

e

ln 



EXAM-STYLE

QUESTIONS

2x−1

6

Given

that

f

(x)

=

e

7

Given

that

f

(x)

=

e

−1

find

f

(x)

and

state

its

domain.

0.25x

,

−2

≤

x

≤

4,

state

the

domain

and

−1

range

of

f

−1

8

Given

that

f

(x)

=

ln 3x,

9

Given

that

f

(x)

=

ln(x

x

>

0,

find

f

(x).

x

−

1),

x

>

1,

and

g(x)

=

2e

find

(g

f °

.

We

Laws

can

of

deduce

logari thms

the

laws

of

logarithms

p

equations,

x

=



=

a

q

and

y

=

and



=

a



then





=









and



=







and



so

=







× 



=



=

 



+ 

=



+ 



and

hence

 



Exponential



 

+



 

and logarithmic functions

from

the

exponential

)(x)

This

equation

is

tr ue

for

logarithms

in

any

base

so

Notice

➔

log

x

+

log

y

=

log

that

xy

log xy

≠

log x

×

log y

x

 

=







÷ 

=



and

that

log

log

x

log

y

≠

 y







so

=



− 







and



hence

=







− 



 



x

➔

log

x

–

log

y

=

log y







=









=









so

=







and



hence



=

 



 

n

➔

We

n

log

can

x

=

also

log

x

derive

this



➔



key

result

from

the

third

law .



 





−1 ×







=

− 



 



All

can

these

be

laws

are

omitted.

Formula

tr ue

Y ou

for

must

logarithms

lear n

these

in

any

laws

base

as

and

they

are

so

the

not

in

bases

the

booklet.

Example



1

Express

log

5

+

log

2

36 2

log

10

as

a

single

logarithm.

2

2

Answer

1

log

5

+

log

2

36

log

2

10 2

2

1 n 2

=

log

5 + log 2

=

log

36

log

2

5 + log 2

10

n log

6 2

log

30 2

= log

x a

10 2

log x = log

x a

2

log

+ log

y = log xy

10 2

x

=

log

3

log x

log y =

2

y

Chapter





Exercise

1

M

Express

as

single

logarithms:

   

a

b

log

24

e

3log

–

log

2

c

2log

f

log

8

–

4log

2



 

d

x

–

2log

y

x

–

log

y

–

log



g

2

  

Express

+

as

   

single

−

  

logarithms:

  



a

 

 − 









b



 





    



 



c

  

 −   



2ln3

d

–

 

ln18



 e

3ln2

–

2

f













 



 



3

Find

the



a

value

of

  

each

expression



log

b



(each

24

–

answer

log

2

is

3

an

c

integer).



  

2





d

  



 

e

   





Example

Given



that

a

=

log

x,

b

=

log

5

⎛

log

write

5

in

⎟ 2

y

and

c

=

log

z, 5

⎞

x

⎜ ⎜

y

5

3

terms

of

a,

b

and

c

log

z

⎟

z

⎝

⎠

Answer

⎛

⎞

x

2

log 5

⎜

=

⎟ 2

⎜ y

3

log

x

log

5

y

3

z

5

⎟

z

⎝

⎠

1

x

log

3

2

2

=

y

(log

5

+

5

)

5

1

=

log

x

− 2log

5

y

− 3log

5

z 5

2

1

=

a − 2b

− 3c

2

Exercise

N

EXAM-STYLE

1

Given

that

QUESTION

p

=

log

a

and

q

=

log

2

of

p

and/or

q

b,

find

an

expression

2

for:  3

a

log

ab

b

2

log

a

c

 

 d



 

e

 





Exponential

 

2

and logarithmic functions

in

 

terms



z

Let

2

x

=

log P,

y

=

log Q

and

z

=

log R.

 

⎛

Express



Write

these

where

a

in

⎟





⎝

3

⎞



⎜

terms

expressions

and

b

of

x,

y

and

z

⎠

are

in

the

form

a

+

blog x

integers.





log10x

a



b



c

 



d





EXAM-STYLE





QUESTIONS





Given

4

that





write



y

in

the

form

y

=

pa

+

q





where

p

and

q

are

integers

to

be

found.



Write

5

in

 

the

form

a

+

blog



x

where

a

and

b

are

3

 

integers.

x xln2

Show

6

Notice

that

that

e

=

2

question

6

in

Exercise

4 N

demonstrates

the

general

result

x

a

xlna

=

e

Change

of

Sometimes

there

is

a

Suppose

base

you

need

formula

y

=

log

to

that

a

change

enables

and

you

the

you

want

base

to

to

do

of

a

logarithm

and

this.

change

the

log

to

base c.

b

y

If

y

=

log

a

then

a

=

b

b

y

Star t

with

a

=

Take

logs

to

b

base

c

of

both

sides:

y

log

a

=

log

a

=

ylog

c

log

b

c

c

b

c



 



= 

 

But

y

=

log

a

so

b

➔

Change

of

base



formula:

 





This

=

formula

is

useful





 as



most

only

Y ou

can

use

this

formula

to

evaluate

a

logarithm

or

to

logarithm

to

any

calculators

logs

to

base

change

10

a

give

or

e.

base.

Chapter





Example

Use

the



change

of

base

formula

to

evaluate

log

9

to

3

4

significant

figures.

Answer

log 9

log

9

For

=

Change

4

the

log

to

base

10

= 1.58 (3 sf)

Use

calculator

to

evaluate

answer.

Example

log

3

=

a



and

log

x

6

=

b.

x

Find

log

6

in

terms

of

a

and

b

3

Answer

log

Use

6

the

change

of

base

for mula.

x

log

6 = 3

log

3 x

b = a

Exercise

1

O

Use

the

to

significant

3

change

of

base

formula

to

evaluate

these

expressions

figures.

⎛  ⎞ a

log

7



b



⎜

c

⎟

log

2

(0.7)

3

⎝



⎠

7

d

log

e

log

e

7

2

Given

7

3

that

log

x

=

y,

express

log

3

EXAM-STYLE

3

If

log

2

log

=

d

log

x

and

log

6

a

y

log

24

e

log

y,

find

Given

terms

c

log

12

f

log

of

y

of

x

and

6

GDC

log

to

x

sketch

b

y

=

that

36

2

3

2

these

2log

4

5

in

2

6

your

=

=

b

a

Use

terms

a

6

2

4

in

QUESTION

a

a

x 9

graphs.

x

5

log

a

=

b

express

y

in

terms

of

b

4

2

a

y

=

log

a

b

y

=

log

4

a

16



c



=











Exponential

and logarithmic functions

d



=









y:

base

10

logs,

10.

log 4

is

omitted.

the

.

Exponential

Solving

Y ou

In

can

exponential

use

Section

numbers

you

are

will

and

logarithms

4.2

you

were

the

lear n

to

solved

same

how

to

logari thmic

equations

solve

exponential

exponential

or

equations

could

solve

be

equations

made

equations

equations.

the

where

where

same.

the

In

base

the

this

base

section

numbers

different.

Example



x

Solve

5

=

9

Answer

x

5

=

9

=

log

9

=

log

9

Choose

base

10

or

x

log

x log

5

5

log 9

x

Take

logs

of

Now

bring

Rear range

both

down

the

sides.

the

natural

exponent.

you

logs

can

use

so

that

your

GDC.

equation.

= log 5

x

=

1.3652…

x

=

1.37

(3

sf)

Check

an

Example

6

question

requires

ln a

x + 1

=

the

answer



x

Solve

whether

exact

3

giving

your

answer

in

the

form ln b

where

a

and

b

are

integers.

Answer

x

x+1

6

=

3

x

ln 6

x

ln 6 −

x +1

=

x

ln 6

=

(x

x

ln 6

=

x

x

ln 3

x (ln 6 − ln 3)

=

=

Take

ln 3

+ 1) ln 3

ln 3 + ln 3

natural

Bring

down

Multiply

Collect

ln 3

the

out

of

both

sides.

exponents.

brackets.

x-ter ms

Factorize

ln 3

logs

and

together.

divide.

ln 3

x

=

(ln 6

ln 3 )

a

ln 3

x

ln a

=

ln 2

− ln b

=

ln

b

Chapter





Example



3x

Solve

1−x

e

=

5

,

giving

an

exact

answer.

Answer

x 3x

=

x (3

=

ln 5

3x

=

(1–

3x

=

ln 5

x ln 5

+

logs

since

ln

e

=

x

1 – x

ln e

+

natural

5

3x

3x

Use

1 – x

e

x)

=

ln 5

=

ln 5

ln 5)

–

ln 5

Bring

down

the

exponents.

x ln 5

Multiply

Collect

out

Leave

brackets.

x-ter ms

in

together.

log

your

form

answer

since

ln 5

Factorize x

and

an

divide.

(3 + ln5)

Exercise

1

Solve

required.

P

these

equations

x

to

find

the

value

of

x

to

x

2

a

=

5

b

3

f

2

3

significant

x

=

50

c

5

g

e

figures.

x+1

=

17

7

d

=

16





  

 2x−1



 



Solve

=

3.2

×

x

10

=

6

h



=









EXAM-STYLE

2

−3





e

QUESTION

these

equations

to

find

the

value

of

x

to

3

significant

figures.



x+2

x −3

2x −5

2−x

a

2

e

e

=

5

=

c



3

4e

f

7

d

= 

x −1

=

(0.5)

−0.001x

3x −2

x

=

4

x

 +



3

b

3x −1

Example

=

244

g

35e

=

95



ln a

x −1 x+2

Solve

3

×

6

=

2

×

3

,

giving

your

answer

in

the

form

x

,

=

ln b

where

a,

b

∈



Answer

x

ln (3

×

–

6

+

ln (6

1

x + 2

)

x

ln 3

=

ln (2

×

3

– 1

)

Take

+

ln 2

+

+

(x

–

1) ln 6

=

ln 2

ln(3

+

(x

both

+

+

x ln 6

x ln 6

x(ln 6

–

–

ln 6

xln 3

–

ln 3)

=

=

=

ln 2

ln 2

ln 2

+

+

x ln 3

ln 9

x-ter ms

2)ln 3

2ln 3

+

sides.

)

and ln 3

logs

2

of =

natural

)

x

Collect ln 3

+

+

+

factorize.

2ln 3

ln 6

ln 6

–

–

ln 3

ln 3

⎛ 108 ⎞ ln

⎜

You

⎟

⎝ x

6

⎞

simplify

ln a

=

⎛

ln 2

fur ther

this

⎟

and logarithmic functions

a



– ln b

⎜

⎝ 3 ⎠

Exponential

can’t

ln 36

⎠

3

=

ln



exact

=

ln

b

any

answer

is

Exercise

Q

EXAM-STYLE

1

Solve

QUESTIONS

these

equations

to

find

the

value

x



a

  

d

5

=



x – 1

2

×

Solve

b

4

e

3

x

2x

2

=

these

3

×

×

7

equations

to

find

3

of

x

the

3

significant

=

figures.

x

5

3

c

x – 1

4

to

2x – 1

×

2

x

=

4

×

5

x + 2

=

2

×

value

7

of

x

in

the

 

form



,

=

where

a,

b

∈



 

x + 2

a

2

c

5

x – 3

=

x

5

3

Solve

=

2

×

6

(6

b

4

−

Solving

=

x

–

8

×

7

1

)(2

x + 2

)

x





Some

d

x

3

=

2(4

)

x

 

a

×

3 – 2x

3

for

5

x

x + 1

×

b



=



logarithmic

logarithmic

x

–

3(2

)

=

0

equations

equations

can

be

solved

by

ensuring

that

both

The

sides

of

the

equation

contain

logarithms

written

to

the

same

argument

expression

Then

you

can

equate

the

is

the

base.

inside

the

arguments

brackets.

Example



2

Solve

log

(x

)

=

log

a

(3 x

+ 4)

a

Answer

2

(x

log

)

=

log

=

3x

4

=

0

1)

=

0

a

(3x

+

4)

a

2

x

+

4

Equate

the

arguments.

2

x

(x

−

−

3x

4)(x

x =

Y ou

−

+

4

or

x

you

Substituting

the

log

Example

Solve

x

that

both

cannot

=

of

the

quadratic.

−1

check

must

Remember

gives

=

Solve

4

a

and

solutions

find

x

=

positive

the

−1

are

possible.

logarithm

into

number

both

so

of

sides

here

a

negative

of

both

the

number.

original

solutions

are

equation

possible.



ln(12 −

x )

=

ln x

+ ln( x

− 5)

Answer

ln(12

−

x)

=

ln x

ln(12

−

x)

=

ln x (x

+

ln(12

−

x)

=

ln(x

=

x

ln(x

−

−

5)

5)

2

−

5x)

2

12

−

x

−

5x

Equate

arguments.

2

x

(x

−

−

4 x

6)(x

x

=

6

−

+

12

2)

or

x

=

0

=

0

=

−2

Solve

the

quadratic.

{

Continued

on

next

page

Chapter





When

ln x

When

ln

x

so

x

=

and

x

=

and

x

=

Check

6

ln(x

6

−

is

−

the

5)

are

only

solution.

 

equations

=





 

 

 









− 



− 

 

 

b

 

d



+ 

=

 −

 



− 

it

Example



log

(x

is

−

+  +



=







− 



=

to

solve

a

+ 

log

equation

using

Since

=

exponents.

3

5

Answer

log

(2x

–

1)

=

3

5

3

b

5

=

2x

–

1

log

x

b

⇒

x

=

a

a

125

=

2x

=

x

Example

Solve

2x

–

1

126

=

63



log

x

+

log

2

(x

−

2)

=

3

2

Answer

x

log

+ log

2

(x

− 2)

=

3

2

[x (x

log

− 2 )] =

3

2

Using

the

first

law

on

page

2

(x

log

2x )

=

3

2

2

x

3

− 2x

=

2

b

Since

log

x

=

b

⇒

a

2

x

− 2x

=

8

2

x

(x

+ 2 )( x

x

x



=

− 2x

4

=

−2

is

Exponential

− 8

− 4)

or

the

x

=

0

=

=

only

0

4

solution

and logarithmic functions

x

must

be

positive.

x

=

a

=



 



easier

2)

  



Sometimes

Solve

x.





e

for





c

negative

QUESTION

these



a

solutions.

positive.

R

EXAM-STYLE

Solve

are

−2

ln(x

Exercise

1

5)

123.

+ 

Exercise

Solve

1

these

log

a

S

(x

equations

−

2)

=

2

for

x.

log

b

9

(2x

−

1)

=

3



c



 −

 

= 

3 

Solve

2

these



a



equations

−  +





log

c



(2x

−

3)

–

(4x

−

8)

–

log

log

(4x

−

5)

=

(x

−

5)

=

4

2

0

QUESTIONS



that

an

Hence

log

b

2

x



+



find



7

EXAM-STYLE

Given

=

x.



7

3

for

expression

or

 x

+ 

=





for

otherwise

A

A 

in

solve

terms

log

x

of

+

x.

log

2

(2x

+

7)

=

2

2

Y ou

will

change

Solve

4





+







need

the

to

base

here

= 



rst.



Solve

5





+





.



= 



Applications

of

exponential

and

Extension

material

Worksheet

4

linear

logarithmic

Exponential

Models

of

exponential

and

growth

and

decay

functions.

areas

just

a

few

applications

of

exponential

of

appear

mathematics

to

decay

be

completely

growth

disconnected and

to

form

use

that are

CD:

decay

T wo

Here

on

Reduction

functions

growth

exponential

-

might

be

exponentials

models.

and

probability.

