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.
Hybrid Block Method for the Solution of First Order Initial Value Problems of...iosrjce
Method of collocation of the differential system and interpolation of the approximate solution which
is a combination of power series and exponential function at some selected grid and off-grid points to generate
a linear multistep method which is implemented in block method is considered in this paper. The basic
properties of the block method which include; consistency, convergence and stability interval is verified. The
method is tested on some numerical experiments and found to have better stability condition and better
approximation than the existing methods
Lesson 24: Areas, Distances, the Integral (Section 041 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.
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.
Hybrid Block Method for the Solution of First Order Initial Value Problems of...iosrjce
Method of collocation of the differential system and interpolation of the approximate solution which
is a combination of power series and exponential function at some selected grid and off-grid points to generate
a linear multistep method which is implemented in block method is considered in this paper. The basic
properties of the block method which include; consistency, convergence and stability interval is verified. The
method is tested on some numerical experiments and found to have better stability condition and better
approximation than the existing methods
Lesson 24: Areas, Distances, the Integral (Section 041 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.
This learner's module discusses or talks about the topic Radical Expressions. It also teaches how to recognize basic radical notation. It also teaches the multiplication and division of Radical Expressions.
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.
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 more good examples.
Lesson 22: Optimization II (Section 041 handout)Matthew Leingang
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.
This learner's module discusses or talks about the topic Radical Expressions. It also teaches how to recognize basic radical notation. It also teaches the multiplication and division of Radical Expressions.
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.
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 more good examples.
Lesson 22: Optimization II (Section 041 handout)Matthew Leingang
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 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 021 slides)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 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.
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.
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.
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.
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.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
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:
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.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
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.
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
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.
Welcome to the first live UiPath Community Day Dubai! Join us for this unique occasion to meet our local and global UiPath Community and leaders. You will get a full view of the MEA region's automation landscape and the AI Powered automation technology capabilities of UiPath. Also, hosted by our local partners Marc Ellis, you will enjoy a half-day packed with industry insights and automation peers networking.
📕 Curious on our agenda? Wait no more!
10:00 Welcome note - UiPath Community in Dubai
Lovely Sinha, UiPath Community Chapter Leader, UiPath MVPx3, Hyper-automation Consultant, First Abu Dhabi Bank
10:20 A UiPath cross-region MEA overview
Ashraf El Zarka, VP and Managing Director MEA, UiPath
10:35: Customer Success Journey
Deepthi Deepak, Head of Intelligent Automation CoE, First Abu Dhabi Bank
11:15 The UiPath approach to GenAI with our three principles: improve accuracy, supercharge productivity, and automate more
Boris Krumrey, Global VP, Automation Innovation, UiPath
12:15 To discover how Marc Ellis leverages tech-driven solutions in recruitment and managed services.
Brendan Lingam, Director of Sales and Business Development, Marc Ellis
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
1. Section 4.5
Optimization Problems
V63.0121.021, Calculus I
New York University
November 23, 2010
Announcements
Turn in HW anytime between now and November 24, 2pm
No Thursday recitation this week
Quiz 5 on §§4.1–4.4 next week in recitation
. . . . . .
2. . . . . . .
Announcements
Turn in HW anytime
between now and
November 24, 2pm
No Thursday recitation this
week
Quiz 5 on §§4.1–4.4 next
week in recitation
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 2 / 31
3. . . . . . .
Objectives
Given a problem requiring
optimization, identify the
objective functions,
variables, and constraints.
Solve optimization
problems with calculus.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 3 / 31
4. . . . . . .
Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 4 / 31
5. . . . . . .
Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 5 / 31
6. . . . . . .
Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 5 / 31
7. . . . . . .
Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
..
ℓ
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 5 / 31
8. . . . . . .
Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
..
