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.
1) The document discusses basic differentiation rules in Calculus I, including the derivative of constant functions, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine.
2) It provides examples of finding the derivative of squaring and cubing functions using the definition of the derivative, and discusses properties of these functions and their derivatives.
3) The document also introduces the concept of the second derivative and notation for it, and provides an example of finding the derivative of the square root function.
Lesson 12: Linear Approximations and Differentials (handout)Matthew Leingang
The line tangent to a curve 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 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 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 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.
Bayesian inversion of deterministic dynamic causal modelskhbrodersen
1. The document discusses various methods for Bayesian inference and model comparison in dynamic causal models, including variational Laplace approximation, sampling methods, and computing model evidence.
2. Variational Laplace approximation involves factorizing the posterior distribution and iteratively optimizing a lower bound on the model evidence called the negative free energy.
3. Sampling methods like Markov chain Monte Carlo generate stochastic approximations to the posterior by constructing a Markov chain with the target distribution as its equilibrium distribution.
Lesson 13: Exponential and Logarithmic Functions (handout)Matthew Leingang
This document contains lecture notes from a Calculus I class covering exponential and logarithmic functions. The notes discuss definitions and properties of exponential functions, including the number e and the natural exponential function. Conventions for rational, irrational and negative exponents are also defined. Examples are provided to illustrate approximating exponential expressions with irrational exponents using rational exponents. The objectives are to understand exponential functions, their properties, and apply laws of logarithms including the change of base formula.
1) The document discusses basic differentiation rules in Calculus I, including the derivative of constant functions, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine.
2) It provides examples of finding the derivative of squaring and cubing functions using the definition of the derivative, and discusses properties of these functions and their derivatives.
3) The document also introduces the concept of the second derivative and notation for it, and provides an example of finding the derivative of the square root function.
Lesson 12: Linear Approximations and Differentials (handout)Matthew Leingang
The line tangent to a curve 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 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 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 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.
Bayesian inversion of deterministic dynamic causal modelskhbrodersen
1. The document discusses various methods for Bayesian inference and model comparison in dynamic causal models, including variational Laplace approximation, sampling methods, and computing model evidence.
2. Variational Laplace approximation involves factorizing the posterior distribution and iteratively optimizing a lower bound on the model evidence called the negative free energy.
3. Sampling methods like Markov chain Monte Carlo generate stochastic approximations to the posterior by constructing a Markov chain with the target distribution as its equilibrium distribution.
Lesson 13: Exponential and Logarithmic Functions (handout)Matthew Leingang
This document contains lecture notes from a Calculus I class covering exponential and logarithmic functions. The notes discuss definitions and properties of exponential functions, including the number e and the natural exponential function. Conventions for rational, irrational and negative exponents are also defined. Examples are provided to illustrate approximating exponential expressions with irrational exponents using rational exponents. The objectives are to understand exponential functions, their properties, and apply laws of logarithms including the change of base formula.
Lesson 12: Linear Approximation and Differentials (Section 41 slides)Matthew Leingang
The line tangent to a curve 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.
