Question

calculate the average rate of change for the function f(x)=3x^4 - 2x^3 - 5x^2 + x + 5, from x=-1 to x=1. (1 point)

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 $f(x)=3x^{4}-2x^{3}-5x^{2}+x + 5$.

Step2: Calculate $f(-1)$

Substitute $x=-1$ into $f(x)$:
\[

$$\begin{align*} f(-1)&=3(-1)^{4}-2(-1)^{3}-5(-1)^{2}+(-1)+5\\ &=3\times1-2\times(-1)-5\times1 - 1+5\\ &=3 + 2-5 - 1+5\\ &=4 \end{align*}$$

\]

Step3: Calculate $f(1)$

Substitute $x = 1$ into $f(x)$:
\[

$$\begin{align*} f(1)&=3(1)^{4}-2(1)^{3}-5(1)^{2}+1 + 5\\ &=3\times1-2\times1-5\times1+1 + 5\\ &=3-2-5 + 1+5\\ &=2 \end{align*}$$

\]

Step4: Calculate the average rate of change

Use the formula $\frac{f(b)-f(a)}{b - a}$ with $a=-1$, $b = 1$, $f(-1)=4$, and $f(1)=2$:
\[

$$\begin{align*} \frac{f(1)-f(-1)}{1-(-1)}&=\frac{2 - 4}{1+1}\\ &=\frac{-2}{2}\\ &=-1 \end{align*}$$

\]

Answer:

$-1$