Biology

But ●

Growth

●

Human

of

micro-organisms

in

a

consider

A

Spread

problem…

group

of

people

go

to

lunch

and

population

afterwards ●

this

culture

of

a

pick

up

their

hats

at

virus

random.

What

is

the

probability

that

Physics

no ●

Nuclear

●

Heat

chain

one

gets

their

own

hat?

reactions

It

can

be

shown

that

this

probability

transfer 1 is

.

Economics Y ou

●

Pyramid

Processing

power

e

to

one

of

you the

basis

have

like

to

studied

explore

this

probability

once

fur ther .)

of

of

Can your

might

these

technolog y as

●

wish

(Y ou pick

Computer

may

schemes

you

think

of

any

other

areas

Mathematical

computers

of

knowledge

that

are

surprisingly

Exploration.

●

Internet

trafc

growth

connected?

Chapter





Exponential

Example

The

growth



population,

A(t ),

in

thousands,

of

a

city

is

modeled

(0.02)t

by

of

the

function

years

after

A(t )

2010.

=

30e

Use

where

this

model

t

to

is

the

number

answer

these

questions:

a

What

b

By

was

what

each

the

population

percentage

is

the

of

the

city

in

population

2010?

of

the

city

increasing

year?

c

What

d

When

will

the

will

population

the

city’s

of

the

population

city

be

be

in

2020?

60 000?

Answers

t

0

a

A(0)

=

30e

=

30

is

so

The

population

in

2010

the

for

number

2010,

t

=

of

years

after

2010,

0

was

30 000.

(0.02)

b

A(1)

=

Write

30e

one

( 0.02 )

an

year

equation

after

for

the

population

2010.

30 e ( 0.02 )

=

e

Calculate

the

multiplying

factor.

30

= 1.0202...

The

at

population

2.02%

each

is

increasing

year. In

2020,

t

=

10

( 0.02 ) ×10

A(10 )

c

=

=

In

30e

36.642...

2020

the

population

will

be

36 642

( 0.02 ) t

d

60

=

30e

When

population

is

60 000,

( 0.02 ) t

2

=

A(t)

e

Take

( 0.02 ) t

ln 2

=

ln e

ln 2

=

0 .02t

=

60

logarithmics

Bring

down

ln 2

t

=

Solve

0

t

The

after

=

34.657...

population

34.65

during



02

will

years,

be

that

60 000

is,

2044.

Exponential

and logarithmic functions

for

t.

the

of

each

exponent.

side.

Exponential

Example

A

decay



casserole

is

removed

from

the

oven

and

cools

according

to

the

model

−0.1t

with

the

equation

What

a

the

If

b

T (t)

temperature

the

85e

,

where

t

is

the

time

in

minutes

and

T

is

°C.

temperature

of

the

casserole

when

it

is

removed

from

oven?

the

the

is

=

in

temperature

casserole

to

of

the

reach

room

room

is

25 °C,

how

long

will

it

take

for

temperature?

Answers

0

T (0)

a

=

85e

=

The

85

the

temperature

casserole

0

is

of

oven,

casserole

t

=

is

removed

from

0

the

85 °C

T

= 25

25 0

the

1t

85e

b

When

=

25

if

the

temperature

room

is

Take

logarithms

of

the

25 °C.

5

1t

e

=

=

85

of

both

sides.

17

5 0

ln

1t

e

=

ln

17

5

0

1t

=

ln

17

1 .22377...

=

t

The

=

12.2

casserole

temperature

Exercise

1

The

(3

will

after

Solve

for

t.

sf)

reach

12.2

room

min.

T

sum

of

€450

is

invested

at

3.2%

interest,

compounded

annually .

a

Write

after

b

2

In

How

i

b

how

formula

many

stages

people

many

after

How

a

for

the

value

of

the

investment

years.

early

infected

a

n

After

the

down

2

long

years

of

and

a

day

were

days

would

take

value

first

epidemic

the

exceed

there

number

rose

€600?

were

by

100

10%.

infected

ii

it

the

measles

each

people

will

for

after

250

a

week?

people

to

be

infected?

Chapter





3

Forest

fire

is

area

If

is

10

how

4

fires

left

spread

to

bur n

exponentially .

unchecked

Ever y

15%

of

hour

the

that

the

remaining

bur nt.

hectares

long

Joseph

will

did

aircraft

are

a

his

it

bur nt

take

until

parachute

velocity

and

at

the

fire

10 000

jump

time

t

for

becomes

hectares

charity .

seconds

out

are

his

control

bur ning?

After

after

of

jumping

out

parachute

of

the

opened

–1

was

v

where

m s

−0.063t

v

=

9

+

a

Sketch

b

What

29e

the

was

graph

of

Joseph’s

v

against

speed

at

t

the

instant

the

parachute

opened?

What

c

great

If

d

he

on

his

lowest

possible

speed

if

he

fell

from

a

ver y

height?

actually

landed

after

45

seconds

what

was

his

speed

landing?

How

e

was

long

when

the

did

it

take

parachute

him

to

reach

half

the

speed

he

had

opened?

b

5

Two

variables

When

of

The

a

n

=

and

x

and

=

32

n

are

and

connected

when

n

=

by

3,

x

the

=

formula

108.

Find

x

=

the

a

×

n

values

b

American

ear thquake

2,

x

geologist

to

Charles

Richter

dened

the

magnitude

of

an

be

I

M

=

log S

where

M

is

the

ear thquake

taken

of

a

100

(measured

km

from

‘standard’

0.001

the

by

(as

the

a

decimal),

amplitude

epicenter

ear thquake.

The

of

the

I

of

is

a

the

intensity

seismograph

ear thquake)

intensity

of

a

and

standard

of

the

reading

S

is

the

the

Richter

Review

Evaluate

Scale

Solve

these



3

equations.

x−1

=

90

Exponential

is

b

5

Richter

Scale

287

2x+3

a

( S)

fur ther .

5

2

mm

Severity

exercise

log

in

intensity

ear thquake

millimetres.

Explore

1

magnitude

3x

=

3

and logarithmic functions

2x

c

2

×

3

x

=

5

Mild

0–4.3

Moderate

4.3–4.8

Intermediate

4.8–6.2

Severe

6.2–7.3

Catastrophic

7.3+

3

Solve

a

b

these

 

log

+

(x

equations.

  

+

6)

–

− 

log

5

(x

= 

+

2)

=

c

ln

d

Solve

(4x

–

7)

=

(

Solve



2





)



=



=

4

The

  

 



EXAM-STYLE

x

5



 

e

log

5

 

QUESTIONS

functions

f

and

g

are

defined

as

2x

f (x)

=

e

for

all

real

x



 (  )

=

 

for

x

>

0



a

State

the

b

Explain

ranges

why

of

both

f

(x)

and

g (x).

functions

have

inverses.

- 1

Find

expressions

c

Find

an

d

Solve

for

expression

the

for

inverse

( f

g)(x)

functions f

and

( g

°

the

equation

( f

f

–1

(x)

and

g

(x).

)(x)

°

g)(x)

=

( g

°

f

)(x)

°

0.08t

5

The

number,

where

t

is

a

Find

b

How

the

n,

the

the

of

insects

number

population

long

does

it

obser vations

Review

of

in

a

days

of

take

after

the

the

colony ,

is

given

by

obser vation

colony

after

population

50

to

n

=

4000e

commences.

days.

double

from

when

commenced?

exercise

✗  +

 

1

Solve



⎛





⎞

= ⎜ ⎝

⎟ 

⎠

 +

2

Find

the

exact

Give

your

value

x

satisfying

the

equation







  +

=





 

answer

in

the

where

form

a,

b

∈



 

⎛  ⎞ 3

Find

the

exact

value

of

 



+

−







⎜ ⎝

EXAM-STYLE



⎟ 

 

⎠

QUESTION

 4

 

Write



+







−  





as

a

single

logarithm.





5

Solve

a

 

c



(

 

− ) =

    

=



b

  +



d





(





− ) =

−  +











− 

= 



EXAM-STYLE

6

If

m

=

log

QUESTION

4

and

n

=

log

x

a

log 4

8

8,

find

expressions

in

terms

of

m

and

n

for

x

b

log x

2

c

log x

16

d

log

32

8

Chapter





3(x−1)

7

The

function

Describe

a

f

is

defined

series

of

for

all

real

values

transformations

of

x

whereby

by

the

f

(x)

=

graph

e

+

2

≠

1.

of

x

y

=

f

(x)

can

EXAM-STYLE

be

obtained

from

the

graph

of

y

=

e

QUESTIONS

−1

8

Find

the

inverse

function

f

(x)

2x

a

f

(x)

=

if

3x

3e

f

b

(x)

=

10

f

c

(x)

=

log

(4x)

2

9

Solve

these

a

b

and

are

simultaneous

positive

real

equations

for a

and

b,

given

that

numbers.

1

log

64 +

log

a

b

=

8

log

a

=

ba

2

CHAPTER

4

SUMMARY

Exponents

Laws

of



●



●



●



exponents



+ 

× 

=









÷ 



=











=





●











 

●





 







●







 

 



























 

●



= 



Exponential

●

functions

An exponential function

is

a

function

of

the

form

x

f

(x)

=

a

where

●

The

domain

●

The

range

is

●

The

graph

of

of

a

is

the

the

a

positive

real

exponential

set

of

all

number

function

positive

real

(that

is

is, a

the

>

set

0)

of

and

all

a

real

numbers.

numbers.

x

the

exponential

function f (x)

=

e

is

a

graph

−x

of

exponential

growth

of

exponential

decay .

and

the

graph

y

of

f

(x)

=

e

is

a

graph

y

x

f(x)

=

e –x

y

=

e

(0, 1) 1 (0, 1)

0

x

0

x

Continued



Exponential

and logarithmic functions

on

next

page

Logarithms

Properties

of

logarithms



●



If





then











●





= 



●



 =





●



b



●



●



is



undefined

is

for

any

base

a

if

b

is

negative

undefined









=





Logarithmic

●

To

find

and

an

then

functions

of

inverse

rearrange

a

to

function

make

y

the



●

Generally



if





switch

x

and

y

subject.







algebraically ,



then









 

x

y

=

log

y

=

ln x

x

is

the

inverse

of

y

=

a

a

x

●

is

the

inverse

of

the

exponential

function y

=

e

x

y

y

=

e

y

=

x

(0, 1) y

=

In x

x (1, 0)

x

●

log

(a

log

)

=

x

and

x a

a

=

x

a

x

ln(e

lnx

)

=

x

and

e

=

x

log (10

Laws

x

log x

)

of

=

x

and

(10

)

=

x

logarithms

●

   

●

 

●

 



=

 



−





=

 



=

  

 ●





  



Change

of

base



formula

 

●





=





 

Chapter





Theory

of

The

“The

knowledge

beauty

greatest

music,

mathematics

standing

on

the

has

of

the

mathematics

simplicity

borderland

of

all

that

beautiful

Herbert

The

Beautiful

and

Have

solved

you

pleased

Was

it

with

just

solution

Look

ever

at

and

+

it

+

Turnbull

two

in

correct,

stylish,

solutions

simplify

=

x²

–

xy

=

x²

–

2yz

=

x²

–

(y²

=

x²

–

(y

–

xz

–

+

+

–

y

–

+

y²

mathematics

(x

+

even

to

y

or

+

the

z)(x

was

it

and

all

and

that

is

199

and

been

because

your

beautiful?

problem:

–

y

–

z)

z)

xy

–

2yz

science,

poetr y

(1885–1961)

Solution

z)(x

in

supreme

ar t.”

Mathematicians,



(x

wonderful

of

solutions

problem

was

efficient,

these

Solution

a

in

is

inevitableness

solution?

because

was

Expand

y

your

simple

Westren

Great

and

–

y²

–

yz

+

xz

–

yz

–

z²

(x

+

=

(x

=

x²

y

+

+



z)(x

(y

+

–

y

z))(x

–

z)

–

(y

+

z))

z²

+

(y

+

z)²

z²)

z)²

“Pure

mathematics

is,

in

its

way,



the

They

the

both

second

give

us

the

solution

same

seems

right

better.

answer

It’s

and

more

yet

somehow

elegant

poetr y

logical

than

the

first

ideas.”

and Albert

insightful

of

Einstein

one. (1879–1955)



Theory

of

knowledge:

The

beauty

of

mathematics

Simple,

beautiful

“The

essence

equations

of

mathematics

but

Stan

to

Gudder,

make

is

not

to

model

make

complicated

Professor

Here

that

of

are

simple

things

mathematics,

some

the

world

things

complicated,

simple.”

University

famous

of

Denver

equations

2

Einstein’s

Newton’s

equation:

second

E

law:

=

F

mc

=

ma

k

Boyle’s

law:

V

= p

Schrödinger’s

equation:

Hψ

=

E ψ

m

m

1

Newton's

law

of

universal

gravitation:

F

=

2

G 2

r

Isn’t

it

using

star tling

that

These

equations

moon

and

inter net

human



the

mathematical

bring

and

universe

equations

have

him

helped

back,

understand

can

such

to

put

develop

the

be

as

described

these?

man

on

the

wireless

workings

of

the

body .

These

are

just

five

equations

–

which

is

your

favorite?



Is

it

one

possible

day

that

discover

mathematics

the

ultimate

and

science

theor y

will

of

ever ything:



A

theor y

together



A

theor y

outcome

carried

Now

fully

known

that

of

has

any

explains

physical

predictive

experiment

and

links

phenomena?

power

that

for

could

the

be

out?

wouldn’t

”

that

all

Boyle's

that

Law

be

wonderful?

explains

why

bubbles

increase

in

size

as

they

rise

to

the

surface.

Chapter





Rational



CHAPTER

functions

OBJECTIVES:

1

The

2.5

reciprocal

x

function



x

≠

0,

its

graph

and

self-inverse

nature

x

The

rational

Ver tical

and

Applying

Before

Y ou

1

e.g.

x

horizontal

rational

you

should

Expand

function

ax

+

b

cx

+

d



and

its

asymptotes

functions

to

real-life

situations

start

know

how

to:

Skills

polynomials.

Multiply

the

graph

1

polynomials

check

Expand

−4(2x

a

2

−2(3x

−

1)

and

−2(3x

−

1)

=

−6x

2

2

(x

+

−

polynomials.

5)

6(2x

b

2

+

1):

2

−

3)

2

c

−x (x

+

e

x (x

3)(x

−

7)

x

d

+

2

(x

+

3)

8)

3

+

3x (x

3x

the

Graph

1)

=

3x

+

3x

horizontal

2

Draw

these

lines

x

=

0,

y

=

0,

x

=

3,

x

=

−2,

y

=

−3,

on

one

graph.

y

x

and

ver tical

e.g.

Graph

lines.

the

=

x,

x

=

−1,

y

y

=

−2

=

−x,

=

=

–x

x

=

2

4

3

lines

2

y

y

y

=

y

=

x

2,

y

=

4

x –2

y

=

3

and

y

on

the

x

same

=

=

–2

–1 –4

graph.

3

Recognize

and

describe

3

Describe

y

the

y 8

a

translation.

transformations y

=

3

x

B

6

e.g.

