# Examples of Average and Instantaneous Rate of Change

__Example__:

Let

**(a)** Find the average rate of change of with respect to over the interval .

**(b)** Find the instantaneous rate of change of with respect to at point .

__Solution__:

**(a) For Average Rate of Change:**

We have

Put

Again put

The average rate of change over the interval is

**(b) For Instantaneous Rate of Change:**

We have

Put

Now, putting then

The instantaneous rate of change at point is

__Example__:

A particle moves on a line away from its initial position so that after seconds it is feet from its initial position.

**(a)** Find the average velocity of the particle over the interval.

**(b)** Find the instantaneous velocity at .

__Solution__:

**(a) For Average Velocity:**

We have

Put

Again put

The average velocity over the interval is

**(b) For Instantaneous Velocity:**

We have

Put

Now putting

The instantaneous velocity at is