IB Diploma Program Course Companion Mathematical Studies Standard Level (SL) [1 ed.] 0198390130, 9780198390138, 9780199129331

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O X

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S t U D ie S C O U R S E

C O M PA N I O N

Peter Blythe

Jim Fensom

Jane Forrest

Paula Waldman de Tokman

S ta N D a R D

Le V e L

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Great

Clarendon

Oxford

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v

8.2

Venn

8.3

Extending

diagrams

334

8.4

Problem-solving

Contents

Chapter



Number

number

and

1.1

The

1.2

Approximations

1.3

Standard

1.4

SI

units

Chapter







sets

and

error

form

of

measurement

Descriptive

2.1

Classification

2.2

Simple

2.3

Grouped

2.4

Measures

2.5

Cumulative

2.6

Box

2.7

Measures

of

discrete

and

algebra

or

central

of



Basic

8.6

Conditional

probability

22

8.7

Two

cases:

47

continuous

data

tendency

cur ves

graphs

dispersion

Geometry

3.1

Gradient

3.2

Equations

3.3

The

sine,

3.4

The

sine

Chapter



of

a

of

and

trigonometry

and

line

cosine

Sample

8.9

Tree

tangent

ratios

r ules

Mathematical

space

Chapter

9.2

Compound

conjunction

73

9.5

Tr uth

tables:

resolving

9.6

Logical

9.7

Compound

9.8

Arguments



the

and

103



10.3

Angles

4.5

Graphs

Chapter



+

bx

GDC

of

+

to

Statistical

The

5.2

Correlation

normal

5.3

The

regression

5.4

The

chi-squared



6.2

The

6.3

Calculating

gradient

given

to

situations

189

tangent

the

6.6

Local

maximum

dierential

of

a

and

the

normal

to

and

Number

Geometric

7.3

Currency

7.4

Compound

minimum

in

Surface

10.5

Volumes

a

397

and

trigonometry

points

two

in

lines,

a

or



solids

solid

of

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426

between

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

429

three-dimensional

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441

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204

11.3

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216

11.4

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228

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233

11.6

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454

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263

2

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267

3

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4

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271

5

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275

6

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7

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algebra

points

455

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256

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283

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modeling:

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482

486

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525

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304

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Number

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CHAPTER

Natural

1.2

Approximation:

numbers,

percentage

and

Y ou

1

Substitute

G

and

F

basic

you

should

are

places,

rational

numbers,

signicant

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numbers,

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estimation;

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standard

form;

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units

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other

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integers,

decimal

numbers

standard

SI

1.4

;

errors

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in



OBJECTIVES:

1.1

1.3

algebra

and

how

formulae,

linked

to:

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e.g.

through

1

the

formula

check

Find

the

y

value

x

and

are

a

y

=

3x

c

y

=

(1

of

linked

y

when

x

through

=

−0.1

the

if

formula

2

F

(x

1

1)

2

G

.

=

F

Find

=

value

of

G

when

(x

−

1)

y

b

= x

+ 2

98

F

the

G

98.

−

x)

(2x

+

1).

1

=

= 9

7 2

Solve

for

x

98 + 2

a 2

Solve

simple

equations

in

one

3x

−

7

=

14

b

2(x

−

6)

=

d

x

1200

b

0.1%

following

inequalities.

4

variable,

1

e.g.

2

(1 − x )

c

a

3

2x

−

8

=

10

2x

=

18

x

=

9

Calculate

x

b

x

=

25

=

5

or

x

=

−5

3

0

=

16

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a

percentages,

=

2

2

8%

of

of

234.

e.g. 4

Solve

the

5

Calculate

5%

of

×

240.

240

=

12

Represent

their

solutions

on

the

100

number 4

Solve

inequalities

solution

2x

+

on

7

≤

10

2x

≤

3

the

and

represent

number

line,

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10

c

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−

x

≤

1

b

3x

−

6

>

12

e.g.

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line.

0

1.5

≤

0

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2

1

x

≤

1.5 a

|−5|

b

2 5

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e.g.

the

|2.5|

absolute

=

2.5,

value

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of

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number,

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12 c

|0|

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7|

8

× 100

d

8

●

The

●

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castle

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100 km

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snow

takes

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weeks

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world, ●

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temperature

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castle’s

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biggest

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build.

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it

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to

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it

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northern

melting.

from

Finland.

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in

it

1996,

built

has

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13 000

to

rebuilt

20 000 m

when ●

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300 000

people

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world

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there

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castle

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walls

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1000 m.

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numbers

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measurement.

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temperature

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winter

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13%.

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of

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world’s

population

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with

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world

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area

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a

circle

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radius

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1 cm

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

1

The

numbers

60,

−45,

,

9.69

π

and

belong

to

different

sets ,

number

which

are

3

described

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the

over

end

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➔

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●

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use

to

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this

next

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numbers

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3

4

5

unit

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Find

the

value

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expressions

when

a

=

5

and

b

=

are

a

+

b

a

ii

×

b

a

iii

−

b

b

iv

−

State

whether

your

answers

to

part

a

are

natural

numbers

or

5

b

i

natural

+

7

Exercise

Find

a

=

12

5

ii

×

7

=

35

natural

ii

iii

5

iii

not

the

2a

State

b

+

value

b

of

these

2(a

ii

whether

your

Investigation

State

a

whether

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−

7

=

−2

natural

iv

7

−

5

=

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2

negative

not

or

each

false?

+

expressions

b)

answers

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a

iii

when a

to

par t

natural

statement

Whenever

is

you

=

2

and

b

=

2

−

are

a

(a

iv

natural

−

4.

b)

numbers

or

not.

numbers

true

add

or

two

false.

If

natural

it

is

false,

number s

give

an

example

will

be

a

natural

a

or

false?

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you

multiply

two

natural

c

product

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or

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false?

be

a

natural

+

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dierence

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and

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1

b

c

=

you

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a

is

c,

we

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say

sum

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a

b

a

×

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we

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subtract

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natural

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of

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numbers

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as

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many

numbers

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even b

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7.

natural i

elements

set.

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origin

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represented

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set

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2

a

x

+

5

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of

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for

x.

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whether

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solution

to

the

equation

+

is

11

b

−3x

=

neither

5

=

11

x

=

6

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b

x

is

an

=

10

integer.

x

=

is

not

an

value

of

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following

j

j

i

ii +

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State

ever yday

at

expressions

when

j

=

4

and

k

=

−2.

least

three.

j

from

k

2

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Brahmagupta

in

whether

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+ 2k

answers

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par t

a

are

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India.

with b

represent

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5k

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

the

k

negative

situations.

3

Find

use

many

10

x

a

nor

10

numbers

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the

is

We

x

to

zero.

not.

Answers

a

integers

negative.

equation

or

the

3



each

integer

to

this:

positive

Solve

line

zero

negative

●

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

integers

placed

right

●

can

this

positive

natural

numbers.

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of

. . .}

●

Any

extension

integers.

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➔

an

The

to

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writing

lived

669

is

CE

credited

the

rst

not.

book

that

included

Answers zero

5( -2 ) a

4

i

4

the

expressions,

and

negative

substituting numbers.

=

=

-2 +

Write

14

-7 the

2

numbers

for

the

letters.

2

4

- ( -2 )

= 1

ii

5

2

4

+

2( -2 )

You

can

use

your

GDC

to

evaluate

this.

When

using

fractional

use

your

GDC

expressions,

brackets

numerator

to

indicate

and

the

to

input

remember

clearly

to

the

denominator,

or GDC

use

the

fraction

template.

Plus

b

i

integer

ii

not

an

integer

help

on

CD:

demonstrations

and

GDCs

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on

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for

the

TI-84

FX-9860GII

the

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Chapter





Exercise

1

B

a

Solve

b

State

a

Solve

b

State

a

Find

the

equation

whether

or

4x

not

+

2

=

your

0.

solution

to

par t a

is

an

integer.

2

2

3

the

equation

whether

the

a

or

value

x

not

of

=

these

expressions

to

par t a

when a

are

=

integers.

−2

and

b

=

4.

2

3a

ii

− b

a + b

State

solutions

9

b

i

b

4.

your

whether

or

Investigation

not

–

your

answers

to

par t a

are

integers.

integers

a

State

whether

each

of

these

statements

is

true

or

false.

If

false,

If

=

then

c

we

b

give

an

example

to

show

why. say

The

a

of

sum

two

integers

is

always

an

that

c

is

quotient

b

The

dierence

c

The

quotient

of

two

integers

is

always

an

two

integers

is

always

an

of

a

and

b

integer Quotient

of

the

integer

means

integer ratio.

The

d

The

In

product

set

the

is

integers

rational

not

quantities

of

two

investigation

integers

set

of

of

you

always

that

rational

cannot

always

an

integer

numbers, 

should

an

is

have

integer.

be

So

found

we

represented

that

need

with

a

the

new

integers.

quotient

set

as

This

of

there

set

two

are

is ,

the



is

an

The

set

of

rational

numbers



⎨

that

be

q

≠

0

as

by

0

is

not

⎫

p

where

p

and

q

are

integers

and

q

≠

q

⎩

This



is

division

⎧

of

numbers.

Note

➔

extension

0

dened.

⎬

⎭

definition

written

examples

as

of

means

the

that

quotient

rational

a

number

of

two

is

rational

integers.

Here

if

it

can

The

are

decimal

rational

numbers.

nite

expansion

number

number

of

may

of

have

decimal

a

a

places

(for

7 ●

7

is

a

rational

number

as

it

can

be

written

as

, example

−1.5)

example

0. 6).

or

may

recur

(for

1

and

both

7

and

1

are

integers. A

number

with

recurring

3 ●

−3

is

a

rational

number

as

it

can

be

written

as

,

digits

has

a

period,

which

is

the

digit

1

and

both

−3

and

1

are

integers.

or

group

of

digits

that

is

repeated

0 ●

0

is

a

rational

number

as

it

can

be

written

as

,

after

the

decimal

point.

For

example,

4

and

both

0

and

4

are

the

integers.

period

of

0.66666...

is

6

3 ●

−1.5

is

a

rational

number

as

it

can

be

written

as

,

period

2

and

both

−3

and

2

are

integers. 6

●

0. 6 = 0.666...

is

a

rational

number

as

it

can

be

written

as

, 9

and



both

Number

6

and

and

9

algebra

are

1

integers.

of

0.767676...

is

76.

and

the

From

these

examples

we

can

see

that

any

integer

is

also

a

rational F ind

number

but

not

all

rational

numbers

are

integers.

Y ou

can

the

some

of

the

rational

numbers

on

the

number

line

like

out

more

about

represent

this:

histor y

numbers

of

on

rational

pages

40–41. –0.5

1

0

4

Example

1

1

8

4

0.5

1

1.25



a

Express 1.3

b

Hence

as

a

fraction.

