L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
The document provides an overview of calculating limits. It begins with announcements about assignments and exams for a Calculus I course. The outline then previews the key topics to be covered, including the concept of a limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The bulk of the document explains the error-tolerance game approach to defining a limit and works through examples of basic limits. It also establishes four limit laws for arithmetic operations: addition of limits, subtraction of limits, scaling of limits, and multiplication of limits.
The document defines key concepts related to limits and derivatives:
1) It defines left-hand and right-hand limits and discusses how the limit of a function is defined if the left and right limits coincide.
2) It provides examples of evaluating limits of functions, including limits involving polynomials, rational functions, and trigonometric functions.
3) It discusses properties of limits, such as the algebra of limits and the sandwich theorem.
4) It introduces the definition of the derivative as the limit of the difference quotient, and defines the derivative of a function at a point. It also discusses the algebra of derivatives and lists some standard derivative rules.
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 basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The document provides an overview of calculating limits. It begins with announcements about assignments and exams for a Calculus I course. The outline then previews the key topics to be covered, including the concept of a limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The bulk of the document explains the error-tolerance game approach to defining a limit and works through examples of basic limits. It also establishes four limit laws for arithmetic operations: addition of limits, subtraction of limits, scaling of limits, and multiplication of limits.
The document defines key concepts related to limits and derivatives:
1) It defines left-hand and right-hand limits and discusses how the limit of a function is defined if the left and right limits coincide.
2) It provides examples of evaluating limits of functions, including limits involving polynomials, rational functions, and trigonometric functions.
3) It discusses properties of limits, such as the algebra of limits and the sandwich theorem.
4) It introduces the definition of the derivative as the limit of the difference quotient, and defines the derivative of a function at a point. It also discusses the algebra of derivatives and lists some standard derivative rules.
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 basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
This document contains notes from a calculus class lecture on linear approximation and differentials. It discusses how to use the tangent line approximation to estimate values of functions near a given point, using the example of estimating sin(61°) by approximating near 0° and 60°. It provides the formulas for finding the equation of the tangent line and illustrates how the approximation improves when taken closer to the point. Another example estimates the square root of 10 by approximating the square root function near 9.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
the derivative measure the instantaneous rate of change of a function, or the slope of the line tangent to its graph. It has countless applications.
[Note: We did not do this entire show in class. We will finish it on Thursday]
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
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 Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Optimization is a killer feature of the derivative. Not only do we often want to optimize some system, nature does as well. We give a procedure and many examples.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
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.
Lesson 17: Interminate forms and L'Hôpital's Rule (worksheet solutions)Matthew Leingang
This document provides solutions to 8 examples of evaluating limits using techniques like L'Hopital's rule and determining the indeterminate form.
The solutions include:
1) Using L'Hopital's rule twice to evaluate a limit of sin^2(x)/xsin(x)+3x^2 as x approaches 0.
2) Identifying a limit of sin(4x)/cos(2x)+1 as a "red herring" since the numerator approaches 0 and denominator approaches a non-zero value.
3) Evaluating a limit without needing L'Hopital's rule by factorizing and simplifying.
4) Applying L'Hopital's rule multiple times and
This document contains examples and explanations of limits involving various functions. Some key points covered include:
- Substitution can be used to evaluate limits, such as substituting 2 into -2x^3.
- Left and right hand limits must agree for the overall limit to exist.
- The limit of a piecewise function exists if the left and right limits are the same.
- Graphs can help verify limit calculations and show discontinuities.
- Special limits involving trigonometric and greatest integer functions are evaluated.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
This document contains notes from a calculus class lecture on linear approximation and differentials. It discusses how to use the tangent line approximation to estimate values of functions near a given point, using the example of estimating sin(61°) by approximating near 0° and 60°. It provides the formulas for finding the equation of the tangent line and illustrates how the approximation improves when taken closer to the point. Another example estimates the square root of 10 by approximating the square root function near 9.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
the derivative measure the instantaneous rate of change of a function, or the slope of the line tangent to its graph. It has countless applications.
[Note: We did not do this entire show in class. We will finish it on Thursday]
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
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 Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Optimization is a killer feature of the derivative. Not only do we often want to optimize some system, nature does as well. We give a procedure and many examples.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
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.
Lesson 17: Interminate forms and L'Hôpital's Rule (worksheet solutions)Matthew Leingang
This document provides solutions to 8 examples of evaluating limits using techniques like L'Hopital's rule and determining the indeterminate form.
