Mean Value Theorem
For function f differentiable in the open interval (a, b) and continuous on the closed interval [a, b], there exists a point c between a and b that satisifies:
f (b) − f (a) = f '(c)(b − a)
Example:
Example:
For f (x) = x2 in the interval [1, 7], determine the value of c, which satisifies the Mean Value Theorem.
Solution:
f (x) = x2
f '(x) = 2x
a = 1
b = 7
f (a) = f (1) =12 = 1
f (b) = f (7) = 72 = 49
Using the Mean Value Theorem:
49 − 1 = 2c (7 − 1)
48 = 2c (6)
c = 4
f '(x) = 2x
a = 1
b = 7
f (a) = f (1) =12 = 1
f (b) = f (7) = 72 = 49
Using the Mean Value Theorem:
49 − 1 = 2c (7 − 1)
48 = 2c (6)
c = 4