Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
This document contains notes from a calculus class section on continuity. Key points include:
- The definition of continuity requires that the limit of a function as x approaches a value exists and is equal to the value of the function at that point.
- Many common functions like polynomials, rational functions, trigonometric functions, exponentials and logarithms are continuous based on properties of limits.
- Functions can fail to be continuous if the limit does not exist or the function is not defined at a point. An example function is given that is not continuous at x=1.
A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
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.
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.
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.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
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.
This document contains notes from a calculus class section on continuity. Key points include:
- The definition of continuity requires that the limit of a function as x approaches a value exists and is equal to the value of the function at that point.
- Many common functions like polynomials, rational functions, trigonometric functions, exponentials and logarithms are continuous based on properties of limits.
- Functions can fail to be continuous if the limit does not exist or the function is not defined at a point. An example function is given that is not continuous at x=1.
A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
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.
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.
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.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
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.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
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.
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.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
The 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.
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 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.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
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.
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.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
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
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
This document provides an introduction to functions and their representations. It defines what a function is and gives examples of functions expressed through formulas, numerically, graphically and verbally. It also covers properties of functions including monotonicity, symmetry, even and odd functions. Key examples are used throughout to illustrate concepts.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
The 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.
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 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.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
Lesson 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 document discusses the concept of a limit in calculus. It provides an outline that covers heuristics, errors and tolerances, examples, pathologies, and a precise definition of a limit. It introduces the error-tolerance game used to determine if a limit exists and illustrates this through examples evaluating the limits of x^2 as x approaches 0 and |x|/x as x approaches 0. It also discusses one-sided limits and evaluates the limit of 1/x as x approaches 0 from the right.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.
This document contains lecture notes on continuity from a Calculus I class at New York University. It begins with announcements about office hours and homework grades. It then reviews the definition of a limit and introduces the definition of continuity as a function having a limit equal to its value at a point. Examples are provided to demonstrate showing a function is continuous. The document states that polynomials, rational functions, and trigonometric functions are continuous based on their definitions and limit properties. It concludes by explaining the continuity of inverse trigonometric functions.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.
Continuity is the property that the limit of a function near a point is the value of the function near that point. An important consequence of continuity is the intermediate value theorem, which tells us we once weighed as much as our height.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and describes special types of discontinuities. The IVT states that if a function is continuous on a closed interval and takes on intermediate values, there exists a number in the interval where the function value is equal to the intermediate value. Examples are provided to illustrate the IVT, including proving the existence of the square root of two.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
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.
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 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.
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.
This document discusses continuity of functions, including definitions of continuity at a point and on an interval. It introduces the intermediate value theorem and extreme value theorem for continuous functions on closed intervals. It also states that differentiability at a point implies continuity at that point.
The document discusses several key concepts regarding derivatives:
(1) It explains how to use the derivative to determine if a function is increasing, decreasing, or neither on an interval using the signs of the derivative.
(2) It provides theorems and rules for finding local extrema (maxima and minima) of functions using the first and second derivative tests.
(3) It also discusses absolute extrema, monotonic functions, and the Rolle's Theorem and Mean Value Theorem which relate the derivative of a function to values of the function.
The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.
The document discusses how to determine if a function is increasing or decreasing on an interval using the derivative. It states that if the derivative is positive on an interval, the function is increasing on that interval, and if the derivative is negative, the function is decreasing. It provides steps to determine where a function is increasing or decreasing: 1) take the derivative, 2) find critical points where the derivative is 0 or undefined, 3) plot critical points to get intervals, 4) check sign of derivative in intervals. An example problem demonstrates finding the intervals where a cubic function is increasing or decreasing.
The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
This document provides an overview of the basic rules for taking derivatives, including:
- The derivative of a linear function f(x) = mx + b is f'(x) = m
- The derivative of a constant function f(x) = c is f'(x) = 0
- The power rule states that for f(x) = xn, the derivative is f'(x) = nxn-1
- The constant multiple rule states that the derivative of c * f(x) is c * f'(x)
- The sum and difference rules state that the derivative of f(x) + g(x) is f'(x) + g'(x) and the derivative
The document discusses limits and continuity of functions. It defines the limit of a function as x approaches a number, and gives examples of limits that do and do not exist. It then discusses properties of limits, including:
- If limits of f(x) and g(x) both exist as x approaches a, then the limit of their sum and product also exist.
