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Find the average rate of change of the given function on the interval [tex]\([0,4]\)[/tex].

[tex]\[ f(x) = 2x^2 - x - 1 \][/tex]

Enter your answer as a reduced improper fraction, if necessary.

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Sagot :

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].