Question

1. (03.04) calculate the average rate of change for the function ( f(x) = x^4 + 3x^3 - 5x^2 + 2x - 2 ), from ( x = -1 ) to ( x = 1 ). (2 points)

options: -1, 1, -5, 5

Explanation:

Step1: Recall the average rate of change formula

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

Step2: Calculate \( f(-1) \)

Substitute \( x=-1 \) into \( f(x)=x^{4}+3x^{3}-5x^{2}+2x - 2 \):
\[

$$\begin{align*} f(-1)&=(-1)^{4}+3(-1)^{3}-5(-1)^{2}+2(-1)-2\\ &=1+3(-1)-5(1)+(-2)-2\\ &=1 - 3-5 - 2-2\\ &=1-(3 + 5+2 + 2)\\ &=1 - 12\\ &=- 11 \end{align*}$$

\]

Step3: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x)=x^{4}+3x^{3}-5x^{2}+2x - 2 \):
\[

$$\begin{align*} f(1)&=(1)^{4}+3(1)^{3}-5(1)^{2}+2(1)-2\\ &=1 + 3-5 + 2-2\\ &=(1 + 3+2)-(5 + 2)\\ &=6 - 7\\ &=-1 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \( \frac{f(1)-f(-1)}{1-(-1)} \), substitute \( f(1)=-1 \) and \( f(-1)=-11 \):
\[

$$\begin{align*} \frac{f(1)-f(-1)}{1-(-1)}&=\frac{-1-(-11)}{1 + 1}\\ &=\frac{-1 + 11}{2}\\ &=\frac{10}{2}\\ &=5 \end{align*}$$

\]

Answer:

5