This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It discusses the informal heuristic definition of a limit, examples to illustrate limits, and outlines the topics to be covered, including heuristics, errors and tolerances, examples, pathologies, and the precise definition of a limit. It also contains announcements about assignments and deadlines.
The document provides an example of using the error-tolerance game to evaluate the limit of x^2 as x approaches 0. Player 1 claims the limit is 0, and is able to show for any error level chosen by Player 2, there exists a tolerance such that the values of x^2 are within the error level when x is within the tolerance of 0, demonstrating that the limit exists and is equal to 0.
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.
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.
The document discusses calculating limits in Calculus I. It covers basic limits like lim x=a and lim c=c as x approaches a. The objectives are to know these basic limits, use limit laws to compute elementary limits, use algebra to simplify limits, understand the Squeeze Theorem, and use it to demonstrate a limit. It also discusses the error-tolerance game to determine if a limit exists and gives examples of limits that do and do not exist.
This document outlines the key rules for differentiation that will be covered in Calculus I class. It introduces the objectives of understanding derivatives of constant functions, the constant multiple rule, sum and difference rules, and derivatives of sine and cosine. It then provides examples of finding the derivatives of squaring and cubing functions using the definition of a derivative. Finally, it discusses properties of the derivatives of these functions.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It discusses the informal heuristic definition of a limit, examples to illustrate limits, and outlines the topics to be covered, including heuristics, errors and tolerances, examples, pathologies, and the precise definition of a limit. It also contains announcements about assignments and deadlines.
The document provides an example of using the error-tolerance game to evaluate the limit of x^2 as x approaches 0. Player 1 claims the limit is 0, and is able to show for any error level chosen by Player 2, there exists a tolerance such that the values of x^2 are within the error level when x is within the tolerance of 0, demonstrating that the limit exists and is equal to 0.
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.
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.
The document discusses calculating limits in Calculus I. It covers basic limits like lim x=a and lim c=c as x approaches a. The objectives are to know these basic limits, use limit laws to compute elementary limits, use algebra to simplify limits, understand the Squeeze Theorem, and use it to demonstrate a limit. It also discusses the error-tolerance game to determine if a limit exists and gives examples of limits that do and do not exist.
This document outlines the key rules for differentiation that will be covered in Calculus I class. It introduces the objectives of understanding derivatives of constant functions, the constant multiple rule, sum and difference rules, and derivatives of sine and cosine. It then provides examples of finding the derivatives of squaring and cubing functions using the definition of a derivative. Finally, it discusses properties of the derivatives of these functions.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
This document contains notes from a Calculus I class lecture on the derivative. The lecture covered the definition of the derivative and examples of how it can be used to model rates of change in various contexts like velocity, population growth, and marginal costs. It also discussed properties of the derivative like how the derivative of a function relates to whether the function is increasing or decreasing over an interval.
The document discusses creating a culture of creativity in the classroom that is collaborative, expressive, learner-centered, congenial, and playful. It emphasizes allowing room for mistakes in an accepting environment where students have freedom of expression and responsibility for their learning within various cultural contexts, with an emphasis on collaborative work in a congenial space.
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.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits and the properties of limits, such as how limits behave under arithmetic operations.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.
The document appears to be lecture slides for a Calculus I class at NYU. It discusses announcements like midterm grades being submitted and an upcoming quiz. It then summarizes student evaluations of the class, including both positive and negative feedback. The remainder of the document outlines and discusses the topics of inverse trigonometric functions, including their definitions, domains, ranges, and derivatives. Graphs are provided to illustrate inverse functions and how to obtain the graph of an inverse from the original function. Specific inverse trig functions like arcsin and arccos are defined.
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.
This document contains lecture notes from a Calculus I class on the topic of continuity. It includes definitions of continuity and the intermediate value theorem. It provides examples of showing functions are continuous and discusses ways continuity can fail. Specifically, it explains a function is not continuous at a point if the limit does not exist there or if the function is not defined at that point.
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.
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 contains lecture notes on limits involving infinity from a Calculus I class at New York University. It reviews the definitions of infinite limits, limits at positive and negative infinity, and vertical asymptotes. Examples are provided of known infinite limits and how to use a number line to determine one-sided limits at points where a function is discontinuous. The objectives are to intuitively evaluate limits involving infinity and use algebraic manipulation to show such limits.
- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions.
- There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24.
- The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.
Lesson 16: Inverse Trigonometric Functions (Section 021 slides)Mel Anthony Pepito
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
This document contains notes from a Calculus I class lecture on the derivative. The lecture covered the definition of the derivative and examples of how it can be used to model rates of change in various contexts like velocity, population growth, and marginal costs. It also discussed properties of the derivative like how the derivative of a function relates to whether the function is increasing or decreasing over an interval.