ℓ
.
w
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 5 / 31
9. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
10. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
11. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
Since p = 2ℓ + 2w, we have ℓ =
p − 2w
2
,
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
12. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
Since p = 2ℓ + 2w, we have ℓ =
p − 2w
2
, so
A = ℓw =
p − 2w
2
· w =
1
2
(p − 2w)(w) =
1
2
pw − w2
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
13. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
Since p = 2ℓ + 2w, we have ℓ =
p − 2w
2
, so
A = ℓw =
p − 2w
2
· w =
1
2
(p − 2w)(w) =
1
2
pw − w2
Now we have A as a function of w alone (p is constant).
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
14. . . . . . .
Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
Since p = 2ℓ + 2w, we have ℓ =
p − 2w
2
, so
A = ℓw =
p − 2w
2
· w =
1
2
(p − 2w)(w) =
1
2
pw − w2
Now we have A as a function of w alone (p is constant).
The natural domain of this function is [0, p/2] (we want to make
sure A(w) ≥ 0).
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 6 / 31
15. . . . . . .
Solution Concluded
We use the Closed Interval Method for A(w) =
1
2
pw − w2
on [0, p/2].
At the endpoints, A(0) = A(p/2) = 0.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 7 / 31
16. . . . . . .
Solution Concluded
We use the Closed Interval Method for A(w) =
1
2
pw − w2
on [0, p/2].
At the endpoints, A(0) = A(p/2) = 0.
To find the critical points, we find
dA
dw
=
1
2
p − 2w.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 7 / 31
17. . . . . . .
Solution Concluded
We use the Closed Interval Method for A(w) =
1
2
pw − w2
on [0, p/2].
At the endpoints, A(0) = A(p/2) = 0.
To find the critical points, we find
dA
dw
=
1
2
p − 2w.
The critical points are when
0 =
1
2
p − 2w =⇒ w =
p
4
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 7 / 31
18. . . . . . .
Solution Concluded
We use the Closed Interval Method for A(w) =
1
2
pw − w2
on [0, p/2].
At the endpoints, A(0) = A(p/2) = 0.
To find the critical points, we find
dA
dw
=
1
2
p − 2w.
The critical points are when
0 =
1
2
p − 2w =⇒ w =
p
4
Since this is the only critical point, it must be the maximum. In this
case ℓ =
p
4
as well.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 7 / 31
19. . . . . . .
Solution Concluded
We use the Closed Interval Method for A(w) =
1
2
pw − w2
on [0, p/2].
At the endpoints, A(0) = A(p/2) = 0.
To find the critical points, we find
dA
dw
=
1
2
p − 2w.
The critical points are when
0 =
1
2
p − 2w =⇒ w =
p
4
Since this is the only critical point, it must be the maximum. In this
case ℓ =
p
4
as well.
We have a square! The maximal area is A(p/4) = p2
/16.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 7 / 31
20. . . . . . .
Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 8 / 31
21. . . . . . .
Strategies for Problem Solving
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Review and extend
György Pólya
(Hungarian, 1887–1985)
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 9 / 31
22. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
23. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
24. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
25. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
26. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
5. If Q is a function of more than one “decision variable”, use the
given information to eliminate all but one of them.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
27. . . . . . .
The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
5. If Q is a function of more than one “decision variable”, use the
given information to eliminate all but one of them.
6. Find the absolute maximum (or minimum, depending on the
problem) of the function on its domain.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 10 / 31
28. . . . . . .
Polya's Method in Kindergarten
Name [_
Problem Solving Strategy
Draw a Picture
Kathy had a box of 8 crayons.
She gave some crayons away.
She has 5 left.
How many crayons did Kathy give away?
UNDERSTAND
•
What do you want to find out?
Draw a line under the question.
You can draw a picture
to solve the problem.
crayons
What number do I
add to 5 to get 8?
8 - = 5
5 + 3 = 8
CHECK
Does your answer make sense?
Explain.
Draw a picture to solve the problem.
Write how many were given away.
I. I had10 pencils.
I gave some away.
I have 3left. How many
pencils did I give away?
~7
What number
do I add to 3
to make 10?