Introductory talk
more technicities in
@inproceedings{schoenauer:inria-00625855,
hal_id = {inria-00625855},
url = {http://hal.inria.fr/inria-00625855},
title = {{A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize}},
author = {Schoenauer, Marc and Teytaud, Fabien and Teytaud, Olivier},
abstract = {{Multi-Modal Optimization (MMO) is ubiquitous in engineer- ing, machine learning and artificial intelligence applications. Many algo- rithms have been proposed for multimodal optimization, and many of them are based on restart strategies. However, only few works address the issue of initialization in restarts. Furthermore, very few comparisons have been done, between different MMO algorithms, and against simple baseline methods. This paper proposes an analysis of restart strategies, and provides a restart strategy for any local search algorithm for which theoretical guarantees are derived. This restart strategy is to decrease some 'step-size', rather than to increase the population size, and it uses quasi-random initialization, that leads to a rigorous proof of improve- ment with respect to random restarts or restarts with constant initial step-size. Furthermore, when this strategy encapsulates a (1+1)-ES with 1/5th adaptation rule, the resulting algorithm outperforms state of the art MMO algorithms while being computationally faster.}},
language = {Anglais},
affiliation = {TAO - INRIA Saclay - Ile de France , Microsoft Research - Inria Joint Centre - MSR - INRIA , Laboratoire de Recherche en Informatique - LRI},
booktitle = {{Artificial Evolution}},
address = {Angers, France},
audience = {internationale },
year = {2011},
month = Oct,
pdf = {http://hal.inria.fr/inria-00625855/PDF/qrrsEA.pdf},
}
Lesson 2: A Catalog of Essential Functions (handout)Matthew Leingang
The document outlines topics to be covered in a Calculus I class, including essential functions such as linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. It provides examples and definitions of these functions and notes how their graphs can be transformed through vertical and horizontal shifts. Assignments include WebAssign problems due January 31st and a written assignment due February 2nd.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
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.
Design of observers for nonlinear systems using the Frobenius theorem. Presentation for the defense of my MSc Thesis at the School of Applied Mathematics, NTU Athens.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
This document summarizes the uses of the Christoffel-Darboux (CD) kernel in the spectral theory of orthogonal polynomials. The CD kernel is defined in terms of orthogonal polynomials and can be interpreted as the integral kernel of a projection operator. It has applications in analyzing the zeros of orthogonal polynomials, Gaussian quadrature, variational principles, and characterizing the absolutely continuous, singular continuous, and pure point spectra of measures. Recent work has expanded its uses in studying universality in the bulk of the spectrum and properties of orthogonal polynomials.
This document defines statistical symbols and their meanings that are used throughout a statistics textbook. It provides the symbol, how it is spoken, and a brief definition of its meaning. Some key symbols defined include population mean (μ), sample mean (x), standard deviation (s), probability (P), normal distribution (N), z-score, confidence level, confidence interval, null hypothesis (H0), and alternate hypothesis (H1). The document is organized by statistical topic and lists over 50 common statistical symbols.
This document provides an overview of data analysis techniques for asteroseismology. It discusses principles of data analysis including Fourier analysis, power spectrum estimation, and goodness-of-fit tests. It describes methods for parameter fitting such as maximum likelihood estimation and least squares fitting. It also discusses challenges in data analysis like irregularly spaced data and closely spaced frequencies. A variety of spectral analysis techniques are introduced such as the Lomb-Scargle periodogram and deconvolution analysis to address these challenges.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
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.
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.
The document reviews concepts related to random variables and random processes. It defines key terms such as:
- Discrete and continuous random variables and their probability distribution and density functions.
- Joint, marginal, and conditional density functions which describe the relationships between multiple random variables.
- Independent and orthogonal random variables, and concepts like inner products, that are used to analyze relationships between random variables.
- Various types of convergence for sequences of random variables such as almost sure, mean square, and in probability which are important for analyzing random processes over time.
The review covers critical foundational concepts for understanding and working with random variables and stochastic processes.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also covers derivatives of sine and cosine. Examples are provided, like finding the derivative of the squaring function x^2, which is 2x. Notation for derivatives is explained, including Leibniz notation. The concept of the second derivative is also introduced.
The document discusses the history and development of perceptrons from 1943 to 1958, including McCulloch and Pitts' work establishing that networks of mathematical units can perform computations, Hebb's theory of learning through synaptic plasticity, and Rosenblatt's perceptron, the first machine learning algorithm that was proven to be able to learn any function that could be programmed into it. The perceptron is a simple learning algorithm for binary classification that works by adjusting the weights between inputs and an output unit to find a separating hyperplane between classes.
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.
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 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.
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.
Lesson 12: Linear Approximation and Differentials (Section 41 slides)Matthew Leingang
The line tangent to a curve 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.
