IDENTIFYING RATES

CAN YOU THINK OF ANY RATES THAT YOU HAVE SEEN OR

HEARD IN YOUR EVERYDAY LIVES?

Rates describe how much one quantity changes with respect to another.

EXAMPLES

A car travels 259 kilometres using 35 litres of petrol. Express this rate in km/L.

TRY THESE 2 EXAMPLES

Example 1 Example 2

Which of the following represent a rate?a 20 m/sb 75 cents per packetc $13

Answer 1 Answer 2

CONSTANT RATE OF CHANGE

if petrol is $1.60 per litre, then every litre of petrol purchased at this rate always costs $1.60. This means 10 litres of petrol would cost $16.00 and 100 litres of petrol would cost $160.00. Calculating the gradient from the graph

W H E N T H E R AT E O F C H A N G E O F O N E Q UA N T I T Y W I T H R E S P E C T T O A N O T H E R

D O E S N O T A LT E R , T H E R AT E I S C O N S TA N T.

EXAMPLE

SOLUTION

EXAMPLE 2

SOLUTION

VARIABLE RATES

IF A RATE IS NOT CONSTANT ( IS CHANGING), THEN IT MUST BE A

VARIABLE RATE.

EXAMPLE

SOLUTION

WHAT IS AN AVERAGE RATE?

If a rate is variable, it is sometimes useful to know the average rate of change over a

specified interval.

EXAMPLE 1

SOLUTION

EXAMPLE 2

SOLUTION

EXAMPLE 3

SOLUTION

SOLUTION

INSTANTANEOUS RATES

WHAT IS AN INSTANTANEOUS RATE?

If a rate is variable, it is often useful to know the rate of change at any given time or point, that is, the instantaneous rate of change.

For example, a police radar gun is designed to give an instantaneous reading of a vehicle's speed. This enables the police to make an immediate decision as to whether a car is breaking the speed limit or not.

CALCULATING INSTANTANEOUS

RATES:1. drawing a tangent to the curve at

the point in question

2. calculating the gradient of the tangent over an appropriate interval (that is, between two points whose coordinates are easily identified).Note: The gradient of the curve at a point, P, is defined as the gradient of the tangent at

that point.

EXAMPLE 1

a Use the following graph to find the gradient of the

tangent at the point where L = 10.

b Hence, find the instantaneous rate of

change of weight, W, with respect to length, L, when

L = 10.

SOLUTION

EXAMPLE 3

SOLUTION

SOULTION

RATES OF CHANGE OF POLYNOMIALS

RATES OF CHANGE OF POLYNOMIALS

We have seen that instantaneous rates of change can be found from a graph by finding the gradient of the tangent drawn through the point in question. The following method uses a series of approximations to find the gradient.

EXAMPLE

EXAMPLE

SOLUTION