Find

the

translations

that

map 4

2

that

map

y

=

x

3

onto

y

=

onto

x

functions 2

A

and

A

is

B

A

and

B

and

write

6 x

0

of

2

a

horizontal

units

to

the

shift

down

right.

of

B =

x

A

and

equations

B –4

2

y

the

2

2

Function A

B

is

a

units

is

y

ver tical

up.



is

y

=

Rational

(x

shift

Function

2

B

=

x

+

3.

functions

−

of

2)

3

A

–6

–2

A

x

0 –4

2

4

6

–8

If

you

have

sounds

and

quality

of

a

rough

8160

an

so

the

idea

MP3

on

you

recording

is

minutes

player,

can

that

of

a

you

on

setting

4GB

music.

do

fit

The

and

MP3

That’s

know

it?

the

player

how

many

answer

length

will

songs,

depends

of

hold

the

136

on

song.

hours

albums,

the

However,

or

approximately

2000

songs

of

4

minutes

or

1000

songs

of

8

minutes

or

4000

songs

of

2

minutes.



This

leads

us

to

the



function

=

where

s

is

the

number

of



songs

and

m

is

the

number

of

minutes

that

a

song

lasts.



This

function

is

an

example

of

the

reciprocal

function,



 

.

= 

In

this

chapter,

reciprocal

you

will

functions

use

and

a

GDC

other



expressed

in

the

form



 

ver tical

domain

and

asymptotes

ranges

of

for

.

the

the

explore

the

functions

graphs

that

can

of

be

+ 

= 

and

to

rational

Y ou

will

examine

horizontal

+ 

graphs

of

these

functions

and

the

functions.

Chapter





.

Reciprocals

Investigation

Think

of

pairs

E.g.

24

×

and

add

1,

of

12

some

–

numbers

×

2,

8

more

×

24

12

8

3

y

1

2

3

8

and

your

0

Now

≤

y

tr y

pairs

≤

the

as

whose

3,

pairs

x

Show

graphing

3

of

×

product

product

8.

Copy

is

pairs

24.

the

table

numbers.

coordinates

on

a

graph

with

0

≤

x

≤

24

24.

same

idea

with

negatives,

e.g.

−12

×

−2 End

and

graph

these

the

Explain

what

behavior

you

notice

appearance

●

the

value

of

x

as

y

gets

bigger

●

the

value

of

y

as

x

gets

bigger

as

fur ther

either

the

end

behavior

of

of

a

about graph

●

is

too.

your

it

is

and

followed

fur ther

in

direction.

graph.

Zero

does

not

have

1

➔

The

of

reciprocal

a

number

is

1

divided

by

that

number.

a

reciprocal

as

is 0

undened.

What

does



For

example,

the

reciprocal

of

2

is your

GDC

show

for



Taking

the

reciprocal

of

a

fraction

tur ns

For

example,

the

reciprocal

of

is

1



reciprocal

A

number

down.

=

1

×



.

The

reciprocal





of

is

or





multiplied

by

its

reciprocal

0?







÷

=





of 

➔

upside



÷



The

it





1

4.



equals

1.



For

example

3

×

=

1



Geometrical

Example

in

the

reciprocal

of

inverse

were

1

Find

quantities

 propor tion

describedas

2 reciprocali

2

translation

in

a1570

of

Euclid’ s

Answer Elements from

=

2

Write

as

an

improper

fraction.

2 5

5

2 Check:

Reciprocal

of

=

Tur n

it

upside

can

find

2

reciprocals

of

algebraic

terms

too.

The



The reciprocal

of

x

is

−1

or

x

is

−1

and

x

×

x

= 1 5

reciprocal

number

➔

2 ×

down.

5

Y ou

BCE.

5

1

2

300

also

or

a

of

a

variable

called

its

=1

 multiplicative



Rational

functions

inverse.

Exercise

Find

1

A

the

reciprocals.

2

a

3

b

e

the

h

3 



reciprocals.

6.5

a

−1



g



Find

d



f



2

−3

c





x

b



y

c

3x

d





4y

e



The



+ 





term

was f

g

h



Multiply

3

i

each

quantity

by

its

6

a

b

is

the

reciprocal

of

the

reciprocal

of

is

the

reciprocal

of

the

reciprocal

of

function

y

b

What

c

Will

the

when

xy

x

=

is

happens

d

Find

e

What

f

Will

.

The

ever

x

The

reach

when

ever

48

the

480

ii

value

y

zero?

is

third

describe

4?

x?

of

y

4800

iii

edition

reach

(1797)

two

to

numbers

whose

product

This

the

is

1.

when

x

gets

48 000

iv

the

zero?

function

of

x

function

used

in

the

Investigation

480

ii

value

is

larger?

Explain.

48

i

to

reciprocal

reciprocal

use

back

24

i

to

happens

x

far

Encyclopaedia

you

y

as

Britannica

What

the

as

working.



What

Find

common

least

c

b

a

your

of

a

For

5

Show

at





4



reciprocal.



in

j





reciprocal

4800

iii

when

y

gets

48 000

iv

page

on

142.

larger?

Explain.

function

is

k

f (x)

= x

where

k

Graphs

is

of

a

constant.

reciprocal

functions

Investigation

Use

your

GDC

to

–

draw

all

have

graphs

all

the

of

graphs

in

similar

reciprocal

this

Draw

a

graph

of

( x)

a

2

=

g ( x)

b

is

the

effect

of

Draw

a

graph

of

changing

( x)

a

the

value

of

is

the

effect

of

= x

the

numerator?

2

=

g ( x)

b

3

=

h( x )

c

=

x

x

What

h( x )

c x

1 2

3

=

x

What

functions

investigation.

1 1

shapes.

changing

the

sign

of

the

x

numerator?

4 3

Copy

a

and

complete

this

table

for

f ( x)

= x

x

0.25

0.4

0.5

1

2

4

8

10

16

f (x)

b

What

c

Draw

do

the

you

notice

graph

of

about

the

the

values

function.

of

x

and

f (x)

d

Draw

f

What

in

the

the

table?

line

y

=

x

on

the

same

graph.

4 e

Reect

f ( x)

in

=

the

line

y

=

x

do

you

notice?

x

1

g

What

does

this

tell

you

about

the

inverse

function

f

?

Chapter





Asymptotes

The

on

graphs

page

closer

of

143

to

the

all

the

functions

consist

axes

but

of

f

(x),

two

never

g(x)

and

cur ves.

actually

h(x)

The

touch

in

the

cur ves

or

Investigation

get

cross

closer

and

them.

The

The

axes

are

asymptotes

to

the

graph.

is

word

derived

Greek

➔

If

a

cur ve

gets

continually

closer

to

a

straight

line

meets

it,

the

straight

line

is

called

=

b

is

an

asymptote

to

the

function

y

=

f

x

→ ∞,

f

(x )

=

symbol

‘not

together’.

f (x)

→ b

y

The

means

(x) y

As

the

an asymptote falling

y

from

asymptotos,

but which

never

asymptote

→means

=

b

‘approaches’.

The

horizontal

line

k

➔

The

graph

of

any

reciprocal

function

of

the

form

y

has

=

y

a

x

ver tical

asymptote

x

=

0

and

a

horizontal

asymptote

y

=

b

is

=

The

graph

of

a

reciprocal

function

is

called

horizontal

of

the

0 graph

➔

a

asymptote

of

y

=

f(x).

a hyperbola

y ●

The

x-axis

is

the

horizontal x

=

0, the

y-axis, 6

asymptote.

is

an

k

asymptote y

= x

●

The

y-axis

is

the

ver tical

4 y

=

–x

asymptote. 2

●

Both

are

the

all

domain

the

except

real

and

range

The

reciprocal

has

many

–4

4

6

=

0, the

x-axis,

in

The

applications

zero. y

●

two

separate

par ts

–4

of

is

an

computer

graph

are

reflections

=

other

in

y

=

–6

related

y

=

−x

and

symmetr y

In

Chapter

1

you

y

=

for

saw

x

are

this

that

to

−x number

●

par ticularly

x

of

those

each

science

asymptote.

algorithms, y

the

function

x

numbers

lines

of

may

function.

to

draw

these

the

inverse

function

of

f

theor y.

wish

to

Y ou

investigate

fur ther .

(x),



you

reflect

its

graph

in

the

line

y

=

x.

If

you

reflect

f

(x)

= 

in

the

line

y

=

x

you

get

the

same

graph

as

for

f

(x).

The

reciprocal 1

➔

The

reciprocal

function

is

a

self-inverse

function function,

f(x)

=

,

is

x

one

The

equation

of

the

function

in

the

Investigation

on

page

142

of

the

simplest

is examples

of

a

function



xy

=

24.

It

can

be

written

as



=

and

is

a

reciprocal



It



has

a

graph

Rational

simil ar

functions

to

the

one

shown

above.

function.

that

is

self-inverse.

The

design

of

the

Yas

Hotel

Asymptote

Architecture)

It

a

also

of

the

has

Formula

1

is

in

Abu

based

Dhabi

on

racetrack

(designed

mathematical

running

through

by

models.

the

center

hotel!

Example



✗ For

●

each

write

function:

down

the

equations

of

the

vertical

and

horizontal

asymptotes

●

sketch

●

state

the

the

graph

domain

and

range.

9

9 a

y

b

=

y

=

+ 2 x

x

Answers

a

Asymptotes

are

x

=

0

and

y

=

0

y

=

2

y

20

15

10

5

x

0 –6

–4

–2

2

4

6

–5

–10

–15

–20

Domain

range

b

y

x

∈

∈

,

,

Asymptotes

y

x

≠

are

≠

0,

0

x

=

0

and

The

y

is

6

f

graph

the

(x)

same

but

of

as

f

(x)

the

shifted

2

+

2

graph

units

of

in

4

the

y-direction.

2

x –30

–20

–10

–2

–4

–6

Domain

range

y

x

∈

∈

,

,

y

x

≠

≠

0,

2

Chapter





Exercise

1

Draw

B

these

on

separate





a

graphs.



=



b

=

xy

c



=

8

Y ou

need

to



On

the

same

graph

show



=





do

questions

4b

and



analytically

algebra a

Sketch

the

graph



of

 

=

and

write

down

its

the

graph



of

 

3b

both

(using

and

sketching



Sketch

c

by

asymptotes.



b

able

=



3

be

 and

2

to



and

using

transformations)

=

+ 

and

write

down

and

its



using

your

GDC.

asymptotes.

4

Identify

and

the

then

horizontal

state

their

and

ver tical

domain

and

asymptotes

of

these

functions

range.

It

=



b

=

+ 

5

The

Corr yvreckan,

world,

the

is

between

coast

west

The

of

and

heard



c

=

the

the

roar

16 km

of

third

largest

Flood

the

to

draw

graphs.



islands

Scotland.

the

speed

the

help

− 





may









a

of

whirlpool

Jura

tides

resulting

and

and

in

Scarba

inflow

maelstrom

the

off

from

can

the

be

away .

of

the

surrounding

water

increases

as

you



approach

the

center

and

is

modeled

by



where

=

s

is

 −1

the

speed

the

center

a

Use

and

of

in

your

0

≤

s

the

water

in

m s

and

d

is

the

distance

from

metres.

GDC

≤

to

sketch

the

function

with

0

≤ d

≤

50

200.

−1

6

b

At

c

What

The

what

is

force

distance

the

(F )

is

speed

the

of

required

speed

the

to

10 m s

water

raise

an

?

100 m

object

from

of

the

mass

center?

1500 kg

is

[



modeled

by



where

=

Archimedes

believed

l

is

the

length

of

the

lever

in

is

to

have

a

place

said

metres “Give

me

to



stand,

and

the

force

is

measured

in

Sketch

the

graph

with



≤



≤

 

≤



How

much

force

would

lever

enough

you

need

to

and

I

≤  shall

b

a

newtons. long

a

and

apply

if

you

had

a

move

the

earth. ”

2 m

N

is

the

symbol

for

lever?

the c

How

long

force

of

would

the

lever

need

to

be

if

you

could

manage

unit

a newton.



Rational

functions

i

1000 N

ii

2000 N

iii

3000 N?

of

force,

the

.

Rational

Have

you

noticed

functions

the

way

the

sound

of

a

siren

changes

as

a Sound

fire

engine

or

police

car

passes

you?

The

obser ved

frequency

frequency

measured

higher

than

the

emitted

frequency

during

the

approach,

it

at

the

instant

of

passing

by ,

and

it

is

lower

during

the

for

it

moves

the

obser ved

toward

you

This

is

frequency

called

of

the

sound

Doppler

when

the

effect.

The

source

is

of

per

second.

equation

traveling

is:





away .

her tz

number

the waves

time

in

is (Hz),

identical

is

is



= 





where

−1

●

330

is

●

f

the

is

the

speed

of

obser ved

sound

in

frequency

m s

in

Hz

1

●

f

is

the

emitted

●

v

is

the

velocity

f

is

a

rational

frequency

of

the

source

toward

you

function.

1

h (x)

 

➔

A

rational

function

is

a

function

of

the

form



 

since

  

where

g

and

h

are

this

of

the

course

form

g(x)

px

+

q

and

so

be

zero

a

value

divided

polynomials. by

In

cannot

=

h(x)

we

will

can

be

restricted

investigate

to

linear

rational

zero

is

undened.

functions

functions f

(x)

where





 

+ 

= 

Example

+ 



The

−1

A

vehicle

is

coming

towards

you

at

96 km h

(60

miles

per

hour)

units

must

sounds

its

hor n

with

a

frequency

of

8000 Hz.

What

is

the

frequency

sound

you

hear

if

the

speed

of

sound

is

330 m s

can

to

−1

−1

=

Conver t

96 000 m h

metres

96 000 −1

96 000 m h

be

the

equation.

same

Y ou

per

hour

to

round

get

an

numbers

approximate

answer .

second.

−1

=

=

26.7 m s

3600

Since

330

Observed

kilometres

per

all

the

?

Answer

96 km h

speed

of

in

−1

the

of

and

frequency

1

hour

=

3600

seconds

f

=

330

v

330 × 8000

=

330

=

26.7

8700 Hz (3 sf )

Chapter





Investigation

–

graphing

rational

functions

1

Use

a

your

GDC

to

show

sketches

of

y

,

=

y

=

y

x

x

1

1

1

=

2

x

+

3

2

and

y

= x

Copy

b

and

+

3

complete

the

table.

Rational

Ver tical

Horizontal

Domain

function

asymptote

asymptote

Range

1

y

= x

1

y

= x

2

1

y

= x

+

3

2

y

= x

What

c

the

+

3

effect

ver tical

does

changing

the

denominator

d

What

do

you

notice

about

the

horizontal

e

What

do

you

notice

about

the

domain

the

range

the

ver tical

What

f

do

on

asymptotes?

and

the

value

of

asymptote?

you

horizontal

have

asymptote?

notice

about

and

the

value

of

the

asymptote?

k

Rational

functions

of

the

form

y

= x

− b

1

is



A

rational

function



=

, where 

will

have

that

is,

a

vertical

when

x

=

k

and

b

are



asymptote

consider

when

the

denominator

equals

zero,

b

detail

horizontal

Example

in

this

the

Knowledge

the

The

undened.

asymptote

will

be

We

will

0

constants,

end

of

in

more

Theor y

section

the

of

at

chapter .

the x-axis.



1 a

Identify

b

State

c

Sketch

the

horizontal

and

ver tical

asymptotes

of

y

= x

the

domain

the

and

function

3

range.

with

the

help

of

your

GDC. Y ou

may

explore

wish

the

Answers

of a

The

x-axis

horizontal

x

=

3

is

( y

=

0)

is

the

asymptote.

the

ver tical

asymptote.

