4.2A Rolle’s Theorem*

Special case of Mean Value TheoremExample of existence theorem

(guarantees the existence of some x = c but does not give value of c)

*Published in 1691… Rolle at one point thought calculus was “a collection of ingenious fallacies”… he later ‘saw the light’…

Rolle’s Theorem

Let f be continuous on [a,b] and differentiable on (a,b). If f(a)=f(b), then a number c (a,b) f’(c)=0.

( means “there exists”, means “in the”, and means “such that”.)

Graphical Representation of Rolle’s Theorem

Ex. Determine if Rolle’s Thm applies. If it does, find x=c guaranteed by it for f(x) = 3x – x3 on [-2,1].

• Solution:– Verify f(x) is continuous on [-2,1]. Yes,b/c polynomial.– f(-2) = 2; f(1) = 2. Therefore f(-2) = f(1). Thus

Rolle’s applies.–

– 1 is not solution b/c not in (-2,1). – therefore c = -1.

2

2

( ) 3 3

0 3(1 )

1, 1

f x x

x

x

Ex. Explain why y = tan(x) does not have a value x=c on [0, π] |f’(c)=0 even though tan(0) = tan(π)=0.

• Solution: y = tan x is not continuous on [0,] b/c vertical asymptote at x = /2.

4.2 B Mean Value Theorem (MVT) for Derivatives

• An existence theorem• Example of MVT in “everyday” context:• You just won States and are flying to Disney World.

The trip is about 800 miles and takes 2 hours. Therefore, the average speed of the plane is 800 miles/2 hours, or 400 mph. There are times when the plane’s speed will be greater than 400 mph and times when it will be less than 400 mph. Thus there must be at least one time when the plane’s instantaneous speed equals the average speed of 400 mph.

If f (x) is continuous over [a,b] and differentiable over (a,b), then

at some point c between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

The Mean Value Theorem only applies over a closed interval.

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

y

x0

A

B

a b

Slope of secant line:

f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to secant line.

c

Ways to think of MVT

• The x=c value where the derivative equals the average rate of change.

• The x=c value where the instantaneous rate of change equals the average rate of change.

• The x=c value where the secant line and tangent line are parallel because their slopes are equal. (a geometric or graphical interpretation)

Examples

• Determine if MVT applies to f(x) = x(2/3) on [0,1]. If so, find the x = c guaranteed by it.

• Find x = c such that the tangent line at x = c is parallel to the secant line for the function, f(x) = x + 1/x on [1,2].

• Why does MVT not apply to f(x) = 1/x - |x-2| on [0, 3]?