4

1

calculate

3 + Hence’

5

Give

your

answer

as

a

fraction.

is

command

is

a

term

frequently

that

used

in

Answers exams.

Let

a

a

= 1 .3 then

‘hence’

a

=

1.3333 . . .

10a

=

13.333 . . .

use

Multiply

number

10a

−

a

=

=

9a

13.333 . . .

=

Divide

4

=

3 +

4

4 +

same

another

the

work

period.

to

preceding

nd

required

the

result.

10a.

to

sides

its

by

9.

simplest

for m.

32 =

3

Exercise

Find

from

both

Simplify

Use

5

a

common

denominator

of

15

15

or

a

the

obtain

to

3

= 5

1

a

to

read

tr y

=

4

1

with

Subtract

12

9

b

1.3333 . . .

10

you

then

12

12

a

−

by

If

your

GDC.

C

the

decimal

expansion

of

these

fractions. 

2

5

2

4

11



÷

3,

3

3

4

9

7

5 use

b

For

i

2

3

a

each

is

fraction

finite

Express

0

in

as

a

whether

fraction.

0

its

decimal

5

+ 1

Express 1

b

8.

Give

your

8

as

a

fraction.

answer

as

a

fraction.

Hence

a

Write

down

a

rational

number

whose

decimal

expansion

is

b

Write

down

a

rational

number

whose

decimal

expansion

recurs.

c

Write

For

down

period

a

that

any

pair

of

number

that

lies

rational

star ts

rational

your

in

number

the

whose

four th

numbers,

you

digit

can

decimal

after

the

always

expansion

decimal

find

a

finite.

has

point.

rational

Express

between

them

on

the

number

line.

For

1

ari thmetic

mean

of

two

numbers

is

halfway

9

as

a

example, fraction.

the

GDC.

expansion

c

a

calculate

state

recurs.

ii

5

a,

between

What

do

you

those notice?

Is

it

true

that

numbers. 9

=

?

Chapter





Example

a

Write



down

a

rational

number

that

lies

on

the

number

line

between

Wri te

down’

is

a

2

and

command

1

term

that

3 means b

Write

down

a

second

rational

number

that

lies

on

the

number

to

2

and

between

Write

show

don’t

much

or

need

any

1 working.

3

c

you

line

down

a

third

rational

number

that

lies

on

the

number

2

line

and

between

1

3

Answers

2 + 1

Find

the

arithmetic

mean

of

5

3 a

=

2

2

6

and

1.

Use

your

GDC

to

3 2

5 +

simplify

3

it.

3

6

b

=

2

How

4

2

many

numbers

3

rational

are

there

+

3

4

17

between

two

24

numbers?

rational

=

c

2

➔

A

number

is

rational

if

●

it

●

its

decimal

expansion

is

finite,

●

its

decimal

expansion

is

non-terminating

can

be

written

as

a

quotient

of

two

integers,

or

or

but

has

a ‘Non-terminating’

recurring

Example

digit

or

patter n

of

digits.

the



x

2

For

each

of

the

expressions

(x

a

+

y)

 5

b y

1 i

Calculate

ii

State

the

value

x

when

=

-4

and

y

= 2

whether

Justify

your

your

answers

to

i

are

rational

numbers

or

not.

answer.

Answers

2

4

i

 2



It

it

is

a







be

 5

i

ii

is

Its

 1

2

2

a

finite

decimal

not



Number

2

It

does

places

and

algebra

1

is

not

number

recur.

and

number.

expansion

1.4142135...

a

the

integers.

rational

decimal

have

as

as

1

1

not

4



number

two



It

2

written

of

49 



rational

can

4





quotient

b

7 

 



ii

2

1 



a

of

does

To

justify

you

know

your

it

is

answer,

rational.

explain

how

opposite

of

is

‘nite’.

Exercise

Write

1

D

down

three

rational

numbers

that

lie

on

the

number

line

9

between

2

and 4

2

Calculate

a

the

value

of

the

expression

2( y

−

when

x )

y

=

3

1

and

x

=

− 8

b

State

whether

a

Write

your

answer

to

part a

is

a

rational

number

or

not.

9 3

down

three

rational

numbers

between

and 5

6

28 b

Write

i

down

three

rational

numbers

between

and

−2.

13

28

How

ii

many

rational

numbers

are

there

between

and

−2?

13

Investigation

State

whether

false,

explain

why

The

dierence

b

The

square

c

The

quotient

d

The

square

In

set

the

root

So

of

a

of

a

rational

need

a

represented

circle

with

is

the

two

of

an

you

is

true

is

always

number

or

false.

If

a

statement

is

is

is

always

a

a

rational

rational

sometimes

always

a

number.

number.

a

rational

rational

number.

number.



have

not

there

is

number s

rational

rational

number s

number

should

as

is

example.

rational

rational

a

numbers

statements

numbers,

set,

radius

area,

two

number

new

with

these

rational

root

real

rational

giving

of

of

investigation

we

What

of

of

by

a

The

a

each

–

found

always

are

a

that

rational

quantities

numbers.

For

the

that

example,

square

number.

cannot

we

be

could

think

of

1 cm.

A,

of

this

circle?

1 cm 2

A

=

π

×

A

=

π

×

A

=

π cm

Is

π

r

2

(1 cm)

2

a

rational

number?

The

decimal

expansion

of

π

from

the

GDC Y ou

is

3.141592654

–

but

these

are

just

the

first

nine

digits

after

ten

decimal

can

nd

the

rst

the thousand

digits

of

point. π

from

this

website:

http://www.joyofpi.

The

decimal

expansion

of

π

has

an

infinite

number

of

com/pi.html.

digits

➔

after

Any

the

decimal

number

number

of

irrational

that

digits

point,

has

after

a

and

no period

decimal

the

(no

expansion

decimal

point

repeating

with

and

no

an

patter n).

infinite

period

is

an

number

Chapter





Irrational

➔

The

numbers

set

of

numbers

rational

complete

numbers,

Natural

include,

for

example, π,

numbers

the

together

number

line

2 ,

with

and

3

the

form

set

the

of

set

irrational

How

of

numbers

real

Can



numbers

1

2

3

4

5

March



the

and –3

–2

–1

0

1

2

14

(or ,

3/14)

world

4

are

of

in

a

lot

the

of

three

Also

people

Pi

Day,

most

March

1

4

2

events

bir thday

are

–1

0

1

numbers

2

so

–2

–1

Calculate

a

is

the

number

line

in

the

is

an

day/month

Day

is

format

July

1

approximation

to

the

2

of

length

l

3

these

of

The

area

A

of

a

a

measurements

and

state

whether

it

is

rational

diagonal

of

a

square

with

side

length

of

1 cm.

circle

with

radius

cm. 

1 cm

Answers

2

a

l

2

=

1

=

2

2

+

Use

1

Pythagoras´

theorem.

2

l

l

2

=

2

is

=

2

an

irrational

It

number

is

1.4142...

not

finite,

not

recur ring.

2

b

A

=

π r

Use

the

for mula

for

the

area

of

2



A

=

π

×

1



a

1









π

=

circle.

×



2

A

1

=

is

1 cm

a

rational

Exercise

1

a

number

E

Calculate

triangle

the

with

length,

sides

h,

2 cm

of

the

and

hypotenuse

of

a

right-angled

1.5 cm. h

2

b

State

a

Calculate

b

State

whether

the

h

is

rational

area,

A,

of

a

or

1.5

irrational.

circle

with

diameter

10 cm.

2



Number

whether

and

or

A

algebra

1

is

rational

or

irrational.

which

value



each

both

Pi

,

/7,

1 b

Alber t

together .

irrational.

The

1

3

complete

0

Example

or

3,

signicant

14

r

–3

around

as

sometimes

celebrated

Approximation

Real

month/day

celebrate

π.

Einstein’ s –5

2

–2

them?



–5

–3

there?

3

digits

Rationals

are

count

6

format,

Integers

we

real



On

0

many

of

π

Example

Solve

a

8

+

x

State

b



this

>

inequality

and

represent

the

solution

on

the

number

line.

Do

we

same

5

whether

p

−π

=

is

a

solution

to

the

inequality

given

in

part

all

use

the

notation

in

mathematics?

a

We

are

using

an

Answers empty

8

a

+

x

>

5

x

>

−3

that

x

dot

=

−3

included.

–2

–1

0

−π

=

−3.142...,

so

−π

is

not

a

solution

of




−1

line.

number

p

is

a

given.

Inequality

x

+

1

>

4

1

≤

x

+

1

≤

8



x

>

1

p



10

π

.

It

is

Approximations

impor tant

value

and

an

Sometimes,

because

we

use

●

The

●

The

the

to

that

understand

approximate

as

in

the

exact

take

you

the

area

height

of

the

examples,

not

known

measurements

approximate

present

are

error

difference

between

an exact

value.

following

values

and

of

the

only

approximate

(maybe

reaches

Ecuador

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we

is

because

a

certain

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the

quantity

instrument

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283 561 km².

Pyramid

of

Giza

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approximately

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weight

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approximately

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

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exact

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the

economy

Rounding

➔

approximate

as

population

Rounding

given

we

value,

more

remaining

to

0.

Example



a

Write

down

247

b

Write

down

1050

correct

to

correct

the

to

nearest

the

ten.

nearest

hundred.

Answers

250

a

Both

10

1100

b

but

Both

of

240

250

1000

100

Exercise

Write

1

2

Write

3

Write

numbers

these

Write

numbers

b

these

b

these

109

to

1100

1050

Because

being

are

closer

and

is

multiples

are

multiples

digit

is

5,

150

b

to

correct

to

the

the

the

10 016.01

d

nearest

10.

269

d

nearest

100.

3062

d

nearest

1000.

9640

952

d

a

number

that

correct

to

the

nearest

100

6

Write

down

a

number

that

correct

to

the

nearest

1000

7

Write

down

a

number

that

correct

to

the

nearest

unit

Rounding

➔

numbers

decimal

is

a

rounding

a

rounding

Rounding

same

to

a

given

is

200.

is

is

3000.

6.

number

(dp)

numbers

to

the

nearest

tenth,

to

the

nearest

etc.

Rounding

as

places

rounding

Rounding

as

the

up.

unit.

1240

to

c

nearest

1015.03

c

1500

the

down

hundredth,

the

after

Write

This

in

round

5

of

of

247.

exactly

the

rounded

108.9

c

correct

numbers

to

c

correct

numbers

105 607

a

correct

24.5

b

140

a

4

these

246.25

a

250

G

358.4

a

is

and

middle.

one

and

as

a

number

correct

it

nearest

to

the

number

correct

it

nearest

to

the

number

rounding

it

correct

to

the

to

one

decimal

place

is

the

same

tenth

to

two

decimal

places

is

the

same

hundredth .

to

three

nearest

decimal

places

is

the

thousandth

Chapter





To

write

3.021

correct

to

1

dp:

Rounded

F irst

digit

the

digit

right

less

NUMBER

to

is

than

5

3

●

0



1

3

●

0

......

