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