Since the numerator will never be 0,

the

graph

touches

The

of

the

x

=

functions

never

x-axis.

is

zero

3.

{

Rational

function

denominator

when



this

Continued

on

next

page

innity.

to

concept

b

Domain

Range

x

y

c

∈

∈

,

,

y

x

≠

≠

0

3

y

8

6

1

4 y

= x

–

3

2

x

0 –2

–4

–2

–4

–6

–8

Exercise

1

Identify

C

the

horizontal

and

ver tical

and

range.

asymptotes

of

these

functions Y ou

and

state

their

domain

should

algebra





a

= 



c

=

+ 







d

= 



=

‘using







e



=



f

+  



=

−  

+ 



g

an

+  

h



=



−  

Sketch

the

each

domain

function

and

with

the

help

question



and

=

+ 



c

your

state

=

+ 

although

answers

to

check

with

a



e

−  





=

1,

want







b

=





GDC

range.



d

your

may

GDC.

 a

of

do

+  you

2

analytic

to



=

+ 

called

+  method’)



is





 b

use

(this

Use

+ 

the



=

− 



f

=

your

GDC

correct

with

viewing

+  window.





+ 







g



=

− 



h

=

+

3

When



i

=

lightning

instantaneously .

strikes,

the

But

sound

the

+  



light

reaches

of

the

your

thunder

eyes



vir tually

travels

at

−1

approximately

the

.

331 m s

temperature

of

the

However,

sound

surrounding

air.

waves

The

are

time

affected

sound

by

takes

to



travel

one

kilometre

is

modeled

by



=

where 

time

a

b

in

seconds

Sketch

If

you

the

are

and

c

graph

one

the

thunder,

On

the

is

of

the

t

is

temperature

for

kilometre

what

the



in

temperatures

away

and

it

temperature

is

of

t

is

the

+ 

degrees

from

3

−20 °C

seconds

the

Celsius.

to

40 °C.

before

you

surrounding

air?

hear

 4

a

same

set

of

axes,

sketch y

=

x +

2

and



= 

Compare

linear

the

two

function

graphs

and

its

and

make

reciprocal

connections

+ 

between

the

function.

 b

Now

do

the

same

for

y

=

x

+

1

and



= 

+ 

Chapter





Rational

functions

of

the

y

form

ax

+ b

cx

+ d

=



➔

Ever y

rational

called

a

function

of

the



form

has 

graph

of

any

rational

function



has

Use

your

GDC

x

y

=

, +

x

Copy

b

a

ver tical

and

a

+ 

asymptote.

Investigation

a

graph

+ 

= 

horizontal

a

+ 

hyperbola.



The

+ 

=

y

graphing

show

x

+ 1

x

+

=

3

and

to

–

sketches

rational

y

and x

complete

2x

=

the

+



of

2x

, 3

functions

y

1

=

3

x

+

3

table.

Rational

Ver tical

Horizontal

function

asymptote

asymptote

Domain

Range

x

y

= x

y

+

3

x

+ 1

x

+

= 3

2x

y

= +

x

2x

y

3

1

= x

+

3

c

What

do

you

notice

about

the

horizontal

d

What

do

you

notice

about

the

domain

asymptotes?

and

the

value

of

the

ver tical

asymptote?

y

➔

The

ver tical

asymptote

occurs

at

the x-value

that

makes

the 4

denominator

zero. 3



➔

The

horizontal

asymptote

is

the

line



= a

 y

2

= c 1

To

find

the

horizontal

asymptote

rearrange

the

equation

to

make

x

0

x

the

–6

subject.

–4

–2

–1

ax

y

+ b

d

=

x –2

cx

y ( cx

cyx

+ d )

+ d

=

ax

− ax

=

b − dy

x

=

b

dy

cy

The

horizontal

that

is,

asymptote

when a

cy

=

a

or

y

=

c



Rational

functions

–3

+ b

a

occurs

when

the

denominator

is

zero,

= c

Example



x

For

the

function

y

+ 1

= 2x

a

sketch

b

find

c

state

the

the

ver tical

the

4

graph

and

domain

horizontal

and

asymptotes

range.

Answers

y

a

4

3

2 x

y

+

1

= 2x

–

4

1

x

0 –2

–4

–1

–2

–3

b

Ver tical

asymptote

x

=

2

When

2x

a

c

−

4

=

0,

1

Horizontal

asymptote

y

x

∈ ,

x

≠

=

2

a

=

=

1,

=

y =

2,

2

Domain

c

x

c

2

1

Range

y ∈ ,

y

≠ 2

Exercise

1

Identify

and

the

then

 a

D



state

the

Match

 b







=

b



i

 

− 

y

c



x

1

x

3

d

= 

ii



8

6

4

2

2

x

–6



y

4

–4

+ 



=

6

–2



=

graph.

+ 

y

–2

functions

 



d

8

–4

these

+ 

= − 

= 



− 





the

of

range.

c

with

 a

and

asymptotes

+ 



function

ver tical

=



the

and

domain

+ 

= 

2

horizontal

0 –4

x

–2

–2

–4

–6

Chapter





y

iii

y

iv

8

8

6

6

4

4

2

2

x

0

x

0

–2

–4

–2

–2

–2

–4

–6

3

Sketch

and

each

using

your

GDC

and

state

the

domain

range.





a

function



+ 

=



b





c

 

+ 





= + 

− 

+ 



= 

+ 





+  Check



d

=



e



=



f

 



 

by

using

graph





g



= 

4

Write

x

5

=

a

−4

Chris

and

a

design

and

in

has

T-shir ts

ver tical

at y

=

surfers

It

that

will

it

to

function.





asymptote

at

3

and

cost

will

the

GDC

= 

a

for

garage.

estimate



− 

asymptote

their

they

that

answer

your

set

up

$450

cost

a

to

$5.50

T-shir t

set

to

up

the

print

T-shir t.

Write

a

Write

linear

a

T-shir t

c

horizontal

Lee

T-shir ts.

b

function

business

equipment

a

i

−

rational

 

=



and

printing

each



h

your

=



What

rational

of

is

function

Remember

to

domain

x

giving

take

function

producing

the

C (x)

of

of

the

A (x)

the

total

cost

of

cost

into

account.

set-up

giving

the

average

producing x

cost

per

them.

A (x)

in

the

context

of

the

problem?

Explain.

d

Write

e

Find

down

the

ver tical

asymptote

of

A (x). Sketch

this

the

value

Exam-Style

6

horizontal

have

rule

is

over

age

of

‘Take

plus

the

12.

the

context

for

of

A(x).

the

What

meaning

does

problem?

Question

Y oung’s

the

in

asymptote

age

a

way

two,

of

the

Multiply

of

calculating

based

child

this

on

in

the

adult

years

number

by

doses

and

the

of

medicine

for

children

dose.

divide

adult

by

their

age

dose.’



This

is

modeled

by

the

function



where

= 

dose,

a

is

the

adult

years.



Rational

functions

dose

in

mg

and

t

c

is

the

child’s

+ 

is

the

age

of

the

child

in

the

function.

a

Make

of

a

table

of

values

for

ages

2

to

draw

to

12

with

an

adult

dose

100 mg.

b

Use

your

c

Use

the

values

from

a

a

graph

of

the

function.



graph

to

estimate

the

dose

for

a

7

-year

old.



d

Write

down

e

What

does

Y oung’s

7

The

a

new

cost

for

cost

a

a

Sketch

a

d

Since

e

Explain

the

A

it

of

the

the

horizontal

horizontal

asymptote.

asymptote

mean

for

electricity

costs

and

function

as

a

per

$550.

that

lasts

year

for

a

refrigerator

Determine

for

15

the

years.

total

is

$92.

annual

Assume

costs

electricity .

that

gives

function

of

the

the

annual

number

cost

of

of

a

years

you

own

refrigerator.

graph

window?

f

of

refrigerator

refrigerator

c

of

purchase

Develop

the

value

refrigerator

include

b

the

equation

r ule?

average

A

the

of

Label

this

is

a

the

that

the

function.

What

is

an

appropriate

scale.

rational

meaning

function,

of

the

determine

horizontal

its

asymptotes.

asymptote

in

terms

of

refrigerator.

company

will

last

difference

Review

offers

at

in

least

a

refrigerator

twenty

years.

that

Is

costs

this

$1200,

but

refrigerator

says

wor th

that

the

cost?

exercise

✗

Extension

material

Worksheet

5

fractions

Exam-Style

1

Match

the



 

function

with

the

ii





 

CD:

aysmptotes

graph.







 

v



a

iii









 

 

=

 

iv

and

on

Continued

Question

 i

-



 













+ 

= 

vi



 



+ 



+ 



y

b

y

8 6

6 4

4 2

2

0

x

–2

–2 0

x

–2

–2 –4

–4 –6

–6

Chapter





Exam-Style

QuestionS

c

d

y

y

6 8

4 6

2 4

x

–2

x

0 –1

–3

3 –4

–2

–6 –4

e

y

f

y

6 6

4 4

2 2

x

0 –2

x

0

–2

–4 –4

–6 –6

 2

Given



a

 

b



 



3

i

Sketch

ii

Determine

iii

Find

For

the

each

domain

the

of

and





=

=



c



 

+ 

= 

+ 



function.

the

ver tical

domain

these

and

and

range

functions,

horizontal

of

write

the

asymptotes

of

the

function.

function.

down

the

asymptotes,

range.

a

b

y

y

6

8

5

4 f (x)

6

=

6 x

+

4

f (x)

=

–

3 4

x

2 x

0 –6

–2

–4

–2 x

0 –6

–4

–2

–4

–2

–6

–4

–8

–6

–8

y

c

y

d

6

8

4

6

2

4

–3 f (x)

2 f (x)

=

– x

+

=

+ x

–

5

1

2

6 2 0

x

x

0 –8



Rational

functions

–6

–4

4

–4

–2

–6

–4

–8

–6

6

8

4

A

group

of

weekend

a

If

c

the

in

want

a

spa.

health

number

terms

b

Draw

c

Explain

The

students

represents

this

5

at

a

of

f

is

given

student

an

=

,

x

∈

,

x

Find

the

ver tical

iii

Write

b

Find

c

Hence

down

the

sketch

the

to

represents

show

the

cost

and

domain

of

−2

asymptote

asymptote

the

of

y

=

f

(x)

graph.

the

point P

at

which

the

of

the

intersection

graph

of

y

=

of

f

the

(x),

graph

with

showing

the

the

axes.

asymptotes

lines.

with

the

help

of

your

− 

State

the

b



 



=

+ 



c

 

=









+ 





−  

e



 

London

to



f



from



=



flies

of

GDC.

range.

=

distance

of



=

airline

graph

intersect.

function

and

 

the

Question

each

 

of

of

coordinates



An

and s

equation

range



2

a

exercise

Exam-Style

domain

the

points

dotted

Review



for

students.

on

≠

ii

d

voucher

+ 

horizontal



a

$300.

by

the

a

teacher

costs



Find

Sketch

each

write

of

their

voucher

function.

limitations

asymptotes

1

the

i

by

for

number

of

give

function.



a

cost

students,

the

any

function

(x)

of

graph



f

the

to

The

New

 

=

+ 

Y ork,

which

is

− 

a

5600 km. 

a

Show

that

this

information

can

be

written

as



=  −1

where

and

b

is

Sketch

and

c

t

s

If

of

0

the

the

the

a

≤

the

is

t

average

time

graph

≤

in

of

speed

of

the

plane

in

km h

hours.

this

function

with

0

≤ s

≤

1200

20.

flight

takes

10

hours,

what

is

the

average

speed

plane?

Chapter





Exam-Style

3

People

with

amount

of





Questions

sensitive

time



skin

spent

in

must

direct

be

careful

sunlight.

about

The

the

relation

+ 

= 

where

the

is

the

spend

a

Sketch

b

Find

s

i

in

=

relation

number

what

mayor

Bangkok.

of

s

this

is

The

the

=

and

that



≤

minutes

40

cost

in

population



s

the

sun

person

skin

with

=

value,

sensitive

gives

skin

damage.

≤  

skin

scale

can

≤



be

≤ 

exposed

when

100

asymptote?

out

(c)

is

a

that

iii

represents

giving

s

without

when

of

horizontal

the

time

sunlight

is

Explain

percent

direct

of

ii

d

city

minutes

10

What

The

in

amount

this

the

c

in

time

maximum

can

4

m

for

face

Thai

is

a

person

masks

baht

given

with

during

for

giving

a

sensitive

flu

skin.

outbreak

masks

to

m

by

  



= 

a

Choose

the

b

the

20%

of

the

The

suitable

scale

and

use

your

GDC

to

help

sketch

cost

of

supplying

50%

ii

iii

90%

population.

Would

this

5

a

function.

Find

i

c



it

be

model?

function

f

possible

Explain

(x)

is

to

supply

your

defined

all

of

the

population

using

answer.

as

 f

(x)

=

,

 + 

a

Sketch

b

Using

your

the

equation

the

value

Rational

value

functions

of

sketch,

ii

the

≠ 

cur ve

i

iii



the

x



of

of

of

the

the

f

for

write

each

−3

≤

x

≤

5,

down

asymptote

x-intercept

y-intercept.

showing

the

asymptotes.

CHAPTER

5

SUMMARY

Reciprocals

●

The

●

A

of

reciprocal

number

a

number

multiplied

by

its

is

1

divided

reciprocal

by

that

equals

number.

1.



For

example

3

×

=

1



 −1

●

The

of

reciprocal

x

is

or

−1

x

and

x

×

x

=1



The

●

If

reciprocal

a

cur ve

never

●

The

gets

meets

graph

continually

it,

of

function

the

any

closer

straight

line

reciprocal

is

to

a

straight

called

function

line

but

an asymptote

of

the

form



y

=

has

a

ver tical

asymptote

x

=

0

and

a

horizontal



asymptote

●

The

graph

y

=

of

0

a

reciprocal

function

is

called

y

a hyperbola x

=

0, the

y-axis,

6

■

The

x-axis

is

the

horizontal

■

The

y-axis

is

the

ver tical

■

Both

the

domain

and

asymptote.

is

an

4

asymptote.

range

are

all

asymptote

y

the

real

=

–x

2

numbers f

except

zero. x –2

–4

■

The

two

separate

par ts

of

the

graph

are

4

y

of

each

other

in

y

=

=

y

●

y

The

=

x

and

y

=

reciprocal

Rational

−x

are

lines

function

is

0, the

x

–4

−x

is

■

6

reflections

a

of

symmetr y

self-inverse

for

this

=

an

asymptote.

x –6

function.

function

functions

  ●

A

rational

function

is

a

function

of

the

form



 

= y

  

where

g

and

h

are

polynomials.

4

 ●

Ever y

rational

function

of

the

form



has 

called

a

+ 

=

a

graph

3

+ 

hyperbola.

a y

●

2

= c

The

the

ver tical

asymptote

denominator

occurs

at

the x-value

that

makes

1

zero. 

x –6

●

The

horizontal

asymptote

is

the

line



–4

–2

= –1



d x –2

= c

–3

Chapter





Theory

of

knowledge

Number

Egyptian

systems

fractions

3

The

ancient

Egyptians

only

1

In

algebra:

= 4x

fractions

with

a

numerator

of

for

1

example:

, 3

etc.

meant

that

each

algebraic

Egyptian

expression

as

fraction.

4

instead

of

they

4

5

7

23

3x

4x

4x

24x

wrote

4

1

Write

an

3

This

4x

1

,

2

+ 2x

1, 

1

1

used

1

+

.

