1. 1. Be able to apply The Mean Value Theorem to various functions.
Objectives:
1. Be able to apply The Mean Value Theorem to various functions.
Critical Vocabulary:
The Mean Value Theorem
Warm Up: Determine if Rolle’s Theorem can be applied. If it
can, find all values of c such that f’(c) = 0.
f(x) = x2 - 5x + 4; [1, 4]

2. If f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that
Secant Line
Tangent Line
Tangent Line
Secant Line
(b, f(b))
(a, f(a)) a c b

3. Example 1: Given f(x) = 5 - (4/x), find all values of c in
the open interval (1,4) such that
1st: Find f’(c):
f(4) = 5 - 1
f(1) = 5 - 4
f’(c) = 1
f(4) = 4
f(1) = 1
2nd: Set f’(x) = f’(c) to determine the x values determined by
the Mean Value Theorem
f(x) = 5 – 4x-1
f(2) = 5 - 2
f’(x) = 4x-2
f(2) = 3
f’(x) = 4/x2
(2, 3) is the point where the tangent is parallel to the secant

4. (-1/3, 14/27) is the point where the tangent is parallel to the secant
Example 2 (page 326 #24): Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that
f(x) = x(x2 - x - 2) ;[-1, 1]
1st: Secant Line
f(1) = -2
f’(c) = -1
f(-1) = 0
2nd: Tangent Line
f(-1/3) = -1/3(1/9 + 1/3 - 2)
f(x) = x3 – x2 - 2x
3x2 – 2x - 2 = -1
f(-1/3) = 14/27
f(x) = 3x2 – 2x - 2
3x2 – 2x - 1 = 0
(3x + 1)(x - 1) = 0
(-1/3, 14/27) is the point where the tangent is parallel to the secant
3x + 1 = 0
x - 1 = 0
x = -1/ x = 1

5. Page #19-27 odds, 33