Find the average rate of change of y = x2 + 2x between x = 2 and x = 5.
- At x = 2: y = 4 + 4 = 8. At x = 5: y = 25 + 10 = 35.
- Change in y = 35 − 8 = 27. Change in x = 5 − 2 = 3.
- Gradient of the chord = 27 ÷ 3 = 9.
Answer: 9
Compound measures and rates of change
Maths · Ratio, proportion and rates of change. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A compound measure is made from two other measures. Speed is distance per unit of time, density is mass per unit of volume and pressure is force per unit of area. The unit tells you the calculation: km/h means kilometres divided by hours. Unit prices work the same way and let you compare value.
On a graph, the gradient is a rate of change: how much y changes for each 1 unit of x. On a distance-time graph the gradient is speed.
On a curve the rate keeps changing. The gradient of a chord between two points gives the average rate. The gradient of a tangent at one point gives the instantaneous rate there.
Rules to remember
- speed = distance ÷ time
- density = mass ÷ volume
- pressure = force ÷ area (N/m2)
- Gradient = change in y ÷ change in x = (y2 − y1)/(x2 − x1)
- Average rate of change between two points = gradient of the chord joining them.
- Instantaneous rate of change at a point = gradient of the tangent at that point.
- A negative gradient means the quantity is decreasing.
Common mistake
Using minutes as a decimal of an hour: 1 hour 30 minutes is 1.5 hours, not 1.30 hours. Change minutes to hours by dividing by 60.
Worked example
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