What is the average value of the function f(x)=−x−5 on the interval [2, 5] ? Round your answer to the nearest hundredth. (1 point) Responses 25.5 25.5 8.5 8.5 5.1 5.1 −25.5 - 25 . 5 −8.5
The average value of a function on an interval can be found by taking the definite integral of the function over the interval and dividing by the length of the interval.
The definite integral of f(x) = -x - 5 can be found as follows:
∫[-x - 5]dx = -x^2/2 - 5x + C
To evaluate the definite integral over the interval [2, 5], we can subtract the value of the antiderivative at the lower limit (5) from the value of the antiderivative at the upper limit (2):
(-2^2/2 - 5(2) + C) - (-5^2/2 - 5(5) + C)
= (-4/2 - 10 + C) - (-25/2 - 25 + C)
= (-2 - 10 + C) - (-25/2 - 25 + C)
= (-12 + C) - (-25/2 + C)
= -12 + C + 25/2 - C
= 25/2 - 12.
The length of the interval [2, 5] is 5 - 2 = 3.
Dividing the definite integral by the length of the interval:
(25/2 - 12) / 3 = (25/2 - 12) * (1/3)
= (25/2 - 12) / 3
= (25/2 - 12) * (1/3)
= (25/6 - 12/3)
= (25/6 - 4)
= 25/6 - 24/6
= 1/6.
Rounded to the nearest hundredth, the average value of the function f(x) = -x - 5 on the interval [2, 5] is approximately 0.17.
The correct response is: 0.17