Question

Average rate of change help!!

Answer 1

Given that the function is $f(x)=x^{3}-6 x^{2}+4 x+7$

We need to determine the average rate of change over the interval $0 \leq x \leq 5$

Value of f(x) when x = 0:

Substituting x = 0 in the function $f(x)=x^{3}-6 x^{2}+4 x+7$, we have;

$f(0)=(0)^{3}-6 (0)^{2}+4 (0)+7$

$f(0)=0-0+0+7$

$f(0)=7$

Thus, the value of f(0) is 7.

Value of f(x) when x = 5:

Substituting x = 5 in the function $f(x)=x^{3}-6 x^{2}+4 x+7$, we have;

$f(5)=(5)^{3}-6 (5)^{2}+4 (5)+7$

$f(5)=125-150+20+7$

$f(5)=2$

Thus, the value of f(5) is 2.

Average rate of change:

The average rate of change can be determined using the formula,

$Rate \ of \ change=\frac{f(b)-f(a)}{b-a}$

where $a=0$ and $b=5$

Thus, we have;

$Rate \ of \ change=\frac{f(5)-f(0)}{5-0}$

$Rate \ of \ change=\frac{2-7}{5-0}$

$Rate \ of \ change=\frac{-5}{5}$

$Rate \ of \ change=-1$

Thus, the average rate of change over the interval $0 \leq x \leq 5$ is -1.