(a) E-ZPass cannot prove that the driver was speeding. E-ZPass records entry and exit times and locations, but does not continuously track speed. It cannot determine the driver's exact speed at any point during the trip, so it cannot prove a specific speeding violation occurred. The best it could show is an average speed that may or may not indicate speeding depending on the specific speed limit(s) along the route.
Quantum Minimax Theorem in Statistical Decision Theory (RIMS2014)tanafuyu
This is almost self-contained explanation of our recent result. The contents are based on our talk in the RIMS2014 conference.
Recently, many fundamental and important results in statistical decision theory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples.
In our recent preprint, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Quantum Minimax Theorem in Statistical Decision Theory (RIMS2014)tanafuyu
This is almost self-contained explanation of our recent result. The contents are based on our talk in the RIMS2014 conference.
Recently, many fundamental and important results in statistical decision theory have been extended to the quantum system. Quantum Hunt-Stein theorem and quantum locally asymptotic normality are typical successful examples.
In our recent preprint, we show quantum minimax theorem, which is also an extension of a well-known result, minimax theorem in statistical decision theory, first shown by Wald and generalized by LeCam. Our assertions hold for every closed convex set of measurements and for general parametric models of density operator. On the other hand, Bayesian analysis based on least favorable priors has been widely used in classical statistics and is expected to play a crucial role in quantum statistics. According to this trend, we also show the existence of least favorable priors, which seems to be new even in classical statistics.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
INDUCTIVE LEARNING OF COMPLEX FUZZY RELATIONijcseit
The objective of this paper to investigate the notion of complex fuzzy set in general view. In constraint to a
traditional fuzzy set, the membership function of the complex fuzzy set, the range from [0.1] extended to a
unit circle in the complex plane. In this article the comprehensive mathematical operations on the complex
fuzzy set are presented. The basic operation of complex fuzzy set such as union, intersection, complement
of complex fuzzy set and complex fuzzy relation are studied. Also vector aggregation and fuzzy relation
over the complex fuzzy set are discussed. Two novel operations of complement and projection for complex
fuzzy relation are introduced.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Chemistry Assignments.
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Lesson 18: Maximum and Minimum Values (Section 021 handout)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 18: Maximum and Minimum Values (Section 021 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Lesson 18: Maximum and Minimum Values (Section 041 slides)Matthew Leingang
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Lesson 19: The Mean Value Theorem (Section 021 slides)
1. Section 4.2
The Mean Value Theorem
V63.0121.021, Calculus I
New York University
November 11, 2010
Announcements
Quiz 4 next week (November 16, 18, 19) on Sections 3.3, 3.4, 3.5,
3.7
. . . . . .
2. . . . . . .
Announcements
Quiz 4 next week
(November 16, 18, 19) on
Sections 3.3, 3.4, 3.5, 3.7
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 2 / 29
3. . . . . . .
Objectives
Understand and be able to
explain the statement of
Rolle’s Theorem.
Understand and be able to
explain the statement of
the Mean Value Theorem.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 3 / 29
4. . . . . . .
Outline
Rolle’s Theorem
The Mean Value Theorem
Applications
Why the MVT is the MITC
Functions with derivatives that are zero
MVT and differentiability
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 4 / 29
5. . . . . . .
Heuristic Motivation for Rolle's Theorem
If you bike up a hill, then back down, at some point your elevation was
stationary.
.
.Image credit: SpringSun
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 5 / 29
6. . . . . . .
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). Then
there exists a point c in
(a, b) such that f′
(c) = 0.
. .
.a
.
.b
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 6 / 29
7. . . . . . .
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). Then
there exists a point c in
(a, b) such that f′
(c) = 0.
. .
.a
.
.b
..c
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 6 / 29
8. . . . . . .
Flowchart proof of Rolle's Theorem
.
.
.
.
Let c be
the max pt
.
.
Let d be
the min pt
.
.
endpoints
are max
and min
.
.
.
is c an
endpoint?
.
.
is d an
endpoint?
.
.
f is
constant
on [a, b]
.
.
f′
(c) = 0 .
.
f′
(d) = 0 .
.
f′
(x) ≡ 0
on (a, b)
.no .no
.yes .yes
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 8 / 29
9. . . . . . .
Outline
Rolle’s Theorem
The Mean Value Theorem
Applications
Why the MVT is the MITC
Functions with derivatives that are zero
MVT and differentiability
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 9 / 29
10. . . . . . .
Heuristic Motivation for The Mean Value Theorem
If you drive between points A and B, at some time your speedometer
reading was the same as your average speed over the drive.
.
.Image credit: ClintJCL
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 10 / 29
11. . . . . . .
