Advanced · 15 min
Average and instantaneous change
Calculate average rates and prepare for the idea of a derivative.
Compare changes over an interval
An average rate of change divides the change in output by the change in input. On a graph, it is the slope of the line joining two points.
Worked example
For f(x) = x² from x = 1 to x = 3: average rate = (9 − 1)/(3 − 1) = 4.
Look at shorter intervals
An instantaneous rate describes change at one input. Average rates over shorter nearby intervals can approach it; a derivative formalises this limiting process.
Worked example
For f(x) = x² near x = 2: rates from 2 to 2.1 and from 2 to 2.01 are 4.1 and 4.01, approaching 4.
Explore a shrinking interval
- For f(x) = x², calculate average rates from x = 3 to x = 3.1 and to x = 3.01.
- Compare the results and predict the instantaneous rate at x = 3.