2

Their

fractions

were

all

in

the

4

Where

1

and

form

are

called

uni t

do

you

think

this

could

be

fractions .

n

useful? 2

Numbers

such

as

were

represented

as

7

What

2

sums

of

unit

fractions

(e.g.

1

= 7

the

twice

(so

same

fraction

+ 4

could

not

1

of

these

).

be

it

possible

to

write

ever y

fraction

used

an

Egyptian

fraction?

1

= 7

limitations

28

as 2

the

fractions?

Is

Also,

are

1

+ 7

was

not

allowed). How

7

5

1

do

you

1

know?

For

example,

would

be

8



Write

these

+ 2

as

unit

8

fractions.

5

5

2

6

6

8

5

7

In

an

Inca

quipu,

the

strings

represent

numbers

The

Rhind

1650

BCE

fractions

200



Theory

of

knowledge:

Number

systems

Mathematical

contains

copied

years

older!

a

from

Papyr us

table

of

another

dated

Eg yptian

papyrus

Is

there

a

dierence

between 25¢

zero

More

had

the

and

than

2000

systems

ninth

for

a

a

circle

this

became

Who



What



Make



Notice



Now



We



rst

some

How

par t

did

on

the

zero

before

of

in

keep

The

Hindu

a

the

name

place

rows’.

sifr

that

of

The

if,

tens,

Arabs

eventually

was

nothing?

the

subsets

and

the

1

be

is

{0}

were

What

not

something.

the

tentative

Mayan

and

equation

BCE.

and

{0,

1,

2,

another

9

3}.

is

{

x

=

3²

about

a

year

sure

+

what

Zeno’ s

use

Inca

of

of

to

}.

and

do

the

equation

3x

=

0.

zero?

with

paradoxes

(a

zero

good

and

they

topic

to

questioned

research)

how

depend

in

zero.

cultures

understand





In

and

remarked

the

cultures

number.

that?

subset

Greeks

could

appears

‘to

of

mathematician

al-Khwarizmi

(empty).

Solve

CE

ancient

nothing



1

Islamic

used

that

all

one

this.

have

The

of

be

and

absence

zero?

used

that

tr y

the

an

zero.

mean

list

Babylonian

number

sifr

used

was

a

no

word

this



CE,

should

circle

our

Does

ago,

Muhammad

calculation,

little

called



years

representing

centur y

philosopher

in

nothing?

zero?

ative?

What

happens

if

you

divide

zero

by



anything?

g



pens

The

Mayans

shell

symbol

represent

if

used

you

divide

zero

by

by

zero?

zero?

a

to

zero.

Chapter





Patterns,

sequences

and

 series

CHAPTER

OBJECTIVES:

Arithmetic

1.1

geometric

series.

sequences

and

sequences

Sigma

and

series;

series;

sum

sum

of

of

nite

arithmetic

nite

and

series;

innite

geometric

notation.

Applications

n

The

1.3

binomial

theorem:

expansion

of

(

a +

b

)

,

n ∈ ;

⎛ n ⎞

Calculation

of

binomial

coefcients

using

Pascal’ striangle

and

⎜

⎝

Before

Y ou

1

you

should

Solve

change

e.g.

the

Solve

and

how

to:

quadratic

subject

the

⎠

start

know

linear

⎟

r

of

a

equation

Skills

equations

and

1

formula.

n(n

–

4)

=

12

check

Solve

each

a

3x

b

p(2

c

2

2

–

5

–

equation.

=

p)

5x

=

+

7

–15

n

n

–

4n

=

12

4n

–

12

=

0

2)

=

0

+

9

=

41

2

–

n

2

(n

–

6)(n

n

e.g.

–2,

Make

ac

b

2

=

+

=

=

b

ac

n

b

–

=

the

for

a

6m

b

2pk

+

k

8k

=

30

6

subject

of

this

3

If

T

3

Substitute

known

e.g.

the

values

into

–

5

=

3

formula.

3

+

Solve

T

=

2x

(x

+

3y),

then

find

the

value

of

when

a

x

=

3

and

b

x

=

4.7

y

=

5

y

=

formulae. and

–2

4

Using

formula

A

=

of

=

2

3p

–

10q,

x

find

the

value

A

if

p

4

A

=

3p

–

–

=

3(2)

A

=

3(16)

A

=

48

10(1.5)

–

q

=

1.5

4

Using

value

10q

4

A

and

15

the

of

formula

m

m

if

a

x

=

5

and

y

=

3

b

x

=

3

and

y

=

–2

c

x

=

–5



A



=

–

15

sequences



= 

33

Patterns,

and

and

series

=

2

3

–

y

,

find

the

The

bacteria

in

this

petri

dish

are

growing

and

reproducing;

in

this

[

Bacteria

petri

case

the

total

measured

mass

The

at

as

mass

of

patter n

after

8

this

help

will

the

make

we

you

predict

●

work

●

predict

●

calculate

the

●

calculate

how

in

be

use

or

how

the

to

24

a

total

it

will

will

dish

the

so

At

10:00

8

a.m.

will

be

the

6

mass

grams,

a

is

the

on.

forms

the

a

numerical

mass

of

patter n.

bacteria

in

the

dish

near

patter ns.

and

distant

Patter ns

future.

can

For

to:

countr y

take

natural

it

and

at

mathematical

a

distance

long

hours.

mass

predict

about

of

two

in

hours.

patter ns

long

long

in

study

population

how

total

grams,

used

will

ever y

the

predictions

●

out

12

hours

can

the

so

bacteria

12

chapter

example,

be

could

hours,

us

doubles

3 grams,

12:00

This

In

mass

growing

dish

to

pay

resource

that

take

in

a

for

20

off

will

years

a

loan

last

bouncing

an

bank

ball

will

investment

to

travel

double

value.

Chapter





. Patterns

Investigation

Joel

He

decides

saves

week,

and

Copy

a

and

to

$20

so

and

how

star t

the

and

–

saving

saving

rst

sequences

money

money.

week,

$25

the

second

week,

$30

the

third

on.

complete

much

he

the

has

table

below

saved

in

to

total,

show

for

Week

Weekly

T otal

number

savings

savings

1

20

20

2

25

45

3

30

75

how

the

much

rst

8

Joel

saves

each

week,

weeks.

4

5

6

7

8

b

How

much

will

c

How

much

money

d

How

long

T ry

e

Let

let

Let

of

In

the

week

as

T

write

the

for

for

total

10th

save

him

week

his

the

to

the

amount

formula

represent

investigation

a

form

number

are

some

8,

11,

14,

400,

1,

4,

9,

in

week?

total

save

a

amount

of

in

the

total

of

money

In

of

the

rst

at

money

he

17th

year?

least

Joel

saves

week?

$1000?

saves

each

each

week,

week.

and

number .

the

total

savings,

5,

10,

15,

amount

and

Patterns,

The

total

is

a

let

n

of

money

represent

Joel

the

has

saved.

number

25,

100,

25,

…

…

sequences

amounts

to

a

sequences.

…

and

series

of

money

of

Joel

money

saves

he

has

each

saved

sequence.

patter n

…

200,

amounts

according

number

20,

the

different

sequence

17,

16,

a

order

Here

800,

above,

sequence.

passes

par ticular



a

the

in

Joel

for

formula

represent

to

will

take

represent

form

A

a

it

save

weeks.

time

➔

write

M

n

T ry

f

to

will

Joel

of

numbers

r ule.

arranged

in

a

➔

Each

individual

called

In

is

the

11,

Y ou

a

can

or

element,

of

a

sequence

is

term

sequence

the

number,

third

also

8,

11,

term

use

is

the

14,

17,

…,

14,

and

notation

so

u

the

first

term

is

8,

the

second

term

on.

to

denote

the

nth

term

of

a

n

sequence,

So

u

for

=

where

8,

11,

8,

u

1

Y ou

n

is

14,

=

a

positive

17,

11,

…

u

2

can

you

=

14,

integer.

could

and

say

so

on.

3

continue

the

patter n

if

you

notice

that

the

value

previous

term:

of

each Sometimes,

term

is

three

greater

than

the

value

of

the

letters

8,

11,

14,

17,

20,

23,

represent

26

a

For

this

sequence,

you

could

write:

=

u

8

and

u

1

This

is

called

a

recursive

formula,

in

=

u

n+1

which

the

+

a

example,

a

,

t

n

on

the

value

of

the

previous

,

of

the

sequence

one-half

the

800,

value

400,

of

the

200,

100,

previous

terms

of

or

we

x

might

n

to

n

term. represent

In

to

term use

depends

the

u

sequence.

For

of

use

than

3

n

value

we

other

…,

the

value

of

each

term

a

the

nth

term

sequence.

is

term.



In

this

case,

=

u

800



and

=



 +

1





Example

Write

a



recursive

a

9,

15,

b

2,

6,

21,

18,

27,

54,

formula

for

the

n th

term

of

each

sequence.

…

…

Answers

u

a

=

9

and

u

1

=

u

n+1

+

6

To

get

add

u

b

=

2

and

u

1

from

one

ter m

to

the

next,

you

from

one

ter m

to

the

next,

you

n

=

3u

n+1

To

6.

get

n

multiply

by

3.

Sometimes

Sometimes

term

of

of

a

In

the

a

term

it

is

more

sequence.

without

useful

With

having

a

to

write

general

to

know

a general formula for the

formula,

the

value

you

of

can

the

find

the

previous

nth

value

called

rule

the

for

1,

4,

9,

16,

25,

…

,

each

term

is

a

perfect

2

first

term

is

term’.

that

n,

square. term

number ,

will

2

,

1

nth

term.

the

The

is

‘general

the

Remember

sequence

this

the

second

is

2

,

and

so

on.

A

general

formula always

be

a

whole

2

for

the

nth

term

of

this

sequence

is

u

=

n number .

n

In

the

sequence

5,

10,

15,

20,

25,

…

,

each

term

is

a

multiple

of

We

could

not

3

5. have

a

‘

th’

term,

or

a

4

The

first

term

is

5

×

1,

the

second

is

5

×

2,

and

so

on.

‘7.5th’

A

general

formula

for

the

nth

term

of

this

sequence

is

u

=

term.

5n.

n

Chapter





Example

Write

a

4,

a

general

8,

1

12,

1

,

,

6

for

the

n th

term

of

each

sequence.

…

1

,

3

formula

16,

1

,

b



9

…

12

Answers

a

u

=

4n

Each

ter m

is

a

multiple

of

4.

n

1 b

u

=

The

denominators

are

the

multiples

n

3n

of

Exercise

1

Write

down

3,

7,

11,

c

3,

4,

6,



 

u

a

15,



…

  , …

down

10

the

and

and

first



=

(





=

a

2,

c

64,

Write

1,

2,

4,

d

5,

–10,

8,

…

20,

–40,

…

6.0,

6.01,

6.012,

6.0123,

…

terms

in

)

each

u

b

sequence.

=

3

and





=

 +

1

+



)



u

d

=

x

and





 

1









recursive

4,

b



(



4

four

 +

a

sequence.



=

Write

each

f

 +

1

3

in





u

terms

…

1

c

three





=

next

13,





Write

the

9,





2

A

a

e

3.

6,

32,

8,

for

each

…

16,

down

formula

sequence.

1,

b

8,

the

…

7,

d

first

four

terms

in

3,

9,

12,

each

27,

17,

…

22,

…

sequence. T o

nd

the

rst

term

n

a

u

c

u

=

3

=

2

u

b

n

6

a

a

2,

c

64,

The

n

=

1;

to

n

u

d

general

4,

6,

32,

2 ,

2

3

=

n

nd

the

second

use

n

2

term

n

Write

1

+

substitute

1

n

e

−6n

n

n

5

=

8,

the

nth

8,

…

term

of

9,

each

b

1,

3,

27,

d

7,

12,

f

x,

2x,

is

known

17,

=

and

so

on.

sequence.

…

22,

…

4

,

,

3

for

…

16,

3 ,

formula

4

…

3x,

4x,

…

5

sequence

1,

1,

2,

3,

5,

8,

13,

…

as

the

Fibonacci

sequence.

a

Find

b

Write

.

the

a

15th

term

recursive

Arithmetic

of

the

formula

Fibonacci

for

the

sequence.

Fibonacci

sequence.

sequences [

Fibonacci,

as

In

the

sequence

8,

11,

14,

17,

…,

the

value

of

each

term

is

than

example



of

Patterns,

the

value

of

the

previous

an ari thmetic sequence ,

sequences

and

series

term.

or

This

sequence

arithmetic

of

known

Pisa

three (Italian

greater

also

Leonardo

is

an

progression.

c.

1175 – c.

1250)

➔

In

an

arithmetic

sequence,

the

terms

increase

or

decrease

by

a

Examples

of

arithmetic

progressions

constant

value.

This

value

is

called

on

or

d.

The

common

difference

can

appear

the common dierence ,

be

a

positive

or

a

the

Ahmes

negative

Papyrus,

which

dates

value. from

For

8,

about

1650

BCE.

example:

11,

14,

17,

…

In

this

sequence, u

=

8

and

d

=

3

=

35

and

d

=

–5

=

4

and

d

=

0.1

=

c

and

d

=

c

1

35,

30,

25,

20,

…

In

this

sequence, u 1

4,

4.1,

4.2,

4.3,

…

In

this

sequence, u 1

c,

2c,

3c,

4c,

…

In

this

sequence,

u 1

For

any

arithmetic

sequence,

u

=

u

n+1

We

can

find

difference,

In

an

d,

to

term

the

arithmetic

=

u

any

the

first

of

the

+

sequence

previous

d

n

by

adding

the

common

term.

sequence:

term

1

u

=

u

2

u

=

u

3

u

d

+

d

=

(u

2

=

+

d

=

=

d)

+

d

=

u

+

+

2d)

+

d

=

u

1

u

+

d

=

+

3d

+

4d

1

(u

4

2d

1

(u

3

5

+

1

u

4

u

+

1

+

3d)

+

d

=

u

1

1

…

…

=

u

u

n

➔

+

(n

–

1)d

1

Y ou

can

find

formula:

Example

u

the

=

u

nth

+

n

1

12th

term

term

(n

–

of

an

arithmetic

sequence

using

the

1) d



a

Find

the

b

Find

an

of

expression

the

for

arithmetic

the

n th

sequence

13,

19,

25,

…

term.

Answers

a

u

=

13

and

d

=

6

Find

these

values

by

looking

at

the

1

u

=

13

+

(12

=

13

+

66

=

79

–

1)6

sequence.

12

u

For

n

=

the

12

12th

into

ter m,

the

substitute

for mula

12

u

=

u

n

b

u

=

13

+

(n

–

1)6

+

(n

–

1) d

1

For

the

nth

ter m,

substitute

the

n

=

13

+

6n

–

6

values

of

u

and

d

into

the

for mula

1

u

= n

6n

+

7

u n

=

u

+

(n

–

1) d

1

Chapter





Example



If

Find

the

number

of

terms

in

the

arithmetic

a

sequence

sequence

continues

84,

81,

78,

…,

and

there

term,

Answer

u

=

84

and

=

84

+

d

indenitely

12.

=

–3

Find

these

values

by

looking

at

it

is

is

no

an

nal

innite

sequence.

the

1

u

(n

–

1)(–3)

=

12

If

sequence.

a

sequence

ends,

or

n

Substitute the values of

u

has

and d into

a

‘last

term’

it

1

the formula u

= u

n

84

–

3n

There

+

3

=

87

3n

=

75

n

=

25

are

25

–

3n

terms

in

=

12

Solve

for

a

+ (n – 1)d

1

n.

the

sequence.

