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
The document is a lecture on finding maximum and minimum values of functions. It begins with announcements about an upcoming quiz. It then outlines the objectives of understanding the Extreme Value Theorem and Fermat's Theorem, and using the Closed Interval Method. The document discusses the Extreme Value Theorem, which states that a continuous function on a closed interval attains absolute maximum and minimum values. It also covers Fermat's Theorem, which states that if a function has a local extremum where it is differentiable, the derivative is equal to 0 at that point. Examples are provided to illustrate the importance of the hypotheses in the Extreme Value Theorem.
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 26: The Fundamental Theorem of Calculus (Section 041 slides)Matthew Leingang
g(x) represents the area under the curve of f(t) from 0 to x. As x increases from 0 to 10, g(x) will increase, representing the accumulating area under f(t) over the interval [0,x].
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
This document summarizes a paper on the comparison-based complexity of multi-objective optimization. It outlines that the paper presents complexity upper and lower bounds for finding the Pareto front and Pareto set in multi-objective optimization problems. For upper bounds, it shows the complexity is O(Nd log(1/e)) for finding the whole Pareto set and O(N+1-d log(1/e)) for finding a single point, where N is the dimension and d is the number of objectives. For lower bounds, it applies a proof technique using packing numbers from the mono-objective case to the multi-objective case to derive tight complexity bounds.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 handout)Matthew Leingang
This document contains lecture notes on the fundamental theorem of calculus from a Calculus I class. The notes discuss:
1) The first and second fundamental theorems of calculus, which relate differentiation and integration as inverse processes.
2) How to use the first fundamental theorem to differentiate functions defined by integrals.
3) Biographies of several mathematicians involved in the development of calculus, including Newton, Leibniz, Gregory and Barrow.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Matthew Leingang
The document outlines a calculus class lecture on the fundamental theorem of calculus, including recalling the second fundamental theorem, stating the first fundamental theorem, and providing examples of differentiating functions defined by integrals. It gives announcements for upcoming class sections and exam dates, lists the objectives of the current section, and provides an outline of topics to be covered including area as a function, statements and proofs of the theorems, and applications to differentiation.
The document summarizes the Wang-Landau algorithm and some of its improvements. The Wang-Landau algorithm is an adaptive Markov chain Monte Carlo method that iteratively estimates the density of states of a system. It partitions the state space into bins and iteratively adjusts estimates of the density within each bin so that the generated samples spend an equal amount of time in each bin. The algorithm has been improved through automatic binning methods, adaptive proposal distributions, and using parallel interacting chains. An example application to variable selection is also discussed.
Lesson 18: Maximum and Minimum Values (Section 021 slides)Matthew Leingang
The document is a lecture on finding maximum and minimum values of functions. It begins with announcements about an upcoming quiz. It then outlines the objectives of understanding the Extreme Value Theorem and Fermat's Theorem, and using the Closed Interval Method. The document discusses the Extreme Value Theorem, which states that a continuous function on a closed interval attains absolute maximum and minimum values. It also covers Fermat's Theorem, which states that if a function has a local extremum where it is differentiable, the derivative is equal to 0 at that point. Examples are provided to illustrate the importance of the hypotheses in the Extreme Value Theorem.
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 26: The Fundamental Theorem of Calculus (Section 041 slides)Matthew Leingang
g(x) represents the area under the curve of f(t) from 0 to x. As x increases from 0 to 10, g(x) will increase, representing the accumulating area under f(t) over the interval [0,x].
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
This document summarizes a paper on the comparison-based complexity of multi-objective optimization. It outlines that the paper presents complexity upper and lower bounds for finding the Pareto front and Pareto set in multi-objective optimization problems. For upper bounds, it shows the complexity is O(Nd log(1/e)) for finding the whole Pareto set and O(N+1-d log(1/e)) for finding a single point, where N is the dimension and d is the number of objectives. For lower bounds, it applies a proof technique using packing numbers from the mono-objective case to the multi-objective case to derive tight complexity bounds.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 handout)Matthew Leingang
This document contains lecture notes on the fundamental theorem of calculus from a Calculus I class. The notes discuss:
1) The first and second fundamental theorems of calculus, which relate differentiation and integration as inverse processes.
2) How to use the first fundamental theorem to differentiate functions defined by integrals.
3) Biographies of several mathematicians involved in the development of calculus, including Newton, Leibniz, Gregory and Barrow.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Matthew Leingang
The document outlines a calculus class lecture on the fundamental theorem of calculus, including recalling the second fundamental theorem, stating the first fundamental theorem, and providing examples of differentiating functions defined by integrals. It gives announcements for upcoming class sections and exam dates, lists the objectives of the current section, and provides an outline of topics to be covered including area as a function, statements and proofs of the theorems, and applications to differentiation.
The document summarizes the Wang-Landau algorithm and some of its improvements. The Wang-Landau algorithm is an adaptive Markov chain Monte Carlo method that iteratively estimates the density of states of a system. It partitions the state space into bins and iteratively adjusts estimates of the density within each bin so that the generated samples spend an equal amount of time in each bin. The algorithm has been improved through automatic binning methods, adaptive proposal distributions, and using parallel interacting chains. An example application to variable selection is also discussed.
