(1 point)
Find the average value of g(t)=3t+5 over the interval [2,8]. Round your answer to the nearest hundredth.
the average value of g(t) over [a,b] is ∫[a,b] g(t) dx / (b-a)
so now you have
∫[2,8] (3t+5) dt / (8-2)
= 120/6
= 20
LOL
the value in the middle. It is linear.
g(5) = 20
To find the average value of a function over an interval, we need to calculate the definite integral of the function over that interval and divide it by the length of the interval.
In this case, we are given the function g(t) = 3t + 5 over the interval [2, 8].
Step 1: Calculate the definite integral of g(t) over the interval [2, 8].
To find the integral of 3t + 5, we can use the power rule of integration. The integral of t^n is (1/n+1)t^(n+1) + C, where C is the constant of integration. Applying this rule to g(t) = 3t + 5, we get:
∫ (3t + 5) dt = (3/2)t^2 + 5t + C
To evaluate this definite integral over the interval [2, 8], we subtract the value of the integral at the lower limit from the value at the upper limit:
[(3/2)(8)^2 + 5(8) + C] - [(3/2)(2)^2 + 5(2) + C]
Simplifying this expression gives us:
[(3/2)(64) + 40 + C] - [(3/2)(4) + 10 + C]
= (96 + 40 + C) - (6 + 10 + C)
= 136 - 16
= 120
So, the definite integral of g(t) over the interval [2, 8] is 120.
Step 2: Calculate the length of the interval.
The length of the interval [2, 8] can be found by subtracting the lower limit from the upper limit:
8 - 2 = 6
So, the length of the interval [2, 8] is 6.
Step 3: Divide the definite integral by the length of the interval.
To find the average value of g(t) over the interval [2, 8], we divide the definite integral (120) by the length of the interval (6):
Average value = 120 / 6 = 20
Therefore, the average value of g(t) = 3t + 5 over the interval [2, 8] is 20.
g(2) = 11
g(8) = 29
average value of g(t) = (29+11)/2 = 20