A bit of help please?

Question

In regards to differential calculus, I'm wanting to know what is the difference between the average rate of change and instantaneous rate of change.....and what would be a practical example that would show this ??

Answer 1

The average rate of change is the rate of change over an interval.

The instantaneous rate of change is the rate of change at a point.

For example... take the function f(x)=x^2.

The rate of change over the interval [0,3] is [f(3)-f(0)]/3=(9-0)/3=3.

The instantaneous rate of change at a point is the same thing as the derivative at a point.

f'(x)=2x
f'(3)=6

So the average rate of change from 0 to 3 is 3, but the instantaneous rate of change at x=3 is 6.