Lesson 19: The Mean Value Theorem
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Transcript of Lesson 19: The Mean Value Theorem
![Page 1: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/1.jpg)
Section 4.2The Mean Value Theorem
V63.0121.002.2010Su, Calculus I
New York University
June 8, 2010
Announcements
I Exams not graded yet
I Assignment 4 is on the website
I Quiz 3 on Thursday covering 3.3, 3.4, 3.5, 3.7
![Page 2: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/2.jpg)
Announcements
I Exams not graded yet
I Assignment 4 is on thewebsite
I Quiz 3 on Thursday covering3.3, 3.4, 3.5, 3.7
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 2 / 28
![Page 3: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/3.jpg)
Objectives
I Understand and be able toexplain the statement ofRolle’s Theorem.
I Understand and be able toexplain the statement of theMean Value Theorem.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 3 / 28
![Page 4: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/4.jpg)
Outline
Rolle’s Theorem
The Mean Value TheoremApplications
Why the MVT is the MITCFunctions with derivatives that are zeroMVT and differentiability
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 4 / 28
![Page 5: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/5.jpg)
Heuristic Motivation for Rolle’s Theorem
If you bike up a hill, then back down, at some point your elevation wasstationary.
Image credit: SpringSunV63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 5 / 28
![Page 6: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/6.jpg)
Mathematical Statement of Rolle’s Theorem
Theorem (Rolle’s Theorem)
Let f be continuous on [a, b]and differentiable on (a, b).Suppose f (a) = f (b). Thenthere exists a point c in (a, b)such that f ′(c) = 0.
a b
c
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 6 / 28
![Page 7: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/7.jpg)
Mathematical Statement of Rolle’s Theorem
Theorem (Rolle’s Theorem)
Let f be continuous on [a, b]and differentiable on (a, b).Suppose f (a) = f (b). Thenthere exists a point c in (a, b)such that f ′(c) = 0.
a b
c
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 6 / 28
![Page 8: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/8.jpg)
Flowchart proof of Rolle’s Theorem
Let c bethe max pt
Let d bethe min pt
endpointsare maxand min
is c anendpoint?
is d anendpoint?
f isconstanton [a, b]
f ′(c) = 0 f ′(d) = 0f ′(x) ≡ 0on (a, b)
no no
yes yes
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 8 / 28
![Page 9: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/9.jpg)
Outline
Rolle’s Theorem
The Mean Value TheoremApplications
Why the MVT is the MITCFunctions with derivatives that are zeroMVT and differentiability
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 9 / 28
![Page 10: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/10.jpg)
Heuristic Motivation for The Mean Value Theorem
If you drive between points A and B, at some time your speedometerreading was the same as your average speed over the drive.
Image credit: ClintJCLV63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 10 / 28
![Page 11: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/11.jpg)
The Mean Value Theorem
Theorem (The Mean Value Theorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c).
a
b
c
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 11 / 28
![Page 12: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/12.jpg)
The Mean Value Theorem
Theorem (The Mean Value Theorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c).
a
b
c
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 11 / 28
![Page 13: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/13.jpg)
The Mean Value Theorem
Theorem (The Mean Value Theorem)
Let f be continuous on [a, b]and differentiable on (a, b).Then there exists a point c in(a, b) such that
f (b)− f (a)
b − a= f ′(c).
a
b
c
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 11 / 28
![Page 14: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/14.jpg)
Rolle vs. MVT
f ′(c) = 0f (b)− f (a)
b − a= f ′(c)
a b
c
a
b
c
If the x-axis is skewed the pictures look the same.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 12 / 28
![Page 15: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/15.jpg)
Rolle vs. MVT
f ′(c) = 0f (b)− f (a)
b − a= f ′(c)
a b
c
a
b
c
If the x-axis is skewed the pictures look the same.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 12 / 28
![Page 16: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/16.jpg)
Proof of the Mean Value Theorem
Proof.
The line connecting (a, f (a)) and (b, f (b)) has equation
y − f (a) =f (b)− f (a)
b − a(x − a)
Apply Rolle’s Theorem to the function
g(x) = f (x)− f (a)− f (b)− f (a)
b − a(x − a).
