the study of change - projectmaths.ie€¦ · rate of change = slope = = =10 run 1 rise run gym...
TRANSCRIPT
![Page 1: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/1.jpg)
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.
![Page 2: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/2.jpg)
Things have changed.........
![Page 3: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/3.jpg)
The Mathematics of
Change
![Page 4: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/4.jpg)
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)
![Page 5: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/5.jpg)
(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
![Page 6: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/6.jpg)
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
![Page 7: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/7.jpg)
Can you see Rates of Change?
![Page 8: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/8.jpg)
Rates of Change Around Us
![Page 9: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/9.jpg)
World in Motion
![Page 10: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/10.jpg)
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.
![Page 11: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/11.jpg)
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
![Page 12: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/12.jpg)
Container A Container B
What about if container B has some water in it already?
What would your graph look like now?
h h
![Page 13: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/13.jpg)
Container C Container D
Now, sketch separate graphs for these two containers.
How are they different to what you have drawn already?
![Page 14: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/14.jpg)
Cylinders Cone
What’s the difference?
![Page 15: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/15.jpg)
100 m
Constant Speed in the Real World
![Page 16: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/16.jpg)
The World’s Fastest Man
![Page 17: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/17.jpg)
Student Activity
![Page 18: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/18.jpg)
Student Activity
![Page 19: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/19.jpg)
How are we going to find the
instantaneous rate of change?
![Page 20: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/20.jpg)
Calculus The study of change – how things
change and how quickly they change
Newton Leibniz
![Page 21: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/21.jpg)
![Page 22: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/22.jpg)
( ) ( )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
![Page 23: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/23.jpg)
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
![Page 24: The Study of Change - projectmaths.ie€¦ · Rate 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](https://reader034.vdocument.in/reader034/viewer/2022043012/5fab90807fac660a334f100e/html5/thumbnails/24.jpg)
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