# Examples of Average and Instantaneous Rate of Change

Example:

Let $y = {x^2} - 2$
(a) Find the average rate of change of $y$ with respect to $x$ over the interval $[2,5]$.
(b) Find the instantaneous rate of change of $y$ with respect to $x$ at point $x = 4$.

Solution:

(a) For Average Rate of Change:
We have
$y = f(x) = {x^2} - 2$

Put $x = 2$
$\therefore f(2) = {(2)^2} - 2 = 4 - 2 = 2$

Again put $x = 5$
$\therefore f(5) = {(5)^2} - 2 = 25 - 2 = 23$

The average rate of change over the interval $[2,5]$ is
$\frac{{f(5) - f(2)}}{{5 - 2}} = \frac{{23 - 2}}{3} = \frac{{21}}{3} = 7$

(b) For Instantaneous Rate of Change:
We have
$y = f(x) = {x^2} - 2$

Put $x = 4$
$\therefore f(4) = {(4)^2} - 2 = 16 - 2 = 14$

Now, putting $x = {x_1}$ then
$\therefore f({x_1}) = {x_1}^2 - 2$

The instantaneous rate of change at point $x = 4$ is

Example:
A particle moves on a line away from its initial position so that after $t$ seconds it is $S = 2{t^2} - t$ feet from its initial position.
(a) Find the average velocity of the particle over the interval$[1,3]$.
(b) Find the instantaneous velocity at $t = 2$.

Solution:

(a) For Average Velocity:
We have
$S(t) = 2{t^2} - t$

Put $t = 1$
$\therefore S(1) = 2{(1)^2} - 1 = 2 - 1 = 1$

Again put $t = 3$
$\therefore S(3) = 2{(3)^2} - 3 = 18 - 3 = 15$

The average velocity over the interval $[1,3]$ is
${V_{ave}} = \frac{{S(3) - S(1)}}{{3 - 1}} = \frac{{15 - 1}}{2} = \frac{{14}}{2} = 7{\text{ }}ft/Sec$

(b) For Instantaneous Velocity:
We have
$S(t) = 2{t^2} - t$

Put $t = 2$
$\therefore S(2) = 2{(2)^2} - 2 = 8 - 2 = 6$

Now putting $t = {t_1}$
$\therefore S({t_1}) = 2{t_1}^2 - {t_1}$

The instantaneous velocity at $t = 2$ is