Then g is continuous on [a, b] and differentiable on (a, b) since f is. Alsog(a) = 0 and g(b) = 0 (check both) So by Rolle’s Theorem there exists apoint c in (a, b) such that
0 = g ′(c) = f ′(c)− f (b)− f (a)
b − a.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 13 / 28
![Page 17: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/17.jpg)
Proof of the Mean Value Theorem
Proof.
The line connecting (a, f (a)) and (b, f (b)) has equation
y − f (a) =f (b)− f (a)
b − a(x − a)
Apply Rolle’s Theorem to the function
g(x) = f (x)− f (a)− f (b)− f (a)
b − a(x − a).
Then g is continuous on [a, b] and differentiable on (a, b) since f is. Alsog(a) = 0 and g(b) = 0 (check both) So by Rolle’s Theorem there exists apoint c in (a, b) such that
0 = g ′(c) = f ′(c)− f (b)− f (a)
b − a.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 13 / 28
![Page 18: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/18.jpg)
Proof of the Mean Value Theorem
Proof.
The line connecting (a, f (a)) and (b, f (b)) has equation
y − f (a) =f (b)− f (a)
b − a(x − a)
Apply Rolle’s Theorem to the function
g(x) = f (x)− f (a)− f (b)− f (a)
b − a(x − a).
Then g is continuous on [a, b] and differentiable on (a, b) since f is.
Alsog(a) = 0 and g(b) = 0 (check both) So by Rolle’s Theorem there exists apoint c in (a, b) such that
0 = g ′(c) = f ′(c)− f (b)− f (a)
b − a.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 13 / 28
![Page 19: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/19.jpg)
Proof of the Mean Value Theorem
Proof.
The line connecting (a, f (a)) and (b, f (b)) has equation
y − f (a) =f (b)− f (a)
b − a(x − a)
Apply Rolle’s Theorem to the function
g(x) = f (x)− f (a)− f (b)− f (a)
b − a(x − a).
Then g is continuous on [a, b] and differentiable on (a, b) since f is. Alsog(a) = 0 and g(b) = 0 (check both)
So by Rolle’s Theorem there exists apoint c in (a, b) such that
0 = g ′(c) = f ′(c)− f (b)− f (a)
b − a.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 13 / 28
![Page 20: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/20.jpg)
Proof of the Mean Value Theorem
Proof.
The line connecting (a, f (a)) and (b, f (b)) has equation
y − f (a) =f (b)− f (a)
b − a(x − a)
Apply Rolle’s Theorem to the function
g(x) = f (x)− f (a)− f (b)− f (a)
b − a(x − a).
Then g is continuous on [a, b] and differentiable on (a, b) since f is. Alsog(a) = 0 and g(b) = 0 (check both) So by Rolle’s Theorem there exists apoint c in (a, b) such that
0 = g ′(c) = f ′(c)− f (b)− f (a)
b − a.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 13 / 28
![Page 21: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/21.jpg)
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3 − x = 100 in theinterval [4, 5].
Solution
I By the Intermediate Value Theorem, the function f (x) = x3 − x musttake the value 100 at some point on c in (4, 5).
I If there were two points c1 and c2 with f (c1) = f (c2) = 100, thensomewhere between them would be a point c3 between them withf ′(c3) = 0.
I However, f ′(x) = 3x2 − 1, which is positive all along (4, 5). So this isimpossible.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 14 / 28
![Page 22: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/22.jpg)
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3 − x = 100 in theinterval [4, 5].
Solution
I By the Intermediate Value Theorem, the function f (x) = x3 − x musttake the value 100 at some point on c in (4, 5).
I If there were two points c1 and c2 with f (c1) = f (c2) = 100, thensomewhere between them would be a point c3 between them withf ′(c3) = 0.
I However, f ′(x) = 3x2 − 1, which is positive all along (4, 5). So this isimpossible.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 14 / 28
![Page 23: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/23.jpg)
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3 − x = 100 in theinterval [4, 5].
Solution
I By the Intermediate Value Theorem, the function f (x) = x3 − x musttake the value 100 at some point on c in (4, 5).
I If there were two points c1 and c2 with f (c1) = f (c2) = 100, thensomewhere between them would be a point c3 between them withf ′(c3) = 0.
