Finding Rates of Change – Part 1

Slideshow 29, MathematicsMr. Richard Sasaki, Room 307

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

MeaningWhat does rate of change mean?The rate of change is the amount a variable changes over time (or in relation to another variable).We looked at rate of change in Grade 8 so we’ll have a review first of a similar example.ExampleA 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 seconds.

𝑓 (𝑥 )=1.5𝑥+10

Number of seconds

Amount of petrol

The rate of change, 1.5

𝑎=¿𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑜𝑓 𝑓 (𝑥 )𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑜𝑓 𝑥

Answers

𝑎=2 𝑎=14

𝑎=−12

𝑎=203

The rate of change…changes?For linear relationships, as we were able to see, the rate of change is constant.

41 4

1 41

The rate of change would change for a relationship.non-linearIf we look at the rate of change for , it’s clear that the rate of change is not constant.

The gradient triangles would all differ.

Dropping ThingsWe’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=12𝑎𝑡 2

Time (seconds)Distance (metres)Acceleration (gravity - )9.8

(2 s.f)

⇒𝑥≈ 4.9𝑡 2

ExampleMao dives off of a platform and it takes her 5 seconds to reach the water. How far did she dive?

𝑥≈ 4.9𝑡 2⇒𝑥 ≈4.9 ∙52¿122.5𝑚

Answers1. 490m

No, it would take less (as the stone speeds up over time)2. 78.4m It would be greater.3. 3 seconds4. 1 second

Slopes and Rolling ThingsAgain, for estimation, we will ignore air resistance and friction to simplify the examples. These will be similar to square proportion examples at the start of Chapter 4.ExampleA 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=𝑘𝑡 2

looks like acceleration⇒ 98=72𝑘⇒𝑘=2⇒𝑥=2𝑡 2

10s metres

Answers1. , 36 metres2. 4.9 would be falling so it must be less3. , 60.5 metres, 20 seconds

Ranges of DistanceAs you know, the speed of something moving changes if its distance travelled is directly proportional to the square of time taken.

For the case , as time increases, distance increases.

Time &

Distan

ce in

creas

ing eve

nly…How is the distance increasing?The distance travelled is increasing more quickly as time passes by.

111

3 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.

Ranges of DistanceAs the speed increases, obviously the range of distance will differ depending on the time.ExampleA child sleds down a straight slope 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 the time taken to travel down the slope.

𝑥=𝑘𝑡 2⇒7.2=32𝑘⇒𝑘=0.8⇒𝑥= 𝑓 (𝑡 )=0.8 𝑡 2Write down the distance travelled from 2 to 5 seconds.𝑥2 5=¿𝑓 (5 )− 𝑓 (2)¿0.8 ∙52−0.8∙22¿20−3.2¿16.8𝑚

Answers

1. 4, 12, 48, 192, is a continuous variable (not discrete)2. 2, 16, 14, 398