Calculating the Average Value of a Function on an Interval: An Example using the f(x) = 2x + 3 Function

average value of f(x) on [a, b]

To find the average value of a function f(x) on the interval [a, b], you need to evaluate the definite integral of f(x) over that interval and then divide it by the length of the interval

To find the average value of a function f(x) on the interval [a, b], you need to evaluate the definite integral of f(x) over that interval and then divide it by the length of the interval. The average value can be calculated using the following formula:

Average value = (1 / (b – a)) * ∫[a to b] f(x) dx

Let’s go through an example to demonstrate how to find the average value of a function on a specific interval:

Example: Find the average value of the function f(x) = 2x + 3 on the interval [1, 4].

1. Calculate the definite integral of f(x) from a to b:
∫[1 to 4] (2x + 3) dx = [x^2 + 3x] evaluated from 1 to 4
= [(4^2 + 3*4) – (1^2 + 3*1)]
= [16 + 12 – 1 – 3]
= 24

2. Determine the length of the interval:
The length of the interval [1, 4] is given by (b – a):
(4 – 1) = 3

3. Plug the values into the average value formula:
Average value = (1 / (4 – 1)) * 24
= (1 / 3) * 24
= 8

Therefore, the average value of the function f(x) = 2x + 3 on the interval [1, 4] is 8.

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