Find the average value of the function f over the interval [0, 8].
f(x) = 5e^-x
that would be
(∫[0,8] 5e^-x dx)/(8-0)
= 5/8 (1-e^-8)
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 then divide the result by the width of the interval.
First, let's calculate the definite integral of f(x) = 5e^(-x) over the interval [0, 8]. The indefinite integral of this function is -5e^(-x). We can now find the definite integral using the Fundamental Theorem of Calculus:
∫(0 to 8) 5e^(-x) dx = [-5e^(-x)](0 to 8)
= -5e^(-8) - (-5e^(-0))
= -5e^(-8) + 5
Now, we need to divide this result by the width of the interval, which is 8 - 0 = 8.
Average value = (-5e^(-8) + 5)/8
Please note that the value is approximate since e is an irrational number.
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 then divide it by the width of the interval. In this case, we want to find the average value of the function f(x) = 5e^(-x) over the interval [0, 8].
Step 1: Calculate the definite integral of f(x) over the interval [0, 8].
∫[0,8] 5e^(-x) dx
Step 2: Integrate the function with respect to x.
∫[0,8] 5e^(-x) dx = [-5e^(-x)] from 0 to 8
Step 3: Evaluate the integral at the upper and lower bounds.
[-5e^(-8)] - [-5e^(-0)]
Step 4: Simplify the expression.
[-5e^(-8)] - [-5e^0] = -5e^(-8) + 5
Step 5: Divide the result by the width of the interval.
The width of the interval [0, 8] is 8 - 0 = 8.
Step 6: Calculate the average value.
Average value = (-5e^(-8) + 5) / 8
And that's how you can find the average value of the function f(x) = 5e^(-x) over the interval [0, 8].