Question

How does instantaneous rate of change differ from average rate of change?

Answer 1

Instantaneous rate of change is essentially the value of the derivative at a point; in other words, it is the slope of the line tangent to that point. Average rate of change is the slope of the secant line passing through two points; it gives the average rate of change across an interval.

Below is a graph showing a function, `f(x)`, and the secant line across an interval `[2, 4]`. The slope of this secant line, which is

`(Deltay)/(Deltax) = (f(4) - f(2))/(4 - 2)`

is the average rate of change of `f(x)`.

Below is a graph showing the function `f(x) = x^2`, as well as the line tangent at `x = 2`. The slope of this line is:

`dy/dx = f'(2) = 2*2 = 4`,

and it is the instantaneous rate of change at the point `(2,4)`.