......

ROUNDED

3.021

=

3.0

(1 dp)

NUMBER

Rounded

digit

Digits

remains

the

unchanged

rounded

are

To

write

10.583

NUMBER

correct

to

2

to

right

Digits

of

the

digit

rounded

deleted

are

to

right

of

digit

deleted

dp:

1

0

●

5

8

3

1

0

●

5

8

......

ROUNDED

NUMBER

10.583

Rounded

digit

Digits

remains

of

write

4.371

to

1

=

10.58

(2 dp)

right

digit

are

deleted

dp:

Rounded

F irst

digit

the

digit

right

more

NUMBER

the

rounded

unchanged

To

to

to

is

than

5

4

●

3

7

1

4

●

4

......

......

ROUNDED

NUMBER

4.371 Rounded

is

digit

Digits

changed

to

1

of

more

to

the

rounded

are

right

Digits

digit

the

deleted

➔

Rounding

●

If

the

keep

rules

digit

the

If

the

add

Example

1

deleted

after

the

one

digit

that

is

being

unchanged

rounded

and

delete

is

all

less

than

5

the

digits.

digit

to

of

digit

decimals

rounded

following

●

for

right

rounded

are

to

after

the

the

one

rounded

that

digit

is

and

being

rounded

delete

all

the

is

5

or

more

following

then

digits.



a

Write

down

10.045

b

Write

down

1.06

correct

correct

to

to

1

2

dp.

dp.

Answers

a

10.045

b

1.06

=

10.05

(2 dp)

10.045

Next

digit

is

5,

so

round

6,

so

round

up:

10.05



=

Number

1.1

and

(1 dp)

algebra

1.06

1

Next

digit

is

up:

1.1

=

4.4

(1dp)

Exercise

Write

1

Write

numbers

these

Write

numbers

these

to

1

numbers

correct

to

2

2.401

correct

to

d

0.09

d

28.0751

dp.

9.6201

c

the

3.9002

b

dp.

c

201.305

b

10.0485

a

correct

301.065

b

0.0047

a

3

these

45.67

a

2

H

nearest

thousandth.

201.7805

c

0.008 41

d

1.8 4

Calculate

;

use

your

GDC.

2

3.08

Give

1

a

your

×

0.012

answer

dp

2

b

correct

dp

3

c

to

dp

nearest

d

100

nearest

e

1000.

3

( p

that

p

=

3.15

and

q

=

0.8,

find

the

value

+ q )

of

giving p

your

2

a

answer

correct

dp

3

b

dp

nearest

c

unit

nearest

d

Write

down

a

number

that

correct

to

2

dp

is

2.37.

7

Write

down

a

number

that

correct

to

1

dp

is

4.1.

of

numbers

significant

➔

The

number

figures

that

to

figures

of

are

q

to

6

Rounding

+

a

given

number

(sf)

significant

known

ten.

figures

with

some

in

a

result

degree

of

is

the

number

of

reliability . 51

0

Given

ni

5

depends

on

the

measurement

that

is

being

31

sometimes

taken.

1

41

This

21

For

example,

if

the

length

of

a

pencil

is

measured

with

a

r uler

the

say:

is

1 mm,

then

the

measurement

is

only

millimetre.

3

8

Y ou

division

nearest

9

to

01

smallest

accurate

2

11

whose

7

can

The

length

of

this

pencil

is

14.6 cm.

cannot

say:

The

length

of

this

pencil

is

5

you

14.63 cm.

4

6

But

4

The

length

of

the

pencil

can

be

given

correct

to

3

sf

but

cannot

be

5

3

correct

to

4

sf.

2

given

1

significant

●

All

non-zero

●

Zeros

digits

between

0

for

figures:

are

signicant.

non-zero

digits

578 kg

has

0 004 km

are

6

mc

Rules

4 sf

has

5 sf

signicant.

Make

●

Zeros

to

the

left

of

the

rst

0.03 g

non-zero

has

sure

you

 sf understand

digit

are

not

digit

●

Zeros

right

placed

of

the

when

a

signicant.

after

other

decimal

digits

point

are

but

to

the

0.100 ml

has

is

signicant.

3 sf

signicant.

Chapter





The

r ules

similar

number

This

for

to

rounding

the

of

ones

for

decimal

example

Example

to

a

given

rounding

number

to

the

of

significant

nearest

10,

figures

1000,

etc.

are

or

to

a

places.

shows

you

the

method.



a

Write

down

24.31

correct

b

Write

down

1005

c

Write

down

0.2981

correct

to

to

correct

2 sf.

3 sf.

to

2 sf.

Answers

24.31

a

=

24

(2 sf)

24

24.25

24.5

Digit

rounded

Rounded

24.75

to

digit

right

is

of

less

number :

25

than

5.

00

4

2

Change Leave

the

the digit

the

right

of

the

1005

=

1010

Digit

(3 sf )

the

to

rounded

unchanged. digit

b

digits

rounded

to

right

rounded

of

rounded

digit.

digit

Change

all

is

equal

digits

to

to

to

0.

5.

the

Add

right

1

of

to

the 9

rounded

digit

to

+

the

Digit

0.2981 = 0.30 (2 sf )

c

1

to

of

➔

Rounding

●

If

the

rules

for

(n+1)th

to

the

the

right

of

rounded

rounded

rounded

significant

figure

is

less

figure

is

5

digit.

digit

digit

Change

to

is

greater

all

digits

than

to

5.

the

Add

right

0.

If

●

the

In

(n+1)th

both

cases

deleted

should

if

be

decimal

Example

all

they

the

are

figures

than

5

more

figures

to

replaced

or

the

by

to

right

zeros

then

keep

the nth

figure

then

the

of

if

add

right

the

they

of

1

to

figure n

decimal

are

to

this

the

should

point

left

figure.

of

be

and

the

point.



3

12.4

Let

t

=

2.1

a

b

+

3

Write

down

Write

the

i

3

the

value

answer

significant

to

of

par t

figures

t

a

giving

the

correct

ii

2

full

calculator

Number

and

algebra

1

display .

to

significant

figures.

{



=

10

Replace

0.

rounded

Add

the

left

digit.

unchanged.

●

1

0.

Continued

on

next

page

1

to

of

digit

the

the

with

digit

to

rounded

Answers

a

497.5466391

b

i

498

ii

500

Exercise

1

Write

a

2

3

a

4

a

the

number

these

significant

2

these

these

numbers

+

2

×

to

1

498

=

500

(3 sf )

(2 sf )

2

correct

to

3

10

c

each

of

these

d

1290

d

0.001 32

d

1560.03

d

0.5006

e

numbers.

1209

sf.

sf.

1.075

c

0.3259

b

of

390.8

correct

0.0801

b

to

c

numbers

2971

correct

0.072

b

figures

0.02

c

numbers

8 .7 5

of

200

b

355

Write

=

I

280

Write

3

7.54

106

Write

a

497. 54

sf.

410

1.6

Calculate

. 4

0

Give

a

6

answer

sf

Write

a

7

1

your

value

nearest

Write

3

b

the

correct

sf

of

π

correct

a

238

a

Calculate

2

b

these

to

1

c

unit

down

(1

3

c

to

4609

b

d

nearest

hundredth.

to

dp

numbers

sf)

dp

the

(3

2

sf

accuracy

sf)

c

d

3

dp.

stated.

2.7002

(3

sf)

3

3 8

375

.

Write

down

the

full

calculator

display .

2

1 .5

b

Give

i

2

your

sf

+ 1 .8

answer

ii

3

to

sf

par t

a

iii

correct

4

to

sf.

Chapter





Often

in

exams

you

need

to

do

multi-step

calculations. The

In

those

situations,

keep

at

least

one

more

significant

digit

in

general

rule

Mathematical

results

than

needed

in

your

final

instance,

if

the

final

answer

needs

to

be

given

correct

to

3

Unless

carr y

at

unrounded

least

4

values

sf

in

in

the

your

intermediate

calculations,

or

store

the

GDC.

The

of

exactly



diagram

wire,

The

to

small

and

are

15 cm

Find

correct

represents

are

The

3

other

length

window

out

Their

two

of

significant

of

grille

the

or

three

question

be

given

correct

significant

‘Congruent’

triangles

hypotenuse

sides

the

made

house.

right-angled

congruent .

total

to

a

pigeons

triangles

long.

the

the

must

figures.

keep

all

in

answers

to

Example

otherwise

sf, stated

then

Studies

answer. is

For

in

inter mediate

are

wire,

L.

equal

Give

exactly

is

shape

the

and

means

same

size.

lengths.

your

answer

figures.

Answers

Let

x

be

the

side

length

of

the

triangles.

2

x

2

+

x

First

find

sides

using

the

length

of

the

shor ter

Pythagoras.

2

=

15

=

225

2

2x

15 cm x 2

x

=

x

112.5

=

112.5

x

x

=

10.6066 ...

Keep

this

more

than

this

L

=

31

L

=

31

L

=

508.804

L

=

×

×

x

+

12

×

10.6066

509 cm

Exercise

In

15

. . .

+

12

×

is

the

just

with

. . .

either

three

an

grille

triangles

15

value

length

inter mediate

there

with

exact

significant

are

length

31

x

or

as

value.

sides

and

with

figures

of

12

sides

The

J

a

area

QUESTIONS

Find

four

b

2

Let

a

the

a

circle

length

significant

Find

to

of

the

two

the

length

the

correct

is

of

p

10.5 cm

its

of

=

its

4

radius.

Give

your

answer

correct

to

circumference.

Give

your

answer

correct

figures.

2

arithmetic

to

.

figures.

significant

numbers

Find

q

and

mean

=

of

10

p

and

q.

Give

your

answer

sf.

2

b

Find

c

Find

Give



Number

the

the

value

area

your

and

forget

the

answers.

2

1

not

down

15.

(3 sf)

EXAM-STYLE

Do

of

of

a

answer

algebra

1

(p

+

q)

.

Give

rectangle

correct

your

whose

to

2

sf.

answer

sides

are p

correct

cm

and

to

q

3

sf.

cm

long.

to

units

write

in

your

Estimation

An

to

estimate

check

➔

To

the

of

estimate

involved

Example

A

theatre

seats

a

in

is

quanti ty

reasonableness

to

the

answer

an

of

to

approximation

an

a

that

is

usually

used

answer.

calculation,

round

all

the

numbers

1 sf.



has

the

98

rows;

each

row

has

23

seats.

Estimate

the

number

of

theatre.

Answer

Exact

100

×

20

=

2000

seats

Round

98

to

1 sf

→

3

Round 23 to 1 sf

Example

Estimate

→

answer

is

98

×

100

=

54

seats.