The solutions include:
1) Using L'Hopital's rule twice to evaluate a limit of sin^2(x)/xsin(x)+3x^2 as x approaches 0.
2) Identifying a limit of sin(4x)/cos(2x)+1 as a "red herring" since the numerator approaches 0 and denominator approaches a non-zero value.
3) Evaluating a limit without needing L'Hopital's rule by factorizing and simplifying.
4) Applying L'Hopital's rule multiple times and
This document contains examples and explanations of limits involving various functions. Some key points covered include:
- Substitution can be used to evaluate limits, such as substituting 2 into -2x^3.
- Left and right hand limits must agree for the overall limit to exist.
- The limit of a piecewise function exists if the left and right limits are the same.
- Graphs can help verify limit calculations and show discontinuities.
- Special limits involving trigonometric and greatest integer functions are evaluated.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
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.
The document provides an introduction to evaluating limits, including:
1. The limit of a constant function is the constant.
2. Common limit laws can be used to evaluate limits of sums, differences, products, and quotients if the individual limits exist.
3. Special techniques may be needed to evaluate limits that involve indeterminate forms, such as 0/0, infinity/infinity, or limits approaching infinity. These include factoring, graphing, and rationalizing.
The document provides information on determining limits of algebraic functions. It discusses different methods for calculating limits, including dividing the numerator and denominator by the highest power term, and multiplying by the conjugate of the numerator and denominator. Examples are provided to illustrate each method and determine limits as the variable approaches a value.
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.
The document discusses limits and rules for calculating limits. It provides examples of estimating limits graphically and calculating one-sided limits. The key rules covered are:
1) The sum and difference rules for adding and subtracting limits.
2) The product rule for multiplying two functions with limits.
3) The quotient rule for dividing two functions with limits, assuming the limit of the denominator is not zero.
This document examines L'Hopital's rule for evaluating limits that result in indeterminate forms. It demonstrates how to apply the rule by taking derivatives of functions
Profº Marcelo Santos Chaves Cálculo I (limites trigonométricos)MarcelloSantosChaves
The document provides solutions to 12 limit problems involving trigonometric functions. Each problem is solved in 3 steps or less. The solutions show that:
1) Many of the limits evaluate to simple numeric values like 1, 0, or constants like a.
2) Trigonometric limits are often solved by factorizing the expressions and applying standard trigonometric limits like lim(sinx/x) = 1 as x approaches 0.
3) More complex problems are broken down into composite limits and simplified through algebraic manipulation and properties of limits.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand the derivative of constants, the constant multiple rule, the sum rule, the difference rule, and derivatives of sine and cosine. Examples are provided to find the derivatives of squaring, cubing, square root, and cube root functions using the definition of the derivative. Graphs and properties of these functions and their derivatives are also discussed.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
Similar to Lesson 18: Indeterminate Forms and L'Hôpital's Rule (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Salesforce Integration for Bonterra Impact Management (fka Social Solutions A...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on integration of Salesforce with Bonterra Impact Management.
Interested in deploying an integration with Salesforce for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
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Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.
6. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x .
lim
x→0 sin(x2 )
. . . . . .
7. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x .
lim =1
x→0 sin(x2 )
. . . . . .
8. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x .
lim =1
x→0 sin(x2 )
sin 3x
lim
x→0 sin x
. . . . . .
9. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x .
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
. . . . . .
10. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
sin2 x .
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different
0
answers in different cases, we say this form is indeterminate.
. . . . . .
11. Outline
Indeterminate Forms
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
. . . . . .
12. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
. . . . . .
13. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
. . . . . .
14. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
. . . . . .
15. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
Limit of a quotient is the quotient of the limits ... whoops!
This is true as long as you don’t try to divide by zero.
. . . . . .
19. Language Note
It depends on what the meaning of the word “is” is
Be careful with the
language here. We are
not saying that the limit
0
in each case “is” , and
0
therefore nonexistent
because this expression
is undefined.
The limit is of the form
0
, which means we
0
cannot evaluate it with
our limit laws.
. . . . . .
21. Outline
Indeterminate Forms
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
. . . . . .
22. The Linear Case
Question
If f and g are lines and f(a) = g(a) = 0, what is
f(x)
lim ?
x→a g(x)
. . . . . .
