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)

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Accepted Answer

# 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:
$-1$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall average rate - of - change formula

Step2: Calculate $f(-1)$

Step3: Calculate $f(1)$

Step4: Calculate the average rate of change

Answer: