1. Finding Rates of Change – Part 1
Slideshow 29, Mathematics
Mr. Richard Sasaki

2. Objectives Recall the meaning of rate of change
Be able to find the rate of change for linear and simple quadratic (square) relationships
Be able to find ranges for such relationships with differing rates of change

3. Meaning What does rate of change mean?
The rate of change is the amount a variable changes over time (or in relation to another variable).
Example
A car contains 10 litres of petrol. It begins to be filled at a constant rate. 40 seconds later, it contains 70 litres. Write down a function for the amount of petrol in the car after 𝑥 seconds.
𝑓 𝑥 =1.5𝑥+10
Amount of petrol 𝑎=
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑓 𝑥 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑓 𝑥
Number of seconds
The rate of change, 𝑎=
1.5

4. Answers
𝑎= 1 4
𝑎=2
𝑎=− 1 2
𝑎= 20 3

5. The rate of change…changes?
For linear relationships, as we were able to see, the rate of change is constant.
The rate of change would change for a relationship.
non-linear
If we look at the rate of change for 𝑦= 𝑥 2 , it’s clear that the rate of change is not constant. 1 4 1 4 1 4
The gradient triangles would all differ.

6. Dropping Things
We’re going to look at mechanics a little. If an object is dropped, what is the formula you use (if we ignore air resistance)?
𝑥= 1 2 𝑎 𝑡 2 ⇒
𝑥≈4.9 𝑡 2
Distance (metres)
Time (seconds)
Acceleration (gravity 𝑚 𝑠 −2 )
9.8
(2 s.f)
Example
Sakura dives off of a platform and it takes her 5 seconds to reach the water. How far did she dive?
𝑥≈4.9 𝑡 2
⇒𝑥≈4.9∙ 5 2
=122.5 𝑚

7. 𝑥=4.9∙ 10 2 =490 𝑚
No, it would take less (as the stone speeds up over time).
𝑥=4.9∙ 4 2 =78.4 𝑚
(1, 4.9)
It would be greater.
44.1=4.9 𝑡 2 ⇒𝑡=3 seconds
4.9=4.9 𝑡 2 ⇒𝑡=1 second
𝑡≈ 𝑥 4.9 𝑜𝑟 10𝑥 7

8. Slopes and Rolling Things
Again, for estimation, we will ignore air resistance and friction to simplify the examples. We will use 𝑦∝ 𝑥 2 in these cases.
Example
A ball rolls down a slope for 7 seconds and is 98 metres in length. If the distance travelled is directly proportional to the square of the time, write an equation for the distance it has fallen and hence, write down how far it travels rolling for a total of 10 seconds.
𝑥=𝑘 𝑡 2
⇒98= 7 2 𝑘
⇒𝑘=2
⇒𝑥=2 𝑡 2
𝑎 looks like acceleration
10 𝑠𝑒𝑐 ⇒𝑥=
200 metres

9. 16=𝑘∙ 4 2 ⇒𝑘=1, ∴𝑥= 𝑡 2
𝑥= 6 2 =36 𝑚
(4, 8)
𝑘=4.9 would be falling so it must be less.
32=𝑘 ∙8 2 ⇒𝑘= 1 2 , ∴𝑥= 𝑡 2 2
(2, 2)
𝑥= =60.5 metres
200= 𝑡 2 2 ⇒𝑡=20 seconds

10. Ranges of Distance
As you know, the speed of something moving changes if its distance travelled is directly proportional to the square of time taken.
For the case 𝑥= 𝑡 2 , as time increases, distance increases.
How is the distance increasing?
The distance travelled is increasing more quickly as time passes by.
Time & Distance increasing evenly…
This is because the speed is increasing (at a constant rate (acceleration)).
Like with these gradient triangles, we can find the average speed for ranges of time. 3 1 1 1

11. Ranges of Distance
As the speed increases, obviously the range of distance will differ depending on the time.
Example
A child sleds down a straight slope 𝑥 metres long and sleds 7.2 metres in 3 seconds. Assuming the distance is directly proportional to the square of time taken, write an equation for the time taken to travel down the slope.
𝑥=𝑘 𝑡 2
⇒7.2= 3 2 𝑘
⇒𝑘=0.8
⇒𝑥=𝑓 𝑡 =0.8 𝑡 2
Write down the distance travelled from 2 to 5 seconds.
𝑥 2~5 =
𝑓 5 −𝑓(2)
=0.8∙ 5 2 −0.8∙ 2 2
=20−3.2
=16.8𝑚

12. 𝑥=2 𝑡 2
𝑥 1~2 =2 ∙2 2 −2∙ 1 2 =8−2=6 𝑚
𝑥 5~7 =2 ∙7 2 −2∙ 5 2 =98−50=48 𝑚
𝑥=𝑘 𝑡 2 ⇒
6.4=𝑘∙ 4 2 ⇒
𝑘=0.4
, ∴𝑥=0.4 𝑡 2
𝑥 4~8 =0.4∙ 8 2 −0.4∙ 4 2 =
19.2 𝑚 4 12 48
192
𝑥 is a continuous variable (not discrete). There are no gaps between valid answers. 2 16 14
398