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