The document discusses creating a culture of creativity in the classroom that is collaborative, expressive, learner-centered, congenial, and playful. It emphasizes allowing room for mistakes in an accepting environment where students have freedom of expression and responsibility for their learning within various cultural contexts, with an emphasis on collaborative work in a congenial space.
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.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits and the properties of limits, such as how limits behave under arithmetic operations.
This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.
The document appears to be lecture slides for a Calculus I class at NYU. It discusses announcements like midterm grades being submitted and an upcoming quiz. It then summarizes student evaluations of the class, including both positive and negative feedback. The remainder of the document outlines and discusses the topics of inverse trigonometric functions, including their definitions, domains, ranges, and derivatives. Graphs are provided to illustrate inverse functions and how to obtain the graph of an inverse from the original function. Specific inverse trig functions like arcsin and arccos are defined.
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.
This document contains lecture notes from a Calculus I class on the topic of continuity. It includes definitions of continuity and the intermediate value theorem. It provides examples of showing functions are continuous and discusses ways continuity can fail. Specifically, it explains a function is not continuous at a point if the limit does not exist there or if the function is not defined at that point.
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.
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 contains lecture notes on limits involving infinity from a Calculus I class at New York University. It reviews the definitions of infinite limits, limits at positive and negative infinity, and vertical asymptotes. Examples are provided of known infinite limits and how to use a number line to determine one-sided limits at points where a function is discontinuous. The objectives are to intuitively evaluate limits involving infinity and use algebraic manipulation to show such limits.
- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions.
- There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24.
- The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.
Lesson 12: Linear Approximation (Section 41 handout)Matthew Leingang
The line tangent to a curve, which is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.
Lesson 16: Inverse Trigonometric Functions (Section 021 slides)Mel Anthony Pepito
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
This document contains lecture notes from a Calculus I class at New York University. It discusses the definition of continuity for functions, examples of continuous and discontinuous functions, and properties of continuous functions like sums, products, and compositions of continuous functions being continuous. It also addresses trigonometric functions like sin, cos, tan, and sec being continuous on their domains.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
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.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
1. Section 1.3
The Concept of Limit
V63.0121.041, Calculus I
New York University
September 13, 2010
Announcements
Let us know if you bought a WebAssign license last year and
cannot login
First written HW due Wednesday
Get-to-know-you survey and photo deadline is October 1
. . . . . .
2. Announcements
Let us know if you bought
a WebAssign license last
year and cannot login
First written HW due
Wednesday
Get-to-know-you survey
and photo deadline is
October 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 2 / 36
3. Guidelines for written homework
Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (041), recitation number, date,
assignment number, book sections.
Explain your work and your reasoning in your own words. Use
complete English sentences.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 3 / 36
4. Rubric
Points Description of Work
3 Work is completely accurate and essentially perfect.
Work is thoroughly developed, neat, and easy to read.
Complete sentences are used.
2 Work is good, but incompletely developed, hard to read,
unexplained, or jumbled. Answers which are not ex-
plained, even if correct, will generally receive 2 points.
Work contains “right idea” but is flawed.
1 Work is sketchy. There is some correct work, but most of
work is incorrect.
0 Work minimal or non-existent. Solution is completely in-
correct.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 4 / 36
5. Examples of written homework: Don't
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 5 / 36
6. Examples of written homework: Do
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 6 / 36
7. Examples of written homework: Do
Written Explanations
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 7 / 36
8. Examples of written homework: Do
Graphs
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 8 / 36
9. Objectives
Understand and state the
informal definition of a limit.
Observe limits on a graph.
Guess limits by algebraic
manipulation.
Guess limits by numerical
information.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 9 / 36
11. Zeno's Paradox
That which is in
locomotion must arrive
at the half-way stage
before it arrives at the
goal.
(Aristotle Physics VI:9, 239b10)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 10 / 36
13. Heuristic Definition of a 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.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 12 / 36
15. The error-tolerance game
A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
Step 1 Dana proposes L to be the limit.
Step 2 Emerson challenges with an “error” level around L.
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not counting a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
Step 4 If Dana’s move is a good one, Emerson can challenge again or
give up. If Emerson gives up, Dana wins and the limit is L.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 14 / 36
16. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
17. The error-tolerance game
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
18. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
19. The error-tolerance game
T
. his tolerance is too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
20. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
21. The error-tolerance game
S
. till too big
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
22. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
23. The error-tolerance game
T
. his looks good
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
24. The error-tolerance game
S
. o does this
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
25. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can still move.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
26. The error-tolerance game
L
.
.
a
.
To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.
