section 4.2: rolle’s theorem and the “mvt” for derivatives · (section 4.2: rolle’s theorem...

(Section 4.2: Rolle’s Theorem and the “MVT” for Derivatives) 4.2.1

SECTION 4.2: ROLLE’S THEOREM and THE “MVT” FOR DERIVATIVES

LEARNING OBJECTIVES

• Know and understand Rolle’s Theorem and when it applies. • Find critical numbers (CNs) related to the conclusion of Rolle’s Theorem. • Know and understand the “MVT” for Derivatives and when it applies. • Find values related to the conclusion of the MVT for Derivatives. • Compare the EVT, the MVT for Derivatives, and Rolle’s Theorem. PART A: THE IMAGE OF THE “ELASTIC ROPE”

• Imagine two people at x = a and x = b a < b( ) holding an elastic rope in the xy-plane. They hold the ends of the rope at the same height; the rope can vary in length. They wiggle the rope vertically in such a way that the Vertical Line Test (VLT) is never violated.

• Take a photo. If the rope has equation y = f x( ) , where Dom f( ) = a, b⎡⎣ ⎤⎦ , then f

is guaranteed to have a critical number (“CN”) in the x-interval a, b( ) . There could be more than one. • If we forbid such points as corners and cusps, where we’d lose differentiability, then Rolle’s Theorem guarantees the existence of a horizontal tangent line to the graph of y = f x( ) . Again, there could be more than one (see Warning 1).

c is a CN c1 and c2 are CNs

′f c( ) = 0 ′f c1( ) = 0 and ′f c2( ) = 0

(Section 4.2: Rolle’s Theorem and the “MVT” for Derivatives) 4.2.2 PART B: ROLLE’S THEOREM

Rolle’s Theorem

(Assume a < b .)

If, for a function f ,

1) f is continuous on a, b⎡⎣ ⎤⎦ ,

2) f is differentiable on a, b( ) , and

3) f a( ) = f b( ) ,

then ∃c ∈ a,b( ) such that ′f c( ) = 0 .

• That is, ′f c( ) = 0 for some c in a, b( ) .

• Such a value of c is a CN of f .

• There is a horizontal tangent line to the graph of

y = f x( ) at c, f c( )( ) .

• WARNING 1: Like the EVT, Rolle’s Theorem is an existence theorem, not a uniqueness theorem. It guarantees (“gives sufficient conditions for”) the existence of such a CN, but it does not guarantee that there is only one.

• For Condition 2), why not require differentiability on a, b⎡⎣ ⎤⎦ ? That would prevent us from applying Rolle’s Theorem to the case below. We would like to apply Rolle’s Theorem to such half-circular graphs; we have differentiability on a, b( ) but not on a, b⎡⎣ ⎤⎦ .


Example 1 (Rolle’s Theorem)

Let f x( ) = x4 −18x2 on the restricted domain −4, 4⎡⎣ ⎤⎦ .

Check the conditions of Rolle’s Theorem.

Can we apply Rolle’s Theorem to f on −4, 4⎡⎣ ⎤⎦ ? Yes, because:

• f is polynomial on −4, 4⎡⎣ ⎤⎦ , so f is continuous on −4, 4⎡⎣ ⎤⎦ and

differentiable on −4, 4( ) . Conditions 1) and 2) check out.

• f −4( ) = −32 and f 4( ) = −32 , so f −4( ) = f 4( ) and Condition 3) checks out. We could have observed that f is a polynomial, even function and that −4 and 4 are opposite x-values, so f −4( ) = f 4( ) .

Find the CNs related to the conclusion of Rolle’s Theorem.

′f x( ) = 4x3 − 36x

Rolle’s Theorem applies, so ′f c( ) = 0 for some c in −4, 4( ) . Find all such values for c; these will be CNs of f . We will solve

′f x( ) = 0 on the x-interval −4, 4( ) .

′f x( ) = 0

4x3 − 36x = 0

• WARNING 2: Do not divide both sides by x without considering the case x = 0 . Factoring is safer.

4x x2 − 9( ) = 0

4x x + 3( ) x − 3( ) = 0

x = 0 or x = −3 or x = 3

• WARNING 3: Interval check. Check to see which of these numbers are in −4, 4( ) . If we had gotten x = 10 , for example, it

would have to be rejected. Here, all three numbers are in −4, 4( ) .

