Question

1. (03.04 mc) calculate the average rate of change for the function f(x)= - 2x^4+x^3 - 3x^2+x - 4, from x = - 1 to x = 0. (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, $a=-1$, $b = 0$, and $f(x)=-2x^{4}+x^{3}-3x^{2}+x - 4$.

Step2: Calculate $f(-1)$

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

$$\begin{align*} f(-1)&=-2(-1)^{4}+(-1)^{3}-3(-1)^{2}+(-1)-4\\ &=-2\times1 - 1-3\times1 - 1-4\\ &=-2-1 - 3-1 - 4\\ &=-11 \end{align*}$$

\]

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$:
\[

$$\begin{align*} f(0)&=-2(0)^{4}+(0)^{3}-3(0)^{2}+(0)-4\\ &=-4 \end{align*}$$

\]

Step4: Calculate the average rate of change

\[

$$\begin{align*} \frac{f(0)-f(-1)}{0-(-1)}&=\frac{-4-(-11)}{0 + 1}\\ &=\frac{-4 + 11}{1}\\ &=7 \end{align*}$$

\]

Answer:

7