rate of change for the function f(x)=x^3 - 2x^2 + x + 1 over the intervals 0, 2 and 2, 4. rate of change…

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate for the interval $[0,2]$

First, find $f(0)$ and $f(2)$. $f(0)=0^{3}-2\times0^{2}+0 + 1=1$. $f(2)=2^{3}-2\times2^{2}+2 + 1=8-8 + 2+1=3$. Then, the average rate of change over $[0,2]$ is $\frac{f(2)-f(0)}{2 - 0}=\frac{3 - 1}{2}=\frac{2}{2}=1$.

Step3: Calculate for the interval $[2,4]$

Find $f(4)$: $f(4)=4^{3}-2\times4^{2}+4 + 1=64-32 + 4+1=37$. The average rate of change over $[2,4]$ is $\frac{f(4)-f(2)}{4 - 2}=\frac{37-3}{2}=\frac{34}{2}=17$.

Answer:

The average rate of change over the interval $[0,2]$ is $1$. The average rate of change over the interval $[2,4]$ is $17$.