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.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
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 an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
This document discusses limits involving infinity in calculus. It defines an infinite limit as a limit where the values of the function can be made arbitrarily large by taking the input variable close to, but not equal to, a particular value. Examples are given of functions that approach infinity as their input approaches a number. The outline indicates the document will cover infinite limits, limits at infinity, and indeterminate forms involving infinity.
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 dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
Infinity is a dangerous place where the rules of arithmetic break down. But it is a useful concept and study both infinite limits and limits at infinity.
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 an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
This document discusses limits involving infinity in calculus. It defines an infinite limit as a limit where the values of the function can be made arbitrarily large by taking the input variable close to, but not equal to, a particular value. Examples are given of functions that approach infinity as their input approaches a number. The outline indicates the document will cover infinite limits, limits at infinity, and indeterminate forms involving infinity.
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.
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
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!
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.
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.
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.
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.
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.
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.
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.
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.
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 basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
This document appears to be lecture notes on limits involving infinity from a Calculus I class at New York University. It begins with announcements about office hours and an upcoming quiz. It then reviews the definition of limits, discusses limits involving infinity like vertical asymptotes, and outlines topics to be covered like infinite limits, limit laws with infinity, and limits as x approaches infinity.
The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
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
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!
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.
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.
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.
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.
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.
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.
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.
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.
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 basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
This document appears to be lecture notes on limits involving infinity from a Calculus I class at New York University. It begins with announcements about office hours and an upcoming quiz. It then reviews the definition of limits, discusses limits involving infinity like vertical asymptotes, and outlines topics to be covered like infinite limits, limit laws with infinity, and limits as x approaches infinity.
The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.
This document discusses double integrals and their use in calculating volumes. It begins by introducing double integrals as a way to calculate the volume of a solid bounded above by a function f(x,y) over a rectangular region. It then discusses using iterated integrals to evaluate double integrals by first integrating with respect to one variable and then the other. Finally, it provides examples of using double integrals and iterated integrals to calculate volumes.
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 defines key concepts related to limits in calculus, including:
- A limit describes the value a function approaches as the input gets closer to a specific value, even if the function is undefined at that point.
- Symbols like x → a indicate the variable x is taking on values near a but not equal to a.
- The formal definition of a limit states the function f(x) approaches the fixed value l as x approaches the fixed value a.
- Important limit properties and formulas are presented, such as limits of sums, constants, and logarithms.
The document discusses limits involving infinity in calculus. It introduces the concept of an infinite limit, where the limit of a function equals infinity as x approaches a. This occurs when the values of the function can be made arbitrarily large by taking x sufficiently close to but not equal to a. Examples of functions with this property are provided. The document also outlines key topics to be covered, including infinite limits, limits at infinity, and properties of infinite limits.
This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.
The document defines Riemann sums and definite integrals. Riemann sums approximate the area under a function curve between two points by dividing the interval into subintervals and evaluating the function at sample points in each. The definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity. Geometrically, the definite integral represents the net area between the function curve and x-axis over the interval.
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.
This document discusses limits of functions. It begins by defining the limit of a function f(x) as x approaches a number c as the value that f(x) approaches as x gets closer to c. It provides examples of limits, including one-sided limits and limits at infinity. Key theorems are presented for computing limits, including properties of limits and the sandwich theorem. The document focuses on conceptual understanding and applying techniques to evaluate a variety of limit examples.
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
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 6: Limits Involving Infinity (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.
How to Manage Reception Report in Odoo 17Celine George
A business may deal with both sales and purchases occasionally. They buy things from vendors and then sell them to their customers. Such dealings can be confusing at times. Because multiple clients may inquire about the same product at the same time, after purchasing those products, customers must be assigned to them. Odoo has a tool called Reception Report that can be used to complete this assignment. By enabling this, a reception report comes automatically after confirming a receipt, from which we can assign products to orders.
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
How to Setup Default Value for a Field in Odoo 17Celine George
In Odoo, we can set a default value for a field during the creation of a record for a model. We have many methods in odoo for setting a default value to the field.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
spot a liar (Haiqa 146).pptx Technical writhing and presentation skills
Lesson 6: Limits Involving Infinity
1. Section 1.6
Limits involving Infinity
V63.0121.006/016, Calculus I
February 3, 2010
Announcements
Office Hours: M,W 1:30–2:30, R 9–10 (CIWW 726)
Written Assignment #2 due today.
WebAssignments due Tuesday.
First Quiz: Friday February 12 in recitation (§§1.1–1.4)
. . . . . .
2. Recall the definition of limit
Definition
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.
. . . . . .
3. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x →0 + x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .
4. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x →0 + x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .
5. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x →0 + x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .
6. Recall the unboundedness problem
1
Recall why lim doesn’t exist.
x →0 + x
y
.
.? .
L
. x
.
No matter how thin we draw the strip to the right of x = 0, we
cannot “capture” the graph inside the box.
. . . . . .
7. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .
8. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
. x
.
. . . . . .
9. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
10. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
11. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
12. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
13. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
14. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
15. Infinite Limits
Definition
The notation y
.
lim f(x) = ∞
x →a
means that values of f(x) can
be made arbitrarily large (as
large as we please) by taking
x sufficiently close to a but
not equal to a.
“Large” takes the place . x
.
of “close to L”.
. . . . . .
16. Negative Infinity
Definition
The notation
lim f(x) = −∞
x →a
means that the values of f(x) can be made arbitrarily large
negative (as large as we please) by taking x sufficiently close to a
but not equal to a.
. . . . . .
17. Negative Infinity
Definition
The notation
lim f(x) = −∞
x →a
means that the values of f(x) can be made arbitrarily large
negative (as large as we please) by taking x sufficiently close to a
but not equal to a.
We call a number large or small based on its absolute value.
So −1, 000, 000 is a large (negative) number.
. . . . . .
18. Vertical Asymptotes
Definition
The line x = a is called a vertical asymptote of the curve y = f(x)
if at least one of the following is true:
lim f(x) = ∞ lim f(x) = −∞
x →a x →a
lim f(x) = ∞ lim f(x) = −∞
x →a + x →a +
lim f(x) = ∞ lim f(x) = −∞
x →a − x →a −
. . . . . .
20. Infinite Limits we Know
y
.
.
.
1
lim
+ x
=∞
x →0 .
1
lim = −∞
x →0 − x . . . . . . . x
.
.
.
.
. . . . . .
21. Infinite Limits we Know
y
.
.
.
1
lim
+ x
=∞
x →0 .
1
lim = −∞
x →0 − x . . . . . . . x
.
1
lim =∞
x →0 x 2 .
.
.
. . . . . .
22. Finding limits at trouble spots
Example
Let
x2 + 2
f(x ) =
x2 − 3x + 2
Find lim f(x) and lim f(x) for each a at which f is not
x →a − x→a+
continuous.
. . . . . .
23. Finding limits at trouble spots
Example
Let
x2 + 2
f(x ) =
x2 − 3x + 2
Find lim f(x) and lim f(x) for each a at which f is not
x →a − x→a+
continuous.
Solution
The denominator factors as (x − 1)(x − 2). We can record the
signs of the factors on the number line.
. . . . . .
25. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
So
. . . . . .
26. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
So
. . . . . .
27. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
. . f
.(x)
1
. 2
.
So
. . . . . .
28. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . . f
.(x)
1
. 2
.
So
. . . . . .
29. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞.
+ . f
.(x)
1
. 2
.
So
lim f(x) = +∞
x →1 −
. . . . . .
30. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞
+ − . f
.(x)
1
. 2
.
So
lim f(x) = +∞
x →1 −
lim f(x) = −∞
x →1 +
. . . . . .
31. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞ .
+ − − . f
.(x)
1
. 2
.
So
lim f(x) = +∞
x →1 −
lim f(x) = −∞
x →1 +
. . . . . .
32. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞ . . ∞ .
+ − − −
f
.(x)
1
. 2
.
So
lim f(x) = +∞ lim f(x) = −∞
x →1 − x →2 −
lim f(x) = −∞
x →1 +
. . . . . .
33. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞ . . ∞ .. ∞
+ − − − +
f
.(x)
1
. 2
.
So
lim f(x) = +∞ lim f(x) = −∞
x →1 − x →2 −
lim f(x) = −∞ lim f(x) = +∞
x →1 + x →2 +
. . . . . .
34. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞ . . ∞ .. ∞
+ − − − +
f
.(x)
1
. 2
.
So
lim f(x) = +∞ lim f(x) = −∞
x →1 − x →2 −
lim f(x) = −∞ lim f(x) = +∞
x →1 + x →2 +
. . . . . .
35. Use the number line
−
.. 0
.. .
+
. x − 1)
(
1
.
−
. 0
.. .
+
. x − 2)
(
2
.
.
+
. x2 + 2 )
(
.
+ . ∞ .. ∞ . . ∞ .. ∞
+ − − − + .
+
f
.(x)
1
. 2
.
So
lim f(x) = +∞ lim f(x) = −∞
x →1 − x →2 −
lim f(x) = −∞ lim f(x) = +∞
x →1 + x →2 +
. . . . . .
