Purpose

(1)To recap on rate of change and

distinguish between average and

instantaneous rates of change.

(2)To introduce the idea of the

derivative.

Things have changed.........

The Mathematics of

Change

21

19

18

20

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0 1 5 6 7 8 9 10 4 3 2

Run=3

Rise=6

𝑆𝑙𝑜𝑝𝑒 =𝑦2 − 𝑦1𝑥2 − 𝑥1

= 2

2

4

Slope

(m)

(0, 50)

(1, 60)

(2, 70)

(3, 80)

(4, 90)

(5, 100)

(6, 110)

Number of Visits (v)

Tota

l Co

st (

c)

Rise 10Rate of change = Slope = = = 10

Run 1

Rise

Run

Gym Membership

Total Cost c = 50 +10v

Days Elapsed

Amount (C)

0 110

1 105

2 100

3 95

4 90

Problem Solving Owen has 22 five-cent coins in his wallet. Each day, he decides to take one coin from his wallet and put it into a money box.

Rise -5Slope = = = -5

Run 1

Can you see Rates of Change?

Rates of Change Around Us

World in Motion

Graphs of Real Life Situations

Depth of

Water

Time

1

2

5

6

7

3

4

Discuss how the depth of water changes over time as Archie takes his bath. Create a word bank of the terms which arise during your discussion.

Using the same diagram, draw a rough sketch of how the height of water in these containers changes with respect to time as they are being filled up. They both are of the same height and are being filled at the same rate.

Container A Container B

h h

Container A Container B

What about if container B has some water in it already?

What would your graph look like now?

h h

Container C Container D

Now, sketch separate graphs for these two containers.

How are they different to what you have drawn already?

Cylinders Cone

What’s the difference?

100 m

Constant Speed in the Real World

The World’s Fastest Man

Student Activity

Student Activity

How are we going to find the

instantaneous rate of change?

Calculus The study of change – how things

change and how quickly they change

Newton Leibniz

( ) ( )Average Rate of Change =

f a h f a

h

0

( ) ( )Instantaneous Rate of Change = lim

h

f a h f a

h

Finding the Derivative using First Principles

Deriving the Slope Function

Using First Principles

2

2

2 2

2 2 2

2

( ) 6

( ) ( ) ( ) 6

( ) 2 6

( )( ) ( ) 2 6 6

( ) ( ) 2

( )

f x x x

f x h x h x h

f x h x xh h x h

f x h f x x xh h x h x x

f x h f x xh h h

f x h f

0

( )2 1

( ) ( )lim 2 1

( ) 2 1

h

xx h

h

f x h f xx

h

f x x

The Central Ideas

Slope as average rate of change

of a function

Successive secants to approximate

the Instant

The Derivative will do this for us

m

aneous rate

ost efficie

of c

ntly.

hange