calculate the average rate of change of f(x) = 3x² + 2x + 1 for 2 ≤ x ≤ 4.\na 10\nb 20\nc 37\nd 40

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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 2$, $b = 4$, and $f(x)=3x^{2}+2x + 1$.

Step2: Calculate $f(4)$

Substitute $x = 4$ into $f(x)$: $f(4)=3\times4^{2}+2\times4 + 1=3\times16+8 + 1=48+8 + 1=57$.

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=3\times2^{2}+2\times2 + 1=3\times4+4 + 1=12+4 + 1=17$.

Step4: Calculate the average rate of change

$\frac{f(4)-f(2)}{4 - 2}=\frac{57 - 17}{2}=\frac{40}{2}=20$.

Answer:

B. 20