Exercise

For

1

each

sequence:

i

Find

the

ii

Find

an

a

3,

c

36,

e

5.6,

Find

2

B

5,

a

the

 



an

9,

the

nth

…

6.8,

…,

term.

25,

b

46,

6.2,

15,

for

…

41,



…

of

terms

in

d

100,

f

x,

each

255

x

55,

87,

+

a,

…

74,

…

x

2a,

+

…

sequence.

b

4.8,

  

d

250,

5m,

8m,

…,

80m

f

x,

5.0,

5.2,

…,

38.4

221,

192,

…,

–156

3x

+

3,

5x

+

6,

…,

19x

+

arithmetic

common

sequence,

u

=

48

and

u

=

75.

Find

the

first

term

and

12

difference.

Answer

u

+

3d

=

u

+

3d

=

75

the

3d

=

27

to

d

=

9

times.

9

48

To

get

from

the

9th

ter m,

12

u

,

to

9

12th

ter m,

u

,

you

would

need

12

u

= 9

u

+

(9

–

1)9

u

+

72

u

48

To

=

48

for mula.

=

–24

1

first

–24,

term

and

difference



the

=

1

is

add

find

Patterns,

of

the

is

the

sequence

common

9.

sequences

and

series

common

the

1

The

27



9

the

40,



Example

In

expression





2m,

e

term.

number

10,

 c

6,

15th

first

dif ference

ter m,

use

the

3

nite

sequence.

is

Exercise

An

1

C

arithmetic

Find

the

common

EXAM-STYLE

In

2

an

sequence

has

first

term

19

arithmetic

sequence,

the

common

3

Find

the

value

of

x

4

Find

the

value

of

m

u

=

term.

or

sequence

This

In

a

2,

The

For

in

the

37

and

u

in

the

=

4.

21

and

the

arithmetic

first

term.

sequence

arithmetic

3,

sequence

x,

m,

8,

…

13,

3m

–

6,

…

sequences

18,

is

54,

an

…,

each

example

sequence ,

geometric

called

31.6.

term

of

is

three

times

the

previous

a geometric sequence ,

progression.

multiplying

is

6,

sequence

geometric

➔

difference

Geometric

the

term

QUESTION

Find

In

15th

difference.

10

.

and

the

the

common

previous

r,

term

ratio ,

common

ratio,

each

can

be

term

by

or

a

can

be

constant

obtained

value.

by

This

value

r

positive

or

negative.

example:

1,

5,

25,

125,

…

u

=

1

and

r

=

5

=

3

and

r

=

–2

=

81

1

3,

–6,

12,

–24,

…

u 1



81,

27,

9,

3,

…

u

and



=

1



2

k,

k

3

,

k

4

,

k

,

…

u

=

k

and

r

=

k

1

For

any

geometric

sequence, u

=

(u

n+1

sequence

For

any

u

by

multiplying

geometric

=

the

=

u

=

u

first

the

)r.

Y ou

can

find

any

term

of

the

n

previous

term

by

the

common

ratio, r.

sequence:

term

1

u 2

×

r

×

r

1

2

u 3

=

(u

2

×

r)

×

r

=

u

1

×

r

1

2

u

=

u

4

×

r

=

(u

3

×

r

3

)

×

r

=

1

u

×

r

×

r

1

3

u

=

u

5

×

r

=

(u

4

×

r

4

)

×

r

=

1

u 1

…

…

n

=

u

u

n

➔

Y ou

×

–

1

r

1

can

find

the

nth

n

formula:

u n

=

u

(r

–

term

of

a

geometric

sequence

using

the

1

)

1

Chapter





Example

Find

the



9th

term

of

the

sequence

1,

4,

16,

64,

…

Answer

u

=

1

and

r

=

4

Find

these

values

by

looking

at

the

=

9

1

sequence.

9

u

–

1

8

=

1(4

)

=

=

1(65 536)

=

65 536

1(4

)

For

the

9th

term,

substitute

n

9

n

into

the

for mula

u

=

u

n

u

–

(r

1

)

1

9

Example

Find

the



12th

term

of

the

sequence

7,

–14,

28,

–56,

…

Answer

u

=

7

and

r

=

7((–2)

=

–2

Find

these

values

by

looking

at

the

1

sequence.

12

u

–

1

11

)

=

7((–2)

For

)

the

12th

ter m,

substitute

12

=

7(–2048)

n

=

–14 336

u

=

12

into

n

u

Exercise

For

each

16,

c

1,

10,

e

2,

6x,

8,

sequence,

4,

find

the

Example

a

…

100,

18x

for mula

)

b

…

d

– 4,

25,

12,

…

f

a

–36,

10,

7

,

4,

6

b,

a

ratio

7th

term.

…

5

,

the

…

2

b

and

a

3

b

,

…



geometric

sequence,

u

=

864

and

u

1

the

the

1

1

common

2

Find

–

(r

D

a

In

u

n

12

1

=

common

=

256

4

ratio.

Answer

4

u

=

u

4

–

1

(r

3

)

=

1

u

(r

)

Substitute

=

864(r

and

u

)

=

256

4

n

u 256

=

u

n

8

–

(r

1

=

=

864

27

8

r

=

3

Solve

27

2

r

= 3



Patterns,

sequences

and

series

for

1

)

3

r

=

4,

u

=

864,

1

3

256

n

1

r.

into

the

for mula

Example

For

the

that



geometric

the

nth

term

sequence

is

greater

5,

15,

than

45,

...

find

the

least

value

of

n

such

50 000.

Answer

u

=

5

and

r

=

3

1

Find n

u

=

5

×

–

u

1

and

r

by

looking

at

the

1

3

n

sequence.

Substitute

u

=

5

and

r

=

3

into

the

1

n

for mula

u

=

u

n

You

can

of

n.

for

into

1

)

1

use

value

–

(r

the

GDC

First,

to

enter

help

the

find

the

for mula

GDC

u

a

function.

Let

help

x

represent

n,

as

shown.

Plus

and

GDCs

look

values

The

n

=

10,

since

u

>

of

9th

ter m

CD:

demonstrations

variable

Now

on

is

at

the TABLE

the

first

ter m

is

n

Alternative

the

n

to

see

Casio

are

on

for

the

TI-84

FX-9860GII

the

CD.

the

ter ms.

32 805,

and

the

10th

98 415.

50 000

10

Exercise

1

A

geometric

Find

2

A

For

that

4

the

the

each

the

a

16,

c

112,

A

sequence

first

geometric

Find

3

E

and

term

and

geometric

nth

24,

term

36,

–168,

that

find

and

sequence

first

geometric

Show

term

the

has

the

252,

greater

...

sequence

two

2nd

are

has

two

term

common

3rd

term

find

than

5th

term

3.2.

and

6th

term

144.

ratio.

least

value

of

n

such

1000.

1,

d

50,

2.4,

values

5.76,

55,

term

possible

possible

and

ratio.

the

b

first

50

–18

common

sequence,

…

there

the

is

has

9

60.5,

and

values

for

the

…

...

third

for

the

second

term

144.

common

ratio,

term.

Chapter





Find

5

the

value

EXAM-STYLE

Find

6

7x

.

–

the

2,

u

,

the

u

1

,

+

4,

+

,

u

1

u

(Σ)

+

sequence

18,

p,

40.5,

…

,

u

ways

of

…,

of

x

in

the

geometric

sequence

…

at

a

u

to

add

sequence

is

a

and

the

series

terms

gives

of

a

sequence.

a series

sequence.

n

+

u

3

Greek

geometric

notation

4

2

the

value

3x,

looks

3

in

QUESTION

terms

u

2

u

The

4x

section

Adding

p

positive

Sigma

This

of

…

+

+

u

4

letter

is

a

series.

n

Σ,

called

‘sigma’,

is

often

used

to

represent

sums

of

values.

When



➔

∑ 



means

the

sum

of

the

first

n

terms

of

a

sequence.

sum



of

Y ou

read

this

‘the

sum

of

all

the

terms u

from

i

=

1

to

i

=

n’.

i

arithmetic

common

sequence

difference

sequence

is

=

u

6n

6.

+

8,

A

14,

20,

general

…

r ule

has

for

first

term

the nth

8

term

and

of

this

( 

+ 

2

n



The

sum

of

the

first

five

terms

of

this

sequence

∑

is



This

To

to

means

‘the

calculate

5

into

sum

this

the

of

sum,

all

the

terms

substitute

expression

6n

+

2,

all

and

6n

the

+

2

from

integer

add

)

= 

n

=

1

values

to

of

n

n

=

5’.

from

them:



( 

∑ 

+ 

)

=

[6(1)

+

2]

+

[6(2)

+

2]

+

32

+

[6(3)

+

2]

+

[6(4)

+

2]

= 

+

=

Example

8

[6(5)

+

14

+

+

2]

20

+

26

=

100



4

2

a

Write

the

expression

∑ ( x

b

Calculate

the

sum

of

x

3

)

as

a

sum

of

terms.

= 1

these

terms.

Answers

4

2

a

∑ ( x

x

3

)

= 1

2

=

(1

2

–

3)

+

(2

–

2

+

=

b



represent

values

in

(3

–2

–2

+

+

1

Patterns,

3)

Substitute

consecutive

integers

2

–

1

+

3)

+

6

+

6

+

(4

+

–

3)

beginning

13

13

=

sequences

with

18

and

series

x

=

with

4

x

=

1

and

ending

1

you

sigma

are

a

this

= 

form,

The

you

using

notation

Example



8

a

Evaluate

the

2 ∑ (

expression

a

=

)

3

‘Evaluate’

Answer

8

Substitute a

3

2 ∑ (

a

=

)

=

4

2

+

5

2

+

6

2

+

consecutive

nd

integers

the

tells

value

you

so

to

the

7

2

+

2 beginning

with

a

=

3

and

nal

ending

answer

will

be

a

8

3

+

2 with

=

8

+

+

=

the

128

+

+

32

+

a

=

8

64

256

504

Example

Write

16

number .



series

3

+

15

+

75

+

375

+

1875

+

9375

using

sigma

notation.

Answer

n

u

=

–

The

1

3(5

ter ms

are

a

geometric

)

n

progression,

common

with

ratio

first

ter m

3

and

5.

6

This n

∑ (

3

series

six

ter ms

of

the

Write

F

an

expression

a

1

+

2

b

9

+

16

c

27

d

240

e

5x

f

4

+

7

+

10

g

1

+

3

+

9

+

+

25

+

+

3

+

+

+

+

h

a

+

Write

2a

+

7x

+

+

+

+

23

120

6x

4

25

2

2

first

progression.

= 1

Exercise

1

the

)) geometric

n

is

1

(5

5

+

+

each

+

21

60

+

+

6

+

+

+

…

+

series

+

+

+

+

+

a

+

using

sigma

8

17

+

7.5

10x

55

59 049

5a

sum

of

terms.

8

7

5

11

r

a

a

 n

 3n

 1

b

notation.

5

4a

as

series

15

4

+

7

19

9x

…

+

each

49

30

8x

13

27

3a

+

36

3

+

for

4  



c

∑ (

5

(

2

n

)

d

 

x



Remember ,

a

 1

r

=

n

3



nd

Evaluate.

7

5

9

 n

 1

 8n

5

b

3 

r

 1

the

need

10

to

tells

you

value,

give

so

to

you

numerical

2

r

a

word

5

evaluate

3

the

 1



c

m  

m

 1



d

 x



7 x

4

 answers.

4

Chapter





.

Arithmetic

series

Carl

Friedrich

Gauss

(1777–1885)

The

sum

of

the

terms

of

a

sequence

is

called

a

series.

The

sum

arithmetic

sequence

is

called

an ari thmetic series

said

the

terms

of

an

is

to

be

the

greatest

mathematician

For

so

example,

5

+

12

+

5,

19

12,

+

19,

26

+

26,

33

33,

+

40

40

is

is

an

an

arithmetic

arithmetic

sequence,

19th

how

series.

the

When

a

series

has

only

a

few

elements,

adding

the

individual

it

not

a

difficult

would

helpful

be

to

find

denotes

S

task.

ver y

a

the

However,

if

a

time-consuming

r ule,

sum

or

of

formula,

the

first

n

series

to

add

for

has

all

50

evaluating

terms

of

a

terms

these

or

terms.

100

It

sum

series.

For

a

of

of

the

F ind

out

worked

the

out

rst

terms integers.

terms

will

arithmetic

centur y.

Gauss

100

is

often

of

be

series.

series

n

Remember

with

n

must

S

=

u

n

For

S

u

+

=

2

u

+

+

u

3

(u

1

we

u

arithmetic

n

If

+

1

an

+

4

d )

+

this

(u

1

reverse

…

+

+

n

order

of

would

be

the

same,

positive

2d )

+

be:

(u

1

the

a

n

would

+

be

integer .

u

5

series

+

u

+

3d )

+

(u

1

the

terms

+

4d )

+

…

+

(u

1

in

the

+

(n

–

1)d)

1

equation,

the

value

of

the

Star t

sum

that

terms,

and

it

would

look

like

with

the

nal

this:

term

u

,

then

the

next-

n

S

=

u

n

+

(u

n

–

d )

+

(u

n

–

2d )

+

(u

n

–

3d )

+

(u

n

–

4d )

+

…

+

n

u to-last

1

term

is

u n

and

Adding

these

two

equations

for

S

ver tically ,

term

by

so

term,

n

2S

=

(u

n

This

is

+

(u

=

)

+

(u

u

)

+

u

1

added

)

+

(u

n

n

+

u

1

times,

)

+

(u

n

+

1

u

)

n

+

(u 1

+

u

)

n

so:

n

n(u

n

+

u

1

Dividing

+

n

1

2S

u

1

)

n

both

sides

by

2

gives:

 

= 

(

+



+





)





Substitute



(

− ) 

for

u

, then

n







=

(



+





+



(  − )  )

Y ou

find

using

the

the

sum

of

the

first

=



+

 

)

or

Patterns,

sequences



=

( 











terms

of

an



(



n

formula:





(  − )  )



can

series

+





➔

( 

=

and

series

+

(  − )  )

arithmetic

+

…

+

(u 1

+

u n

)

on.

–

d,

Example

Calculate

29

+

21



the

+

13

sum

+

of

the

first

15

terms

of

the

series

…

Answer

u

=

29

and

d

=

–8

1

15

S

=

( 2 ( 29 )

15

+

(15

− 1)

8)

(

For

)

the

sum

of

15

ter ms,

2

substitute

=

7.5(58

=

–405

–

112)

n

=

15

into

the

for mula

n

S

=

2u

(

n

+

(n

1

− 1

d

)

)

2

Example

a

b



Find

the

14

15.5

+

Find

the

number

+

17

sum

of

+

of

terms

18.5

the

+

in

…

the

+

series

50

terms.

Answers

a

u

=

14

and

d

=

1.5

Find

these

values

by

looking

1

at

u

= 50

the

To

sequence.

find

n,

substitute

the

n

u

=

14

+

(n

–

1)(1.5)

=

12.5 + 1.5n

values

you

know

into

the

n

12.5 + 1.5n = 50

1.5n

=

for mula

37.5

u

=

u

n

n

=

25

+

(n

–

1)d

1

Solve

for

n.

