Mean Value of a Function
What is the mean value of a function?
- The mean value of a function may be thought of as the ‘average’ value of a function over a given interval
- For a function f(x), the mean value of the function over the interval [a, b] is given by
- Note that the mean value is simply a real number – it is not a function
- The mean value depends on the interval chosen – if the interval [a, b] changes, then the mean value may change as well
- Because is a real number, the graph of is a horizontal line
- This gives a geometrical interpretation of the mean value of a function over a given interval
- If A is the area bounded by the curve y = f(x), the x-axis and the lines x = a and x = b, then the rectangle with its base on the interval [a, b] and with height also has area A
- i.e.
What are the properties of the mean value of a function?
- If is the mean value of a function f(x) over the interval [a, b], and k is a real constant, then:
- f(x) + k has mean value over the interval [a, b]
- kf(x) has mean value over the interval [a, b]
- -f(x) has mean value over the interval [a, b]
- If then the area that is above the x-axis and under the curve is equal to the area that is below the x-axis and above the curve
Worked example
Let be the function defined by .
a)
Find the exact mean value of over the interval .
b)
Write down the exact mean value of each of the following functions over the interval :
(i)
(ii)
(iii)