Question

find the average rate of change of $f(x)=x^{3}+2x^{2}+3x - 1$ from - 1 to 1

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 1$, and $f(x)=x^{3}+2x^{2}+3x - 1$.

Step2: Calculate $f(1)$

Substitute $x = 1$ into $f(x)$:
$f(1)=1^{3}+2\times1^{2}+3\times1 - 1=1 + 2+3 - 1=5$.

Step3: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$:
$f(-1)=(-1)^{3}+2\times(-1)^{2}+3\times(-1)-1=-1 + 2-3 - 1=-3$.

Step4: Calculate the average rate of change

Use the formula $\frac{f(1)-f(-1)}{1-(-1)}$.
$\frac{5-(-3)}{1 - (-1)}=\frac{5 + 3}{2}=\frac{8}{2}=4$.

Answer:

$4$