25 b

S

=

(14

25

+ 50

) Substitute

2

the =

12.5(64)

=

800

of

last

n

the

ter m

into

the

first

and

ter m,

the

value

for mula

n

S

=

(u

n

+

u

1

n

)

2

Exercise

1

Find

3

2

+

2.6

3

Find

–

3

+

94

the

3.4

sum

+

of

+

the

first

12

terms

of

the

arithmetic

series

the

first

18

terms

of

the

arithmetic

series

first

27

terms

of

the

arithmetic

series

first

16

terms

of

the

series

(3

...

of

88

sum

+

of

...

sum

+

the

5x)

sum

9

the

+

Find

(2

+

+

100

4

the

6

Find

G

+

of

–

the

...

the

4x)

+

(4

–

3x)

+

...

Chapter





EXAM-STYLE

5

6

Consider

QUESTION

the

a

Find

the

b

Find

the

Find

the

Write

120

number

sum

sum

Example

a

series

of

of

of

+

terms

the

the

116

+

in

112

the

...

+

+

28.

series

terms.

series

15

+

for

the

22

+

29

+

…

+

176



an

expression

S

,

sum

of

the

first

n

terms,

of

the

series

n

64

b

+

60

Hence,

+

56

find

+

…

the

value

of

n

for

which

S

=

0

n

Answers

a

u

=

64

and

d

=

–4

Substitute

the

values

for

1

u n

S

=

d

into

the

for mula

1

( 2 ( 64 )

n

and

+

( n − 1) ( −4 ) )

n

S

2

=

(

n

2u 1

+ (n − 1)d

)

2

n

=

(128 − 4 n + 4 ) 2

n

=

(132 − 4 n ) 2

2

S

=

66 n − 2 n

n

Set

2

b

66n

−

2n

=

S

0

=

2n(33

–

n)

=

can

=

0

or

n

=

two

=

33

your

the

positive

1

An

series

has

u

=

4

and

S

1

the

value

EXAM-STYLE

2

a

n.

The

equation

the

GDC.)

the

When

equation

we

solve

usually

by

has

to

word

hence

question

use

your

answer

in

number

integer,

of

we

ter ms

must

disregard

n

be

=

a

0

Write

of

the

=

1425

30

common

difference.

QUESTION

an

expression

for

S

,

for

the

series

1

+

7

+

13

+

…

n

b

Hence,

find

the

value

of

n

for

which

S

=

833

n

3

a

Write

an

expression

for

S

,

for

an

arithmetic

series

n

with

u

=

–30

and

d

=

3.5

1

b

Hence,

find

the

value

of

n

for

which

S

=

105

n

4

In

Januar y

they



sell

2012,

600

a

How

b

Calculate

Patterns,

a

new

drinks,

many

then

drinks

the

total

sequences

coffee

and

will

700

shop

in

they

number

series

sells

March,

expect

of

and

to

drinks

500

sell

they

drinks.

so

in

on

In

in

Febr uar y ,

an

arithmetic

December

expect

to

sell

2012?

in

2012.

progression.

in

tells

you

previous

this

H

arithmetic

Find

for

this

solutions.

Since

Exercise

solve

solve

33

factoring,

n

and

also

0

using n

0,

n

(You

par t.

5

In

an

and

the

6

In

arithmetic

the

of

common

an

ten

sum

the

first

the

ten

2nd

terms

term

is

is

–20.

four

Find

times

the

the

first

5th

term

term,

and

difference.

arithmetic

times

find

sequence,

the

the

series,

sum

common

of

the

the

sum

first

difference

3

of

the

terms.

and

the

first

If

12

the

value

terms

first

of

is

term

equal

is

to

5,

S 20

.

Just

Geometric

as

an

arithmetic

sequence,

a

a

the

following

series

geometric

geometric

Adding

series

is

the

is

series

sum

the

of

the

sum

of

terms

the

of

an

terms

arithmetic

of

sequence.

terms

of

a

geometric

sequence

gives

the

equation: Multiply

this 2

=

S

u

n

+

u

1

r

+

u

1

+

=

u

n

r

+

u

1

u

r

+

+

+

u

u

1

4

r

+

u

1

–

2

n

r

+

u

1

3

r

n

…

1

2

rS

3

r

1

r

…

+

n

–

S

n

=

–

u

n

+

u

1

r

by

of

r

1

r

u

–1

n

r

+

u

1

Subtract

r

the

rst

1

equation

rS

sides

1

n

+

1

–

both

equation

from

the

both

sides

n

=

u

1

r

–

u

1

second.

1

n

S

(r

–

1)

=

u

n

(r

–

1)

1

Factorize

of



 



(



the



)

equation.

= 





Y ou

➔

Y ou

can

find

the

sum

of

the

first

n

terms

of

a

geometric

may

nd

convenient

using

the







more

to

use

the

formula:

rst



it

series

formula

when



(



)

 

or

=







 r

,



where

r

≠

>

1,

as

it

avoids

1













using

a

negative

denominator

Example

Calculate



the

sum

of

the

first

12

terms

of

the

series

1

+

3

+

9

+

...

Answer

u

=

1

and

r

=

3

Substitute

the

values

of

1

12

1

(3

S

1

)

u

,

r

and

n

into

the

for mula

1

=

12

n

3

1

u 1

S

(r

1

)

= n

531 440

r

1

= 2

=

265 720

Chapter





Example



Geometric a

Find

the

8192

b

+

number

6144

Calculate

+

the

of

terms

4608

sum

…

+

of

the

in

+

the

series

series are

often

the

study

seen

in

1458. of

fractals,

terms. such

as

the

Koch

snowake.

Answers 6144 a

u

=

8192

r

and

3

=

=

Find

r

by

dividing

u

1

n

3

⎛

1

Substitute

the

values

⎠

n

the

for mula

u

=

u

n

n

729

3

⎛

=

1

⎟ 4

⎝

⎠

6

3

⎛

=

3

6

3

6

729

=

729

and

4

=

4096

⎞

= ⎜

6

4096

4

⎝

⎟ 4

You

could

=

also

solve

this

equation

⎠

using

logarithms

6 (see

=

Example

19).

7

Substitute

7

⎛ ⎛ 8192 ⎜ 1

3

⎜

⎜ ⎝

values

of

u

,

r

1

⎟

⎟ 4

the

⎞ ⎞

and

n

into

the

for mula

⎟ ⎠

⎝

S

into

)

1

6

n

know

1

⎞

⎜ 4096

1

–

(r

=

8192

–

you

⎟ 4

⎝

1458

1

⎞

⎜

b

u

4

1458 = 8192

n

by

2

8192

⎠ n

=

u

7

1

3

S

1

(r

r

4

)

1 [

You

⎛ 14 197 ⎞ 8192

1

= n

⎜

can

also

calculate

sums

Koch

snowake

using

⎟ 16 384

⎝

the

seq

(and

sum)

functions

on

⎠

= 1

your

GDC.

4

=

28 394

Exercise

1

I

Calculate

the

value

of

S

for

each

geometric

series.

12

a

c

2

0.5

64

+

–

1.5

32

Calculate

+

+

4.5

16

the

–

+

8

value

…

0.3

b

…

+

d

of

S

for



(

each

+

0.6

+ ) +

+

1.2

(

+

+ 

)

…

+

(

 

+



)

+



series.

20



0.25

a

+

0.75

+

2.25



…

+

b

+

+





+

…





c

3

–

6

+

12

EXAM-STYLE

–

24

+

…

 

d

+



(



)

+



(



)

+



(



)

+



QUESTION So

3

For

each

geometric

at

i

find

the

number

ii

calculate

far

we

of

arithmetic

1024

b

2.7

+







1536

10.8

+

2304

43.2

+

 +



590.49

Patterns,

+

 +

c

d

+

+

sequences

sum and

a

looked

and

terms geometric

the

have

series:

…

…

+

+

26 244

2764.8

there

of

other

Are

types

mathematical

 +



sequences



196.83

and

and

+



sequences

+

series.

+

65.61

series

series?

+

…

+

0.01

used?

How

are

they

Example



GDC

For

the

geometric

series

3 + 3

2

+ 6 + 6

2

+

... ,

determine

the

Plus

value

of

n

for

which

>

S

help

on

CD:

demonstrations

least

and

Casio

Alternative

for

the

TI-84

FX-9860GII

500 GDCs

n

are

on

the

CD.

Answer

An

u

=

3

and

r

=

2

Substitute

the

known

old

Indian

fable

values

1

tells

into

n

the

S

us

that

a

prince

for mula.

n

3

2

(

1

) Enter

S

=

>

the

S

500

was

so

new

game

taken

that

he

with

the

equation

n

n

2

1

into

the

of

chess

GDC.

asked

its

Remember:

inventor

On

the

the

number

GDC,

the

X

represents

ter ms,

and

reward.

The

man

f1(x)

said

represents

choose

‘n’,

his

of

to

he

would

like

S n

one

Look

at

the TABLE

to

grain

rst

the

chess

sums

of

the

first

n

four

etc.,

number

This

to

of

the

first

of

the

first

13

12

456.29,

ter ms

the

ter ms

is

and

on

ask

the

doubling

each

seemed

that

the

time.

so

the

little

prince

is agreed

approximately

on

ter ms.

third

sum

on

of

board,

grains

second,

The

rice

square

see

two

the

of

the

the

straight

away.

sum Ser vants

star ted

bring

rice

to

approximately the

–

and

648.29 to

the

prince’ s

surprise

soon

n

=

13,

since

S

>

the

great

grain

overowed

the

500 chess

13

board

to

ll

the

palace.

When

the

sum

of

a

geometric

series

includes

an

exponent n, How

you

can

use

logarithms.

rice

many

did

have

Example

A

the

the

of

prince

give

the

man?



geometric

Find

to

grains

progression

value

of

n

such

has

that

first

S

term

=

of

0.4

and

common

ratio

2.

26 214

n

Answer

n

0

S

4

(2

1

)

=

= 26 214

n

2

1

n

0

4

(2

1

=

)

26 214

n

2

–

1

=

65 535

n

2

n

=

65 536

=

log

(65 536)

Express

this

using

logarithms.

2

log

n

65 536

log

n

Use

the

change-of-base

rule

and

your

=

=

2

GDC

to

find

this

value.

16

Chapter





Exercise

For

1

J

each

series,

determine

the

least

value

of

n

for

which

S

>

400

n

25.6

a



57.6

…

+

b

14

d

0.02



+



–

42

+

126

–

378

…

+

+

0.2

+

2

+

…



geometric

Find

+



+



A

38.4



+

c

2

+

the

series

common

has

third

ratio

term

and

the

1.2

and

value

of

eighth

term

291.6

S 10

In

3

a

geometric

series,

S

=

20

and

S

4

Find

the

common

EXAM-STYLE

=

546.5

7

ratio,

if

r

>

1

QUESTION 

4

Find

a

the

common

ratio

for

the

geometric

 +

series 

Hence,

b

find

the

least

value

of

n

such

that

S

>



‘Hence’

+



+

tells



you



to

use

previous

800

your

answer

in

n

this

In

5

6

a

geometric

sum

of

In

geometric

a

the

.

the

sum

series,

first

of

6

terms

series,

the

first

Investigation

2

a

are

three

+

+

1

240

c

For

1

–

60

each

i

F ind

ii

Use

+

+

of

–

Find

the

the

If

r

three

sum

first

>

1,

and

terms

of

four

find

the

GDC

Do

2

you

Now

3

the

full

notice

use

ten

to

the

terms.

times

Extension

material

Worksheet

6

-

on

CD:

Finance

ratio.

infinity

series

series.

75

+

30

+

12

...

+

...

+

to

ratio,

r

calculate

values

any

your

is

and

seven

common

the

values

of

S

,

S

10

Write

304,

first

terms

the

sums

is

series:

common

your

first

converging

3.75

these

the

the

of

terms.

b

–

sum

series

…

15

of

1330.

the

geometric

0.5

sum

is

two

Convergent

Here

the

par t.

you

see

patterns?

GDC

to

on

Why

calculate

your

do

the

GDC

you

think

value

of

,

S

15

.

20

screen.

this

S

is

for

happening?

each

series.

50

Do

For

you

each

think

of

the

your

calculator

series

in

the

is

correct?

investigation

Explain

you

why

or

why

not.

should Paradox

have

noticed

that

the

values

of

S

,

S

10

close.

This

series

has

is

because

when

a

and

S

15

are

ver y

20

Suppose

30-metre

a

common

ratio

of

|r|




0

for

all

x

in

If

f

′′(x)




2,

f

a

graph

for

f

x

′′(x)

is




when




0

⇒

relative

minimum

first

derivative

test

could

also

be

used.

3

12

48

y

=

48

⇒

y

=

=

4 Find

The

numbers

Example

A

second

number.

four th

enclose

are

12

and

4.



rectangular

The

the

12

x

plot

side

the

of

of

farmland

the

maximum

plot

is

area.

is

enclosed

bordered

Find

the

by

by

a

180 m

stone

maximum

of

wall.

fencing

Find

material

the

on

three

dimensions

of

sides.

the

plot

that

area.

Answer

Make

to w

be

a

sketch

and

assign

variables

to

the

quantities

deter mined.

w

l

Write A

=

an

equation

for

the

area,

the

quantity

to

be

lw

maximized. 2w

+

l

=

180

⇒

l

=

180

–

2w

2

A

=

(180

–

2w)w

=

180w

–

Use

the

other

equation

A′(w )

= 180 − 4 w

180 − 4 w

=

0

Find

and

the

then

for

=

the

infor mation

area

derivative

find

derivative w

given

to

rewrite

the

2w

the

equals

using

of

the

critical

only

two

equation

numbers,

variables.

to

be

maximized

where

the

0.

45

{

Continued

on

next

Chapter

page





Use A′′(w )

=

the

critical A′′( 45 )

=

−4




(Note:

=

press

x

×

the

displays

enter

to

enter

3

.

The

retur ns

you

to

exponent.)

the

cur ve

with

the

default

axes.

{

Continued

on

next

Chapter

page





Pan

the

For

axes

help

see

the

cur ve

For

with

your

Grab

fit

get

and

screen

with

better

view

of

the

cur ve.

manual.

x-axis

see

a

panning ,

GDC

the

help

axes,

to

change

it

to

make

the

exponential

better.

changing

your

GDC

manual.

.

Finding

Example

a

horizontal

asymptote



x

Find

the

horizontal

asymptote

to

the

graph

of

y

=

3

+

2

x

First

Y ou

draw

can

using

a

Press

(or

the

look

split

the

press

The

values

x

0.

→

at

of

y

=

graph

3

+

and

2

a

(see

table

Example

of

the

10).

values

by

screen.

2:View

menu

simply

graph

of

the

|

ctrl

9:Show

T

Table

)

function

are

clearly

decreasing

as

{



Using

a

graphic

display

calculator

Continued

on

next

page

Press

The

and

hold

table

smaller,

to

shows

f1(x)

the

inspection,

you

We

actual

can

The

x

value

can

value

say

line

of

that

=

scroll

as

2

a

of

see,

table.

of

x

get

2.

at

is

→

the

values

f1(x)

f1(x)

f1(x)

is

up

the

approaches

Eventually ,

the

that

2

reaches

the

2.

bottom

On

of

closer

the

screen,

that

2.000 001 881 6...

as

x

horizontal

→

−∞.

asymptote

to

the

x

cur ve

y

=

3

+

2.

Logarithmic

.

functions

Evaluating

Example

Evaluate

logarithms



log

3.95,

ln 10.2

and

log

10

Open

a

Press

ctrl

Enter

For

new

to

base

natural

method,

document

log

the

and

open

and

the

the

base

the

that

without

.

