Average Value

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Definition : Rolle's Theorem

f is a continuous function defined on the closed interval [a,b], is differentiable on the open interval (a,b), and f(a)=f(b)

Then, there exists a value "c" in the interval (a,b)such that f'(c)=0


If f is a continuous function defined on the interval [a,b]

The average of f(x) is : Average =baf(x)dx

Example 1 :

Find the average value of f(x) = for 0 greater than equal to x and x less than equal to 2.
Solution :
Average = 20f(x)dx
=20dx
= []20
= []
= []

Example 2 :

Find the average value of f(x) = given the interval 0 greater than equal to x and x less than equal to 9.
Solution :
Average = 90dx
=90dx
= () |90
= ( - 0)
= ( x 27)
=2

Example 3 :

Find the consumer's surplus for p = 50 - 0.06, the demand curve, with x = 20.
Solution :
The price is:
B = 50 - 0.06
= 50 - 24 = 26
The consumer's surplus is:
200dx
=∫200dx
= |200
=24(20) -
= 480 - 160
= 320