I However, f ′(x) = 3x2 − 1, which is positive all along (4, 5). So this isimpossible.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 14 / 28
![Page 24: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/24.jpg)
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3 − x = 100 in theinterval [4, 5].
Solution
I By the Intermediate Value Theorem, the function f (x) = x3 − x musttake the value 100 at some point on c in (4, 5).
I If there were two points c1 and c2 with f (c1) = f (c2) = 100, thensomewhere between them would be a point c3 between them withf ′(c3) = 0.
I However, f ′(x) = 3x2 − 1, which is positive all along (4, 5). So this isimpossible.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 14 / 28
![Page 25: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/25.jpg)
Using the MVT to estimate
Example
We know that |sin x | ≤ 1 for all x . Show that |sin x | ≤ |x |.
Solution
Apply the MVT to the function f (t) = sin t on [0, x ]. We get
sin x − sin 0
x − 0= cos(c)
for some c in (0, x). Since |cos(c)| ≤ 1, we get∣∣∣∣sin x
x
∣∣∣∣ ≤ 1 =⇒ |sin x | ≤ |x |
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 15 / 28
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Using the MVT to estimate
Example
We know that |sin x | ≤ 1 for all x . Show that |sin x | ≤ |x |.
Solution
Apply the MVT to the function f (t) = sin t on [0, x ]. We get
sin x − sin 0
x − 0= cos(c)
for some c in (0, x). Since |cos(c)| ≤ 1, we get∣∣∣∣sin x
x
∣∣∣∣ ≤ 1 =⇒ |sin x | ≤ |x |
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 15 / 28
![Page 27: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/27.jpg)
Using the MVT to estimate II
Example
Let f be a differentiable function with f (1) = 3 and f ′(x) < 2 for all x in[0, 5]. Could f (4) ≥ 9?
Solution
By MVT
f (4)− f (1)
4− 1= f ′(c) < 2
for some c in (1, 4). Therefore
f (4) = f (1) + f ′(c)(3) < 3 + 2 · 3 = 9.
So no, it is impossible that f (4) ≥ 9.
x
y
(1, 3)
(4, 9)
(4, f (4))
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 16 / 28
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Using the MVT to estimate II
Example
Let f be a differentiable function with f (1) = 3 and f ′(x) < 2 for all x in[0, 5]. Could f (4) ≥ 9?
Solution
By MVT
f (4)− f (1)
4− 1= f ′(c) < 2
for some c in (1, 4). Therefore
f (4) = f (1) + f ′(c)(3) < 3 + 2 · 3 = 9.
So no, it is impossible that f (4) ≥ 9.
x
y
(1, 3)
(4, 9)
(4, f (4))
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 16 / 28
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Food for Thought
Question
A driver travels along the New Jersey Turnpike using E-ZPass. The systemtakes note of the time and place the driver enters and exits the Turnpike.A week after his trip, the driver gets a speeding ticket in the mail. Whichof the following best describes the situation?
(a) E-ZPass cannot prove that the driver was speeding
(b) E-ZPass can prove that the driver was speeding
(c) The driver’s actual maximum speed exceeds his ticketed speed
(d) Both (b) and (c).
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 17 / 28
![Page 30: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/30.jpg)
Food for Thought
Question
A driver travels along the New Jersey Turnpike using E-ZPass. The systemtakes note of the time and place the driver enters and exits the Turnpike.A week after his trip, the driver gets a speeding ticket in the mail. Whichof the following best describes the situation?
(a) E-ZPass cannot prove that the driver was speeding
(b) E-ZPass can prove that the driver was speeding
(c) The driver’s actual maximum speed exceeds his ticketed speed
(d) Both (b) and (c).
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 17 / 28
![Page 31: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/31.jpg)
Outline
Rolle’s Theorem
The Mean Value TheoremApplications
Why the MVT is the MITCFunctions with derivatives that are zeroMVT and differentiability
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 18 / 28
![Page 32: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/32.jpg)
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f ′(x) = 0 on (a, b).
I The limit of difference quotients must be 0
I The tangent line to a line is that line, and a constant function’s graphis a horizontal line, which has slope 0.