• The desired c-values are 0, −3 , and 3.


Here is the graph of y = x4 −18x2 on the x-interval −4, 4⎡⎣ ⎤⎦ :

(Axes are scaled differently.)

• The fact that f is even leads to symmetries about the y-axis. §

PART C: MEAN VALUE THEOREM (“MVT”) FOR DERIVATIVES

Example 2 (Motivating the “MVT” for Derivatives; Revisiting Section 3.1, Example 6)

A car is driven due north 100 miles during a two-hour trip.

• Let t = the time (in hours) elapsed since the beginning of the trip.

• Let y = s t( ) , where s is the position function for the car (in miles). s gives the signed distance of the car from the starting position.

• Let s 0( ) = 0 , meaning that y = 0 corresponds to the starting position.

Then, s 2( ) = 100 (miles). Then, the average velocity of the car on 0, 2⎡⎣ ⎤⎦ is:

s 2( )− s 0( )2− 0

= 100− 02

= 50 mileshour

or mihr

or mph⎛⎝⎜

⎞⎠⎟


The average velocity is 50 mph on 0, 2⎡⎣ ⎤⎦ for the three different s functions below. The slope of the orange secant line segment is 50 mph.


The velocity is constant (50 mph).

(We are not requiring the car to slow down to a stop at the end.)

The velocity is increasing; the car is accelerating.

The car “breaks the rules,” backtracks, and goes south.

• The Mean Value Theorem (“MVT”) for Derivatives implies that the car must be going exactly 50 mph at some t = c in 0, 2( ) .

• At that time, ′s c( ) = 50 mph . The red tangent line at c, s c( )( ) is parallel to the orange secant line segment; they have the same slope. That is, the instantaneous velocity (or rate of change of s with respect to t) at t = c is equal to the average velocity (or rate of change) on 0, 2⎡⎣ ⎤⎦ . The theorem is a key link between the realms of Calculus and Precalculus. • The theorem applies to all three scenarios above, because each s function is continuous on 0, 2[ ] and differentiable on 0, 2( ) . §


Mean Value Theorem (“MVT”) for Derivatives

(Assume a < b .)

If, for a function f ,

1) f is continuous on a, b⎡⎣ ⎤⎦ , and

2) f is differentiable on a, b( )

then ∃c ∈ a,b( ) such that ′f c( ) = f b( )− f a( )

b− a.

• That is, the slope of the tangent line to the graph of y = f x( )

at c, f c( )( ) is equal to the slope of the secant line [segment] on

a, b⎡⎣ ⎤⎦ for some c in a, b( ) . The lines are parallel.

MVT applies to f on a, b⎡⎣ ⎤⎦ ; Both Rolle’s Theorem and MVT

Rolle’s Theorem does not. apply to f on a, b⎡⎣ ⎤⎦ ;

• WARNING 4: Like the EVT and Rolle’s Theorem, the MVT for Derivatives is an existence theorem, not a uniqueness theorem. There might be more than one real value c in a,b( ) that satisfies the conclusion. • WARNING 5: c is not a CN of f , unless the slope of the secant line is 0.

• The MVT shares Conditions 1) and 2) with Rolle’s Theorem, but we now remove Condition 3) f a( ) = f b( ) .

• If it is true that f a( ) = f b( ) , then Rolle’s Theorem also applies, the slope of the secant line is 0, and c is a CN of f . Rolle’s Theorem is a special case of the MVT. The MVT is a generalization of Rolle’s Theorem. In fact, Rolle’s Theorem is used to prove the MVT, as we will see.


Example 3 (MVT for Derivatives)

Let f x( ) = x4 − x on the restricted domain 2, 5⎡⎣ ⎤⎦ .

Check the conditions of the MVT for Derivatives.

Can we apply the MVT to f on 2, 5⎡⎣ ⎤⎦ ? Yes, because:

• f is polynomial on 2, 5⎡⎣ ⎤⎦ , so f is continuous on 2, 5⎡⎣ ⎤⎦ and

differentiable on 2, 5( ) . Conditions 1) and 2) check out.

Find the c-values related to the conclusion of the MVT.

′f x( ) = 4x3 −1

• The MVT applies, so ′f c( ) =

f b( )− f a( )b− a

=f 5( )− f 2( )

5− 2

for some c in 2, 5( ) .