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
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 11 / 29
12. . . . . . .
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
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 11 / 29
13. . . . . . .
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
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 11 / 29
14. . . . . . .
Rolle vs. MVT
f′
(c) = 0
f(b) − f(a)
b − a
= f′
(c)
. .
.a
.
.b
..c
. .
.a
.
.b
..c
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 12 / 29
15. . . . . . .
Rolle vs. MVT
f′
(c) = 0
f(b) − f(a)
b − a
= f′
(c)
. .
.a
.
.b
..c
. .
.a
.
.b
..c
If the x-axis is skewed the pictures look the same.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 12 / 29
16. . . . . . .
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)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 13 / 29
17. . . . . . .
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).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 13 / 29
18. . . . . . .
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.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 13 / 29
19. . . . . . .
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.
Also g(a) = 0 and g(b) = 0 (check both)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 13 / 29
20. . . . . . .
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.
Also g(a) = 0 and g(b) = 0 (check both) So by Rolle’s Theorem there
exists a point c in (a, b) such that
0 = g′
(c) = f′
(c) −
f(b) − f(a)
b − a
.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 13 / 29
21. . . . . . .
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3
− x = 100 in the
interval [4, 5].
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 14 / 29
22. . . . . . .
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3
− x = 100 in the
interval [4, 5].
Solution
By the Intermediate Value Theorem, the function f(x) = x3
− x
must take the value 100 at some point on c in (4, 5).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 14 / 29
23. . . . . . .
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3
− x = 100 in the
interval [4, 5].
Solution
By the Intermediate Value Theorem, the function f(x) = x3
− x
must take the value 100 at some point on c in (4, 5).
If there were two points c1 and c2 with f(c1) = f(c2) = 100, then
somewhere between them would be a point c3 between them with
f′
(c3) = 0.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 14 / 29
24. . . . . . .
Using the MVT to count solutions
Example
Show that there is a unique solution to the equation x3
− x = 100 in the
interval [4, 5].
Solution
By the Intermediate Value Theorem, the function f(x) = x3
− x
must take the value 100 at some point on c in (4, 5).
If there were two points c1 and c2 with f(c1) = f(c2) = 100, then
somewhere between them would be a point c3 between them with
f′
(c3) = 0.
However, f′
(x) = 3x2
− 1, which is positive all along (4, 5). So this
is impossible.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 14 / 29
25. . . . . . .
Using the MVT to estimate
Example
We know that |sin x| ≤ 1 for all x, and that sin x ≈ x for small x. Show
that |sin x| ≤ |x| for all x.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 15 / 29
26. . . . . . .
Using the MVT to estimate
Example
We know that |sin x| ≤ 1 for all x, and that sin x ≈ x for small x. Show
that |sin x| ≤ |x| for all 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|
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 15 / 29
27. . . . . . .
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?
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 16 / 29
28. . . . . . .
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.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 16 / 29
29. . . . . . .
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))
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 16 / 29
30. . . . . . .
Food for Thought
Question
A driver travels along the New Jersey Turnpike using E-ZPass. The
system takes 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. Which of 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).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 17 / 29
31. . . . . . .
Food for Thought
Question
A driver travels along the New Jersey Turnpike using E-ZPass. The
system takes 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. Which of 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).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 17 / 29
32. . . . . . .
Outline
Rolle’s Theorem
The Mean Value Theorem
Applications
Why the MVT is the MITC
Functions with derivatives that are zero
MVT and differentiability
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 18 / 29
33. . . . . . .
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f′
(x) = 0 on (a, b).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 19 / 29
34. . . . . . .
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f′
(x) = 0 on (a, b).
The limit of difference quotients must be 0
The tangent line to a line is that line, and a constant function’s
graph is a horizontal line, which has slope 0.
Implied by the power rule since c = cx0
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 19 / 29
35. . . . . . .
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f′
(x) = 0 on (a, b).
The limit of difference quotients must be 0
The tangent line to a line is that line, and a constant function’s
graph is a horizontal line, which has slope 0.
Implied by the power rule since c = cx0
Question
If f′
(x) = 0 is f necessarily a constant function?
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 19 / 29
36. . . . . . .
Functions with derivatives that are zero
Fact
If f is constant on (a, b), then f′
(x) = 0 on (a, b).
The limit of difference quotients must be 0
The tangent line to a line is that line, and a constant function’s
graph is a horizontal line, which has slope 0.
Implied by the power rule since c = cx0
Question
If f′
(x) = 0 is f necessarily a constant function?
It seems true
But so far no theorem (that we have proven) uses information
about the derivative of a function to determine information about
the function itself
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 19 / 29
37. . . . . . .