36. In English, now
To explain the limit, you can say:
“As x → 1− , the numerator approaches 3, and the denominator
approaches 0 while remaining positive. So the limit is +∞.”
. . . . . .
37. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
38. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
39. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
40. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
41. The graph so far
y
.
. . . . . .
x
−
. 1 1
. 2
. 3
.
. . . . . .
42. Limit Laws (?) with infinite limits
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x →a x→a x →a
That is,
∞ .
. +∞=∞
If lim f(x) = −∞ and lim g(x) = −∞, then
x →a x →a
lim (f(x) + g(x)) = −∞. That is,
x →a
− .
. ∞ − ∞ = −∞
. . . . . .
43. Rules of Thumb with infinite limits
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x →a x→a x →a
That is,
∞ .
. +∞=∞
If lim f(x) = −∞ and lim g(x) = −∞, then
x →a x →a
lim (f(x) + g(x)) = −∞. That is,
x →a
− .
. ∞ − ∞ = −∞
. . . . . .
44. Rules of Thumb with infinite limits
If lim f(x) = L and lim g(x) = ±∞, then
x →a x →a
lim (f(x) + g(x)) = ±∞. That is,
x →a
L+∞=∞
. .
L − ∞ = −∞
. . . . . .
48. Dividing by zero is still not allowed
1 .
. =∞
0
There are examples of such limit forms where the limit is ∞, −∞,
undecided between the two, or truly neither.
. . . . . .
49. Indeterminate Limit forms
L
Limits of the form are indeterminate. There is no rule for
0
evaluating such a form; the limit must be examined more closely.
Consider these:
1 −1
lim =∞ lim = −∞
x→0 x2 x →0 x 2
1 1
lim
+ x
=∞ lim = −∞
x →0 x →0 − x
1 L
Worst, lim is of the form , but the limit does not
x→0 x sin(1/x) 0
exist, even in the left- or right-hand sense. There are infinitely
many vertical asymptotes arbitrarily close to 0!
. . . . . .
50. Indeterminate Limit forms
Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate.
Example
1
The limit lim sin x · is of the form 0 · ∞, but the answer is
x →0 + x
1.
1
The limit lim sin2 x · is of the form 0 · ∞, but the answer is
x →0 + x
0.
1
The limit lim sin x · 2 is of the form 0 · ∞, but the answer is
x →0 + x
∞.
Limits of indeterminate forms may or may not “exist.” It will
depend on the context.
. . . . . .
52. Outline
Infinite Limits
Vertical Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
. . . . . .
54. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f(x) if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
. . . . . .
55. Definition
Let f be a function defined on some interval (a, ∞). Then
lim f(x) = L
x→∞
means that the values of f(x) can be made as close to L as we
like, by taking x sufficiently large.
Definition
The line y = L is a called a horizontal asymptote of the curve
y = f(x) if either
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞
y = L is a horizontal line!
. . . . . .
56. Basic limits at infinity
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ xn
1
lim n = 0
x→−∞ x
. . . . . .
63. Solution
Again, factor out the largest power of x from the numerator and
denominator. We have
x x(1) 1 1
= 2 = ·
x2 + 1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
. . . . . .
65. Another Example
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0.
y
.
. x
.
Notice that the graph does cross the asymptote, which
contradicts one of the heuristic definitions of asymptote.
. . . . . .
66. Solution
Again, factor out the largest power of x from the numerator and
denominator. We have
x x(1) 1 1
= 2 = ·
x2 + 1 x (1 + 1/x2 ) x 1 + 1/x2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2
1
=0· = 0.
1+0
Remark
Had the higher power been in the numerator, the limit would
have been ∞.
. . . . . .
71. Rationalizing to get a limit
Example (√ )
Compute lim 4x2 + 17 − 2x .
x→∞
Solution
This limit is of the form ∞ − ∞, which we cannot use. So we
rationalize the numerator (the denominator is 1) to get an
expression that we can use the limit laws on.
(√ ) (√ ) √4x2 + 17 + 2x
lim 4x2 + 17 − 2x = lim 4x2 + 17 − 2x · √
x→∞ x→∞ 4x2 + 17 + 2x
(4x 2 + 17) − 4x2
= lim √
x→∞ 4x2 + 17 + 2x
17
= lim √ =0
x→∞ 4x 2 + 17 + 2x
. . . . . .
72. Kick it up a notch
Example (√ )
Compute lim 4x2 + 17x − 2x .
x→∞
. . . . . .
74. Summary
Infinity is a more complicated concept than a single number.
There are rules of thumb, but there are also exceptions.
Take a two-pronged approach to limits involving infinity:
Look at the expression to guess the limit.
Use limit rules and algebra to verify it.
. . . . . .