A brief introduction to Hartree-Fock and TDDFTJiahao Chen
The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
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.
This document summarizes Fan XUE's PhD viva voce presentation titled "SPOT: A suboptimum- and proportion-based heuristic generation method for combinatorial optimization problems". The presentation introduces the SPOT algorithm, which uses sampled proportions and suboptima from problem instances to generate heuristics on-the-fly through supervised learning. The SPOT algorithm aims to make use of instance-specific characteristics while avoiding the no-free-lunch theorems. It formalizes the problem domain as an unconstrained equinumerous assignments model to facilitate the learning. The algorithm proceeds in three main phases: sampling and transformation, attribute population, and supervised learning from a library of methods.
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small ...Hector Zenil
Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other. This work presents our work on a beautiful connection between the fractal dimension of space-time diagrams of Turing machines and their time complexity. Presented at Machines, Computations and Universality (MCU) 2013, Zurich, Switzerland.
Fractal dimension versus Computational ComplexityHector Zenil
We investigate connections and tradeoffs between two important complexity measures: fractal dimension and computational (time) complexity. We report exciting results applied to space-time diagrams of small Turing machines with precise mathematical relations and formal conjectures connecting these measures. The preprint of the paper is available at: http://arxiv.org/abs/1309.1779
The document discusses conditional random fields (CRFs), which are probabilistic models used for structured prediction problems. CRFs define a conditional probability distribution p(y|x) via an exponential family form using feature functions. Maximum likelihood, maximum entropy, and MAP estimation techniques can be used to learn the parameters of a CRF by minimizing the negative conditional log-likelihood of labeled training data. Gradient descent or other numerical optimization methods are then required to perform the actual minimization. CRFs provide a principled probabilistic approach to learning the relationships between inputs x and structured outputs y.
The document presents a new approach called FPERT (Fuzzy PERT) for project network analysis that accounts for uncertainty in activity times. It begins with an overview of FPERT and its advantages over conventional PERT. It then discusses key concepts needed for FPERT like fuzzy sets, membership functions, and α-cuts. The document outlines the steps of the proposed FPERT method and provides an example calculation. It concludes by introducing notation that will be used to calculate earliest start, earliest finish, latest start and latest finish times for activities.
Inversion Theorem for Generalized Fractional Hilbert Transforminventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Fractional Fourier Transform Moments Based On Ambiguity FunctionCSCJournals
The fractional Fourier transform can be considered as a rotated standard Fourier transform in general and its benefit in signal processing is growing to be known more. Noise removing is one application that fractional Fourier transform can do well if the signal dilation is perfectly known. In this paper, we have computed the first and second order of moments of fractional Fourier transform according to the ambiguity function exactly. In addition we have derived some relations between time and spectral moments with those obtained in fractional domain. We will prove that the first moment in fractional Fourier transform can also be considered as a rotated the time and frequency gravity in general. For more satisfaction, we choose five different types signals and obtain analytically their fractional Fourier transform and the first and second-order moments in time and frequency and fractional domains as well.
A Numerical Method for the Evaluation of Kolmogorov Complexity, An alternativ...Hector Zenil
We present a novel alternative method (other than using compression algorithms) to approximate the algorithmic complexity of a string by calculating its algorithmic probability and applying Chaitin-Levin's coding theorem.
This document discusses finding the maximum and minimum values of functions. It introduces the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then it attains both a maximum and minimum value on that interval. It also discusses Fermat's Theorem, which relates local extrema of a differentiable function to its derivative. Examples are provided to illustrate these concepts.
We define what it means for a function to have a maximum or minimum value, and explain the Extreme Value Theorem, which indicates these maxima and minima must be there under certain conditions.
Fermat's Theorem says that at differentiable extreme points, the derivative should be zero, and thus we arrive at a technique for finding extrema: look among the endpoints of the domain of definition and the critical points of the function.
There's also a little digression on Fermat's Last theorem, which is not related to calculus but is a big deal in recent mathematical history.
The document provides information on the current status of SPIRAL (System for Programming Integrated Algorithm Libraries). SPIRAL is a system developed by researchers at various universities to automatically generate highly optimized implementations of digital signal processing algorithms. It lists the main researchers and students involved in the project. It is supported by a research grant from DARPA. The document then provides background information on challenges in scientific computing due to Moore's Law and outlines SPIRAL's approach to address these challenges through automating implementation, optimization, and platform adaptation of DSP algorithms.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 handout)Matthew Leingang
This document outlines lecture notes on the fundamental theorem of calculus. It begins with announcements about upcoming lecture dates and exam schedules. It then provides objectives of explaining the fundamental theorems of calculus and using them to find derivatives and integrals. The lecture covers the first and second fundamental theorems, including proofs and examples. It also profiles several important mathematicians who contributed to the development of calculus, such as Newton, Leibniz, Gregory and Barrow.
Calculating Projections via Type CheckingDaisuke BEKKI
Bekki Daisuke and Miho Sato (2015).
