Find the average value of the function on the given interval:

f(x) = e^x/5, [0,5]

you are looking at dividing the definite integral of f(x) from 0 to 5 by the length of the interval, namely (5-0)=5.

∫(1/5)exdx
=(1/5)[ex] from 0 to 5
=(1/5)(e5-e0)
=(1/5)(148.413-1)
=29.483
The average over the interval
=29.483/5
=5.8965

To find the average value of a function on a given interval, you need to compute the definite integral of the function over that interval, and then divide the result by the length of the interval.

In this case, the function is f(x) = e^x/5, and the interval is [0,5].

Step 1: Compute the definite integral of the function over the interval [0,5].

The definite integral of f(x) over the interval [0,5] can be computed as follows:

∫[0,5] (e^x/5) dx

To evaluate this integral, we can use the power rule for integration, which states that the integral of e^x with respect to x is equal to e^x.

So, the integral becomes:

(1/5) ∫[0,5] (e^x) dx

Now, we integrate e^x with respect to x:

(1/5) * e^x | [0,5]

Plugging in the upper and lower limits of integration:

(1/5) * (e^5 - e^0)

Step 2: Calculate the length of the interval [0,5].

The length of the interval [0,5] can be calculated by subtracting the lower limit from the upper limit:

5 - 0 = 5

Step 3: Divide the value from step 1 by the length from step 2.

(1/5) * (e^5 - e^0) / 5

Simplifying the expression:

(1/25) * (e^5 - e^0)

So, the average value of the function f(x) = e^x/5 on the interval [0,5] is (1/25) * (e^5 - e^0).