The

y

to

Finding

inverse

y

GDC

having

values.

line

the

it

=

of

a

will

use

an

log

is

a

Calculator

page.

template.

then

possible

equal

to

e,

to

but

press

use

it

is

del

enter

the

far

same

less

time

ln

ctrl

Note

add

argument

logarithms

with

2.

5

evaluate

the

change

inverse

function

Geometrically

can

this

logarithms

of

base

with

any

base

formula.

function

be

can

found

be

by

done

interchanging

by

reflecting

the x

points

and

in

the

x

Chapter





Example



x

Show

that

the

inverse

of

the

function

y

=

10

x

is

y

=

log

x

by

reflecting

y

=

10

in

the

line

10

y

=

x

Open

First

a

new

we

recognised

plotted

Press

document

will

as

draw

the

a

7:

This

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than

between

greater

and

5:Probability

E:Binomial

Enter

document

less

bound

of

in

trials

this

and

case

the

is

0

probability

and

the

of

upper

success

bound

10.

Click

a

P

on

( X

b

P

( 5

c

P

( X

OK

≤

≤

10)

X

>

Note:

≤

11)

the

=

0.751

15)

=

lower

A

(to

bound

random

sf

(to

3

sf

is

)

3

sf

)

)

12

normal

Calculating

Example

3

0.980

0.131

Calculating

5.3

=

(to

here.

probabi lities

normal

probabilities

from X-values



variable

X

is

normally

distributed

with

a

mean

of

195

and

a

standard

deviation

of

20,

or

2

X

~

N(195, 20

).

Calculate

a

the

probability

that

X

is

less

b

the

probability

that

X

is

greater

c

the

probability

that

X

lies

Open

a

new

document

and

5:Probability

menu

|

than

190

than

between

add

a

194

187

and

Calculator

5:Distributions

|

196.

page.

2:Normal

Cdf

Press

Y ou

μ

enter

need

and

σ

to

in

enter

the

the

values

dialogue

Lower

Bound,

Upper

Bound,

box.

999

For

the

is

the

smallest

used

the

Lower

in

key

Bound,

number

place

of

marked

E

enter

that

− ∞.

−9

can

To

be

enter

×

10

as

entered

the

E,

−9E999.

in

you

the

This

GDC,

need

to

so

is

it

press

E

{



Using

a

graphic

display

calculator

Continued

on

next

page

P(X

a




194)

=

0.520

(to

3




0

and(1

−

2x)

>

0

for

2

x

+ 1 3

11

3( 2 x

3

(x

all

+ 3)

x

h

is

defined,

1)

the x 4

3

10x

12

where

+

5

2

12x

−

3x

−

18x

−

e

gradient

of

h

is

always

4x

;

4x;

4e

15

positive.

2

1

3

(

3

y

13

=

−

(x

− 1)

6

x

;

ln x

y

=

x

14

a

6

b

8

ln x ;

e

14

)

x

−

1 2

6 3

7 15

−

9n

+

x

; 9x

+ 2;

3.5

1

Exercise

7M

3

(9 x

+ 2)

2

4πr

16

3

1 x

7

17

x

2

4

x ;

8

2x

+ 3; 3

2

(2x

Investigation

–

finding

4

9

the

2

4

4

18

derivative

of

5x

+ 3

)

2

180x

3

3e

3

;

+

3x;

3

3

20(x

24x

−3n

x

a

+

+

(6n

+

5)

2

3x)

(3x

+

3) 8

composite

function 4

3

x

10

3

e

;

4x

2

; 12 x

3

4 x

x

e

3

1

f

a

(x)

=

(2

−

x)

2

=

8

−

+

12x

3

6x

−

x

3

Exercise

7L 5 2

x 2 2

f

′ (x)

=

−12

f

′ (x)

=

3(2

+

12x

−

1

3x

8x

3

(2x

−

3)

4

+

2x (2x

−

3)

3

or

6x (2x

−

1)(2x

−

3)

6

1

7

equals

2

b

−

x)

·

(−1)

2

2

2

f

a

(x)

=

(2x

+

−x

1)

dy

x

x

8

e

2

=

4x

+

4x

+

0

+ 2x

2

x

= e

− e

1 dx

8x

f

′ (x)

=

8x

+

4

2

x 2

2

(x

f

b

′ (x)

=

2(2x1)·

y

d

3

x

= e

+ 3)

+ e

2

dx

2

3

x 2

3

f

a

(x)

=

1

x

+ 1

y

d

x

2

(3x

+

+

4

1)

or

3

1

3

2

2

2

x

= e

− e

3

dx 4

=

(2 x

2

9x

+

6x

+

+ 1)

(2x

+ 1)

(2x

+ 1)

1

4

y

d

x

x

= e 3

f

′ (x)

=

+

36x

2 x

12x e

dx

e

5

1

2

f

b

′ (x)

=

2(3x

+

1)

·

+ e

4

2 x

(6x)

2 x

2 x 2

(e

+ e

) When

4

The

derivative

of

a

n

is

odd

composite 2

6x

function

is

the

derivative

n

of

d

6

y x

3

= e

2x

the

outside

respect

to

function

the

inside

1

by

the

and

when

dx

1

function

n

7

multiplied

− e

n

with

x

derivative

is

even

ln x

n

d

of

the

inside

x x

x

−2(e

4

5

f

(x)

=

(x

=

x

2

+ x

− e

x

8

+ 3x

x

+ e

4

3(x

2

+ x

+ 24x

2

2 x

2

)

2

(e

+ 1)

·

(4x

1 = 2

dx

9

+ 6x

dy

+ 3

5

2

x

2

− 3x

− 2)

2

y

d

3

)

2

+ 2x)

= 2 1

8

=

6

+ 2x

3(x

4

+ x

3

)(4x

5

+ 2x)

10

x

2

(x

1

3

2

 3)

 4x

2

(x

9

+ 10x

7

+ 8x

5

5

+ 2x

5x

)

3

dx

x

2

 3)

3

y

d

11

= 3(4x

+ e

n

dx

9

7

(x

′ (x) =

x

= e − 1)

6

9

+ 30x

2 x

(e

+ x

−2 x 11

f

−2 e

or (e

10

+ 3x

′ (x) = 12x

x

)

8

3

)

12

f

y

function.

6 =

3 3

 12 x

4

dx

x

or 1 4

11

=

9

+ 30x

12x

7

+ 24x

5 2

+ 6x (x

y

d

2

 3)

24 =

4

5

dx

x

2

Exercise

x

7K

11

a

(2 x

2 x n

2 )e

n

y

d

(

1)

n !

= 5

1

x

4

;

3x

n

+

2x;

4

5(3x

2x)

dx

2

b

4

+

n +1

x

3

(12x

+

2) y

c

−

1

=

2(x

−

2) 18

3

2

4x

;

2

2x

+

3x

+

10

1;

8

1

2

12(2x

5

2

+

3x

+

1)

3

(4x

+

3)

12

e

25 x

Answers



2

Exercise

1

7N

2

1.4 m;

a

a

4 ft

b

s (2)

1

9.8 m s

The

1

;

ball

0 m s

1 s,

at

=

−16(2)

is

−9.8 m s

moving

rest

downward

at

at

2 s

=

−64

+

+

80

40(2)

+

4

=

T ime

20 ft

be

2 m s

Speed

Velocity

(s)

2

1

;

4

c

;

−16t

i

+

40t

+

4

=

20

1

upward t

ii

at

acceleration

1

+

9.8 m s

c

Let

2

21 m

1

b

d

=

1

)

(m s

)

0

−10

10

1

−8

8

2

−6

6

3

−4

4

4

−2

2

,2 s 2

and

(m s

ds

3 s. d

=

i

−32t

+ 40

dt

2

4000

a

litres;

1778

litres 1

40 ft s

ii

−111

b

litres/min;

During 2

5

the

time

inter val

0

4

20

minutes,

water

is

an

out

average

111

litres

of

the

rate

per

Speeding

up

b

Slowing

down

c

Speeding

d

Slowing

being iv

pumped

a

s

iii

to

tank

29 ft

up

at

3

of

a

v(t)

=

s′(t)

t

minute.

t

(1)

e

t (e

3

)

a

down

Speeding

up

= 2

t

−89

c

20

litres/min;

minutes,

(e

at

water

is

)

b

being

e

(1

out

of

the

tank

down

t )

Exercise

=

pumped

Slowing

t

7P

2t

at

e 3

1

an

average

rate

of

89

1

litres v (t )

a

v (t)

=

8 t

−

12t,

t

≥

0

t

= 2

t

per

a (t)

e

minute.

=

−

24 t

12,

t

≥

0

2

V ′ (t )

d

is

negative

b

for

1

b

second

84 cm s

;

Velocity

is

1

0

≤

t




″

9

(2)

>

f

′(2);

f

is

0;

since

concave

the

up,

f

graph

f

′ (2)

graph

2

of




3

1

relative

6

4

a

2

i

4x

–12x

ii

12x

i

(0, 0),

ii

(3, −27)

iii

(0, 0),

3

b

x

−1 ,− ⎟

⎜ e

⎝

2

3 ⎠

–24x

12

5

relative

minimum

(1,

b

c

0)

(4, 0)

5

x

6

relative

maximum

(0,

1)

4

d

3

10 x

 4 x

2

 3x

 2x

 1 (2, −16)

11

Exercise

7W

y

c

e 2

+ 7)

(x

1

A

–

neither;

B

–

relative

20

and 4 x

f

absolute

15

4e

minimum; 10 2

C

–

absolute

A

–

neither;

maximum

g

12 x

3

(x

3

 1) 5 (4, 0) (0, 0)

2

B

–

relative

2

h 0

minimum;

C

–

relative

and

2x

–3

+ 3

–2

x

–1 5

absolute

maximum;

10

1

2 ln x

i

D

–

absolute

minimum

3

15

x

3

absolute

maximum

20

8;

4

j

(2, –16)

1

x 25

absolute



Answers

minimum

−8

3

3

3

⎞

⎟

3

non–GDC

2

relative

″ (2)

and

1

4

⎞

units

of

Review

0);

thousand

–75) or

3

3

7V c

(–1,

2

x

x

2

9 ,

2

2

1

3

⎜

− 4 3

2

dx

Exercise

2

4

b

(3, –27)

⎠

y stneduts

b

100

7

a

v (t )

=

20 − t

t

b




0

and

t

>

0,

2

Exercise

30

>

0.

h




0

cos

c

−1

t

m

2

sin

11

x

–e

a

sin

sin

x+

e

x

x cos x

2

cos

x 

3

a

3

d

2

i sin x

sin

x

e

b

+

2

C

2

2

 1

12

a

f

'( x )





t

  sin x 

x

+

2xcos

x

1 2

1

2 or

f

b

sin x

a(t)

=

–

e

2

tan x cos

sin t

sin

t

x

sin x cos x

+

 sin t

e

cos x



sin



cos x



x

e

3

ii

2

cos

sin x

t g

(ln x )(cos x )  x

 tan x

sin t

c

s(t)

=

e

+

3 2

h 1

–

2sin

2

x

+

2cos

x

2 4

 ln(cos x )   

b

 C 4

a



2

( 4 sin t

2cos

 3 cos t ) dt

2x

0

4

2 b Exercise

4.34

a

x

+

cos

x

+

C

m

14F

1

b

–2

5

a

i

–2.52

sin

m s

(3 x ) + C

3

1

3 ; 1.73

ii

speeding

up 1

c

cos 2

4;

4

b

2.51

s

and

3.54

(4 x

+ 1) + C

s 4

3

c

7.37

m

a

5.82

m s

3

1 2

;

3 4



d

1.30

Answers

sin

–2

6

4

(2 x

) + C

or

Chapter

1

e

d

c

5

n

P(N

=

n)

p

P( p )

1

1

2

2

+ C 2 cos

( 2t

+ 1)

Skills

11

check 1

36

–

f

cos

(ln

x)

+

C

a

1

5.5

9 1

8516

2

sin x

g

e

 14.6

b

+ C

2

(3sf)

36 36

39

2

3

a

2

2

7

15

6

3

h

 C 36 2  sin x

b

56

c

0.267

4

3

a

0

b

2

c

2

d

2

+

3

5 36

4

π

36

3

a

1.71875

b

2.98

c

8.68

5

2

3 36 5

4

x



5

=

2

36

2

3



Exercise

36

6

15A

 3

36

2



4

1





6



8



1

a

Discrete

b

Continuous

2

36 1 2

6

y



x

 cos x

1

9

1

2

c

Discrete 36

7

p

a

=

2,

q

=

2 d

3π

b

+

2

Continuous

a

2

10

2

b

36

12

s

Review

exercise

P( S

=

s)

n

=

4

n)

GDC 1 2

1

P(N

25 15

0

2

4.53

a

36

36

2

10

36

1.36

b

3

2

16

1

1

a

4.93

36

b

45.0

3

36 36

4

1.23

4

a

=

–

10

sin

(5t)

2

36

36

20

cos (5t)

s ′(t)

i

18

2 36

3

1

2

4

e 5

36

36

s ′′

ii

(t)

=

–

10

sin

(5t) 24

)

cos

+

(–sin

(5t))(5)]

36

6

36

(5t)

e

2

5

cos (5t)

[e

×

[–10(cos

(5t))

(5)]

25

1

6 2

=

50

sin

cos(5t)

(5t)[e

36

7 36 cos (5t)

–

50

cos

(5t)(e

30

)]

2

5 cos (5t)

=

50

e

2

(sin

8

(5t)

36 36

–

cos

1

(5t))

4

36

9

iii 36

 



 















s





5

0

and

s 

 18.4 5

 0 3

 10

3

a

36

Therefore

by

derivative

test

the

s

second

has

T

P(T

11

relative

minimum

2

3

4

5

1

4

10

12

36

36

36

36

6

2

a

=

t)

9

36

at

36



t

1

 12

5

21

36

b

14.2

m

b

P(T

>

4)

=

7

 36

12

Answers



4

5

a

Same

mean

10

a

P(Z

=

b

E(Z )

0)

=

0.7489

35

s

1

2

3

6

6

10

=

70,

The

expected

18

amount P(S

s)

=

1

1

3

6

1

1

Exercise

6

6

to

be

won

on

1

a

15B

ticket

6

91

1

1

c

Lose

$0.30

 15.2 (3 sf) 6

b 2

3

a

1

Investigation:

1

2

1

5

x



, y

The

binomial



b 8

8

quiz

2 6 1 1

3

6

5

1

T

T

2

F

3

4

F

5

3 36

7

0.2

4

2

Y ou

3

questions

would

expect

to

get

2.5

5 right

1 27

5

8

a

k

= 25

Probability that

you

get

right

0.3125

40

E(X)

b 1

9

a

a



=

5

5

, b

out

of

5

=



8

24

6

a

Exercise 25

X

1

2

15C

3

b

1

1

1

96 P(X

=

x)

0.2

1

−

k

k

−

a

b

0.2 16

4

b

10

15

5

c

2

3

4

5

b

6

0 .2

≤

k

c

≤ 1 ,

d 16

16

P(C

=

c)

k

c

1

5

6

5

1

18

18

18

18

18

+

1.6 2

Investigation

–

dice

scores

7

0.2

8

a

P(R

r

1

1

d

0

1

2

3

4

=

r )

18

5

3

a

0.329

b

0.351

c

0.680

d

0.649

a

0.0389

b

0.952

c

0.00870

d

0.932

P(X