• Evaluate the difference quotient

f 5( )− f 2( )5− 2

. This is the slope of

the relevant secant line and the average rate of change of f on 2, 5⎡⎣ ⎤⎦ .

f 5( )− f 2( )5− 2

= 620−143

= 202

• Solve ′f x( ) = 202 on the x-interval 2, 5( ) .

′f x( ) = 202

• WARNING 6: The right-hand side is 202, not 0.

4x3 −1= 2024x3 = 203

x3 = 2034

x = 2034

3

x ≈ 3.702


• WARNING 7: Cube root calculations. Entering “ (203÷ 4) ^1/ 3= ” in a calculator will not work. Instead, “ (203÷ 4) ^ (1/ 3) = ” will work.

• WARNING 8: Interval check. Check to see that 3.702 is in 2, 5( ) .

It is. By the MVT, there had to be at least one solution in 2, 5( ) . If x = 10 , say, were our sole “solution,” an error must have been made. • The desired c-value is about 3.702.

Here is the graph of y = x4 − x on the x-interval 2, 5⎡⎣ ⎤⎦ :


• WARNING 9: Slopes are visually distorted because of the different scaling of the axes. The slopes of the red tangent line and orange secant line segment are both 202. §


Example 4 (An Application of the MVT for Derivatives)

• Assume a function f is differentiable everywhere on .

• Pick any two distinct points on the graph of y = f x( ) . Name their x-coordinates a and b, where a < b . Basically, you choose a and b!

• The MVT for Derivatives will apply to f on a, b⎡⎣ ⎤⎦ . There must be at least

one value c ∈ a, b( ) such that the tangent line at c, f c( )( ) is parallel to the

secant line [segment] on a, b⎡⎣ ⎤⎦ . y

a b

PART D: COMPARING THEOREMS

Assume a < b ; f is a function.

“Nickname” Theorem Requirements on f Guarantees

EVT Extreme

Value Theorem

continuous on a, b⎡⎣ ⎤⎦

f has an absolute maximum and an absolute minimum on

a, b⎡⎣ ⎤⎦ .

MVT (for Derivatives)

Mean Value Theorem for Derivatives

… and differentiable on a, b( )

∃c ∈ a,b( ) such that

′f c( ) = f b( )− f a( )

b− a.

Rolle Rolle’s Theorem

… and

f a( ) = f b( ) ∃c ∈ a,b( ) such that

′f c( ) = 0 .

(Section 4.2: Rolle’s Theorem and the “MVT” for Derivatives) 4.2.10 PART E: “PROOF IDEAS” FOR THE MVT FOR DERIVATIVES

Rolle’s Theorem is used to prove the MVT for Derivatives.

• For now, let’s say a function is “nice” ⇔ it is continuous on a, b⎡⎣ ⎤⎦ and

differentiable on a, b( ) . • Assume that f is a nice function.

• Let the function g correspond to the secant line on a, b⎡⎣ ⎤⎦ . g is nice.

f a( ) = g a( ) , and f b( ) = g b( ) .

• Let h be the “gap function” between f and g. That is, let h x( ) = f x( )− g x( ) . Since f and g are nice, then so is h. Also,

h a( ) = f a( )− g a( ) = 0

h b( ) = f b( )− g b( ) = 0

• Since h is nice, and h a( ) = h b( ) , then Rolle’s Theorem applies to h on a, b⎡⎣ ⎤⎦ .

Therefore, ∃c ∈ a,b( ) such that ′h c( ) = 0 .

• By linearity of differentiation, ′h x( ) = ′f x( )− ′g x( ) .

• Thus, ∃c ∈ a,b( ) such that…

′h c( ) = 0

′f c( )− ′g c( ) = 0

′f c( ) = ′g c( )

• ′g c( ) =

g b( )− g a( )b− a

=f b( )− f a( )

b− a, the slope of the secant line.


• Thus, ∃c ∈ a,b( ) such that…

′f c( ) = ′g c( )′f c( ) = f b( )− f a( )

b− a

Q.E.D. • Note that c is a CN for the h “gap function,” so it is no surprise that we get a local maximum for the gap between the f and g functions at x = c in our figure. Locally, the vertical distance between the graphs of y = f x( ) and y = g x( ) is maximized at x = c .

section 4.2: rolle’s theorem and the “mvt” for derivatives · (section 4.2: rolle’s theorem...

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