I Implied by the power rule since c = cx0
Question
If f ′(x) = 0 is f necessarily a constant function?
I It seems true
I But so far no theorem (that we have proven) uses information aboutthe derivative of a function to determine information about thefunction itself
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 19 / 28
![Page 33: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/33.jpg)
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f ′(x) = 0 on (a, b).
I The limit of difference quotients must be 0
I The tangent line to a line is that line, and a constant function’s graphis a horizontal line, which has slope 0.
I Implied by the power rule since c = cx0
Question
If f ′(x) = 0 is f necessarily a constant function?
I It seems true
I But so far no theorem (that we have proven) uses information aboutthe derivative of a function to determine information about thefunction itself
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 19 / 28
![Page 34: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/34.jpg)
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f ′(x) = 0 on (a, b).
I The limit of difference quotients must be 0
I The tangent line to a line is that line, and a constant function’s graphis a horizontal line, which has slope 0.
I Implied by the power rule since c = cx0
Question
If f ′(x) = 0 is f necessarily a constant function?
I It seems true
I But so far no theorem (that we have proven) uses information aboutthe derivative of a function to determine information about thefunction itself
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 19 / 28
![Page 35: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/35.jpg)
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f ′(x) = 0 on (a, b).
I The limit of difference quotients must be 0
I The tangent line to a line is that line, and a constant function’s graphis a horizontal line, which has slope 0.
I Implied by the power rule since c = cx0
Question
If f ′(x) = 0 is f necessarily a constant function?
I It seems true
I But so far no theorem (that we have proven) uses information aboutthe derivative of a function to determine information about thefunction itself
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 19 / 28
![Page 36: Lesson 19: The Mean Value Theorem](https://reader034.fdocuments.us/reader034/viewer/2022042714/55508e9bb4c905a85c8b4f87/html5/thumbnails/36.jpg)
Why the MVT is the MITCMost Important Theorem In Calculus!
Theorem
Let f ′ = 0 on an interval (a, b).
Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x < y . Then f is continuous on[x , y ] and differentiable on (x , y). By MVT there exists a point z in (x , y)such that
f (y)− f (x)
y − x= f ′(z) = 0.
So f (y) = f (x). Since this is true for all x and y in (a, b), then f isconstant.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.2 The Mean Value Theorem June 8, 2010 20 / 28
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Why the MVT is the MITCMost Important Theorem In Calculus!
Theorem
Let f ′ = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x < y . Then f is continuous on[x , y ] and differentiable on (x , y). By MVT there exists a point z in (x , y)such that
f (y)− f (x)
y − x= f ′(z) = 0.
So f (y) = f (x). Since this is true for all x and y in (a, b), then f isconstant.
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Why the MVT is the MITCMost Important Theorem In Calculus!
Theorem
Let f ′ = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x < y . Then f is continuous on[x , y ] and differentiable on (x , y). By MVT there exists a point z in (x , y)such that
f (y)− f (x)
y − x= f ′(z) = 0.
So f (y) = f (x). Since this is true for all x and y in (a, b), then f isconstant.
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Functions with the same derivative
Theorem
Suppose f and g are two differentiable functions on (a, b) with f ′ = g ′.Then f and g differ by a constant. That is, there exists a constant C suchthat f (x) = g(x) + C .
Proof.
I Let h(x) = f (x)− g(x)
I Then h′(x) = f ′(x)− g ′(x) = 0 on (a, b)
I So h(x) = C , a constant
I This means f (x)− g(x) = C on (a, b)
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Functions with the same derivative
Theorem
Suppose f and g are two differentiable functions on (a, b) with f ′ = g ′.Then f and g differ by a constant. That is, there exists a constant C suchthat f (x) = g(x) + C .
Proof.
I Let h(x) = f (x)− g(x)
I Then h′(x) = f ′(x)− g ′(x) = 0 on (a, b)
I So h(x) = C , a constant
I This means f (x)− g(x) = C on (a, b)
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MVT and differentiability
Example
Let
f (x) =
{−x if x ≤ 0
x2 if x ≥ 0
Is f differentiable at 0?
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MVT and differentiability
Example
Let
f (x) =
{−x if x ≤ 0
x2 if x ≥ 0
Is f differentiable at 0?
