Question
Find the average value of the function f(x)=4−x3 on the interval [2,4] . (1 point) Responses −52 −26 − 13 22 54
Answers
GPT 3.5
To find the average value of a function, we need to evaluate the definite integral of the function over the given interval and then divide that value by the length of the interval.
The definite integral of f(x) = 4 - x^3 over the interval [2,4] is:
∫[2,4] (4 - x^3) dx
To evaluate this integral, we can use the power rule for integration:
∫x^n dx = (1/(n+1)) * x^(n+1) + C
Applying the power rule to our integral:
∫[2,4] (4 - x^3) dx = [4x - (1/4)x^4] evaluated from 2 to 4
= (4(4) - (1/4)(4^4)) - (4(2) - (1/4)(2^4))
= (16 - (1/4)(256)) - (8 - (1/4)(16))
= 16 - 64 + 8 - 4
= -52
Therefore, the average value of the function f(x) = 4 - x^3 on the interval [2,4] is -52.
The definite integral of f(x) = 4 - x^3 over the interval [2,4] is:
∫[2,4] (4 - x^3) dx
To evaluate this integral, we can use the power rule for integration:
∫x^n dx = (1/(n+1)) * x^(n+1) + C
Applying the power rule to our integral:
∫[2,4] (4 - x^3) dx = [4x - (1/4)x^4] evaluated from 2 to 4
= (4(4) - (1/4)(4^4)) - (4(2) - (1/4)(2^4))
= (16 - (1/4)(256)) - (8 - (1/4)(16))
= 16 - 64 + 8 - 4
= -52
Therefore, the average value of the function f(x) = 4 - x^3 on the interval [2,4] is -52.