Introductory talk
more technicities in
@inproceedings{schoenauer:inria-00625855,
hal_id = {inria-00625855},
url = {http://hal.inria.fr/inria-00625855},
title = {{A Rigorous Runtime Analysis for Quasi-Random Restarts and Decreasing Stepsize}},
author = {Schoenauer, Marc and Teytaud, Fabien and Teytaud, Olivier},
abstract = {{Multi-Modal Optimization (MMO) is ubiquitous in engineer- ing, machine learning and artificial intelligence applications. Many algo- rithms have been proposed for multimodal optimization, and many of them are based on restart strategies. However, only few works address the issue of initialization in restarts. Furthermore, very few comparisons have been done, between different MMO algorithms, and against simple baseline methods. This paper proposes an analysis of restart strategies, and provides a restart strategy for any local search algorithm for which theoretical guarantees are derived. This restart strategy is to decrease some 'step-size', rather than to increase the population size, and it uses quasi-random initialization, that leads to a rigorous proof of improve- ment with respect to random restarts or restarts with constant initial step-size. Furthermore, when this strategy encapsulates a (1+1)-ES with 1/5th adaptation rule, the resulting algorithm outperforms state of the art MMO algorithms while being computationally faster.}},
language = {Anglais},
affiliation = {TAO - INRIA Saclay - Ile de France , Microsoft Research - Inria Joint Centre - MSR - INRIA , Laboratoire de Recherche en Informatique - LRI},
booktitle = {{Artificial Evolution}},
address = {Angers, France},
audience = {internationale },
year = {2011},
month = Oct,
pdf = {http://hal.inria.fr/inria-00625855/PDF/qrrsEA.pdf},
}
Lesson 2: A Catalog of Essential Functions (handout)Matthew Leingang
The document outlines topics to be covered in a Calculus I class, including essential functions such as linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. It provides examples and definitions of these functions and notes how their graphs can be transformed through vertical and horizontal shifts. Assignments include WebAssign problems due January 31st and a written assignment due February 2nd.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
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.
Design of observers for nonlinear systems using the Frobenius theorem. Presentation for the defense of my MSc Thesis at the School of Applied Mathematics, NTU Athens.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
This document summarizes the uses of the Christoffel-Darboux (CD) kernel in the spectral theory of orthogonal polynomials. The CD kernel is defined in terms of orthogonal polynomials and can be interpreted as the integral kernel of a projection operator. It has applications in analyzing the zeros of orthogonal polynomials, Gaussian quadrature, variational principles, and characterizing the absolutely continuous, singular continuous, and pure point spectra of measures. Recent work has expanded its uses in studying universality in the bulk of the spectrum and properties of orthogonal polynomials.
This document defines statistical symbols and their meanings that are used throughout a statistics textbook. It provides the symbol, how it is spoken, and a brief definition of its meaning. Some key symbols defined include population mean (μ), sample mean (x), standard deviation (s), probability (P), normal distribution (N), z-score, confidence level, confidence interval, null hypothesis (H0), and alternate hypothesis (H1). The document is organized by statistical topic and lists over 50 common statistical symbols.
This document provides an overview of data analysis techniques for asteroseismology. It discusses principles of data analysis including Fourier analysis, power spectrum estimation, and goodness-of-fit tests. It describes methods for parameter fitting such as maximum likelihood estimation and least squares fitting. It also discusses challenges in data analysis like irregularly spaced data and closely spaced frequencies. A variety of spectral analysis techniques are introduced such as the Lomb-Scargle periodogram and deconvolution analysis to address these challenges.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)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.
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.
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.
The document reviews concepts related to random variables and random processes. It defines key terms such as:
- Discrete and continuous random variables and their probability distribution and density functions.
- Joint, marginal, and conditional density functions which describe the relationships between multiple random variables.
- Independent and orthogonal random variables, and concepts like inner products, that are used to analyze relationships between random variables.
- Various types of convergence for sequences of random variables such as almost sure, mean square, and in probability which are important for analyzing random processes over time.