13
i
ft
ill
:i
i ?
11
ft
I
'•'
«
I
I
ft A
H 11
M i l
U U U U> U U
I I
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 11 / 31
29. . . . . . .
Recall: The Closed Interval Method
See Section 4.1
The Closed Interval Method
To find the extreme values of a function f on [a, b], we need to:
Evaluate f at the endpoints a and b
Evaluate f at the critical points x where either f′
(x) = 0 or f is not
differentiable at x.
The points with the largest function value are the global maximum
points
The points with the smallest/most negative function value are the
global minimum points.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 12 / 31
30. . . . . . .
Recall: The First Derivative Test
See Section 4.3
Theorem (The First Derivative Test)
Let f be continuous on (a, b) and c a critical point of f in (a, b).
If f′
changes from negative to positive at c, then c is a local
minimum.
If f′
changes from positive to negative at c, then c is a local
maximum.
If f′
does not change sign at c, then c is not a local extremum.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 13 / 31
31. . . . . . .
Recall: The First Derivative Test
See Section 4.3
Theorem (The First Derivative Test)
Let f be continuous on (a, b) and c a critical point of f in (a, b).
If f′
changes from negative to positive at c, then c is a local
minimum.
If f′
changes from positive to negative at c, then c is a local
maximum.
If f′
does not change sign at c, then c is not a local extremum.
Corollary
If f′
< 0 for all x < c and f′
(x) > 0 for all x > c, then c is the global
minimum of f on (a, b).
If f′
< 0 for all x > c and f′
(x) > 0 for all x < c, then c is the global
maximum of f on (a, b).
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 13 / 31
32. . . . . . .
Recall: The Second Derivative Test
See Section 4.3
Theorem (The Second Derivative Test)
Let f, f′
, and f′′
be continuous on [a, b]. Let c be in (a, b) with f′
(c) = 0.
If f′′
(c) < 0, then f(c) is a local maximum.
If f′′
(c) > 0, then f(c) is a local minimum.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 14 / 31
33. . . . . . .
Recall: The Second Derivative Test
See Section 4.3
Theorem (The Second Derivative Test)
Let f, f′
, and f′′
be continuous on [a, b]. Let c be in (a, b) with f′
(c) = 0.
If f′′
(c) < 0, then f(c) is a local maximum.
If f′′
(c) > 0, then f(c) is a local minimum.
Warning
If f′′
(c) = 0, the second derivative test is inconclusive (this does not
mean c is neither; we just don’t know yet).
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 14 / 31
34. . . . . . .
Recall: The Second Derivative Test
See Section 4.3
Theorem (The Second Derivative Test)
Let f, f′
, and f′′
be continuous on [a, b]. Let c be in (a, b) with f′
(c) = 0.
If f′′
(c) < 0, then f(c) is a local maximum.
If f′′
(c) > 0, then f(c) is a local minimum.
Warning
If f′′
(c) = 0, the second derivative test is inconclusive (this does not
mean c is neither; we just don’t know yet).
Corollary
If f′
(c) = 0 and f′′
(x) > 0 for all x, then c is the global minimum of f
If f′
(c) = 0 and f′′
(x) < 0 for all x, then c is the global maximum of f
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 14 / 31
35. . . . . . .
Which to use when?
CIM 1DT 2DT
Pro – no need for
inequalities
– gets global
extrema
automatically
– works on
non-closed,
non-bounded
intervals
– only one derivative
– works on
non-closed,
non-bounded
intervals
– no need for
inequalities
Con – only for closed
bounded intervals
– Uses inequalities
– More work at
boundary than CIM
– More derivatives
– less conclusive
than 1DT
– more work at
boundary than CIM
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 15 / 31
36. . . . . . .
Which to use when?
