Question

For the function f(x) = x23, the average rate of change to the nearest hundredth over the interval 2 d x d 4 is

Answer 1

Given:

Consider the function is:

$f(x)=\dfrac{x^2}{3}$

To find:

The average rate of change over the interval 2 ≤ x ≤ 4.

Solution:

We have,

$f(x)=\dfrac{x^2}{3}$

At $x=2$,

$f(2)=\dfrac{2^2}{3}$

$f(2)=\dfrac{4}{3}$

At $x=4$,

$f(4)=\dfrac{4^2}{3}$

$f(4)=\dfrac{16}{3}$

The average rate of change of a function f(x) over the interval [a,b] is:

$m=\dfrac{f(b)-f(a)}{b-a}$

So, the average rate of change over the interval 2 ≤ x ≤ 4 is:

$m=\dfrac{f(4)-f(2)}{4-2}$

$m=\dfrac{\dfrac{16}{3}-\dfrac{4}{3}}{2}$

$m=\dfrac{\dfrac{16-4}{3}}{2}$

On further simplification, we get

$m=\dfrac{12}{3\times 2}$

$m=\dfrac{12}{6}$

$m=2$

Therefore, the average rate of change over the interval 2 ≤ x ≤ 4 is 2.