- If the limit of f(x) exists as x approaches a, then the limit of kf(x) is k times the limit of f(x) for any constant k.
Finally, it discusses continuity, defining a function to be continuous at a point if the limit exists there and equals the function value. It gives
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.
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 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.
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.
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Lesson 5: Continuity
1. Section 1.5
Continuity
V63.0121.006/016, Calculus I
February 2, 2010
Announcements
Office Hours: M,W 1:30–2:30, R 9–10 (CIWW 726)
Written Assignment #2 due Thursday
First Quiz: Friday February 12 in recitation (§§1.1–1.4)
. . . . . .
3. Hatsumon
Here are some discussion questions to start.
True or False
At some point in your life you were exactly three feet tall.
True or False
At some point in your life your height (in inches) was equal to
your weight (in pounds).
. . . . . .
4. Hatsumon
Here are some discussion questions to start.
True or False
At some point in your life you were exactly three feet tall.
True or False
At some point in your life your height (in inches) was equal to
your weight (in pounds).
True or False
Right now there are a pair of points on opposite sides of the
world measuring the exact same temperature.
. . . . . .
5. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
6. Recall: Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of
f, then
lim f(x) = f(a)
x →a
This property is so useful it’s worth naming.
. . . . . .
7. Definition of Continuity
Definition
Let f be a function
defined near a. We say
that f is continuous at a
if
lim f(x) = f(a).
x →a
. . . . . .
8. Definition of Continuity
Definition y
.
Let f be a function
defined near a. We say
that f is continuous at a f
.(a ) .
if
lim f(x) = f(a).
x →a
A function f is
continuous if it is
continuous at every . x
.
point in its domain. a
.
. . . . . .
9. Scholium
Definition
Let f be a function defined near a. We say that f is continuous at
a if
lim f(x) = f(a).
x→a
There are three important parts to this definition.
The function has to have a limit at a,
the function has to have a value at a,
and these values have to agree.
. . . . . .
10. Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it is
continuous on R = (−∞, ∞).
(b) Any rational function is continuous wherever it is defined;
that is, it is continuous on its domain.
. . . . . .
12. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
. . . . . .
13. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
. . . . . .
14. At which other points?
√
For reference: f(x) = 4x + 1
If a > −1/4, then lim (4x + 1) = 4a + 1 > 0, so
x →a
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 4a + 1 = f(a)
x→a x →a x →a
and f is continuous at a.
. . . . . .
15. At which other points?
√
For reference: f(x) = 4x + 1
If a > −1/4, then lim (4x + 1) = 4a + 1 > 0, so
x →a
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 4a + 1 = f(a)
x→a x →a x →a
and f is continuous at a.
If a = −1/4, then 4x + 1 < 0 to the left of a, which means
√
4x + 1 is undefined. Still,
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 0 = 0 = f(a)
x →a + x→a+ x →a +
so f is continuous on the right at a = −1/4.
. . . . . .
16. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞).
. . . . . .
17. Showing a function is continuous
Example
√
Let f(x) = 4x + 1. Show that f is continuous at 2.
Solution
We want to show that lim f(x) = f(2). We have
x →2
√ √ √
lim f(x) = lim 4x + 1 = lim (4x + 1) = 9 = 3 = f(2).
x →a x →2 x →2
Each step comes from the limit laws.
Question
At which other points is f continuous?
Answer
The function f is continuous on (−1/4, ∞). It is right continuous
at −1/4 since lim f(x) = f(−1/4).
x→−1/4+
. . . . . .
18. The Limit Laws give Continuity Laws
Theorem
If f(x) and g(x) are continuous at a and c is a constant, then the
following functions are also continuous at a:
(f + g)(x)
(f − g)(x)
(cf)(x)
(fg)(x)
f
(x) (if g(a) ̸= 0)
g
. . . . . .
