# 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 =∫
^{b}_{a}f(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 = ∫
^{2}_{0}f(x)dx- = ∫
^{2}_{0}dx - = []
^{2}_{0} - = []
- = []

- = ∫

- Average = ∫

**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 = ∫
^{9}_{0}dx- = ∫
^{9}_{0}dx - = () |
^{9}_{0} - = ( - 0)
- = ( x 27)
- =2

- = ∫

- Average = ∫

**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:
- ∫
^{20}_{0}dx- =∫
^{20}_{0}dx - = |
^{20}_{0} - =24(20) -
- = 480 - 160
- = 320

- =∫

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