20



the

average

speed

of

a

car

that

travels

527 km

in

6

hours.

Answer

distance

Average

speed

traveled

=

527 time

→

500

(1 sf) Exact

taken

Round

6

down

to

5

to

make

answer

is

the 57 1

500

=

1

= 100

km

division

h

calculation

87.8 km h

(3 sf)

easier. 6

5

Exercise

1

K

Estimate

the

answers

to

these

calculations.

147 2

a

298

×

10.75

b

3.8

c

103

d

11.02

2

A

lorr y

in

each

is

is

carr ying

container.

210

containers

Estimate

the

with

pipes.

number

of

There

pipes

are

that

18

the

pipes

lorr y

carr ying.

Population 3

Japan

covers

an

area

of

approximately

377 835 km²

and

total

March

2009

Japan’s

population

was

127 076 183.

4

A

tree

the

5

population

yields

number

Mizuki

average

on

of

r uns

average

reams

33 km

speed.

density

in

in

9000

that

1.8

can

=

population

Estimate land

Japan’s

density

in

area

2009.

copy

be

hours.

pages.

made

Estimate

from

Estimate

one

tree.

A

ream

has

500

pages.

Mizuki’s

Average

distance

time

speed =

traveled

taken

Chapter





The

6

Badaling

Scenic

Area

visitors

Peter

7

per

Section

of

the

day .

and

Great

Estimate

calculates

the

area

the

Ming

Wall

the

of

are

Mausoleums

limited

number

this

of

to

53 000

visitors

per

year.

square

2

as

1020.01 m

.

Use

estimation

to

decide 100.1 m

whether

Peter

is

correct.

[

Percentage

errors

Sometimes

need

value

➔

and

The

and

you

the

exact

difference

the

exact

to

know

the

difference

between

an

The

=

v

−

between

is

value

an

estimated

called

the

Why

or approximated value

do

What

error:

errors

kind

the

approximated

value

and

v

the

is

the

exact

‘error’

In

the

and

Ramesh

that

that

concer t

Ramesh

the

each

went

Olivia

there

that

estimated

Calculate

and

value

have

the

meaning?



concer t

estimated

do

E

same

In

errors

know?

‘mistake’

Olivia

of

arise?

E

is

A

Example

China

value.

Do

v

of

v

A

where

Wall

estimated

you

Error

Great

were

that

a

different

there

concer t.

were

1450

people

and

Olivia

1300.

Ramesh

errors

to

attended

attended

there

Olivia

were

and

there

were

1950

people

and

1800.

Ramesh

made

in

their

estimations.

Answer

Olivia:

Error

=

1450

–

v

1300

v

A

Error

=

150

is

negative,

so

use

v

E

|v

v

E

value,

Ramesh:

In

=

1950

Error

=

150

Example

However,

a

Error

smaller

Using

16,

positive

v

A

and

Ramesh

than

is

150

both

more

out

of

E

made

accurate

the

as

same

150

out

error,

of

150.

1950

is

1450.

percentages :

150

150

× 100 %

=

10 .3 %

and

(3 sf)

×

Olivia’s

error

Ramesh’s

These

100 %

=

7 .69 %

(3 sf)

1950

1450

the

the

or

v

of

1800

estimate

propor tion

is

people

Olivia

Ramesh’s

–

|

E

modulus,

instead.

people

v

A

A

represents

error

10.3%

represents

percentages

estimations.

help

They

us

are

of

7.69%

to

the

of

have

called

a

total.

the

total.

better

idea

percentage

of

the

accuracy

of

errors

Sometimes

v

Percentage

error

we

don’t

v A

E

=

have

× 100%

the

exact

value.

➔ v E

where

v

represents

In

approximated

value

A

or

estimated

these

replace

cases

v

with

we

the

E

value

and

v

represents

E



Number

and

algebra

1

the

exact

value

accepted

value

Example

The

as

size

126°.



of

angle

Find

the

M

is

125.7°.

percentage

Salomon

error

he

measures

made

in

M

with

measuring

a

protractor

angle

M

Answer

Percentage

126

Percentage

error

125.7

v

v A

=



100%

error

E

=

× 100%

125.7

v E

Percentage

error

with

v

=

126,

v

A

=

0.239%

Exercise

Let

a

=

125.7

your

GDC.

Round

to

3

sf.

L

EXAM-STYLE

1

Use

(3 sf)

=

E

QUESTIONS

5.2

and

b

=

4.7.

3

a

Find

Xena

b

2

the

estimates

Find

the

Ezequiel’s

grade

a

in

the

error

wrote

the

in

are

final

three

final

c

Calculate

the

percentage

grade

in

measurements

kitchen

are

8.3,

of

par t a

by

6.8

in

is

140.

Xena

and

these

grade

marks

the

The

to

in

9.4

three

her

out

estimation.

of

10.

His

final

marks.

Biology .

correct

to

the

nearest

unit

to

find

Biology .

Calculate

final

b

made

mean

Ezequiel’s

grade

+

answer

marks

is

3a

b

his

3

Biology

of

the

percentage

Calculate

final

value

that

Biology

Ezequiel

his

exact

5.34 m

grade

that

Ezequiel

error

made

found.

by

Ezequiel

when

finding

Biology .

of

the

and

length

3.48 m

and

width

of

a

rectangular

respectively .

2

a

Calculate

b

Write

c

Calculate

using

in

down

the

m

the

both

the

exact

the

length

percentage

length

and

area

the

of

and

error

width

the

the

kitchen.

width

made

if

correct

the

to

1

correct

area

to

was

1

dp.

calculated

dp.

2

4

The

a

area

Find

three

b

Find

José

c

of

the

a

circular

radius

decimal

the

by

your

José.

that

of

the

answer

Give

the

is

89 m

garden.

.

Give

your

answer

correct

to

places.

perimeter

estimates

Use

of

garden

to

your

the

garden.

perimeter

par t

b

answer

to

of

find

the

the

correct

to

garden

is

30 m.

percentage

two

error

significant

made

figures.

Chapter





.

Standard

form

If ●

The

number

of

inter net

users

in

the

world

up

to

we

did

not

use

standard

form,

we

June would

write

the

mass

of

the

Ear th

as



2010

was



×

 5 970 000 000 000 000 000 000 000 kg 

●

The

●

An

mass

of

the

Ear th

is

about

.

kg.

×



a

human

−

These

estimate

for

numbers

the

are

average

either

mass

ver y

large

of

or

ver y

cell

is

about 

g.

small.

When

They

are

written

in

form :

standard

a

way

of

writing

ver y

large

numbers

written

ver y

small

numbers

without

writing

a

lot

of

are

or

in

standard

zeros. form

it

is

easier

to

k

➔

A

number

where

1

is

≤

written

a




3

enter

cur ve

equation

the

the

key

and

for

GDCs,

Casio

plotter,

are

and

on

using

the

a

CD.

to

get

back

to

exponent.)

appears,

of

the

3

press

from

for

Instructions

GDC. graph

T o

CD:

are

x

3

Change

on

instructions

the

instead

of

a

straight

line.

cur ve.

3

the

gradient

function

of

y

=

equation

of

the

x

Have

Once

you

have

the

3

gradient

function

of

y

=

x

cur ve,

nd

,

y

=

3x

the

,

a

guess

cur ve.

down

your

answers

in

the

worksheet

copy

of

the

3

y

=

3

x

y

=

the

equation

your

guess

of

to

the

3

x

y

=

3

3x

y

=

function.

Adjust

your

equation

table.

until

Cur ve

Enter

…

gradient

Write

at

the

3

it

ts.

Then

3

4x

y

=

delete

it.

1

3

−x

y

=

−x

y

=

3

x 

Gradient

Extend

function

your

investigation

so

that

you

can

nd

the

Worksheet

gradient

function

of

any

is

Be

systematic,

so

tr y

simple

cubic

3

Cur ve

y

=

x

cur ves

4

y

=

3

x

−

3

y

Worksheet

CD:

6.3

This

on

table

the

CD.

rst…

3

+

on

cubic.

=

x

3

+

5x

y

=

x

3

−

x

y

=

+

1

3



x

x

y

=

x

+



x 

Gradient

function

Then

move

on

to

more

complicated

3

Cur ve

y

=

x

cubic



+

3x

cur ves...

3

+



y

=

x



+

4x

3

+

3x

y

=

x



+

5x

3

−

4x

+

1

y

=

x



–

x

−

Continued

on

5x

–

4

Gradient

function

Generalize

your

results

to

determine

the

3

for

the

Y ou

general

now

have

quadratic

of

the

cubic

results

functions

table

with

y

cur ve

for

and

the

=

ax

bx

gradient

cubic

formula

for

the

gradient

function



+

+

cx

+

functions

functions.

d

of

linear

Complete

the

functions,

worksheet

copy

these.

Function

Formula

Constant

y

Linear

y

=

=

ax

Gradient

function

a

+

b



Quadratic

y

=

ax

+

3

Cubic



Introducing

y

=

dierential

ax

bx

+

c



+

bx

calculus

+

cx

+

d

next

page

Investigation –

In

this

investigation

Again,

take

a

the

you

gradient

nd

systematic

the

function

gradient

function

of

of

any

any

curve

cur ve.

approach. GDC

instructions

These

on

instructions

CD:

are

for

the

4



F ind

the

gradient

function

of

y

=

x



F ind

the

gradient

function

of

y

=

x



Generalize

5

TI-Nspire

GDC.

the

Plus

TI-84

FX-9860GII

Instructions

and

GDCs,

for

Casio

and

using

a

n

Up

to

this

these

point,

all

results

the

to

nd

powers

in

the

cur ves

y

,

=

your

gradient

cur ve

y

,

=

y

=

have

been

of

y

=

x

graph

plotter,

are

enter

the

CD.

posi tive

,

...

as

well.

x

x

1

T o

on

3



x

function

1

1

1

Consider

the

Remember

on

your

GDC

use

the

key

and

+

select

from

1

x

1

that

the

template

x

=

menu. x

Check The

final

this

result

with

result

your

teacher .

Do

not F inding

Function

Gradient

function

go

on

until

you

this

result

investigation

done

n

y

=

ax

same

process

of

nding

dierentiation.

yourself

how

to

In

the

these

gradient

function

investigations,

of

you

a

cur ve

have

is

known

learned

as

as

true.

The

proof,

for

that

differentiate.

gradient

by

function

Calculus

the Dierentiation

is

the

algebraic

process

used

do

a

was

same

the

gradient

function

of

a

given

forms

of

notation

differentiation.

depend

➔

To

on

the

The

notation

differentiate

are

used

notation

by

a

used

in

function,

know

arrived

building

at

is

true?

both

Isaac

and

mathematician

use

will

(1646–1716).

the

question.

find

to

at

the

almost

British

Newton

function.

you

the

it

to

for

that

the

without

discovered

time

(164–177),

Two

we

result

pattern

mathematician

find

not

proving

How,

always

.

is

so.

be

The

by

have

the

gradient

rival

the

German

Gottfried

The

claims

Leibniz

controversy

lasted

for

over

decades.

dy

function: The

notation dx

was

Function

Gradient

developed

by

function

Leibniz.