19. Why a sum of continuous functions is continuous
We want to show that
lim (f + g)(x) = (f + g)(a).
x →a
We just follow our nose:
lim (f + g)(x) = lim [f(x) + g(x)] (def of f + g)
x →a x →a
= lim f(x) + lim g(x) (if these limits exist)
x →a x →a
= f(a) + g(a) (they do; f and g are cts.)
= (f + g)(a) (def of f + g again)
. . . . . .
22. Trigonometric functions are continuous
t
.an
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .
23. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
. . . . . .
24. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { k π | k ∈ Z }.
c
. ot
. . . . . .
25. Trigonometric functions are continuous
t
.an s
. ec
sin and cos are continuous
on R.
sin 1
tan = and sec =
cos cos c
. os
are continuous on their
domain, which is
{π } .
R + kπ k ∈ Z . s
. in
2
cos 1
cot = and csc =
sin sin
are continuous on their
domain, which is
R { k π | k ∈ Z }.
c
. ot . sc
c
. . . . . .
28. Exponential and Logarithmic functions are continuous
For any base a > 1, .x
a
the function x → ax is .oga x
l
continuous on R
the function loga is
continuous on its .
domain: (0, ∞)
In particular ex and
ln = loge are continuous
on their domains
. . . . . .
31. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. .
. in−1
s
.
−
. π/2
. . . . . .
32. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
.
.
π
. os−1
c . . ec−1
s
. /2
π
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
33. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
34. Inverse trigonometric functions are mostly continuous
sin−1 and cos−1 are continuous on (−1, 1), left continuous
at 1, and right continuous at −1.
sec−1 and csc−1 are continuous on (−∞, −1) ∪ (1, ∞), left
continuous at −1, and right continuous at 1.
tan−1 and cot−1 are continuous on R.
.
.
π
. ot−1
c
. os−1
c . . ec−1
s
. /2
π
.an−1
t
. sc−1
c
. .
. in−1
s
.
−
. π/2
. . . . . .
36. Continuity FAIL
Example
Let {
x2 if 0 ≤ x ≤ 1
f(x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
. . . . . .
37. Continuity FAIL: The limit does not exist
Example
Let {
x2 if 0 ≤ x ≤ 1
f(x ) =
2x if 1 < x ≤ 2
At which points is f continuous?
Solution
At any point a in [0, 2] besides 1, lim f(x) = f(a) because f is
x →a
represented by a polynomial near a, and polynomials have the
direct substitution property. However,
lim f(x) = lim x2 = 12 = 1
x →1 − x →1 −
lim f(x) = lim 2x = 2(1) = 2
x→1+ x →1 +
So f has no limit at 1. Therefore f is not continuous at 1.
. . . . . .
39. Continuity FAIL
Example
Let
x2 + 2x + 1
f(x ) =
x+1
At which points is f continuous?
. . . . . .
40. Continuity FAIL: The function has no value
Example
Let
x2 + 2x + 1
f(x ) =
x+1
At which points is f continuous?
Solution
Because f is rational, it is continuous on its whole domain. Note
that −1 is not in the domain of f, so f is not continuous there.
. . . . . .
42. Continuity FAIL
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
. . . . . .
43. Continuity FAIL: function value ̸= limit
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f continuous?
Solution
f is not continuous at 1 because f(1) = π but lim f(x) = 7.
x →1
. . . . . .
45. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a.
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different.
. . . . . .
51. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a. By re-defining f(a) = lim f(x), f can be made
x →a
continuous at a
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different.
. . . . . .
52. Special types of discontinuites
removable discontinuity The limit lim f(x) exists, but f is not
x →a
defined at a or its value at a is not equal to the limit
at a. By re-defining f(a) = lim f(x), f can be made
x →a
continuous at a
jump discontinuity The limits lim f(x) and lim f(x) exist, but
x →a − x→a+
are different. The function cannot be made
continuous by changing a single value.
