Mean Value of a Function (Edexcel A Level Further Maths: Core Pure): Revision Note

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

  $\overline{\mathrm{f}}=\frac{1}{b-a}{\int }_a^b\mathrm{f}\left(x\right)dx$
  

  • Note that the mean value $\overline{\mathrm{f}}$ 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 $\overline{\mathrm{f}}$ is a real number, the graph of  $y=\overline{\mathrm{f}}$  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. $(b-a)\overline{\mathrm{f}}={\int }_a^b\mathrm{f}\left(x\right)dx$

What are the properties of the mean value of a function?

• If $\overline{\mathrm{f}}$ 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 $\overline{\mathrm{f}}+k$ over the interval [a, b]

  • kf(x) has mean value $k\overline{\mathrm{f}}$ over the interval [a, b]

  • -f(x) has mean value $-\overline{\mathrm{f}}$ over the interval [a, b]

• If $\overline{\mathrm{f}}=0$ 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