A presentation in TYpe Theory and LExical Semantics (TYTLES) in the 27th European Summer School in Logic, Language and Information (ESSLLI 2015), Barcelona, Spain.
This document discusses representing signals as tempered distributions to provide a unified framework for signal processing theory. It introduces:
- Tempered distributions, which allow discrete-time signals to be expressed as distributions involving the Dirac delta function.
- Conditions for continuous- and discrete-time signals to be considered tempered distributions. Continuous signals must satisfy certain growth conditions, while discrete signals must be bounded by a polynomial.
- Existing definitions of multiplication and convolution of distributions have limitations for signal processing. The paper proposes new definitions of multiplication and convolution of distributions appropriate for unified signal processing theory.
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.
The document discusses the concept of "Lean IT", which aims to deliver maximum value to customers at minimum cost by focusing IT resources on high-value activities and reducing waste. It describes how CA's Enterprise IT Management (EITM) solutions can uniquely enable Lean IT by visualizing, automating and optimizing IT systems and processes across areas such as application performance management, service management, project and portfolio management, infrastructure management, and security management.
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.
This document introduces unit vectors and how they can be used to write the components of a vector. It defines a unit vector as having a magnitude of 1 and lists the standard unit vectors i, j, k. It then shows how any vector can be written as the sum of its components in the direction of each unit vector, using examples like 3i + 4j + 7k. Finally, it provides additional examples of writing vectors in component form using unit vectors and calculating the sum of two vectors.
A brief introduction to Hartree-Fock and TDDFTJiahao Chen
The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
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.
This document summarizes Fan XUE's PhD viva voce presentation titled "SPOT: A suboptimum- and proportion-based heuristic generation method for combinatorial optimization problems". The presentation introduces the SPOT algorithm, which uses sampled proportions and suboptima from problem instances to generate heuristics on-the-fly through supervised learning. The SPOT algorithm aims to make use of instance-specific characteristics while avoiding the no-free-lunch theorems. It formalizes the problem domain as an unconstrained equinumerous assignments model to facilitate the learning. The algorithm proceeds in three main phases: sampling and transformation, attribute population, and supervised learning from a library of methods.
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small ...Hector Zenil
Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other. This work presents our work on a beautiful connection between the fractal dimension of space-time diagrams of Turing machines and their time complexity. Presented at Machines, Computations and Universality (MCU) 2013, Zurich, Switzerland.
Fractal dimension versus Computational ComplexityHector Zenil
We investigate connections and tradeoffs between two important complexity measures: fractal dimension and computational (time) complexity. We report exciting results applied to space-time diagrams of small Turing machines with precise mathematical relations and formal conjectures connecting these measures. The preprint of the paper is available at: http://arxiv.org/abs/1309.1779
The document discusses conditional random fields (CRFs), which are probabilistic models used for structured prediction problems. CRFs define a conditional probability distribution p(y|x) via an exponential family form using feature functions. Maximum likelihood, maximum entropy, and MAP estimation techniques can be used to learn the parameters of a CRF by minimizing the negative conditional log-likelihood of labeled training data. Gradient descent or other numerical optimization methods are then required to perform the actual minimization. CRFs provide a principled probabilistic approach to learning the relationships between inputs x and structured outputs y.
The document presents a new approach called FPERT (Fuzzy PERT) for project network analysis that accounts for uncertainty in activity times. It begins with an overview of FPERT and its advantages over conventional PERT. It then discusses key concepts needed for FPERT like fuzzy sets, membership functions, and α-cuts. The document outlines the steps of the proposed FPERT method and provides an example calculation. It concludes by introducing notation that will be used to calculate earliest start, earliest finish, latest start and latest finish times for activities.
Inversion Theorem for Generalized Fractional Hilbert Transforminventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Fractional Fourier Transform Moments Based On Ambiguity FunctionCSCJournals
The fractional Fourier transform can be considered as a rotated standard Fourier transform in general and its benefit in signal processing is growing to be known more. Noise removing is one application that fractional Fourier transform can do well if the signal dilation is perfectly known. In this paper, we have computed the first and second order of moments of fractional Fourier transform according to the ambiguity function exactly. In addition we have derived some relations between time and spectral moments with those obtained in fractional domain. We will prove that the first moment in fractional Fourier transform can also be considered as a rotated the time and frequency gravity in general. For more satisfaction, we choose five different types signals and obtain analytically their fractional Fourier transform and the first and second-order moments in time and frequency and fractional domains as well.
A Numerical Method for the Evaluation of Kolmogorov Complexity, An alternativ...Hector Zenil
We present a novel alternative method (other than using compression algorithms) to approximate the algorithmic complexity of a string by calculating its algorithmic probability and applying Chaitin-Levin's coding theorem.
This document discusses finding the maximum and minimum values of functions. It introduces the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then it attains both a maximum and minimum value on that interval. It also discusses Fermat's Theorem, which relates local extrema of a differentiable function to its derivative. Examples are provided to illustrate these concepts.
We define what it means for a function to have a maximum or minimum value, and explain the Extreme Value Theorem, which indicates these maxima and minima must be there under certain conditions.