23. The Linear Case
Question
If f and g are lines and f(a) = g(a) = 0, what is
f(x)
lim ?
x→a g(x)
Solution
The functions f and g can be written in the form
f(x) = m1 (x − a)
g(x) = m2 (x − a)
So
f (x ) m1
=
g (x ) m2
. . . . . .
24. The Linear Case, Illustrated
y
. y
. = g(x)
y
. = f(x)
g
. (x)
a
. f
.(x)
. . . x
.
x
.
f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m1
= = =
g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2
. . . . . .
25. What then?
But what if the functions aren’t linear?
. . . . . .
26. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear
function?
. . . . . .
27. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear
function?
What would be the slope of that linear function?
. . . . . .
28. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear
function?
What would be the slope of that linear function? The
derivative!
. . . . . .
29. Theorem (L’Hopital’s Rule)
Suppose f and g are differentiable functions and g′ (x) ̸= 0 near a
(except possibly at a). Suppose that
lim f(x) = 0 and lim g(x) = 0
x→a x→a
or
lim f(x) = ±∞ and lim g(x) = ±∞
x→a x→a
Then
f(x) f′ (x)
lim = lim ′ ,
x→a g(x) x→a g (x)
if the limit on the right-hand side is finite, ∞, or −∞.
. . . . . .
30. Meet the Mathematician: L’Hôpital
wanted to be a military
man, but poor eyesight
forced him into math
did some math on his
own (solved the
“brachistocrone
problem”)
paid a stipend to Johann
Bernoulli, who proved
this theorem and named
it after him! Guillaume François Antoine,
Marquis de L’Hôpital
(1661–1704)
. . . . . .
35. Revisiting the previous examples
Example . in x → 0
s
sin2 x H 2 sin x.cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x
lim
x→0 sin x2
. . . . . .
36. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x.
lim
x→0 sin x2
. . . . . .
37. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x.
lim .
x→0 sin x2
. enominator → 0
d
. . . . . .
38. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x. H sin x cos x
2
lim 2.
= lim
x→0 sin x x→0 (cos x2 ) (x )
2
. enominator → 0
d
. . . . . .
39. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x H .
sin x cos x
2
lim = lim
x→0 sin x2 x→0 (cos x2 ) (x )
2
. . . . . .
40. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x H .
sin x cos x
2
lim = lim .)
x→0 sin x2 x→0 (cos x2 ) (x
2
. enominator → 0
d
. . . . . .
41. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 0
n
sin2 x H . cos2 x − sin2 x
sin x cos x H
2
lim = lim lim
.) = x→0 cos x2 − 2x2 sin(x2 )
x→0 sin x2 x→0 (cos x2 ) (x
2
. enominator → 0
d
. . . . . .
42. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 1
n
sin2 x H sin x cos x H
2 cos2 x − sin2 x.
lim = lim = lim
x→0 sin x2 x→0 (cos x2 ) (x )
2 x→0 cos x2 − 2x2 sin(x2 )
. . . . . .
43. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
. umerator → 1
n
sin2 x H sin x cos x H
2 cos2 x − sin2 x.
lim = lim = lim .
x→0 sin x2 x→0 (cos x2 ) (x )
2 x→0 cos x2 − 2x2 sin(x2 )
. enominator → 1
d
. . . . . .
44. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x2 x→0 (cos x2 ) (x )
2 x→0 cos x2 − 2x2 sin(x2 )
. . . . . .
45. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim = lim =1
x→0 sin x2 x→0 (cos x2 ) (x )
2 x→0 cos x2 − 2x2 sin(x2 )
Example
sin 3x H 3 cos 3x
lim = lim = 3.
x→0 sin x x→0 cos x
. . . . . .
47. Beware of Red Herrings
Example
Find
x
lim
x→0 cos x
Solution
The limit of the denominator is 1, not 0, so L’Hôpital’s rule does
not apply. The limit is 0.
. . . . . .
48. Outline
Indeterminate Forms
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
. . . . . .
50. Theorem
Let r be any positive number. Then
ex
lim = ∞.
x→∞ xr
Proof.
If r is a positive integer, then apply L’Hôpital’s rule r times to the
fraction. You get
ex H H ex
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
For example, if r = 3, three invocations of L’Hôpital’s Rule give us
ex H ex H ex H ex
lim = lim = lim = lim =∞
x→∞ x3 x→∞ 3 · x2 x→∞ 3 · 2x x→∞ 3 · 2 · 1
. . . . . .