Newton’ s

dy n

n

y

=

ax

=

notation

1

used

n

f (x)

=

ax

How

n–1

f ′(x)

is

now

=

in

physics.

impor tant

process

is

valid

for all

values

of

is

nax

mathematical

The

only

nax

dx

n,

both

positive

and

negative.

in

enhancing

notation

your

understanding

of

a

subject?

Chapter





Example



dy 7

Given

y

=

4x

,

find

dx

Answer

n

dy 7

=

y

1

=

a x

7 × 4 x

dx

dy n

=

1

na x

dx

dy 6

=

28 x

dx

a

Example

=

4,

n

=

7



5

Given

f

(x)

=

3x

,

find

f

′(x).

The

is

Answer

5–1

f

′(x)

=

5

×

n

3x

f

(x)

=

′(x)

=

f

′(x)

Euler

who

=

1

perhaps

the

greatest

n a x

mathematician

a = 3,

Example

f

of

all.

n = 5



2

Given

was

a x

n

15x

from

notation

(1707–83),

4

f

f ′(x)

(x)

=

3x

–

4x

×

4x

3

+

x

,

find

f

′(x).

Answer 1

Remember 1–1

f

′(x)

=

2–1

–

3x

2

that

x

=

3–1

+

3

×

x

Dif ferentiate

each

ter m

separately. 0

and

that

x

=

1.

2

f

′(x)

=

3

–

Exercise

8x

+

3x

6A

dy 1

Find dx

2

a

y

=

4x

e

y

=

x

3

4

b

y

=

6x

c

y

=

f

y

=

5x

g

y

=

3

7x

d

y

=

5x

x

h

y

=

12x

l

y

=

4

1 3

2

i

y

=

3

1

9x

y

j

=

4

2

x

y

k

=

x

4

2

2

x

Differentiate 1

y

=

7

y

b

=

2

4

3

a

–3x

y

c

=

3

x

–

d

y

=

4

x

– 3

6

e

y

=

i

y

=

–x

f

y

=

j

y

=

–3

x

f

y

=

f

y

=

l

y

=

–7x

3

2 9

x

x

4

3

(x)

=

3x

3

+

5x

4

b

f

(x)

=

5x

3

c

h

′(x).

2

a

k

9

5x

4

2

f

=

12

8

Find

y

3

1

3

g

(x)

=

9x

−

11x

−

4x

4

d

f

(x)

=

x

+

3x

+

2

y ′

is

another

dy

4

Find

writing

y ′

dx

1 6

a

y

=

8

−

c

y

=

7x

5x

+

4x

2

b

y

=

9x

−

5x

+ 2

5



Introducing

+

4x

−

dierential

101

calculus

d

y

=

x(2x

+

3)

way

of

x

Y ou

the

can

use

letters

notation

Example

but

other

not

the

v

x

and

y

for

the

variables.

This

changes



dv

8

Given

than

process.

=

3.5t

,

find

dt

Answer

n

dv 8

=

v

1

=

a t

8 × 3.5t

dt

dv n

=

1

nat

dt

dv 7

=

28t a

dt

Example

=

3.5,

n

=

8



4

3z

f

Given

( z )

=

,

find

f

′(z).

2

Answer

4

3z

3

n

4

f

(z)

f

× z

=

=

(z)

=

az

n

f

′(z)

=

1

naz

3 4

f

′( z )

=

4 ×

1

3

z a

2

=

,

n

=

4

2

3

f

′(z)

=

6z

Example

Given

f



(t)

=

(3t

–1)(t

+

4),

find

f

′(t).

Answer

2

f

(t)

=

3t

f

(t)

=

3t

+ 12t –

t

–

4

Multiply

out

the

brackets.

2

f

′(t)

=

+

11t

6t +

Exercise

–

4

11

Dif ferentiate

each

ter m

separately.

6B

dA 1

Find

dt

2

a

A

=

4t(9

−

t

)

b

A

=

6(2t

+

5)

d

A

=

(t

f

A

=

(6t

+

7)(3t

h

A

=

3(t

+

3)(t

2

c

A

=

t

(t

e

A

=

(5

−

−

5)

t)(3

+

2t)

+

2)(2t

−

3)

−

5)

2

g

2

A

Find

=

f

(t

+

3)(t

−

1)

−

4)

′(r).

1 2

a

f

(r)

=

(r

+

3)(2r

−

6)

b

f

(r)

=

(r

+

d

f

(r)

=

(5

f

f

(r)

=

5(7

3)

2

2

c

f

(r)

=

(2r

−

3)

2

−

2r)

2

e

f

(r)

=

3(r

+

5)

2

−

r)

Chapter





Y ou

can

also

differentiate

denominator

negative

of

a

First

which

you

have

must

powers

write

these

of x

in

terms

the

using

indices.

Example



4

y

Given

functions

fraction.

dy

=

,

find

2

x

dx

Answer

1

Write

2

y

=

4 ×

=

the

function

in

index

for m:

4x 1

2

2

x

=

.

x

2

x dy 2

=

1

−2 × 4 x a

dx

=

4

and

n

Remember

dy

=

the

–2

rules

for

multiplying

3

=

−8 x negative

dx

dy

numbers.

8

=

Rewrite

in

the

original

for m.

3

dx

x

Example



12

f

Given

( x ) =

,

find

f

′(x).

3

Answer

12

1

12

Write

3

f

(

x

)

=

×

=

the

function

in

index

for m.

x 12

3

5

5

x

a

=

and

n

=

–3

5 12 3

f

′( x ) =

−3

×

×

1

x

Be

very

careful

with

minus

signs.

5

36 4

f

′( x ) =

×

x

Simplify.

5

36

f

(x ) =

Rewrite

in

the

original

for m.

4

5x

Exercise

6C

Differentiate

the

following

with

respect

Remember

to x.

same

y

2



f

(x )

3



(x )

question.

x

2

5

2

f



x

x

4

y

4

2

5



y



6

y

 9 

7

8

x

x

x

5

4

3 3

2

7

f

(x )

 7x

8



y

 7  4x



9

g(x )



x



2

5

2

2x

x

x

4

y



4x

x

1

3 10

3

3

11



g

12

 x   5x

y

 8

4

2

x

x

4x

4

x

3

5

3

2

13

y



 3x

14



g

 x   2x

8



Introducing

3

2

 2 

15

A ( x )

=

x

−

+

2

2x

6x

dierential

5

2

 x

4

calculus

2

2x

4 x

use

notation

7

2

3 1

to

as

the

the

.

Calculating

at

a

given

the

gradient

of

a

curve

point y

10

➔

Y ou

can

use

the

gradient

function

to

determine

the

exact

value 8 A

of

the

gradient

at

any

specific

3

Here

is

the

cur ve

y

=

2x

point

on

the

cur ve.

4

2

–

x

–

4x

+

5

with

–2

domain

≤

x

≤

2.

The B 2

cur ve

At

x

intersects

=

–2

the

the

y-axis

function

at

has

a

(0,

5).

negative

x

0

value.

–2

–1

1

2

3

–2

It

increases

to

a

point

A,

then

decreases

to

a

point

B

and

after x

=

1

–4

it

increases

The

again.

gradient

function

of

the

cur ve

will

be

negative

between

points Will

A

and

B

and

positive

the

gradient

elsewhere. function

be

positive

dy 2

Differentiating,

the

gradient

function

is



6x

 2x

or

 4

negative

at

point

A

dx

At

the

y-intercept

(0, 5)

the

value

is

0.

and

at

point

Y ou

can

2

into

:

at

x

=

0,

6(0 )



 2(0 )  4

=

– 4

dx

dx

–4

at

is

the

the

Move

the

gradient

point

the

cur ve

gradient

(0,

point

to

at

5).

on

along

Chapter

nd

your

6.1,

the

check

GDC.

1,

GDC

Plus

use

this

algebraic

method

to

find

Section

Example

help

on

CD:

demonstrations

can

this

See

33.

other

points.

Y ou

B?

Substituting

dy

dy

this

x-coordinate

y

the

and

GDCs

Casio

are

on

Alternative

for

the

TI-84

FX-9860GII

the

CD.

10

gradient

of

the

cur ve

at

other

points.

For

example, 8 A

dy 2

at

x

=



–1,

6( 1)

 2( 1)  4

dx

4

B

dy



2

4

The

dx

–2

This

result

agrees

with

what

can

be

seen

from

–1

1

2

3

cur ve

graph.

using

=

of

−1

the

is

4

at

x

=

0

it

is

−4

6D

questions

a

x

–4

Exercise

These

at

–2

and

the

gradient

x

0

GDC.

can

Make

be

answered

sure

you

can

using

do

the

algebraic

method

or

both.

dy 2

1

If

y

=

x

−

3x,

find

when

x

=

4.

dx

dy 3

2

If

y

=

6x

−

x

+

4,

find

when

x

=

0.

dx

4

3

If

y

=

11

−

2x

dy

3

−

3x

,

find

when

x

=

−3.

dx

Chapter





dy

If

4

y

=

2x(5x

+

4),

find

the

value

of

when

x

=

−1.

dx

3

5

Find

6

Find

the

gradient

of

the

cur ve

y

=

x

−

5x

1

the

gradient

of

the

curve y

=

10

−

at

the

point

where

x

=

6.

4

x

at

the

point

where x

=

−2.

2

2

7

Find

the

gradient

of

8

Find

the

gradient

of

9

s

the

cur ve

y

=

3x

(7

−

4x

)

at

the

point

(1,

9).

2

the

cur ve

y

=

3x

−

5x

+

6

at

the

point

(−2,

2

=

40t

−

5t

s

10

=

t (35

when

t

=

0.

Find

=

when

80t

+

7

v

12

=

0.7t

when

t

=

−4.

−

Find

11.9

when

3

=

3.

t

=

0.7.

dt

dt

A

=

dv

dv

Find

13

t

dt

dt

v

6t)

ds

ds

Find

11

+

3

14h

W

14

=

7.25p

2

dA

Find

when

h

=

dh

dW

.

Find

at

3

p

=

−2.

dp

8

18 2

15

V



4r

.



A

16

 5r

 2

r

r

dA

dV

Find

at

r

=

3.

Find

at

V



  r

A

18

r

=

2.

Find

at

r

dr

15

6r

1.

dA

at

dr



=

 r

dV

Find

V

4.

2

 7r

r

19

=

2

8 3

17

r

dr

dr

12



20

C

=

45r

+ 3

2r

r

dV

dC

Find

at

r

=

5.