CIM 1DT 2DT
Pro – no need for
inequalities
– gets global
extrema
automatically
– works on
non-closed,
non-bounded
intervals
– only one derivative
– works on
non-closed,
non-bounded
intervals
– no need for
inequalities
Con – only for closed
bounded intervals
– Uses inequalities
– More work at
boundary than CIM
– More derivatives
– less conclusive
than 1DT
– more work at
boundary than CIM
Use CIM if it applies: the domain is a closed, bounded interval
If domain is not closed or not bounded, use 2DT if you like to take
derivatives, or 1DT if you like to compare signs.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 15 / 31
37. . . . . . .
Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 16 / 31
38. . . . . . .
Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 17 / 31
40. . . . . . .
Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 19 / 31
41. . . . . . .
Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
Known: amount of fence used
Unknown: area enclosed
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 19 / 31
42. . . . . . .
Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
Known: amount of fence used
Unknown: area enclosed
Objective: maximize area
Constraint: fixed fence length
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 19 / 31
44. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 20 / 31
45. . . . . . .
Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
.
.
..
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 21 / 31
46. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 22 / 31
47. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 22 / 31
48. . . . . . .
Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
.
.
..
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 23 / 31
49. . . . . . .
Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
.
.
...
w
.
ℓ
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 23 / 31
50. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 24 / 31
51. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 24 / 31
52. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 24 / 31
53. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
6.
dQ
dw
= p − 4w, which is zero when w =
p
4
.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 24 / 31
54. . . . . . .
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
6.
dQ
dw
= p − 4w, which is zero when w =
p
4
. Q(0) = Q(p/2) = 0, but
Q
(p
4
)
= p ·
p
4
− 2 ·
p2
16
=
p2
8
= 80, 000m2
so the critical point is the absolute maximum.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 24 / 31
55. . . . . . .
Your turn
Example (The shortest fence)
A 216m2
rectangular pea patch is to be enclosed by a fence and
divided into two equal parts by another fence parallel to one of its
sides. What dimensions for the outer rectangle will require the smallest
total length of fence? How much fence will be needed?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 25 / 31
56. . . . . . .
Your turn
Example (The shortest fence)
A 216m2
rectangular pea patch is to be enclosed by a fence and
divided into two equal parts by another fence parallel to one of its
sides. What dimensions for the outer rectangle will require the smallest
total length of fence? How much fence will be needed?
Solution
Let the length and width of the pea patch be ℓ and w. The amount of
fence needed is f = 2ℓ + 3w. Since ℓw = A, a constant, we have
f(w) = 2
A
w
+ 3w.
The domain is all positive numbers.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 25 / 31
57. . . . . . .
Diagram
....
ℓ
.
w
f = 2ℓ + 3w A = ℓw ≡ 216
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 26 / 31
58. . . . . . .
Solution (Continued)
We need to find the minimum value of f(w) =
2A
w
+ 3w on (0, ∞).
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 27 / 31
59. . . . . . .
Solution (Continued)
We need to find the minimum value of f(w) =
2A
w
+ 3w on (0, ∞).
We have
df
dw
= −
2A
w2
+ 3
which is zero when w =
√
2A
3
.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 27 / 31
60. . . . . . .
Solution (Continued)
We need to find the minimum value of f(w) =
2A
w
+ 3w on (0, ∞).
We have
df
dw
= −
2A
w2
+ 3
which is zero when w =
√
2A
3
.
Since f′′
(w) = 4Aw−3
, which is positive for all positive w, the
critical point is a minimum, in fact the global minimum.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 27 / 31
61. . . . . . .
Solution (Continued)
We need to find the minimum value of f(w) =
2A
w
+ 3w on (0, ∞).
We have
df
dw
= −
2A
w2
+ 3
which is zero when w =
√
2A
3
.
Since f′′
(w) = 4Aw−3
, which is positive for all positive w, the
critical point is a minimum, in fact the global minimum.
So the area is minimized when w =
√
2A
3
= 12 and
ℓ =
A
w
=
√
3A
2
= 18. The amount of fence needed is
f
(√
2A
3
)
= 2 ·
√
3A
2
+ 3
√
2A
3
= 2
√
6A = 2
√
6 · 216 = 72m
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 27 / 31
62. . . . . . .