Even if Emerson shrinks the error, Dana can still move.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 15 / 36
28. Example
Find lim x2 if it exists.
x→0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
29. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
30. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
31. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
32. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
33. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
34. Example
Find lim x2 if it exists.
x→0
Solution
Dana claims the limit is zero.
If Emerson challenges with an error level of 0.01, Dana needs to
guarantee that −0.01 < x2 < 0.01 for all x sufficiently close to
zero.
If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizes
this, Emerson must give up.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 17 / 36
35. Example
|x|
Find lim if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
36. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 18 / 36
37. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
38. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
39. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
40. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
41. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph
. 1.
− inside blue is not
inside green
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
42. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
43. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
44. The error-tolerance game
y
.
.
Part of graph
inside blue is not . .
1
inside green
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
45. The error-tolerance game
y
.
.
Part of graph
inside blue is not . .
1
inside green
. x
.
. 1.
−
These are the only good choices; the limit does not exist.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 19 / 36
46. One-sided limits
Definition
We write
lim f(x) = L
x→a+
and say
“the limit of f(x), as x approaches a from the right, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L as
we like) by taking x to be sufficiently close to a and greater than a.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
47. One-sided limits
Definition
We write
lim f(x) = L
x→a−
and say
“the limit of f(x), as x approaches a from the left, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L as
we like) by taking x to be sufficiently close to a and less than a.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 20 / 36
48. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
49. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
50. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
51. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
52. The error-tolerance game
y
.
. .
1
. x
.
.
Part of graph
. 1.
− inside blue is
inside green
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
53. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
54. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
55. The error-tolerance game
y
.
. .
1
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
56. The error-tolerance game
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
57. The error-tolerance game
y
.
.
Part of graph . .
1
inside blue is
inside green
. x
.
. 1.
−
So lim+ f(x) = 1 and lim f(x) = −1
x→0 x→0−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 21 / 36
58. Example
|x|
Find lim if it exists.
x→0 x
Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0
What would be the limit?
The error-tolerance game fails, but
lim f(x) = 1 lim f(x) = −1
x→0+ x→0−
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 22 / 36
59. Example
1
Find lim+ if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 23 / 36
60. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
61. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
62. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
63. The error-tolerance game
y
.
.
The graph escapes
the green, so no good
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
64. The error-tolerance game
y
.
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
65. The error-tolerance game
y
.
E
. ven worse!
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
66. The error-tolerance game
y
.
.
The limit does not ex-
ist because the func-
tion is unbounded near
0
.?.
L
. x
.
0
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 24 / 36
67. Example
1
Find lim+ if it exists.
x→0 x
Solution
The limit does not exist because the function is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 25 / 36
68. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 26 / 36
70. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
71. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
f(x) = 0 when x =
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
72. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =
f(x) = −1 when x =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
73. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
f(x) = −1 when x =
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
74. Weird, wild stuff
Example
(π )
Find lim sin if it exists.
x→0 x
1
f(x) = 0 when x = for any integer k
k
2
f(x) = 1 when x = for any integer k
4k + 1
2
f(x) = −1 when x = for any integer k
4k − 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 28 / 36
75. Weird, wild stuff continued
Here is a graph of the function:
y
.
. .
1
. x
.
. 1.
−
There are infinitely many points arbitrarily close to zero where f(x) is 0,
or 1, or −1. So the limit cannot exist.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 29 / 36
77. What could go wrong?
Summary of Limit Pathologies
How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 31 / 36
78. Meet the Mathematician: Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis, number
theory
created the definition of
limit we use today but
didn’t understand it
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 32 / 36
80. Precise Definition of a Limit
No, this is not going to be on the test
Let f be a function defined on an some open interval that contains the
number a, except possibly at a itself. Then we say that the limit of f(x)
as x approaches a is L, and we write
lim f(x) = L,
x→a
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 34 / 36
81. The error-tolerance game = ε, δ
L
.
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
82. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
a
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
83. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
84. The error-tolerance game = ε, δ
T
. his δ is too big
L
. +ε
L
.
. −ε
L
.
. − δ. . + δ
a aa
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
85. The error-tolerance game = ε, δ
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
86. The error-tolerance game = ε, δ
T
. his δ looks good
L
. +ε
L
.
. −ε
L
.
. −. δ δ
a . a+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
87. The error-tolerance game = ε, δ
S
. o does this δ
L
. +ε
L
.
. −ε
L
.
. .− δ δ
aa .+
a
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 35 / 36
88. Summary: Many perspectives on limits
Graphical: L is the value the function “wants to go to” near a
Heuristical: f(x) can be made arbitrarily close to L by taking x
sufficiently close to a.
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic/Formulaic: next time
. . . . . .
V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 36 / 36