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
- = []
- = []
- Average = ∫20f(x)dx
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
- Average = ∫90dx
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
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