Find

dr

By

working

point

on

a

backwards

cur ve

Example

at

r

=

1.

dr

with

a

you

can

find

par ticular

the

coordinates

of

a

specific

gradient.



2

Point

at

A

A

is

lies

1.

on

Find

the

the

cur ve

y

=

5x

coordinates

–

of

x

and

the

gradient

of

the

cur ve

A.

Answer

dy

dy

=

5 − 2x

First

find

dx

dx

dy

at

A

= 1

so

5

–

2x

=

1

x

=

2

Solve

the

equation

to

find

x.

dx

2

y

=

A



5(2)

is

(2,

–

(2)

=

6

Substitute

6)

Introducing

the

dierential

calculus

cur ve

x

to

=

2

find

into

y.

the

equation

of

28).

Exercise

6E

2

1

Point

P

is

P

lies

equal

on

to

the

cur ve

y

=

x

+

3x

–

4.

The

gradient

of

the

cur ve

at

7.

a

Find

the

gradient

function

b

Find

the

x-coordinate

of

P .

c

Find

the

y-coordinate

of

P .

of

the

cur ve.

2

2

Point

Q

is

Q

lies

equal

on

to

the

cur ve

y

=

2x

–

x

+

1.

The

gradient

of

the

cur ve

at

–9.

a

Find

the

gradient

b

Find

the

x-coordinate

function

of

Q.

of

c

Find

the

y-coordinate

of

Q.

lies

on

y

the

cur ve.

2

3

Point

at

R

R

is

equal

a

Find

b

The

the

the

to

cur ve

4

+

3x

–

x

and

the

gradient

of

the

cur ve

–3.

gradient

coordinates

EXAM-STYLE

=

function

of

R

are

of

(a,

the

b),

cur ve.

find

the

value

of

a

and

of

b

QUESTIONS

2

4

Point

R

is

lies

equal

Find

The

R

the

on

to

the

cur ve

y

=

x

–

6x

and

the

gradient

of

the

cur ve

at

6.

gradient

coordinates

function

of

of

a

R

Find

the

value

Find

the

coordinates

are

and

of

(a,

of

the

cur ve.

b)

b.

2

5

which

the

gradient

of

of

the

the

point

cur ve

on

is

the

cur ve y

=

3x

the

cur ve y

=

5x

+

x

–

5

at

4.

2

6

Find

the

which

coordinates

the

gradient

of

of

the

the

point

cur ve

on

is

–

2x

–

3

at

9.

3

7

There

the

are

gradient

Find

the

points

two

of

the

on

the

cur ve

coordinates

of

is

cur ve

y

=

x

+

3x

+

4

at

which

6.

these

two

points.

3

8

There

are

gradient

points

two

of

the

cur ve

on

is

the

y

=

x

–

6x

+

1

at

which

the

–3.

Find

the

coordinates

of

Find

the

equation

the

of

cur ve

these

two

straight

points.

line

that

passes

through

these

two

points.

EXAM-STYLE

QUESTION

3

9

There

the

are

two

gradient

points

of

the

on

the

cur ve

Find

the

coordinates

of

Find

the

equation

the

of

is

cur ve

y

=

x

–

12x

+

5

at

which

zero.

these

two

straight

points.

line

that

passes

through

these

two

points.

Chapter





EXAM-STYLE

QUESTIONS

2

10

Point

P

(1,

b)

lies

on

a

Find

the

value

b

Find

the

gradient

c

Show

d

Q

(c,

that

d)

cur ve

is

is

at

of

P

the

the

function

the

to

y

=

x

–

4x

+

1.

b

of

gradient

point

equal

cur ve

on

–2.

the

the

of

cur ve

Show

cur ve.

the

that

cur ve

at

d

is

which

=

also

the

equal

to b

gradient

of

the

–2.

2

11

Point

P

(5,

b)

lies

on

the

a

Find

the

value

b

Find

the

gradient

c

Show

d

Q

(c,

that

d)

cur ve

is

is

Show

at

P

the

d

function

the

to

is

y

=

x

–

3x

–

3.

b

of

gradient

point

equal

that

of

cur ve

on

the

the

of

cur ve.

the

cur ve

cur ve

at

is

which

also

the

equal

to b

gradient

of

the

–3.

also

equal

to

–3.

2

12

Consider

the

function

a

Write

down

b

Show

that

c

Find

for

the

f

at

f

(x)

f

4x

–

x

–

1.

′(x).

x

=

5,

f

(x)

coordinates

which

=

(x)

=

f

of

=

a

f

′(x).

second

point

on

the

cur ve y

=

f

on

the

cur ve y

=

f

(x)

′(x).

2

13

Consider

the

function

a

Write

down

b

Show

that

c

Find

for

the

f

at

f

(x)

f

2x

–

x

+

1.

′(x).

x

=

2,

f

(x)

coordinates

which

=

(x)

=

f

of

=

a

f

′(x).

second

point

(x)

′(x).

2

14

Consider

the

function

a

Write

down

b

Show

that

c

Find

for

the

f

at

f

(x)

=

3x

f

(x)

=

f

f

x

–

1.

′(x).

x

=

1,

coordinates

which

–

(x)

=

f

of

a

′(x).

second

point

on

the

cur ve y

=

f

(x)

′(x).

2

15

Consider

a

Write

b

Find

the

function

down

the

which

f

f

f

(x)

=

2x

–

x

–

1.

′(x).

coordinates

(x)

=

f

of

the

points

on

the

cur ve y

=

f

(x)

for

′(x).

2

16

Consider

a

Write

b

Find

the

down

the

which

function

f

f

f

(x)

=

x

+

5x

–

5.

′(x).

coordinates

(x)

=

f

of

the

points

on

the

cur ve y

=

f

(x)

′(x).

2

17

Consider

Find

the

which



Introducing

f

the

function

coordinates

(x)

=

f

f

of

′(x).

dierential

calculus

(x)

=

the

x

+

point

4x

+

on

5.

the

cur ve y

=

f

(x)

for

for

y

.

The

Here

is

tangent

and

the

normal

to

a

=

f (x)

y

curve

Tangent

➔

a

cur ve

The

y

=

tangent

to

f

(x)

the

with

a

cur ve

point,

at

any

P ,

on

point

the

P

cur ve.

is

the

at

straight

line Normal

which

the

The

The

at

to

the

through

the

with

gradient

equal

to

the

gradient

at

P

and

P

of

P

cur ve

that

the

x-coordinate

the

P

P .

the

through

tangent

of

●

cur ve

normal

passes

●

passes

is

at

P

is

the

straight

to

perpendicular

cur ve

are

closely

line

the

90

degrees

which

tangent.

related

P

x

because,

at

P:

of

the

tangent

is

equal

to

the x-coordinate

of

the

tangent

is

equal

to

the y-coordinate

cur ve

y-coordinate

of

the

cur ve

●

the

Y ou

any

gradient

can

use

cur ve

equation

➔

To

find

the

a

point,

cur ve

the

the

equation

b,

and

the

is

to

P(a, b),

the

Calculate



tangent

differentiation

at

of

of

find

to

the

provided

the

of

equal

the

equation

that

you

x-coordinate,

the

tangent

y-coordinate

gradient

to

of

P ,

a,

of

the

the

know

of

the

cur ve

using

of

the

the

cur ve.

tangent

both

the

point

at

to

P .

P(a, b):

equation

of

cur ve.

dy 

Find

the

gradient



Substitute

function

dx dy

a,

the

x-coordinate

of

P ,

into

to

calculate,

m, For

more

on

the

dx

the

value

of

the

gradient

at

P . equation

Use



Example

Point

P

the

equation

of

a

straight

line

(y

–

b)

=

m

(x

–

a).

line,

see

of

a

straight

Chapter

3.



has

an

x-coordinate

2.

Find

the

equation

of

the

tangent

to

the

3

cur ve

Give

y

=

x

your

–

3

at

answer

P.

in

the

form

y

=

mx

+

c

Answer

3

At

x

=

2,

y

=

(2)

3

–

3

=

5

Use

y

=

x

–

3

y-coordinate

to

of

calculate

the

P .

dy

dy 2

=

3x

Find

the

gradient

function dx

dx

dy 2

At

x

=

=

2,

3(2)

= 12

Substitute

2,

the

x-coordinate

at

P ,

dx dy

m

=

12 into

to

calculate

m,

the

value

of

dx

the

At

(y

P

–

y

(2,

5)

–

=

5)

gradient

Use

12(x

–

2)

5

=

12x

–

y

=

12x

–19

24

the

(y

with

at

P .

Y ou

equation

b)

=

m(x

a

=

2,

Simplify.

b

=

5,

can

check

equation

a)

m

=

12.

tangent

of

the

the

using

your

GDC.

Chapter





Exercise

Find

1

the

6F

the

equation

stated

point,

of

P .

the

Give

tangent

your

to

the

answers

given

in

2

y

a

=

x

the

cur ve

form y

;

c

y

=

6x

e

y

=

2x

P(3, 9)

y

b

=

2x

–

x

;

y

11

=

;

P(2, 8)

5x

–

2x

+

4;

P(3, 7)

d

y

=

3x

f

y

=

10x

–

10;

–

y

4x

=

x

P(1, –7)

–

x

+

;

P(3, –7)

y

h

=

5

–

x

+

–

6x;

P(2, 13)

;

P(4, 0)

y

j

=

5x

–

3x

;

2x

P(–1, –8)

2

;

P(2, 8)

y

l

=

60x

–

5x

+

7;

1

y

=

x

P(2, 107)

2

4

m

P(2, 17)

2

3

6x

5;

2

3

2

k

c.

3

–

2

i

+

P(1, 2)

2

=

mx

2

2

y

=

3

2

g

at

− 7;

y

n

P( 4,121)

=

17

–

3x

+

5x

;

P(0, 17)

2 1 3

y

o

=

2x

(5

–

x);

P(0,

0)

p

y

=

x

− 4 x;

P

( 2, −6 )

4

3

2

2

q

y

=

1

+ 3;

P( −2, 6 )

r

y

=

x

1 ⎞

⎛

3

x

+

;

P

−1 , − ⎜

4

3

3

⎟ 3

⎝

⎠

1 3

s

y

=

x

2

− 7x

+ 5;

P( −2, − 25)

4

Find

2

the

point.

equation

Give

your

of

the

tangent

answers

in

the

to

the

given

form ax

+

y

;

=

by

+

c

=

at

the

stated

0

6

12 a

cur ve

(2,

3)

y

b

= 5 

;

(1,

11)

3

2

x

x 8

6 3

y

c

;

= 6x

(–2,

–14)

d

y

=

x

;



2

(–1,

5)

2

x

x

8 e

y

;

= 5x

(4,

18)

x

To

find

need

➔

the

to

do

The

equation

one

extra

normal

is

of

the

normal

to

a

cur ve

at

a

given

point

you

step.

per pendicular

to

the

tangent

so

its

gradient, m′,

1

is

found

using

the

formula

,

m =

where

m

is

the

gradient

of

m

the

tangent.