Why the MVT is the MITC
Most Important Theorem In Calculus!
Theorem
Let f′
= 0 on an interval (a, b).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 20 / 29
38. . . . . . .
Why the MVT is the MITC
Most Important Theorem In Calculus!
Theorem
Let f′
= 0 on an interval (a, b). Then f is constant on (a, b).
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 20 / 29
39. . . . . . .
Why the MVT is the MITC
Most 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 is
constant.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 20 / 29
40. . . . . . .
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
such that f(x) = g(x) + C.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
41. . . . . . .
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
such that f(x) = g(x) + C.
Proof.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
42. . . . . . .
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
such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
43. . . . . . .
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
such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
Then h′
(x) = f′
(x) − g′
(x) = 0 on (a, b)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
44. . . . . . .
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
such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
Then h′
(x) = f′
(x) − g′
(x) = 0 on (a, b)
So h(x) = C, a constant
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
45. . . . . . .
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
such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
Then h′
(x) = f′
(x) − g′
(x) = 0 on (a, b)
So h(x) = C, a constant
This means f(x) − g(x) = C on (a, b)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 21 / 29
46. . . . . . .
MVT and differentiability
Example
Let
f(x) =
{
−x if x ≤ 0
x2
if x ≥ 0
Is f differentiable at 0?
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 22 / 29
47. . . . . . .
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
lim
x→0−
f(x) − f(0)
x − 0
= lim
x→0−
−x
x
= −1
lim
x→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.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 22 / 29
48. . . . . . .
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
lim
x→0+
f′
(x) = 0 and lim
x→0−
f′
(x) = −1,
the limit lim
x→0
f′
(x) does not exist and so f is not differentiable at 0.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 22 / 29
49. . . . . . .
Why only “sort of"?
This solution is valid but
less direct.
We seem to be using the
following fact: If lim
x→a
f′
(x)
does not exist, then f is not
differentiable at a.
equivalently: If f is
differentiable at a, then
lim
x→a
f′
(x) exists.
But this “fact” is not true!
. .x
.y .f(x)
.
.
.f′
(x)
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 24 / 29
50. . . . . . .
Differentiable with discontinuous derivative
It is possible for a function f to be differentiable at a even if lim
x→a
f′
(x)
does not exist.
Example
Let f′
(x) =
{
x2
sin(1/x) if x ̸= 0
0 if x = 0
. Then when x ̸= 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) = lim
x→0
f(x) − f(0)
x − 0
= lim
x→0
x2 sin(1/x)
x
= lim
x→0
x sin(1/x) = 0
So f′
(0) = 0. Hence f is differentiable for all x, but f′
is not continuous
at 0!
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 25 / 29
51. . . . . . .
Differentiability FAIL
. .x
.f(x)
This function is differentiable at
0.
. .x
.f′
(x)
.
But the derivative is not
continuous at 0!
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 26 / 29
52. . . . . . .
MVT to the rescue
Lemma
Suppose f is continuous on [a, b] and lim
x→a+
f′
(x) = m. Then
lim
x→a+
f(x) − f(a)
x − a
= m.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 27 / 29
53. . . . . . .
MVT to the rescue
Lemma
Suppose f is continuous on [a, b] and lim
x→a+
f′
(x) = m. Then
lim
x→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:
lim
x→a+
f(x) − f(a)
x − a
= lim
x→a+
f′
(cx) = lim
x→a+
f′
(x) = m.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 27 / 29
54. . . . . . .
Theorem
Suppose
lim
x→a−
f′
(x) = m1 and lim
x→a+
f′
(x) = m2
If m1 = m2, then f is differentiable at a. If m1 ̸= m2, then f is not
differentiable at a.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 28 / 29
55. . . . . . .
Theorem
Suppose
lim
x→a−
f′
(x) = m1 and lim
x→a+
f′
(x) = m2
If m1 = m2, then f is differentiable at a. If m1 ̸= m2, then f is not
differentiable at a.
Proof.
We know by the lemma that
lim
x→a−
f(x) − f(a)
x − a
= lim
x→a−
f′
(x)
lim
x→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 sides
agree.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 28 / 29
56. . . . . . .
Summary
Rolle’s Theorem: under suitable conditions, functions must have
critical points.
Mean Value Theorem: under suitable conditions, functions must
have an instantaneous rate of change equal to the average rate of
change.
A function whose derivative is identically zero on an interval must
be constant on that interval.
E-ZPass is kinder than we realized.
V63.0121.021, Calculus I (NYU) Section 4.2 The Mean Value Theorem November 11, 2010 29 / 29