Fermat's Theorem says that at differentiable extreme points, the derivative should be zero, and thus we arrive at a technique for finding extrema: look among the endpoints of the domain of definition and the critical points of the function.
There's also a little digression on Fermat's Last theorem, which is not related to calculus but is a big deal in recent mathematical history.
The document provides information on the current status of SPIRAL (System for Programming Integrated Algorithm Libraries). SPIRAL is a system developed by researchers at various universities to automatically generate highly optimized implementations of digital signal processing algorithms. It lists the main researchers and students involved in the project. It is supported by a research grant from DARPA. The document then provides background information on challenges in scientific computing due to Moore's Law and outlines SPIRAL's approach to address these challenges through automating implementation, optimization, and platform adaptation of DSP algorithms.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 handout)Matthew Leingang
This document outlines lecture notes on the fundamental theorem of calculus. It begins with announcements about upcoming lecture dates and exam schedules. It then provides objectives of explaining the fundamental theorems of calculus and using them to find derivatives and integrals. The lecture covers the first and second fundamental theorems, including proofs and examples. It also profiles several important mathematicians who contributed to the development of calculus, such as Newton, Leibniz, Gregory and Barrow.
Calculating Projections via Type CheckingDaisuke BEKKI
Bekki Daisuke and Miho Sato (2015).
A presentation in TYpe Theory and LExical Semantics (TYTLES) in the 27th European Summer School in Logic, Language and Information (ESSLLI 2015), Barcelona, Spain.
This document discusses representing signals as tempered distributions to provide a unified framework for signal processing theory. It introduces:
- Tempered distributions, which allow discrete-time signals to be expressed as distributions involving the Dirac delta function.
- Conditions for continuous- and discrete-time signals to be considered tempered distributions. Continuous signals must satisfy certain growth conditions, while discrete signals must be bounded by a polynomial.
- Existing definitions of multiplication and convolution of distributions have limitations for signal processing. The paper proposes new definitions of multiplication and convolution of distributions appropriate for unified signal processing theory.
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.
The document discusses the concept of "Lean IT", which aims to deliver maximum value to customers at minimum cost by focusing IT resources on high-value activities and reducing waste. It describes how CA's Enterprise IT Management (EITM) solutions can uniquely enable Lean IT by visualizing, automating and optimizing IT systems and processes across areas such as application performance management, service management, project and portfolio management, infrastructure management, and security management.
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.
This document introduces unit vectors and how they can be used to write the components of a vector. It defines a unit vector as having a magnitude of 1 and lists the standard unit vectors i, j, k. It then shows how any vector can be written as the sum of its components in the direction of each unit vector, using examples like 3i + 4j + 7k. Finally, it provides additional examples of writing vectors in component form using unit vectors and calculating the sum of two vectors.
1. The document discusses techniques for finding extrema of functions, including absolute and local extrema. Critical points, endpoints, and the first and second derivative tests are covered.
2. The mean value theorem and Rolle's theorem are summarized. The mean value theorem relates the average and instantaneous rates of change over an interval.
3. Optimization problems can be solved by setting the derivative of the objective function equal to zero to find critical points corresponding to maxima or minima.
4. Newton's method is presented as an iterative process for approximating solutions to equations, using tangent lines to generate a sequence of improving approximations.
5. Anti-derivatives are defined as functions whose derivatives are a given
This document discusses key concepts related to finding local extrema and inflection points of functions including using the first and second derivatives to determine concavity, where local maxima and minima occur based on changes from increasing to decreasing or vice versa, and how inflection points involve a change in concavity. It also covers using critical numbers found by setting the derivative equal to zero to help identify local extrema and defines absolute extrema as the highest or lowest values over a given interval.
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
The document discusses concepts related to calculus including tangent planes, normal lines, and linear approximations. It provides definitions and equations for calculating the tangent plane to a surface, the normal line to a curve or surface, and using the tangent line as a linear approximation near a given point on a function. Examples are given to demonstrate finding the total derivative of a function and using the tangent line as a linearization.
Lesson 18: Maximum and Minimum Values (Section 021 slides)Mel Anthony Pepito
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.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
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 26: The Fundamental Theorem of Calculus (Section 041 slides)Mel Anthony Pepito
The document discusses the fundamental theorem of calculus. It begins by outlining the topics to be covered, including a review of the second fundamental theorem of calculus, an explanation of the first fundamental theorem of calculus, and examples of differentiating functions defined by integrals. It then provides more details on these topics, such as defining the integral as a limit, stating the second fundamental theorem, using integrals to represent concepts like distance traveled and mass, and working through an example of finding the derivative of a function defined by an integral.
The document is a lecture note on the fundamental theorem of calculus from a Calculus I class at New York University. It provides announcements about upcoming exams and assignments. It then outlines the key topics to be covered, including the first fundamental theorem of calculus and how to differentiate functions defined by integrals. Examples are provided to illustrate using integrals to find the area under a curve and how this relates to the derivative of the area function.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Mel Anthony Pepito
The document provides an overview of Section 5.4 on the Fundamental Theorem of Calculus from a Calculus I course at New York University. It outlines topics to be covered, including recalling the Second Fundamental Theorem, stating the First Fundamental Theorem, and differentiating functions defined by integrals. Examples are provided to illustrate using the theorems to find derivatives and integrals.
