Question

what is the average rate of change from x = -1 to x = 0? (1 point) use the following graph of the function f(x)=3x^4 - x^3+3x^2 + x - 3 to answer this question.

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 $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 0$, and $f(x)=3x^{4}-x^{3}+3x^{2}+x - 3$.

Step2: Calculate $f(-1)$

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

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

\]

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$:
\[
f(0)=3(0)^{4}-(0)^{3}+3(0)^{2}+(0)-3=-3
\]

Step4: Calculate the average rate of change

\[

$$\begin{align*} \frac{f(0)-f(-1)}{0-(-1)}&=\frac{-3 - 3}{0 + 1}\\ &=\frac{-6}{1}\\ &=-6 \end{align*}$$

\]

Answer:

$-6$