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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
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10
9
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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
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