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
(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.
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.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
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.
- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions.
- There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24.
- The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.
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.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also covers derivatives of sine and cosine. Examples are provided, like finding the derivative of the squaring function x^2, which is 2x. Notation for derivatives is explained, including Leibniz notation. The concept of the second derivative is also introduced.
This document contains lecture notes from a Calculus I class at New York University. It discusses the definition of continuity for functions, examples of continuous and discontinuous functions, and properties of continuous functions like sums, products, and compositions of continuous functions being continuous. It also addresses trigonometric functions like sin, cos, tan, and sec being continuous on their domains.
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 handout version to take notes on.
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.
Similar to Lesson 18: Maximum and Minimum Values (Section 041 slides) (20)
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
Taking AI to the Next Level in Manufacturing.pdfssuserfac0301
Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
1. How quickly AI is being implemented in manufacturing.
2. Which barriers stand in the way of AI adoption.
3. How data quality and governance form the backbone of AI.
4. Organizational processes and structures that may inhibit effective AI adoption.
6. Ideas and approaches to help build your organization's AI strategy.
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
Dandelion Hashtable: beyond billion requests per second on a commodity serverAntonios Katsarakis
This slide deck presents DLHT, a concurrent in-memory hashtable. Despite efforts to optimize hashtables, that go as far as sacrificing core functionality, state-of-the-art designs still incur multiple memory accesses per request and block request processing in three cases. First, most hashtables block while waiting for data to be retrieved from memory. Second, open-addressing designs, which represent the current state-of-the-art, either cannot free index slots on deletes or must block all requests to do so. Third, index resizes block every request until all objects are copied to the new index. Defying folklore wisdom, DLHT forgoes open-addressing and adopts a fully-featured and memory-aware closed-addressing design based on bounded cache-line-chaining. This design offers lock-free index operations and deletes that free slots instantly, (2) completes most requests with a single memory access, (3) utilizes software prefetching to hide memory latencies, and (4) employs a novel non-blocking and parallel resizing. In a commodity server and a memory-resident workload, DLHT surpasses 1.6B requests per second and provides 3.5x (12x) the throughput of the state-of-the-art closed-addressing (open-addressing) resizable hashtable on Gets (Deletes).
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
Your One-Stop Shop for Python Success: Top 10 US Python Development Providersakankshawande
Simplify your search for a reliable Python development partner! This list presents the top 10 trusted US providers offering comprehensive Python development services, ensuring your project's success from conception to completion.
Generating privacy-protected synthetic data using Secludy and MilvusZilliz
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For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/how-axelera-ai-uses-digital-compute-in-memory-to-deliver-fast-and-energy-efficient-computer-vision-a-presentation-from-axelera-ai/
Bram Verhoef, Head of Machine Learning at Axelera AI, presents the “How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-efficient Computer Vision” tutorial at the May 2024 Embedded Vision Summit.
As artificial intelligence inference transitions from cloud environments to edge locations, computer vision applications achieve heightened responsiveness, reliability and privacy. This migration, however, introduces the challenge of operating within the stringent confines of resource constraints typical at the edge, including small form factors, low energy budgets and diminished memory and computational capacities. Axelera AI addresses these challenges through an innovative approach of performing digital computations within memory itself. This technique facilitates the realization of high-performance, energy-efficient and cost-effective computer vision capabilities at the thin and thick edge, extending the frontier of what is achievable with current technologies.
In this presentation, Verhoef unveils his company’s pioneering chip technology and demonstrates its capacity to deliver exceptional frames-per-second performance across a range of standard computer vision networks typical of applications in security, surveillance and the industrial sector. This shows that advanced computer vision can be accessible and efficient, even at the very edge of our technological ecosystem.
Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
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What is an RPA CoE? Session 1 – CoE VisionDianaGray10
In the first session, we will review the organization's vision and how this has an impact on the COE Structure.
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• How do the organization’s priorities determine CoE Structure?
Speaker:
Chris Bolin, Senior Intelligent Automation Architect Anika Systems
Lesson 18: Maximum and Minimum Values (Section 041 slides)
1. Section 4.1
Maximum and Minimum Values
V63.0121.041, Calculus I
New York University
November 8, 2010
Announcements
Quiz 4 on Sections 3.3, 3.4, 3.5, and 3.7 next week (November
16, 18, or 19)
. . . . . .
2. Announcements
Quiz 4 on Sections 3.3,
3.4, 3.5, and 3.7 next week
(November 16, 18, or 19)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 2 / 34
3. Objectives
Understand and be able to
explain the statement of
the Extreme Value
Theorem.
Understand and be able to
explain the statement of
Fermat’s Theorem.
Use the Closed Interval
Method to find the extreme
values of a function defined
on a closed interval.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 3 / 34
4. Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)
Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
. . . . . .
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 4 / 34
6. Why go to the extremes?
Rationally speaking, it is
advantageous to find the
extreme values of a
function (maximize profit,
minimize costs, etc.)
