Tutorial Examples of Average and Instantaneous Rate of Change


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 the 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 the 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


Example:
Use the formula for the area of a circle to find,
(a) the average rate of which the area of a circle changes with as the radius increases form to .
(b) the instantaneous rate at which the area of a circle changes with when .

Solution:

(a) For Average Velocity:
We have

Put

Again Put

The average rate of change from to is

(b) For Instantaneous Rate of Change:
We have

Put

Now putting

The instantaneous rate of change at the point is

.


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