To find the average rate of change of the function [tex]\( f(x) = 2x^2 - x - 1 \)[/tex] on the interval [tex]\([0, 4]\)[/tex], follow these steps:
1. Evaluate the function at the endpoints of the interval:
- For [tex]\( x = 4 \)[/tex],
[tex]\[
f(4) = 2(4)^2 - 4 - 1
= 2 \cdot 16 - 4 - 1
= 32 - 4 - 1
= 27
\][/tex]
- For [tex]\( x = 0 \)[/tex],
[tex]\[
f(0) = 2(0)^2 - 0 - 1
= 0 - 0 - 1
= -1
\][/tex]
2. Calculate the difference in the function values:
[tex]\[
f(4) - f(0) = 27 - (-1) = 27 + 1 = 28
\][/tex]
3. Calculate the difference in the [tex]\( x \)[/tex]-values:
[tex]\[
4 - 0 = 4
\][/tex]
4. Compute the average rate of change using the formula:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a} = \frac{28}{4} = 7
\][/tex]
Thus, the average rate of change of the function [tex]\( f(x) = 2x^2 - x - 1 \)[/tex] on the interval [tex]\([0, 4]\)[/tex] is [tex]\( 7 \)[/tex].