Pierre-Louis Maupertuis
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values
(1698–1759) 8, 2010
November 6 / 34
7. Design
.
Image credit: Jason Tromm .
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 7 / 34
8. Why go to the extremes?
Rationally speaking, it is
advantageous to find the
extreme values of a
function (maximize profit,
minimize costs, etc.)
Many laws of science are
derived from minimizing
principles.
Pierre-Louis Maupertuis
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values
(1698–1759) 8, 2010
November 8 / 34
10. Why go to the extremes?
Rationally speaking, it is
advantageous to find the
extreme values of a
function (maximize profit,
minimize costs, etc.)
Many laws of science are
derived from minimizing
principles.
Maupertuis’ principle:
“Action is minimized
through the wisdom of
God.”
Pierre-Louis Maupertuis
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values
(1698–1759) 8, 2010
November 10 / 34
11. Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)
Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 11 / 34
12. Extreme points and values
Definition
Let f have domain D.
.
.
Image credit: Patrick Q
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 12 / 34
13. Extreme points and values
Definition
Let f have domain D.
The function f has an absolute
maximum (or global maximum)
(respectively, absolute minimum) at c if
f(c) ≥ f(x) (respectively, f(c) ≤ f(x)) for all
x in D
.
.
Image credit: Patrick Q
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 12 / 34
14. Extreme points and values
Definition
Let f have domain D.
The function f has an absolute
maximum (or global maximum)
(respectively, absolute minimum) at c if
f(c) ≥ f(x) (respectively, f(c) ≤ f(x)) for all
x in D
The number f(c) is called the maximum
value (respectively, minimum value) of f
on D.
.
.
Image credit: Patrick Q
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 12 / 34
15. Extreme points and values
Definition
Let f have domain D.
The function f has an absolute
maximum (or global maximum)
(respectively, absolute minimum) at c if
f(c) ≥ f(x) (respectively, f(c) ≤ f(x)) for all
x in D
The number f(c) is called the maximum
value (respectively, minimum value) of f
on D.
.
An extremum is either a maximum or a
minimum. An extreme value is either a
.mage credit: Patrick Q value or minimum value.
I
maximum
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 12 / 34
16. The Extreme Value Theorem
Theorem (The Extreme Value Theorem)
Let f be a function which is continuous on the closed interval [a, b].
Then f attains an absolute maximum value f(c) and an absolute
minimum value f(d) at numbers c and d in [a, b].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 13 / 34
17. The Extreme Value Theorem
Theorem (The Extreme Value Theorem)
Let f be a function which is continuous on the closed interval [a, b].
Then f attains an absolute maximum value f(c) and an absolute
minimum value f(d) at numbers c and d in [a, b].
.
.
. .
a
. b
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 13 / 34
18. The Extreme Value Theorem
Theorem (The Extreme Value Theorem)
Let f be a function which is continuous on the closed interval [a, b].
Then f attains an absolute maximum value f(c) and an absolute
minimum value f(d) at numbers c and d in [a, b].
.
maximum .(c)
f
.
value
. .
minimum .(d)
f
.
value
. . ..
a
. d c
b
.
minimum maximum
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 13 / 34
19. No proof of EVT forthcoming
This theorem is very hard to prove without using technical facts
about continuous functions and closed intervals.
But we can show the importance of each of the hypotheses.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 14 / 34
20. Bad Example #1
Example
Consider the function
{
x 0≤x<1
f(x) =
x − 2 1 ≤ x ≤ 2.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 15 / 34
21. Bad Example #1
Example
Consider the function .
{
x 0≤x<1
f(x) = . .
| .
x − 2 1 ≤ x ≤ 2. 1
.
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 15 / 34
22. Bad Example #1
Example
Consider the function .
{
x 0≤x<1
f(x) = . .
| .
x − 2 1 ≤ x ≤ 2. 1
.
.
Then although values of f(x) get arbitrarily close to 1 and never bigger
than 1, 1 is not the maximum value of f on [0, 1] because it is never
achieved.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 15 / 34
23. Bad Example #1
Example
Consider the function .
{
x 0≤x<1
f(x) = . .
| .
x − 2 1 ≤ x ≤ 2. 1
.
.
Then although values of f(x) get arbitrarily close to 1 and never bigger
than 1, 1 is not the maximum value of f on [0, 1] because it is never
achieved. This does not violate EVT because f is not continuous.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 15 / 34
24. Bad Example #2
Example
Consider the function f(x) = x restricted to the interval [0, 1).
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 16 / 34
25. Bad Example #2
Example
Consider the function f(x) = x restricted to the interval [0, 1).
.
. .
|
1
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 16 / 34
26. Bad Example #2
Example
Consider the function f(x) = x restricted to the interval [0, 1).
.
. .
|
1
.
There is still no maximum value (values get arbitrarily close to 1 but do
not achieve it).
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 16 / 34
27. Bad Example #2
Example
Consider the function f(x) = x restricted to the interval [0, 1).
.
. .
|
1
.
There is still no maximum value (values get arbitrarily close to 1 but do
not achieve it). This does not violate EVT because the domain is not
closed.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 16 / 34
28. Final Bad Example
Example
1
Consider the function f(x) = is continuous on the closed interval
x
[1, ∞).
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 17 / 34
29. Final Bad Example
Example
1
Consider the function f(x) = is continuous on the closed interval
x
[1, ∞).
.
. .
1
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 17 / 34
30. Final Bad Example
Example
1
Consider the function f(x) = is continuous on the closed interval
x
[1, ∞).
.
. .
1
.
There is no minimum value (values get arbitrarily close to 0 but do not
achieve it).
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 17 / 34
31. Final Bad Example
Example
1
Consider the function f(x) = is continuous on the closed interval
x
[1, ∞).
.
. .
1
.
There is no minimum value (values get arbitrarily close to 0 but do not
achieve it). This does not violate EVT because the domain is not
bounded.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 17 / 34
32. Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)
Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 18 / 34
33. Local extrema
.
Definition
A function f has a local maximum or relative maximum at c if f(c) ≥ f(x)
when x is near c. This means that f(c) ≥ f(x) for all x in some open interval
containing c.
Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c.
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 19 / 34
34. Local extrema
.
Definition
A function f has a local maximum or relative maximum at c if f(c) ≥ f(x)
when x is near c. This means that f(c) ≥ f(x) for all x in some open interval
containing c.
Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c.
.
.
.
.
.|.
. . .
|
a
. local local b
.
maximum minimum
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 19 / 34
35. Local extrema
.
So a local extremum must be inside the domain of f (not on the end).
A global extremum that is inside the domain is a local extremum.
.
.
.
.
.|.
. . ..
|
a
. local local and global . global
b
maximum min max
.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 19 / 34
36. Fermat's Theorem
Theorem (Fermat’s Theorem)
Suppose f has a local extremum at c and f is differentiable at c. Then
f′ (c) = 0.
.
.
.
.
...
| . .
|
a
. local local b
.
maximum minimum
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 21 / 34
37. Fermat's Theorem
Theorem (Fermat’s Theorem)
Suppose f has a local extremum at c and f is differentiable at c. Then
f′ (c) = 0.
.
.
.
.
...
| . .
|
a
. local local b
.
maximum minimum
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 21 / 34
38. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
39. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
If x is slightly greater than c, f(x) ≤ f(c). This means
f(x) − f(c)
≤0
x−c
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
40. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
If x is slightly greater than c, f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≤ 0 =⇒ lim+ ≤0
x−c x→c x−c
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
41. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
If x is slightly greater than c, f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≤ 0 =⇒ lim+ ≤0
x−c x→c x−c
The same will be true on the other end: if x is slightly less than c,
f(x) ≤ f(c). This means
f(x) − f(c)
≥0
x−c
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
42. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
If x is slightly greater than c, f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≤ 0 =⇒ lim+ ≤0
x−c x→c x−c
The same will be true on the other end: if x is slightly less than c,
f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≥ 0 =⇒ lim ≥0
x−c x→c − x−c
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
43. Sketch of proof of Fermat's Theorem
Suppose that f has a local maximum at c.
If x is slightly greater than c, f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≤ 0 =⇒ lim+ ≤0
x−c x→c x−c
The same will be true on the other end: if x is slightly less than c,
f(x) ≤ f(c). This means
f(x) − f(c) f(x) − f(c)
≥ 0 =⇒ lim ≥0
x−c x→c − x−c
f(x) − f(c)
Since the limit f′ (c) = lim exists, it must be 0.
x→c x−c
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 22 / 34
44. Meet the Mathematician: Pierre de Fermat
1601–1665
Lawyer and number
theorist
Proved many theorems,
didn’t quite prove his last
one
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 23 / 34
45. Tangent: Fermat's Last Theorem
Plenty of solutions to
x2 + y2 = z2 among
positive whole numbers
(e.g., x = 3, y = 4, z = 5)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 24 / 34
46. Tangent: Fermat's Last Theorem
Plenty of solutions to
x2 + y2 = z2 among
positive whole numbers
(e.g., x = 3, y = 4, z = 5)
No solutions to
x3 + y3 = z3 among
positive whole numbers
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 24 / 34
47. Tangent: Fermat's Last Theorem
Plenty of solutions to
x2 + y2 = z2 among
positive whole numbers
(e.g., x = 3, y = 4, z = 5)
No solutions to
x3 + y3 = z3 among
positive whole numbers
Fermat claimed no
solutions to xn + yn = zn
but didn’t write down his
proof
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 24 / 34
48. Tangent: Fermat's Last Theorem
Plenty of solutions to
x2 + y2 = z2 among
positive whole numbers
(e.g., x = 3, y = 4, z = 5)
No solutions to
x3 + y3 = z3 among
positive whole numbers
Fermat claimed no
solutions to xn + yn = zn
but didn’t write down his
proof
Not solved until 1998!
(Taylor–Wiles)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 24 / 34
49. Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)
Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 25 / 34
50. Flowchart for placing extrema
Thanks to Fermat
Suppose f is a continuous function on the closed, bounded interval
[a, b], and c is a global maximum point.
.
. . c is a
start
local max
. . .
Is c an Is f diff’ble f is not
n
.o n
.o
endpoint? at c? diff at c
y
. es y
. es
. .
c = a or
f′ (c) = 0
c = b
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 26 / 34
51. The Closed Interval Method
This means to find the maximum value of f on [a, b], we need to:
Evaluate f at the endpoints a and b
Evaluate f at the critical points or critical numbers 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 or most negative function value are
the global minimum points.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 27 / 34
52. Outline
Introduction
The Extreme Value Theorem
Fermat’s Theorem (not the last one)
Tangent: Fermat’s Last Theorem
The Closed Interval Method
Examples
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 28 / 34
53. Extreme values of a linear function
Example
Find the extreme values of f(x) = 2x − 5 on [−1, 2].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 29 / 34
54. Extreme values of a linear function
Example
Find the extreme values of f(x) = 2x − 5 on [−1, 2].
Solution
Since f′ (x) = 2, which is never zero, we have no critical points and we
need only investigate the endpoints:
f(−1) = 2(−1) − 5 = −7
f(2) = 2(2) − 5 = −1
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 29 / 34
55. Extreme values of a linear function
Example
Find the extreme values of f(x) = 2x − 5 on [−1, 2].
Solution
Since f′ (x) = 2, which is never zero, we have no critical points and we
need only investigate the endpoints:
f(−1) = 2(−1) − 5 = −7
f(2) = 2(2) − 5 = −1
So
The absolute minimum (point) is at −1; the minimum value is −7.
The absolute maximum (point) is at 2; the maximum value is −1.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 29 / 34
56. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
57. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
58. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) =
f(0) =
f(2) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
59. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) = 0
f(0) =
f(2) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
60. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) = 0
f(0) = − 1
f(2) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
61. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) = 0
f(0) = − 1
f(2) = 3
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
62. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) = 0
f(0) = − 1 (absolute min)
f(2) = 3
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
63. Extreme values of a quadratic function
Example
Find the extreme values of f(x) = x2 − 1 on [−1, 2].
Solution
We have f′ (x) = 2x, which is zero when x = 0. So our points to check
are:
f(−1) = 0
f(0) = − 1 (absolute min)
f(2) = 3 (absolute max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 30 / 34
64. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
65. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
66. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
67. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
68. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4
f(0) = 1
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
69. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4
f(0) = 1
f(1) = 0
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
70. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4
f(0) = 1
f(1) = 0
f(2) = 5
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
71. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4 (global min)
f(0) = 1
f(1) = 0
f(2) = 5
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
72. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4 (global min)
f(0) = 1
f(1) = 0
f(2) = 5 (global max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
73. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4 (global min)
f(0) = 1 (local max)
f(1) = 0
f(2) = 5 (global max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
74. Extreme values of a cubic function
Example
Find the extreme values of f(x) = 2x3 − 3x2 + 1 on [−1, 2].
Solution
Since f′ (x) = 6x2 − 6x = 6x(x − 1), we have critical points at x = 0 and
x = 1. The values to check are
f(−1) = − 4 (global min)
f(0) = 1 (local max)
f(1) = 0 (local min)
f(2) = 5 (global max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 31 / 34
75. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
76. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0.
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
77. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
78. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
79. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341
f(0) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
80. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341
f(0) = 0
f(2) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
81. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341
f(0) = 0
f(2) = 6.3496
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
82. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341
f(0) = 0 (absolute min)
f(2) = 6.3496
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
83. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341
f(0) = 0 (absolute min)
f(2) = 6.3496 (absolute max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
84. Extreme values of an algebraic function
Example
Find the extreme values of f(x) = x2/3 (x + 2) on [−1, 2].
Solution
Write f(x) = x5/3 + 2x2/3 , then
5 2/3 4 −1/3 1 −1/3
f′ (x) = x + x = x (5x + 4)
3 3 3
Thus f′ (−4/5) = 0 and f is not differentiable at 0. So our points to
check are:
f(−1) = 1
f(−4/5) = 1.0341 (relative max)
f(0) = 0 (absolute min)
f(2) = 6.3496 (absolute max)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 32 / 34
85. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
86. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.)
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
87. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
88. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) = 0
f(0) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
89. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) = 0
f(0) = 2
f(1) =
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
90. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) = 0
f(0) = 2
√
f(1) = 3
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
91. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) = 0 (absolute min)
f(0) = 2
√
f(1) = 3
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
92. Extreme values of another algebraic function
Example
√
Find the extreme values of f(x) = 4 − x2 on [−2, 1].
Solution
x
We have f′ (x) = − √ , which is zero when x = 0. (f is not
4 − x2
differentiable at ±2 as well.) So our points to check are:
f(−2) = 0 (absolute min)
f(0) = 2 (absolute max)
√
f(1) = 3
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 33 / 34
93. Summary
The Extreme Value Theorem: a continuous function on a closed
interval must achieve its max and min
Fermat’s Theorem: local extrema are critical points
The Closed Interval Method: an algorithm for finding global
extrema
Show your work unless you want to end up like Fermat!
V63.0121.041, Calculus I (NYU) Section 4.1 Maximum and Minimum Values November 8, 2010 34 / 34