Solution (from the definition)
We have
limx→0−
f (x)− f (0)
x − 0= lim
x→0−
−x
x= −1
limx→0+
f (x)− f (0)
x − 0= lim
x→0+
x2
x= lim
x→0+x = 0
Since these limits disagree, f is not differentiable at 0.
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MVT and differentiability
Example
Let
f (x) =
{−x if x ≤ 0
x2 if x ≥ 0
Is f differentiable at 0?
Solution (Sort of)
If x < 0, then f ′(x) = −1. If x > 0, then f ′(x) = 2x. Since
limx→0+
f ′(x) = 0 and limx→0−
f ′(x) = −1,
the limit limx→0
f ′(x) does not exist and so f is not differentiable at 0.
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Why only “sort of”?
I This solution is valid but lessdirect.
I We seem to be using thefollowing fact: If lim
x→af ′(x)
does not exist, then f is notdifferentiable at a.
I equivalently: If f isdifferentiable at a, thenlimx→a
f ′(x) exists.
I But this “fact” is not true!
x
y f (x)
f ′(x)
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Differentiable with discontinuous derivative
It is possible for a function f to be differentiable at a even if limx→a
f ′(x)
does not exist.
Example
Let f ′(x) =
{x2 sin(1/x) if x 6= 0
0 if x = 0. Then when x 6= 0,
f ′(x) = 2x sin(1/x) + x2 cos(1/x)(−1/x2) = 2x sin(1/x)− cos(1/x),
which has no limit at 0. However,
f ′(0) = limx→0
f (x)− f (0)
x − 0= lim
x→0
x2 sin(1/x)
x= lim
x→0x sin(1/x) = 0
So f ′(0) = 0. Hence f is differentiable for all x , but f ′ is not continuous at0!
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Differentiability FAIL
x
f (x)
This function is differentiable at0.
x
f ′(x)
But the derivative is notcontinuous at 0!
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MVT to the rescue
Lemma
Suppose f is continuous on [a, b] and limx→a+
f ′(x) = m. Then
limx→a+
f (x)− f (a)
x − a= m.
Proof.
Choose x near a and greater than a. Then
f (x)− f (a)
x − a= f ′(cx)
for some cx where a < cx < x . As x → a, cx → a as well, so:
limx→a+
f (x)− f (a)
x − a= lim
x→a+f ′(cx) = lim
x→a+f ′(x) = m.
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MVT to the rescue
Lemma
Suppose f is continuous on [a, b] and limx→a+
f ′(x) = m. Then
limx→a+
f (x)− f (a)
x − a= m.
Proof.
Choose x near a and greater than a. Then
f (x)− f (a)
x − a= f ′(cx)
for some cx where a < cx < x . As x → a, cx → a as well, so:
limx→a+
f (x)− f (a)
x − a= lim
x→a+f ′(cx) = lim
x→a+f ′(x) = m.
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Theorem
Supposelim
x→a−f ′(x) = m1 and lim
x→a+f ′(x) = m2
If m1 = m2, then f is differentiable at a. If m1 6= m2, then f is notdifferentiable at a.
Proof.
We know by the lemma that
limx→a−
f (x)− f (a)
x − a= lim
x→a−f ′(x)
limx→a+
f (x)− f (a)
x − a= lim
x→a+f ′(x)
The two-sided limit exists if (and only if) the two right-hand sidesagree.
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Theorem
Supposelim
x→a−f ′(x) = m1 and lim
x→a+f ′(x) = m2
If m1 = m2, then f is differentiable at a. If m1 6= m2, then f is notdifferentiable at a.
Proof.
We know by the lemma that
limx→a−
f (x)− f (a)
x − a= lim
x→a−f ′(x)
limx→a+
f (x)− f (a)
x − a= lim
x→a+f ′(x)
The two-sided limit exists if (and only if) the two right-hand sidesagree.
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Summary
I Rolle’s Theorem: under suitable conditions, functions must havecritical points.
I Mean Value Theorem: under suitable conditions, functions must havean instantaneous rate of change equal to the average rate of change.
I A function whose derivative is identically zero on an interval must beconstant on that interval.
I E-ZPass is kinder than we realized.
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