Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
§3.2–The Mean Value Theorem
Tom Lewis
Spring Semester
2014
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Outline
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Theorem (Rolle’s Theorem)
Let f be a function that satisfies the following three hypotheses:
1. f is continuous on the closed interval [a, b];
2. f is differentiable on the open interval (a, b);
3. f (a) = f (b) = 0
There there exists at least one point c ∈ (a, b) such that f 0 (c) = 0.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Rolle’s Theorem in pictures
Here is a sketch of a function satisfying the conditions of Rolle’s
Theorem. In this example there are three points, c, d, and e,
satisfying the conclusion of the theorem; the theorem tells us that
there must be at least one such point.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
An application of Rolle’s theorem
Problem
Show that f (x ) = x 5 + 2x 3 + 5x + 9 has only one real root.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Theorem (The Mean Value Theorem)
Let f be a function that satisfies the following two hypotheses:
1. f is continuous on the closed interval [a, b];
2. f is differentiable on the open interval (a, b).
There there exists at least one point c ∈ (a, b) such that
f 0 (c) =
f (b) − f (a)
b−a
or, equivalently,
f (b) − f (a) = f 0 (c)(b − a).
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
A geometric interpretation of the MVT
At the point (c, f (c)), the slope of the tangent line is equal to the
slope of the secant line passing through (a, f (a)) and (b, f (b)).
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
A physical interpretation of the MVT
• A man travels from Greenville, SC to Knoxville, TN (180
miles) in 3 hours for an average velocity of 60 mph; thus, at
some point in his trip his speedometer must have read 60 mph.
• The Mean Value Theorem connects average velocity and
instantaneous velocity. If s is a displacement function, then
the according to the MVT
s(b) − s(a)
= s 0 (c)
| {z }
−a }
| b {z
Avg.Vel.
Inst .Vel.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Problem
Demonstrate the conclusion of the Mean Value Theorem for
f (x ) = x 2 on the interval [−1, 3].
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Problem
Let f be a continuous function on [1, 5] such that f (1) = 6 and
7 6 f 0 (x ) 6 9 for x ∈ (1, 5). Show that 34 6 f (5) 6 42.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Theorem
If f 0 (x ) = 0 for all x in an interval (a, b), then f is constant on
(a, b).
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Theorem
If f 0 (x ) = g 0 (x ) for all x in an interval (a, b), then there exists a
constant K such that f (x ) = g(x ) + K for all x ∈ (a, b).
Note
We say that f and g differ by a constant.
Rolle’s Theorem
The Mean Value Theorem
Some consequences of the Mean Value Theorem
Problem
Use calculus to show that sin2 (x ) and − cos2 (x ) differ by a
constant.