51. If r is not an integer, let m be the smallest integer greater than r.
Then if x 1, xr xm , so
ex ex
m.
xr x
The right-hand side tends to ∞ by the previous step.
. . . . . .
52. If r is not an integer, let m be the smallest integer greater than r.
Then if x 1, xr xm , so
ex ex
m.
xr x
The right-hand side tends to ∞ by the previous step. For
example, if r = 1/2, r 1 so for x 1
ex ex
x1/2 x
which gets arbitrarily large.
. . . . . .
54. Theorem
Let r be any positive number. Then
ln x
lim = 0.
x→∞ xr
Proof.
One application of L’Hôpital’s Rule here suffices:
ln x H 1 /x 1
lim = lim r−1 = lim r = 0.
x→∞ xr x→∞ rx x→∞ rx
. . . . . .
55. Outline
Indeterminate Forms
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
. . . . . .
56. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
. . . . . .
57. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then
apply L’Hôpital’s Rule:
√
lim x ln x
x→0+
. . . . . .
58. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then
apply L’Hôpital’s Rule:
√ ln x
lim x ln x = lim
x→0+ x→0+ 1/√x
. . . . . .
59. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then
apply L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim √ = lim 1
x→0+ x→0+ 1/ x x→0+ − 2 x−3/2
. . . . . .
60. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then
apply L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim √ = lim 1
x→0+ x→0+ 1/ x x→0+ − 2 x−3/2
√
= lim −2 x
x→0+
. . . . . .
61. Indeterminate products
Example
Find √
lim x ln x
x→0+
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then
apply L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim √ = lim 1
x→0+ x→0+ 1/ x x→0+ − 2 x−3/2
√
= lim −2 x = 0
x→0+
. . . . . .
62. Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
. . . . . .
63. Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x)
lim
x→0+ x sin(2x)
. . . . . .
64. Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
. . . . . .
65. Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
. . . . . .
66. Indeterminate differences
Example
( )
1
lim − cot 2x
x→0+ x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim = lim+
x→0+ x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
The limit is +∞ becuase the numerator tends to 1 while the
denominator tends to zero but remains positive.
. . . . . .
67. Checking your work
.
tan 2x
lim = 1, so for small x,
x→0 2x
1
tan 2x ≈ 2x. So cot 2x ≈ and
. 2x
1 1 1 1
− cot 2x ≈ − = →∞
x x 2x 2x
as x → 0+ .
. . . . . .
70. Indeterminate powers
Example
Find lim (1 − 2x)1/x
x→0+
Take the logarithm:
( ) ( )
1/x ln(1 − 2x)
ln lim (1 − 2x) = lim+ ln (1 − 2x)1/x = lim+
x→0 + x→0 x→0 x
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H 1−2x
lim = lim+ = −2
x→0+ x x→0 1
. . . . . .
71. Indeterminate powers
Example
Find lim (1 − 2x)1/x
x→0+
Take the logarithm:
( ) ( )
1/x ln(1 − 2x)
ln lim (1 − 2x) = lim+ ln (1 − 2x)1/x = lim+
x→0 + x→0 x→0 x
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H 1−2x
lim = lim+ = −2
x→0+ x x→0 1
This is not the answer, it’s the log of the answer! So the answer
we want is e−2 .
. . . . . .
73. Example
lim (3x)4x
x→0
Solution
ln lim (3x)4x = lim ln(3x)4x = lim 4x ln(3x)
x→0+ x→0+ x→0+
ln(3x) H 3/3x
= lim+ = lim+
x→0 1/4x x→0 −1/4x2
= lim (−4x) = 0
x→0+
So the answer is e0 = 1.
. . . . . .
74. Outline
Indeterminate Forms
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
Summary
. . . . . .
75. Summary
Form Method
0
0 L’Hôpital’s rule directly
∞
∞ L’Hôpital’s rule directly
0 ∞
0·∞ jiggle to make 0 or ∞.
∞−∞ factor to make an indeterminate product
00 take ln to make an indeterminate product
∞0 ditto
1∞ ditto
. . . . . .
76. Final thoughts
L’Hôpital’s Rule only works on indeterminate quotients
Luckily, most indeterminate limits can be transformed into
indeterminate quotients
L’Hôpital’s Rule gives wrong answers for non-indeterminate
limits!
. . . . . .