. . . . . .
54. The greatest integer function
[[x]] is the greatest integer ≤ x.
y
.
. .
3
x [[x]] y
. = [[x]]
0 0 . .
2 . .
1 1
1.5 1 . .
1 . .
1.9 1
2.1 2 . . . . . . x
.
−0.5 −1 −
. 2 −
. 1 1
. 2
. 3
.
−0.9 −1 .. 1 .
−
−1.1 −2
. .. 2 .
−
This function has a jump discontinuity at each integer.
. . . . . .
55. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
56. A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
. . . . . .
58. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
.(x )
.
.
. x
.
. . . . . .
59. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b]
f
.(x )
f
.(b ) .
f
.(a ) .
. a
. x
.
b
.
. . . . . .
60. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b).
f
.(x )
f
.(b ) .
N
.
f
.(a ) .
. a
. x
.
b
.
. . . . . .
61. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
. .
f
.(a ) .
. a
. c
. x
.
b
.
. . . . . .
62. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
.
f
.(a ) .
. a
. x
.
b
.
. . . . . .
63. Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let N
be any number between f(a) and f(b), where f(a) ̸= f(b). Then
there exists a number c in (a, b) such that f(c) = N.
f
.(x )
f
.(b ) .
N
. . . .
f
.(a ) .
. a c
. .1 x
.
c
.2 c b
.3 .
. . . . . .
64. What the IVT does not say
The Intermediate Value Theorem is an “existence” theorem.
It does not say how many such c exist.
It also does not say how to find c.
Still, it can be used in iteration or in conjunction with other
theorems to answer these questions.
. . . . . .
65. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
. . . . . .
66. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2].
. . . . . .
67. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
. . . . . .
68. Using the IVT
Example
Suppose we are unaware of the square root function and that it’s
continuous. Prove that the square root of two exists.
Proof.
Let f(x) = x2 , a continuous function on [1, 2]. Note f(1) = 1 and
f(2) = 4. Since 2 is between 1 and 4, there exists a point c in
(1, 2) such that
f(c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the method
of bisections.
. . . . . .
75. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
. . . . . .
76. Using the IVT
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f.
Solution
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
(More careful analysis yields 1.32472.)
. . . . . .
77. Outline
Continuity
The Intermediate Value Theorem
Back to the Questions
. . . . . .
78. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
. . . . . .
79. Question 1: True!
Let h(t) be height, which varies continuously over time.
Then h(birth) < 3 ft and h(now) > 3 ft.
So by the IVT there is a point c in (birth, now) where
h(c) = 3.
. . . . . .
80. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
. . . . . .
81. Question 2: True!
Let h(t) be height in inches and w(t) be weight in pounds,
both varying continuously over time.
Let f(t) = h(t) − w(t).
For most of us (call your mom), f(birth) > 0 and f(now) < 0.
So by the IVT there is a point c in (birth, now) where
f(c) = 0.
In other words,
h(c) − w(c) = 0 ⇐⇒ h(c) = w(c).
. . . . . .
82. Back to the Questions
True or False
At one point in your life you were exactly three feet tall.
True or False
At one point in your life your height in inches equaled your
weight in pounds.
True or False
Right now there are two points on opposite sides of the Earth
with exactly the same temperature.
. . . . . .
83. Question 3
Let T(θ) be the temperature at the point on the equator at
longitude θ.
How can you express the statement that the temperature on
opposite sides is the same?
How can you ensure this is true?
. . . . . .
84. Question 3: True!
Let f(θ) = T(θ) − T(θ + 180◦ )
Then
f(0) = T(0) − T(180)
while
f(180) = T(180) − T(360) = −f(0)
So somewhere between 0 and 180 there is a point θ where
f(θ) = 0!
. . . . . .
85. What have we learned today?
Definition: a function is continuous at a point if the limit of
the function at that point agrees with the value of the
function at that point.
We often make a fundamental assumption that functions we
meet in nature are continuous.
The Intermediate Value Theorem is a basic property of real
numbers that we need and use a lot.
. . . . . .