Try this one
Example
An advertisement consists of a rectangular printed region plus 1 in
margins on the sides and 1.5 in margins on the top and bottom. If the
total area of the advertisement is to be 120 in2
, what dimensions should
the advertisement be to maximize the area of the printed region?
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 28 / 31
63. . . . . . .
Try this one
Example
An advertisement consists of a rectangular printed region plus 1 in
margins on the sides and 1.5 in margins on the top and bottom. If the
total area of the advertisement is to be 120 in2
, what dimensions should
the advertisement be to maximize the area of the printed region?
Answer
The optimal paper dimensions are 4
√
5 in by 6
√
5 in.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 28 / 31
64. . . . . . .
Solution
Let the dimensions of the
printed region be x and y, P
the printed area, and A the
paper area. We wish to
maximize P = xy subject to
the constraint that
A = (x + 2)(y + 3) ≡ 120
Isolating y in A ≡ 120 gives
y =
120
x + 2
− 3 which yields
P = x
(
120
x + 2
− 3
)
=
120x
x + 2
−3x
The domain of P is (0, ∞).
..
Lorem ipsum dolor sit amet,
consectetur adipiscing elit. Nam
dapibus vehicula mollis. Proin nec
tristique mi. Pellentesque quis
placerat dolor. Praesent a nisl diam.
Phasellus ut elit eu ligula accumsan
euismod. Nunc condimentum
lacinia risus a sodales. Morbi nunc
risus, tincidunt in tristique sit amet,
ultrices eu eros. Proin pellentesque
aliquam nibh ut lobortis. Ut et
sollicitudin ipsum. Proin gravida
ligula eget odio molestie rhoncus
sed nec massa. In ante lorem,
imperdiet eget tincidunt at, pharetra
sit amet felis. Nunc nisi velit,
tempus ac suscipit quis, blandit
vitae mauris. Vestibulum ante ipsum
primis in faucibus orci luctus et
ultrices posuere cubilia Curae;
.
1.5 cm
.
1.5 cm
.1cm.
1cm
.
x
.
y
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 29 / 31
65. . . . . . .
Solution (Concluded)
We want to find the absolute maximum value of P.
dP
dx
=
(x + 2)(120) − (120x)(1)
(x + 2)2
− 3 =
240 − 3(x + 2)2
(x + 2)2
There is a single (positive) critical point when
(x + 2)2
= 80 =⇒ x = 4
√
5 − 2.
The second derivative is
d2
P
dx2
=
−480
(x + 2)3
, which is negative all
along the domain of P.
Hence the unique critical point x =
(
4
√
5 − 2
)
cm is the absolute
maximum of P.
This means the paper width is 4
√
5 cm, and the paper length is
120
4
√
5
= 6
√
5 cm.
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 30 / 31
66. . . . . . .
Summary
Remember the checklist
Ask yourself: what is the
objective?
Remember your geometry:
similar triangles
right triangles
trigonometric functions
Name [_
Problem Solving Strategy
Draw a Picture
Kathy had a box of 8 crayons.
She gave some crayons away.
She has 5 left.
How many crayons did Kathy give away?
UNDERSTAND
•
What do you want to find out?
Draw a line under the question.
You can draw a picture
to solve the problem.
crayons
What number do I
add to 5 to get 8?
8 - = 5
5 + 3 = 8
CHECK
Does your answer make sense?
Explain.
Draw a picture to solve the problem.
Write how many were given away.
I. I had10 pencils.
I gave some away.
I have 3left. How many
pencils did I give away?
~7
What number
do I add to 3
to make 10?
13
i
ft
ill
:i
i ?
11
ft
I
'•'
«
I
I
ft A
H 11
M i l
U U U U> U U
I I
V63.0121.021, Calculus I (NYU) Section 4.5 Optimization Problems November 23, 2010 31 / 31