1) The document is notes from a Calculus I class covering optimization problems.
2) It includes examples of maximizing the area of a rectangle with a fixed perimeter and maximizing the area that can be enclosed by a fence of fixed length.
3) The document reviews strategies for solving optimization problems, including identifying objectives and constraints, drawing diagrams, introducing variables, and using calculus techniques like finding critical points and applying the first and second derivative tests.
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.
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Open newton cotes quadrature with midpoint derivative for integration of al...eSAT Journals
Abstract Many methods are available for approximating the integral to the desired precision in Numerical integration. A new set of numerical integration formula of Open Newton-Cotes Quadrature with Midpoint Derivative type is suggested, which is the modified form of Open Newton-Cotes Quadrature. This new midpoint derivative based formula increase the two order of precision than the classical Open Newton-Cotes formula and also gives more accuracy than the existing formula. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed algorithm is illustrated by means of a numerical example.
Key Words: Numerical Integration, Open Newton-Cotes formula, Midpoint Derivative, Numerical Examples
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
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 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.
This Presentation Is Specially Made For Those Engineering Students Who are In Gujarat Technological University. This Presentation Clears Your All Doubts About Basics Fundamentals of Numerical Integration. Also You Will Learn Different Types Of Error Formula To Solve the Numerical Integration Sum.
Open newton cotes quadrature with midpoint derivative for integration of al...eSAT Journals
Abstract Many methods are available for approximating the integral to the desired precision in Numerical integration. A new set of numerical integration formula of Open Newton-Cotes Quadrature with Midpoint Derivative type is suggested, which is the modified form of Open Newton-Cotes Quadrature. This new midpoint derivative based formula increase the two order of precision than the classical Open Newton-Cotes formula and also gives more accuracy than the existing formula. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed algorithm is illustrated by means of a numerical example.
Key Words: Numerical Integration, Open Newton-Cotes formula, Midpoint Derivative, Numerical Examples
Subject Title: Engineering Numerical Analysis
Subject Code: ID-302
Contents of this chapter:
Mathematical preliminaries,
Solution of equations in one variable,
Interpolation and polynomial Approximation,
Numerical differentiation and integration,
Initial value problems for ordinary differential equations,
Direct methods for solving linear systems,
Iterative techniques in Matrix algebra,
Solution of non-linear equations.
Approximation theory;
Eigen values and vector;
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.
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
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solut...qyjewyvu
Full download : http://alibabadownload.com/product/applied-statistics-and-probability-for-engineers-6th-edition-montgomery-solutions-manual/
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solutions Manual
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.
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.
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.
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
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solut...qyjewyvu
Full download : http://alibabadownload.com/product/applied-statistics-and-probability-for-engineers-6th-edition-montgomery-solutions-manual/
Applied Statistics and Probability for Engineers 6th Edition Montgomery Solutions Manual
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.
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.
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 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 good examples.
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 18: Maximum and Minimum Values (Section 041 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 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.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
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.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
How to Get CNIC Information System with Paksim Ga.pptxdanishmna97
Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
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.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
GraphSummit Singapore | The Art of the Possible with Graph - Q2 2024Neo4j
Neha Bajwa, Vice President of Product Marketing, Neo4j
Join us as we explore breakthrough innovations enabled by interconnected data and AI. Discover firsthand how organizations use relationships in data to uncover contextual insights and solve our most pressing challenges – from optimizing supply chains, detecting fraud, and improving customer experiences to accelerating drug discoveries.
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.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...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.
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.
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.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
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.
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...
Lesson 22: Optimization (Section 021 handout)
1. Section 4.4
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 4 on §§4.1–4.4 next week in recitation
Announcements
Turn in HW anytime
between now and November
24, 2pm
No Thursday recitation this
week
Quiz 4 on §§4.1–4.4 next
week in recitation
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 2 / 31
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 4.4 Optimization Problems November 23, 2010 3 / 31
Notes
Notes
Notes
1
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
2. Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 4 / 31
Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
w
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 5 / 31
Solution Continued
Let its length be and its width be w. The objective function is area
A = w.
This is a function of two variables, not one. But the perimeter is fixed.
Since p = 2 + 2w, we have =
p − 2w
2
, so
A = w =
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 4.4 Optimization Problems November 23, 2010 6 / 31
Notes
Notes
Notes
2
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
3. 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 =
p
4
as well.
We have a square! The maximal area is A(p/4) = p2
/16.
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 7 / 31
Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 8 / 31
Strategies for Problem Solving
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Review and extend
Gy¨orgy P´olya
(Hungarian, 1887–1985)
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 9 / 31
Notes
Notes
Notes
3
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
4. 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 4.4 Optimization Problems November 23, 2010 10 / 31
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 4.4 Optimization Problems November 23, 2010 12 / 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 4.4 Optimization Problems November 23, 2010 13 / 31
Notes
Notes
Notes
4
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
5. 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
fV63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 14 / 31
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 4.4 Optimization Problems November 23, 2010 15 / 31
Outline
Leading by Example
The Text in the Box
More Examples
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 16 / 31
Notes
Notes
Notes
5
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
6. 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 4.4 Optimization Problems November 23, 2010 17 / 31
Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are and w. Length of wire used is p.
4. Q = area = w.
5. Since p = + 2w, we have = 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 4.4 Optimization Problems November 23, 2010 18 / 31
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
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 21 / 31
Notes
Notes
Notes
6
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
7. 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
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 25 / 31
Diagram
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 26 / 31
Solution (Continued)
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 27 / 31
Notes
Notes
Notes
7
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
8. 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 4.4 Optimization Problems November 23, 2010 28 / 31
Solution
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 29 / 31
Solution (Concluded)
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 30 / 31
Notes
Notes
Notes
8
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010
9. Summary
Remember the checklist
Ask yourself: what is the
objective?
Remember your geometry:
similar triangles
right triangles
trigonometric functions
V63.0121.021, Calculus I (NYU) Section 4.4 Optimization Problems November 23, 2010 31 / 31
Notes
Notes
Notes
9
Section 4.4 : Optimization ProblemsV63.0121.021, Calculus I November 23, 2010