3. (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

Question

Accepted Answer

# 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

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the average rate of change formula

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

Step3: Calculate \( f(1) \)

Step4: Calculate the average rate of change

Answer: