The document is a lecture on calculus limits from a Calculus I class. It contains announcements about homework due dates and rubrics. It then discusses the informal definition of a limit, using an "error-tolerance game" to determine if a limit exists for a given function. Examples are provided to demonstrate how this game would be played for specific functions, such as determining the limit of x^2 as x approaches 0. Objectives and an outline of topics are also included.
The yearly lesson plan outlines topics in mathematics to be covered over the year for primary 4 students. It includes 9 topics: whole numbers, fractions, decimals, money, time, length, mass, volume of liquid, and shape and space. Each topic is broken down into learning areas and weeks, with objectives, outcomes, and exams scheduled throughout the year. The plan provides a comprehensive overview of the mathematical concepts primary 4 students will learn in a given year.
The document discusses normal and misere play of impartial games. It begins with an example of the game 0.123 in both normal and misere play. It then explores why there is no misere analogue to the Sprague-Grundy theory for normal play. The document proposes using indistinguishability quotients and computing presentations to generalize the Sprague-Grundy theory to misere play. It provides examples of computing outcomes in misere 0.123 using relations in the commutative semigroup. The document also discusses using the Knuth-Bendix process to obtain a confluent rewriting system to simplify computations in the semigroup.
This document provides an overview of solving linear inequalities and their applications. It discusses key concepts like the definition of a solution to an inequality, determining if a number is a solution, writing interval notation for graphs of inequalities on number lines, and the addition and multiplication properties of inequalities. Examples are provided to demonstrate how to solve inequalities algebraically and graph their solutions. One example solves and graphs the inequality 126t + 1293 > 3000 to determine the years after 2000 that the average cost of tuition and fees at two-year public institutions will be more than $3000.
This document discusses methods for solving inequalities, including:
1) Determining if a number is a solution of an inequality.
2) Graphing inequalities on a number line.
3) Using the addition principle and multiplication principle to solve inequalities algebraically, such as isolating the variable.
4) Applying the addition principle and multiplication principle together to solve more complex inequalities.
This document is a summer math review packet for students entering 8th grade. It contains 50 math problems covering various topics like order of operations, integers, algebraic expressions, fractions, decimals, percents, ratios, proportions, mean, median, mode, range, coordinate system, and transformations. The packet is designed to review these essential math concepts over summer break to prepare students for 8th grade level work.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
The document discusses limits and continuity, explaining what limits are, how to evaluate different types of limits using techniques like direct substitution, dividing out, and rationalizing, and how limits relate to concepts like derivatives, continuity, discontinuities, and the intermediate value theorem. Special trig limits, properties of limits, and how limits can be used to find derivatives are also covered.
The yearly lesson plan outlines topics in mathematics to be covered over the year for primary 4 students. It includes 9 topics: whole numbers, fractions, decimals, money, time, length, mass, volume of liquid, and shape and space. Each topic is broken down into learning areas and weeks, with objectives, outcomes, and exams scheduled throughout the year. The plan provides a comprehensive overview of the mathematical concepts primary 4 students will learn in a given year.
The document discusses normal and misere play of impartial games. It begins with an example of the game 0.123 in both normal and misere play. It then explores why there is no misere analogue to the Sprague-Grundy theory for normal play. The document proposes using indistinguishability quotients and computing presentations to generalize the Sprague-Grundy theory to misere play. It provides examples of computing outcomes in misere 0.123 using relations in the commutative semigroup. The document also discusses using the Knuth-Bendix process to obtain a confluent rewriting system to simplify computations in the semigroup.
This document provides an overview of solving linear inequalities and their applications. It discusses key concepts like the definition of a solution to an inequality, determining if a number is a solution, writing interval notation for graphs of inequalities on number lines, and the addition and multiplication properties of inequalities. Examples are provided to demonstrate how to solve inequalities algebraically and graph their solutions. One example solves and graphs the inequality 126t + 1293 > 3000 to determine the years after 2000 that the average cost of tuition and fees at two-year public institutions will be more than $3000.
This document discusses methods for solving inequalities, including:
1) Determining if a number is a solution of an inequality.
2) Graphing inequalities on a number line.
3) Using the addition principle and multiplication principle to solve inequalities algebraically, such as isolating the variable.
4) Applying the addition principle and multiplication principle together to solve more complex inequalities.
This document is a summer math review packet for students entering 8th grade. It contains 50 math problems covering various topics like order of operations, integers, algebraic expressions, fractions, decimals, percents, ratios, proportions, mean, median, mode, range, coordinate system, and transformations. The packet is designed to review these essential math concepts over summer break to prepare students for 8th grade level work.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
The document discusses limits and continuity, explaining what limits are, how to evaluate different types of limits using techniques like direct substitution, dividing out, and rationalizing, and how limits relate to concepts like derivatives, continuity, discontinuities, and the intermediate value theorem. Special trig limits, properties of limits, and how limits can be used to find derivatives are also covered.
The document defines the limit of a function as x approaches a value a. Specifically:
1) The limit of f(x) as x approaches a, written as lim f(x) = L, means that the values of f(x) can be made arbitrarily close to L by making x sufficiently close to a, without x being equal to a.
2) For the limit to exist, the left and right hand limits must be equal. The function must also be bounded near a and cannot oscillate with increasingly high frequency.
3) Formally, the limit is L if for any positive epsilon, there corresponds a positive delta such that if the absolute value of x - a is less than
This document discusses limits involving infinity in calculus. It begins with definitions of infinite limits, both positive and negative infinity. It provides examples of common infinite limits, such as the limit of 1/x as x approaches 0. It also discusses vertical asymptotes and rules for infinite limits, such as the limit of a sum being infinity if both terms have infinite limits. The document contains examples evaluating limits at points where functions are not continuous.
The document discusses the concept of a limit in calculus. It provides an outline that covers heuristics, errors and tolerances, examples, pathologies, and a precise definition of a limit. It introduces the error-tolerance game used to determine if a limit exists and illustrates this through examples evaluating the limits of x^2 as x approaches 0 and |x|/x as x approaches 0. It also discusses one-sided limits and evaluates the limit of 1/x as x approaches 0 from the right.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
Jane and Jenny, rival Pokemon trainers, get trapped in the forest during a battle between their Pokemon. Jenny's Pokemon, Revoloo, keeps attacking and disturbing the peace, so Jane challenges Jenny to a battle to stop her. During the battle, Revoloo and Jane's Pokemon, Gralinte, are asked mathematical integration questions about solids of revolution that test their abilities. After a long battle involving multiple questions, Revoloo is finally defeated.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It explains the limit laws for addition, subtraction, multiplication, division and nth roots of functions. It uses the error-tolerance game framework to justify the limit laws.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document provides an overview of key concepts related to graphing polynomials, including:
1. Definitions of terms like intervals of increase/decrease, odd/even functions, zeros, and multiplicities.
2. Steps for graphing polynomials which include determining behavior, finding intercepts and zeros, and joining points based on multiplicities.
3. Examples are provided to demonstrate finding zeros and their multiplicities, and graphing a polynomial based on the identified features.
4. Information that can be determined from a polynomial graph, such as degree, leading coefficient, end behavior, intercepts, and intervals of increase/decrease.
Lesson 25: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document discusses indeterminate forms and L'Hopital's rule. It introduces indeterminate forms as limits that can have different values depending on the approach, such as 0/0 or infinity/infinity forms. It then presents L'Hopital's rule, which states that if the limit of the numerator and denominator of a quotient both approach 0, infinity, or negative infinity, the limit can be evaluated by taking the derivative of the numerator and denominator and rearranging terms. Examples are provided to demonstrate how L'Hopital's rule can be used to evaluate indeterminate forms. The document also provides biographical information about Guillaume de l'Hopital, after whom the rule is named.
This document discusses key concepts related to rates of change and derivatives:
1) It defines average rate of change (ARC) as the slope of a secant line on a graph or using the slope formula algebraically, and instantaneous rate of change (IRC) as the slope of the tangent line.
2) It introduces the difference quotient as a way to define ARC and IRC algebraically without a graph by taking the limit as h approaches 0.
3) A derivative is defined as a function that gives the IRC, allowing it to be evaluated at any point without graphing by taking the limit of the difference quotient.
This document provides a summary of key concepts for sequences and series covered on the AP Calculus BC/BCD exam. It begins with definitions of sequences, series, and common notation used. It then drills the reader on topics like finding limits of sequences, identifying geometric and telescoping series, and various convergence tests including ratio, root, integral, comparison, and power series tests. It also covers Taylor and Maclauren series approximations and formulas for common infinite series involving trigonometric, logarithmic, and exponential functions. The document aims to efficiently prepare readers for questions on sequences and series that may appear on the AP exam.
Advanced functions ppt (Chapter 1) part iiTan Yuhang
This document provides definitions and examples to explain how to graph polynomial functions. It discusses determining the degree of the polynomial, the sign of the leading coefficient, end behavior, x- and y-intercepts, and intervals of increase and decrease. Examples are provided to demonstrate how to find zeros, or x-intercepts, and their multiplicities in order to properly graph the polynomial based on these features.
This document discusses key concepts for graphing polynomial functions including:
1. The degree of the polynomial determines the end behavior and maximum number of turns.
2. The sign of the leading coefficient indicates whether the graph faces up or down on both ends.
3. X-intercepts and y-intercepts are found by setting the polynomial equal to 0 or the variable equal to 0.
4. The multiplicity of intercepts determines whether the graph crosses or touches the x-axis at that point.
The document discusses infinite series and sequences. It begins by introducing the concept of an infinite sum and examines whether a sum like 1/2 + 1/4 + 1/8 + ... can be assigned a numerical value. It then defines an infinite series as a sum that continues indefinitely, and a sequence as the individual terms in a series. The key points are:
- An infinite series can be assigned a value by taking the limit of the corresponding sequence of partial sums.
- Common examples like 0.333... and π are actually infinite series.
- Sequences are functions with domain the natural numbers, and their limit is defined similarly to limits of functions.
- Monotonic and bounded sequences are important for
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
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.
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The document defines the limit of a function as x approaches a value a. Specifically:
1) The limit of f(x) as x approaches a, written as lim f(x) = L, means that the values of f(x) can be made arbitrarily close to L by making x sufficiently close to a, without x being equal to a.
2) For the limit to exist, the left and right hand limits must be equal. The function must also be bounded near a and cannot oscillate with increasingly high frequency.
3) Formally, the limit is L if for any positive epsilon, there corresponds a positive delta such that if the absolute value of x - a is less than
This document discusses limits involving infinity in calculus. It begins with definitions of infinite limits, both positive and negative infinity. It provides examples of common infinite limits, such as the limit of 1/x as x approaches 0. It also discusses vertical asymptotes and rules for infinite limits, such as the limit of a sum being infinity if both terms have infinite limits. The document contains examples evaluating limits at points where functions are not continuous.
The document discusses the concept of a limit in calculus. It provides an outline that covers heuristics, errors and tolerances, examples, pathologies, and a precise definition of a limit. It introduces the error-tolerance game used to determine if a limit exists and illustrates this through examples evaluating the limits of x^2 as x approaches 0 and |x|/x as x approaches 0. It also discusses one-sided limits and evaluates the limit of 1/x as x approaches 0 from the right.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
Jane and Jenny, rival Pokemon trainers, get trapped in the forest during a battle between their Pokemon. Jenny's Pokemon, Revoloo, keeps attacking and disturbing the peace, so Jane challenges Jenny to a battle to stop her. During the battle, Revoloo and Jane's Pokemon, Gralinte, are asked mathematical integration questions about solids of revolution that test their abilities. After a long battle involving multiple questions, Revoloo is finally defeated.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It explains the limit laws for addition, subtraction, multiplication, division and nth roots of functions. It uses the error-tolerance game framework to justify the limit laws.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document provides an overview of key concepts related to graphing polynomials, including:
1. Definitions of terms like intervals of increase/decrease, odd/even functions, zeros, and multiplicities.
2. Steps for graphing polynomials which include determining behavior, finding intercepts and zeros, and joining points based on multiplicities.
3. Examples are provided to demonstrate finding zeros and their multiplicities, and graphing a polynomial based on the identified features.
4. Information that can be determined from a polynomial graph, such as degree, leading coefficient, end behavior, intercepts, and intervals of increase/decrease.
Lesson 25: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document discusses indeterminate forms and L'Hopital's rule. It introduces indeterminate forms as limits that can have different values depending on the approach, such as 0/0 or infinity/infinity forms. It then presents L'Hopital's rule, which states that if the limit of the numerator and denominator of a quotient both approach 0, infinity, or negative infinity, the limit can be evaluated by taking the derivative of the numerator and denominator and rearranging terms. Examples are provided to demonstrate how L'Hopital's rule can be used to evaluate indeterminate forms. The document also provides biographical information about Guillaume de l'Hopital, after whom the rule is named.
This document discusses key concepts related to rates of change and derivatives:
1) It defines average rate of change (ARC) as the slope of a secant line on a graph or using the slope formula algebraically, and instantaneous rate of change (IRC) as the slope of the tangent line.
2) It introduces the difference quotient as a way to define ARC and IRC algebraically without a graph by taking the limit as h approaches 0.
3) A derivative is defined as a function that gives the IRC, allowing it to be evaluated at any point without graphing by taking the limit of the difference quotient.
This document provides a summary of key concepts for sequences and series covered on the AP Calculus BC/BCD exam. It begins with definitions of sequences, series, and common notation used. It then drills the reader on topics like finding limits of sequences, identifying geometric and telescoping series, and various convergence tests including ratio, root, integral, comparison, and power series tests. It also covers Taylor and Maclauren series approximations and formulas for common infinite series involving trigonometric, logarithmic, and exponential functions. The document aims to efficiently prepare readers for questions on sequences and series that may appear on the AP exam.
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This document provides definitions and examples to explain how to graph polynomial functions. It discusses determining the degree of the polynomial, the sign of the leading coefficient, end behavior, x- and y-intercepts, and intervals of increase and decrease. Examples are provided to demonstrate how to find zeros, or x-intercepts, and their multiplicities in order to properly graph the polynomial based on these features.
This document discusses key concepts for graphing polynomial functions including:
1. The degree of the polynomial determines the end behavior and maximum number of turns.
2. The sign of the leading coefficient indicates whether the graph faces up or down on both ends.
3. X-intercepts and y-intercepts are found by setting the polynomial equal to 0 or the variable equal to 0.
4. The multiplicity of intercepts determines whether the graph crosses or touches the x-axis at that point.
The document discusses infinite series and sequences. It begins by introducing the concept of an infinite sum and examines whether a sum like 1/2 + 1/4 + 1/8 + ... can be assigned a numerical value. It then defines an infinite series as a sum that continues indefinitely, and a sequence as the individual terms in a series. The key points are:
- An infinite series can be assigned a value by taking the limit of the corresponding sequence of partial sums.
- Common examples like 0.333... and π are actually infinite series.
- Sequences are functions with domain the natural numbers, and their limit is defined similarly to limits of functions.
- Monotonic and bounded sequences are important for
The document summarizes reflections from Patrick Hyatt and Quincie McCalla on a project where they created complex math problems to review concepts they had learned. Both note that while the project provided some review, it took a significant amount of time to develop the problems and stretched their understanding too far at times. They suggest modifying the format of the project to have students work in pairs and take responsibility for specific concepts, then present problems to each other, for a more focused review that clarifies misunderstandings.
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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.
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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.
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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.
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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.
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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.
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- 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.
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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.
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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.
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Chris Bolin, Senior Intelligent Automation Architect Anika Systems
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Dynamic. Modular. Productive.
BoxLang redefines development with its dynamic nature, empowering developers to craft expressive and functional code effortlessly. Its modular architecture prioritizes flexibility, allowing for seamless integration into existing ecosystems.
Interoperability at its Core
With 100% interoperability with Java, BoxLang seamlessly bridges the gap between traditional and modern development paradigms, unlocking new possibilities for innovation and collaboration.
Multi-Runtime
From the tiny 2m operating system binary to running on our pure Java web server, CommandBox, Jakarta EE, AWS Lambda, Microsoft Functions, Web Assembly, Android and more. BoxLang has been designed to enhance and adapt according to it's runnable runtime.
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Empowering Transition with Transpiler Support
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Unlocking Creativity with IDE Tools
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"NATO Hackathon Winner: AI-Powered Drug Search", Taras KlobaFwdays
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In this session, we'll share insights on how we used PostgreSQL to facilitate precise searches across multiple fields in our mobile application. The techniques include using LIKE and ILIKE operators and integrating a trigram-based search to handle potential misspellings, thereby increasing the search accuracy.
We'll also discuss how the azure_ai extension on PostgreSQL databases in Azure and Azure AI Services were utilized to create vectors from user input, a feature beneficial when users wish to find specific items based on text prompts. While our application's case study involves a drug search, the techniques and principles shared in this session can be adapted to improve search functionality in a wide range of applications. Join us to learn how PostgreSQL and Azure AI can be harnessed to enhance your application's search capability.
Session 1 - Intro to Robotic Process Automation.pdfUiPathCommunity
👉 Check out our full 'Africa Series - Automation Student Developers (EN)' page to register for the full program:
https://bit.ly/Automation_Student_Kickstart
In this session, we shall introduce you to the world of automation, the UiPath Platform, and guide you on how to install and setup UiPath Studio on your Windows PC.
📕 Detailed agenda:
What is RPA? Benefits of RPA?
RPA Applications
The UiPath End-to-End Automation Platform
UiPath Studio CE Installation and Setup
💻 Extra training through UiPath Academy:
Introduction to Automation
UiPath Business Automation Platform
Explore automation development with UiPath Studio
👉 Register here for our upcoming Session 2 on June 20: Introduction to UiPath Studio Fundamentals: https://community.uipath.com/events/details/uipath-lagos-presents-session-2-introduction-to-uipath-studio-fundamentals/
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Topics covered:
• The role of a steering committee
• How do the organization’s priorities determine CoE Structure?
Speaker:
Chris Bolin, Senior Intelligent Automation Architect Anika Systems
"$10 thousand per minute of downtime: architecture, queues, streaming and fin...Fwdays
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As part of the talk, we will consider the architectural strategies necessary for the development of highly loaded fintech solutions. We will focus on using queues and streaming to efficiently work and manage large amounts of data in real-time and to minimize latency.
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"$10 thousand per minute of downtime: architecture, queues, streaming and fin...
Lesson 3: The Limit of a Function (slides)
1. Sec on 1.3
The Limit of a Func on
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
January 31, 2011
Announcements
First wri en HW due Wednesday February 2
. Get-to-know-you survey and photo deadline is February 11
2. Announcements
First wri en HW due
Wednesday February 2
Get-to-know-you survey
and photo deadline is
February 11
3. Guidelines for written homework
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (011), recita on number,
date, assignment number, book sec ons.
Explain your work and your reasoning in your own words. Use
complete English sentences.
4. Rubric
Points Descrip on of Work
3 Work is completely accurate and essen ally perfect.
Work is thoroughly developed, neat, and easy to read.
Complete sentences are used.
2 Work is good, but incompletely developed, hard to
read, unexplained, or jumbled. Answers which are
not explained, even if correct, will generally receive 2
points. Work contains “right idea” but is flawed.
1 Work is sketchy. There is some correct work, but most
of work is incorrect.
0 Work minimal or non-existent. Solu on is completely
incorrect.
9. Objectives
Understand and state the
informal defini on of a
limit.
Observe limits on a
graph.
Guess limits by algebraic
manipula on.
Guess limits by numerical
informa on.
11. Yoda on teaching course concepts
You must unlearn
what you have
learned.
In other words, we are
building up concepts and
allowing ourselves only to
speak in terms of what we
personally have produced.
12. Zeno’s Paradox
That which is in locomo on must
arrive at the half-way stage before
it arrives at the goal.
(Aristotle Physics VI:9, 239b10)
13. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
14. Heuristic Definition of a Limit
Defini on
We write
lim f(x) = L
x→a
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L
as we like) by taking x to be sufficiently close to a (on either side of
a) but not equal to a.
15. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
16. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
17. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
18. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not coun ng a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
19. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not coun ng a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
Step 4 If Dana’s move is a good one, Emerson can challenge again
or give up. If Emerson gives up, Dana wins and the limit is L.
22. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
23. The error-tolerance game
This tolerance is too big
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
24. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
25. The error-tolerance game
S ll too big
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
26. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
27. The error-tolerance game
This looks good
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
28. The error-tolerance game
So does this
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
29. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can s ll move.
30. The error-tolerance game
L
.
a
To be legit, the part of the graph inside the blue (ver cal) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can s ll move.
31. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
32. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
33. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana claims the limit is zero.
34. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs
to guarantee that −0.01 < x2 < 0.01 for all x sufficiently close
to zero.
35. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs
to guarantee that −0.01 < x2 < 0.01 for all x sufficiently close
to zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
36. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
37. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
A tolerance of 0.01 works because
|x| < 10−2 =⇒ x2 < 10−4 .
38. Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0
Solu on
Dana has a shortcut: By se ng tolerance equal to the square
root of the error, Dana can win every round. Once Emerson
realizes this, Emerson must give up.
49. A piecewise-defined function
Example
|x|
Find lim if it exists.
x→0 x
Solu on
The func on can also be wri en as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
50. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
. x
−1
51. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit is 1
. x
−1
52. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit is 1
. x
Can you fit an error of 0.5?
−1
53. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- . x
erance?
−1
54. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- .
No. Part of x
erance? graph inside
−1 blue is not
inside green
55. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
Oh, I guess 1
the limit isn’t
1 .
No. Part of x
graph inside
−1 blue is not
inside green
56. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit
is −1 . x
−1
57. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit
is −1 . Can you fit xan
error of 0.5?
−1
58. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
. Can you fit xan
this for a tol-
error of 0.5?
erance?
−1
59. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
No. Part of
graph inside
1 blue is not
How about inside green
this for a tol- . x
erance?
−1
60. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
No. Part of
graph inside
Oh, I guess 1 blue is not
the limit isn’t inside green
−1 . x
−1
61. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit
is 0 . x
−1
62. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I think the limit
is 0 . Can you fit xan
error of 0.5?
−1
63. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
. Can you fit xan
this for a tol-
error of 0.5?
erance?
−1
64. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
How about
this for a tol- . No. None of x
erance? graph inside
−1 blue is inside
green
65. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
Oh, I guess
the limit isn’t . No. None of x
0 graph inside
−1 blue is inside
green
66. The E-T game with a piecewise
function
|x|
Find lim if it exists.
x→0 x y
1
I give up! I
guess there’s . x
no limit!
−1
67. One-sided limits
Defini on
We write
lim f(x) = L
x→a+
and say
“the limit of f(x), as x approaches a from the right, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L as
we like) by taking x to be sufficiently close to a and greater than a.
68. One-sided limits
Defini on
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the le , equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L
as we like) by taking x to be sufficiently close to a and less than a.
77. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
Part of graph 1
inside blue is
inside green
. x
−1
78. The error-tolerance game
|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y
Part of graph 1
inside blue is
inside green
. x
−1
79. A piecewise-defined function
Example
|x|
Find lim if it exists.
x→0 x
Solu on
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−
87. The error-tolerance game with 1/x
y
The limit does not exist
because the func on is
unbounded near 0
1
Find lim+ if it exists. L?
x→0 x
. x
0
88. Another Example
Example
1
Find lim+ if it exists.
x→0 x
Solu on
The limit does not exist because the func on is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x→0 x
112. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =
113. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =
114. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
f(x) = −1 when x =
115. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
Solu on
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
2
f(x) = −1 when x = for any integer k
4k − 1
116. Graph
Here is a graph of the func on:
y
1
. x
−1
There are infinitely many points arbitrarily close to zero where f(x) is
0, or 1, or −1. So the limit cannot exist.
117. What could go wrong?
Summary of Limit Pathologies
How could a func on fail to have a limit? Some possibili es:
le - and right- hand limits exist but are not equal
The func on is unbounded near a
Oscilla on with increasingly high frequency near a
118. Meet the Mathematician
Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contribu ons in geometry,
calculus, complex analysis,
number theory
created the defini on of limit
we use today but didn’t
understand it
119. Outline
Heuris cs
Errors and tolerances
Examples
Precise Defini on of a Limit
120. Precise Definition of a Limit
No, this is not going to be on the test
Let f be a func on defined on an some open interval that contains
the number a, except possibly at a itself. Then we say that the limit
of f(x) as x approaches a is L, and we write
lim f(x) = L,
x→a
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
128. Summary
Many perspectives on limits
Graphical: L is the value the func on
“wants to go to” near a y
Heuris cal: f(x) can be made arbitrarily 1
close to L by taking x sufficiently close
to a. . x
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a
−1
corresponding δ > 0 such that if
0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic: next me
FAIL