Example



Y ou

Point

P

has

x-coordinate

learned

about

–4. 12

Find

the

equation

of

the

normal

to

the

curve

y

gradient

=

at

of

a

P.

x perpendicular

Give

your

answer

in

the

form

Chapter

ax

+

by

+

c

=

0,

where

a,

b,

c

∈



Answer 12

12

At

x

=

−4,

y

=

=

(

4

−3

Use

y

=

)

to

y-coordinate dy

the

of

P .

12 =

dy

− 2

dx

calculate

x

Find

x

the

gradient

function

. dx

1

(Remember,

y

{



Introducing

dierential

calculus

=

12x

)

Continued

on

next

page

3.

line

in

dy

At

x

=

12

−4,

=

dy

3

−

=

−

Substitute

the

value

of

x

into

to

2

dx

4

(

dx

4

)

calculate,

The

gradient

of

the

m,

the

value

of

tangent,

the

gradient

at

P .

3

m

= 4

Hence,

the

gradient

of

nor mal

The

the

is

per pendicular

to

the The

gradient

of

a

line

4

m′

normal,

tangent.

=

perpendicular

to

a

3

line

The

equation

of

the

normal

Use

to

the

equation

of

a

straight

whose

gradient

is

line 1

12

y

(y at

=

P (–4,

–3)

−

b)

=

m(x

a)

with

a

=

−4,

m

is

m

4

b

4

( y

−

is

x

− ( −3))

=

(x

=

m

−3,

= 3

− ( −4))

3

3(y

+

3)

3y

+

9

4x

−

3y

=

=

4(x

4x

+

7

Exercise

Find

the

point

P .

+

4)

Y ou

Simplify.

+

16

Rearrange to the form ax + bx + c = 0,

equation

=

0

where

directly

a,

b,

c

∈

ℤ

y

equation

Give

=

2x

nd

of

a

from

the

normal

the

GDC.

6G

of

your

the

normal

answers

in

to

the

the

given

form ax

+

cur ve

by

+

c

2

1

cannot

at

=

the

stated

0

3

;

P(1,

2)

2

( 2, −3 )

4

y

=

3

+

4x

;

P(0.5,

3.5)

2

x

3x

2

3

y

=

− x

;

P

y

=

+

2

x;

P( − 2, 4 )

2

2

5

y

=

7

y



(x

+

2)

(5

−

x);

P(0,

10)

6

y

=

(x

+

2)

;

P(0, 4)

6

4

;

P(2,

2)

8

y



;

P(−1, 6)

2

x

x

8

3 4

y

9



6x



;

P(1, 14)

10

y



x

;

P(−1, 4)

;

P(3, 13.5)

3

x

x

1 11

y



4  2x

9

;



P(0.5, 1)

12

y

 5x



x

Example

2x



2

The

is

gradient

30.

Find

of

the

the

tangent

values

of

a

to

and

the

cur ve

y

=

ax

at

the

point

P

(3, b)

b

Answer

As

dy

=

the

gradient

of

the

tangent

is

2ax dy

dx

given,

find dx

2a(3)

=

30 dy

⇒

a

=

5

When

x

=

3,

=

30

to

find

dx

The

equation

of

the

cur ve

is

2

y

=

5x

b

=

5(3)

2

⇒

b

=

45

Substitute

x

=

3

b.

Chapter





Exercise

6H

2

1

Find

point

the

equation

where

EXAM-STYLE

x

=

of

the

tangent

to

the

cur ve y

=

(x

−

tangent

to

the

cur ve y

=

x (x

4)

at

the

5.

QUESTIONS

2

2

Find

the

the

equation

point

where

x

of

=

the

−

3)

at

–2.

6 3

Find

the

equation

of

the

normal

to

the

cur ve

y



x



at

the

x

point

where

x

=

4.

1 2

4

Find

the

equation

of

the

normal

to

the

cur ve

y



x

at

the

2

x

point

where

x

=

–1.

2

5

Find

the

6

equations

points

Find

the

the

the

where

y

of

=

equations

points

where

y

the

to

the

cur ve y

=

3x

tangents

to

the

cur ve y

=

2x

−

2x

at

8.

of

=

tangents

the

(3

−

x)

at

−20.

3

7

Find

the

the

equation

point

where

of

it

the

normal

intersects

the

to

the

cur ve y

=

7

cur ve y

=

x

–

5x

−

2x

at

x-axis.

3

8

Find

the

9

a

the

equation

point

Find

where

the

y

of

=

value

the

normal

to

the

+

3x

−

2

at

tangent

to

tangent

to

tangent

to

tangent

to

tangent

to

– 6.

of

x

for

which

the

gradient

of

the

2

the

b

cur ve

Find

the

EXAM-STYLE

10

a

Find

y

=

(4x

−

equation

is

3)

of

the

zero.

tangent

at

this

point.

QUESTION

the

value

of

x

for

which

the

gradient

of

the

16 2

the

y

cur ve

=

x

+

is

zero.

x

b

11

a

Find

Find

the

the

equation

value

of

of

x

the

for

tangent

which

at

the

this

point.

gradient

of

the

2

x

the

y

cur ve



 x

 3

is

5.

2

b

12

a

Find

Find

the

the

equation

value

of

of

x

the

for

tangent

which

at

the

this

point.

gradient

of

the

4

the

13

cur ve

y

=

x

+

3x

−

3

is

3.

b

Find

the

equation

of

the

tangent

c

Find

the

equation

of

the

normal

a

Find

the

value

of

x

for

which

at

at

the

this

this

point.

point.

gradient

of

the

3

the

cur ve

y



4x



is

16.

4

x



b

Find

the

equation

of

the

tangent

c

Find

the

equation

of

the

normal

Introducing

dierential

calculus

at

at

this

this

point.

point.

3

There

14

at

are

which

two

the

equations

EXAM-STYLE

points

on

gradient

of

the

the

of

cur ve y

the

tangents

cur ve

to

the

=

is

2

2x

+

equal

cur ve

at

9x

to

−

36.

these

24x

+

Find

5

the

points.

QUESTION

2

The

15

P

gradient

(3,

b)

Find

is

the

of

the

tangent

to

the

cur ve y

=

x

+

kx

at

the

point

cur ve y

=

x

+

kx

at

the

point

cur ve y

=

kx

cur ve y

=

4

cur ve y

=

px

cur ve y

=

px

7.

value

of

k

and

the

value

of

b

2

The

16

P

gradient

(–2,

Find

b)

the

is

of

the

tangent

to

the

1.

value

of

k

and

that

of

b

2

The

17

gradient

point

Find

P

(4,

the

of

b)

the

is

value

tangent

to

the

−

2x

+

3

at

the

2.

of

k

and

that

of

b

3

The

18

gradient

point

Find

P

(–2,

the

of

b)

the

is

value

tangent

to

the

+

kx

−

x

at

the

–5.

of

k

and

that

of

b

2

The

19

gradient

point

Find

P

(2,

the

of

5)

the

is

value

tangent

to

the

+

qx

at

the

7.

of

p

and

that

of

tangent

to

q

2

The

20

gradient

point

Find

.

P

(–3,

the

gradient

f

changes

(x)

rate

of

the

13)

value

Rates

The

of

of

is

of

change

x

and

qx

−

5

at

the

that

of

q

change

f

′(x),

increases.

of

+

6.

p

function,

as

the

f

wi th

of

a

We

function

say

respect

to

that

f

f

(x)

′(x)

is

a

measure

measures

of

how

the

x

In

general,

of

the

change

of

rate

one

dy

➔

For

the

graph

y

=

f

(x),

the

gradient

=

function

f 

(x )

gives

variable

with

respect

dx

the

rate

of

change

of

y

with

respect

to

to

x

another

gradient

Other

variables

can

also

be

used,

for

is

the

function.

example:

dA

if

A

=

f

(t),

then

=

f

′ (t )

measures

the

rate

of

change

of

A

wi th

dt

respect

If

the

the

t

variable

rate

This

to

is

of

an

changes

represents

change

with

impor tant

as

situations

t

time

that

is

are

time,

respect

concept.

passing

to

If

then

dynamic

–

then

to

the

gradient

the time

you

you

that

measure

are

how

that

measures

passes.

applying

situations

function

a

variable

mathematics

are

to

moving.

Chapter





For

example,

day-to-day

if

C

basis)

represents

we

can

the

say

value

that

C

is

of

a

a

car

(measured

function

of

time:

on

C

a

=

f

(t).

dC



Then,



f

t 

represents

the

rate

at

which

the

value

of

the

dt

car

t,

is

changing

the

rate

of

–

it

measures

inflation

Similarly ,

if

s

point

moving

or

the

rate

deflation

represents

the

of

of

change

the

distance

price

of

of

measured

C

with

the

respect

to

car.

from

a

time:

s

fixed If

to

a

object

then

s

is

a

function

of

=

g

v

an

ds

measures

the

rate

of

change

of

this

distance, s,

velocity

of

object,

what

does

represent?

dt

with

the

dv

g ( t )

=

and

is

(t)

dt

respect

to

t

ds

measures

the

of

veloci ty

the

object

at

time

t

dt

Example



3

The

volume

of

water

in

a

container,

V

cm

,

is

given

by

the

formula

2

V

=

300

+

2t

−

t

,

where

t

is

the

time

measured

in

seconds.

dV a

What

does

represent? dt

dV b

What

units

are

used

for

? dt

dV

Find

c

the

value

of

when

t

=

3.

dt

d

What

does

the

answer

to

c

tell

you?

Answers

dV

represents

a

the

rate

of

The

rate

at

which

the

water

is

dt

entering change

water

of

in

the

the

volume

(or

leaving)

the

container.

of

container.

dV 3

is

b

measured

in

cm

per

3

The

volume

is

measured

in

cm

and

dt

3

second

(cm

time

–1

s

is

measured

in

seconds.

).

dV

=

c

2

− 2t

dt How

At

t

=

would

you

3, decide

by

considering

dV dV

=

2

− 2(3)

=

is

−4

negative,

so

dv

whether

dt dt

water d

Since

this

value

is

negative,

the

volume

is

water

is

leaving



Introducing

at

4 cm

container?

dierential

per

second.

calculus

entering

leaving

the

3

container

was

decreasing.

or

the

the

dt

the

Example

A



company

The

mines

company’s

copper,

profit,

P,

where

the

measured

mass

in

of

copper,

millions

of

x,

is

dollars,

measured

depends

in

on

thousands

the

amount

of

tonnes.

of

copper

2

mined.

a

The

Find

profit

P (0)

is

and

given

P (6)

and

Find

the

function

inter pret

P (x)

these

=

2.3x

−

0.05x

–

12

results.

dP

dP b

by

.

What

does

Y ou

represent?

can

graph

any

dx

dx

function

on

the

GDC.

dP c

Find

the

value

of

P

and

when

x

=

20

and

when

x

=

This

25.

could

give

you

dx fur ther d

Inter pret

e

Find

the

answers

to

value

of

x

for

into

the

c

dP

the

insight

problem.

which

=

0.

dx

f

Determine

P

for

this

value

of

x,

and

inter pret

this

value.

Answers

a

P (0)

=

−12;

P (6)

=

0;

this

is

a

loss

there

the

is

of

no

12

million

profit

break-even

and

dollars.

no

Substitute

x

=

0

in

to

P (x).

loss,

point.

dP

=

b

− 0.1x

+

2.3

represents

the

dx

dP

dP

rate

of

change

of

measures

the

the

rate

of

change

of

P

with

dx

dx

profit

as

the

amount

of

copper

respect

mined

to

x

increases.

dP dP c

At

x

=

20,

P

=

14

and

=

Substitute

0.3

x

=

20

and

x

=

25

into

P (x)

and dx

dx

dP

At

x

=

25,

P

=

14.25

and

=

0.2

dx

d

At

both

At

x

points

the

company

is

profitable.

dP

=

20,

>

0

so

a

fur ther

increase

in

At

x

=

20,

P (x)

is

increasing.

At

x

=

25,

P (x)

is

decreasing.

dx

production

will

make

the

company

more

profi table

dP

At

x

=

25,




0,

is

V

=

10

+

6t

+

t

a

Find

the

rate

at

which

the

volume

is

increasing

when t

b

Find

the

rate

at

which

the

volume

is

increasing

when

3

are

5

Water

65 m

is

of

water

flowing

out

in

of

the

pool.

a

tank.

The

is

given

by

depth

of

the

water,

3

y

a

cm,

Find

2

b



at

time

the

the

Introducing

seconds

rate

seconds

Find

t

at

and

time

which

at

at

dierential

3

the

y

depth

=

is

500

−

calculus

the

tank

is

−

decreasing

seconds.

which

4t

empty .

t

at

=

1.

there

t

=

0

2

3t

t

2

6

The

area,

A

cm

,

of

a

blot

of

ink

is

growing

so

that,

after t

seconds,

A





4 a

Find

the

rate

at

which

the

area

is

increasing

after

2

b

Find

the

rate

at

which

the

area

is

increasing

when

2

seconds.

2

7

The

weight

of

oil

in

a

storage

tank,

W,

varies

the

according

area

of

the

blot

is

30

cm

to

135

the

formula

 10 t



where

 4

W

is

measured

in

tonnes

t ²

and

a

t

is

the

time

Find

the

rate

measured

at

which

in

the

hours,

1

weight

≤

is

t

≤

10.

changing

after

2

hours.

dW

Find

b

the

value

of

t

for

which

=

0.

d t

8

The

angle

tur ned

through

by

a

is

9

given

by

the

a

Find

the

rate

b

Find

the

value

A

small

θ

relation

of

=

of

company’s

t

at

−

P,

in

time

when

the

t

=

P

is

profit

can

measured

a

Find

b

Find

c

Find

P (0)

be

in

modeled

thousands

and

P (5)

and

seconds

body

depends

on

2.

changes

the

direction.

amount

x

of

‘product’

3

This

t

t

θ

of

which

profit,

degrees,

2

4t

increase

body , θ

rotating

3

by

the

function P (x)

of

dollars

inter pret

and x

these

is

=

+

−10x

measured

it

makes.

2

in

40x

+

10x

−

15.

tonnes.

results.

dP

dx dP

the

value

of

P

and

when

x

i

=

2

x

ii

=

3.

dx

d

Inter pret

e

Find

your

answers

to

c

dP

the

value

of

x

and

of

P

for

which

=

0.

What

is

the

impor tance

of

this

point?

dx

f (x)

.

Local

maximum

and

minimum

points

10

8

(turning

points) 6

Here

is

the

graph

of

the

4

function

Q 2

1

f

 x   4x

,



x

≠

0 x

0

x

–1

1 –2

The

graph

has

two

branches,

because

the

function

P

is not

–4

at

defined

the

point

x

=

0.

–6

–8

First,

look

at

the

left-hand

branch

of

the

graph,

for

the –10

domain

x




the

it

the

cur ve

‘tur ns’

three

Where

and

gradients

the

star ts

occur

gradient

right-hand

stops

is

branch

and

decreasing.

in

zero

of

increasing

the

is

the

So,

order:

the

changes

as x

positive,

maximum

graph,

with

zero,

point.

the

0.

increases,

cur ve

that

the

e v it is o P

direction

maximum,

the

cur ve

increases.

Q

decreases

is

said

to

to

be

a

the

point

Q.

After

10

Q,

C

8

local minimum point

D

6

Y ou

can

determine

that

Q

is

a

local

minimum

point

because 4 Q

just

before

Q

(for

example,

at

C)

the

gradient

of

the

curve

is

negative

2

and

just

after

Q

(for

example,

at

D)

the

gradient

of

the

curve

is

positive.

0

At

Q

➔

itself,

At

a

the

local

Where

Local

‘tur ns’

At

cur ve

in

zero

local

is

stops

decreasing

increasing.

the

is

order:

the

minimum

x 1

zero.

So,

and

as x

negative,

zero,

minimum

point.

points

known

are

changes

increases,

positive.

Zero

as

or turning points

points

any

local

and

is

cur ve

star ts

occur

gradient

maximum

the

and

gradients

the

stationary

➔

it

the

e vi ti s o P

three

of

minimum,

direction;

the

gradient

stationar y

minimum

–

or

f

tur ning

′(x)

is

point

–

either

local

maximum

or

At

a

stationar y

point,

dy

zero. y

=

f (x)

then

=

0.

dx

To

find

the

coordinates

of

P (the

local

maximum)

and

of

1

Q

(the

local

minimum)

for

the

function

f

 x   4x

,

 x

use

the

fact

that

at

each

of

these

1

f

 x   4x



points f

′(x)

is

zero.

1

,

so



f

 x   4  2

x

x

1

Remember

that

=

x

x

1

Set

f

′(x)

=

0

which

gives

4 



0

2

x 1

1

Adding

:

4



2

2

x

x

2

Multiplying

by

Dividing

4:

x

2

:

4x

=

1

1 2

by

x

 4

1

1

Taking

square

roots:

x

x

or







tur ning

each

⎛

x

into

f

(x)

to

find

the

,

=

1 ⎞

f

⎛ 1

= ⎜

2 ⎝

2

y-coordinate

of

each

can

+

⎟

⎜

⎠

⎝

=

⎟ 2





f

− ⎜

2



Introducing

⎝

2

using

differentiation.

)

1 ⎞

4

1

=

+

⎟

⎜

⎠

⎝

dierential

without

2

⎛

=

GDC,

4

1

(

⎠

1 ⎞

⎛

,

points

a

Chapter

Section

1

x

local

1

⎞

4

local

and

using

See

At

nd

maximum

minimum

point.

1

At

x-value

Y ou

2

2

Substitute

⎟ 2

calculus

⎠

1

(

) 2

−4

6.3.

1,

if

1

⎛

So,

the

coordinates

of

the

tur ning

points

⎜ 2

To

determine

which

is

the

local

maximum

and

look

at

the

graph

of

the

local

minimum

and

−

find

zero

tur ning

and

tur ning

Exercise

solve

the

which

maximum

is

the

simply

the

⎟ 2

by

looking

at

the

⎠

coordinates.

, − 4

the

local

maximum.

⎟ 2

⎝

To

and

local

⎞

⎜

➔

is

, 4

⎝ 1

⎠

is

⎞

function: ⎜

⎛

the

which

minimum

1

⎛

minimum,

is

decide

⎟ 2

⎝

which

cannot

, − 4

⎜

⎠

Y ou

⎞

−

and

⎟

⎝

1

⎛

⎞

, 4

are

⎠

points,

this

first

set

equation.

the

This

gradient

gives

function

equal

the x-coordinate

of

to

the

point.

6J

dy

Find

the

values

of

x

for

= 0 .

which

Verify

your

answers

by

dx

using

your

GDC.

2

2

1

y

=

x

3

y

=

x

5

y

=

x

7

y

=

4x

9

y

=

2x

11

y

=

x

−

6x

2

y

=

12x

+

10x

4

y

=

3x

−

27x

6

y

=

24x

8

y

=

3x

7

10

y

=

5

11

12

y

=

12x

2

−

2x

2

+

15x

3

3

−

2x

3

3

−

3x

−

9x

3

=

2

+

12x

−

+

2

−

−

45x

+

f

2x

(x )

−

=

6x

x



4x

+

7

14

y

=

y



17

+

x



30x

= 8x

you

or

Example

Find

the

−

5x

1

y

20

have

maximum

8

2x



calculate

−

3

+



x



2

then

36x

1

y

18

2

x

Once

+

x



27 x

x



4

y

x

4 16

+

x

19

+

2

x

y

3

6x

3

+

9 17

+

2

1 15

9x

2

3x

3

y

16x

2

3

13

−

2x

found

the

the

x-coordinate

y-coordinate

of

the

of

any

point

tur ning

and

point,

decide

if

it

you

is

can

a

minimum.



coordinates

of

the

tur ning

points

of

the

cur ve ‘Determine

4

y

=

3x

3

−

8x

30x

+

72x

+

5.

Determine

the

these

nature’

means

nature decide

of

the

2

−

whether

the

point

is

a

local

points. maximum

or

a

local

minimum.

Answer

4

y

=

3x

3

−

8x

2

−

30x

+

72x

+

5

dy 3

= 12 x

2

− 24 x

− 60 x

Dif ferentiate.

+ 72

dx

dy

dy 3

12x

2

−

24x

−

60x

+

72

=

0

At

each

tur ning

point

= 0 dx

dx

{

Continued

on

next

Chapter

page





x

=

At

−2,

x

=

x

=

1,

x

=

=

so

3

3(−2)

(−2,

−

so

x

=

−95)

(1,

1,

y

42)

is

a

−

30(−2)

turning

so

x

=

=

is

(3,

3,

with

your

y

−

8(1)

turning

=

−22)

a

−

+

5

=

−95

2

−

30(1)

+

72(1)

+

5

=

Substitute

42

find

point.

3

3(3)

is

72(−2)

3

3(1)

a

+

point.

4

At

equation

2

8(−2)

4

At

this

GDC.

4

y

Solve

3

−2,

8(3)

turning

the

30(3)

+

72(3)

+

5

–

1

3

0

0

0

=

or

decide

the

x

=

0

for

–2

–2

f

′(–10)

=

fill

x