The review covers critical foundational concepts for understanding and working with random variables and stochastic processes.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also covers derivatives of sine and cosine. Examples are provided, like finding the derivative of the squaring function x^2, which is 2x. Notation for derivatives is explained, including Leibniz notation. The concept of the second derivative is also introduced.
The document discusses the history and development of perceptrons from 1943 to 1958, including McCulloch and Pitts' work establishing that networks of mathematical units can perform computations, Hebb's theory of learning through synaptic plasticity, and Rosenblatt's perceptron, the first machine learning algorithm that was proven to be able to learn any function that could be programmed into it. The perceptron is a simple learning algorithm for binary classification that works by adjusting the weights between inputs and an output unit to find a separating hyperplane between classes.
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.
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 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.
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.
1) The document is a set of lecture notes for a Calculus I class covering limits.
2) It introduces the concept of limits and the error-tolerance game used to determine limits. Common limit rules and examples of limits are presented, including direct substitution, limits of polynomials, and trigonometric limits.
3) The notes cover determining limits using algebra, such as limit laws for addition, subtraction, multiplication, division and powers. Exceptions and edge cases are discussed.
The derivative of a function is another function. We look at the interplay between the two. Also, new notations, higher derivatives, and some sweet wigs
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 key concepts related to rates of change and derivatives:
1) It defines average rate of change (ARC) as the slope of a secant line on a graph or using the slope formula algebraically, and instantaneous rate of change (IRC) as the slope of the tangent line.
2) It introduces the difference quotient as a way to define ARC and IRC algebraically without a graph by taking the limit as h approaches 0.
3) A derivative is defined as a function that gives the IRC, allowing it to be evaluated at any point without graphing by taking the limit of the difference quotient.
This document contains lecture notes from a Calculus I class at New York University on the topic of continuity. The notes cover the definition of continuity, examples of continuous and discontinuous functions, and properties of continuous functions. Specifically, it discusses how the limit laws can be used to show that combinations of continuous functions are also continuous. It also examines ways a function can fail to be continuous, such as if the limit does not exist or if the function is not defined at a point. The overall document provides an introduction to and overview of the concept of continuity in calculus.
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.
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.
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!
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.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
This document provides a review of key calculus concepts for MATH 31A, including continuity, the squeeze theorem, the intermediate value theorem, differentiability, the chain rule, related rates, extreme values, optimization, the fundamental theorem of calculus, integration by substitution, and finding areas and volumes of revolution. It emphasizes understanding concepts, showing work, and not panicking if steps are inconclusive.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It explains the limit laws for addition, subtraction, multiplication, division and nth roots of functions. It uses the error-tolerance game framework to justify the limit laws.
The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.
This document provides a review of key calculus concepts for MATH 31A including continuity, the squeeze theorem, the intermediate value theorem, differentiability, chain rule, related rates, optimization, the fundamental theorem of calculus, integration by substitution, areas, volumes, and general tips for the final exam. Key points covered include the definition of continuity, how to apply the squeeze theorem and intermediate value theorem, the relationship between differentiability and continuity, how to set up and solve related rates problems, how to find critical points and extreme values using derivatives, and how to integrate using the fundamental theorem of calculus and substitution.
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 provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
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.
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.
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.
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. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Sec on 1.5
Con nuity
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
.
February 7, 2011
.
.
Notes
Announcements
Get-to-know-you extra
credit due Friday
February 11
Quiz 1 February 17/18 in
recita on
.
.
Notes
Objectives
Understand and apply the defini on of
con nuity for a func on at a point or
on an interval.
Given a piecewise defined func on,
decide where it is con nuous or
discon nuous.
State and understand the Intermediate
Value Theorem.
Use the IVT to show that a func on
takes a certain value, or that an
equa on has a solu on
.
.
. 1
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2. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Last time
Defini on
We write
lim f(x) = L
x→a
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L
as we like) by taking x to be sufficiently close to a (on either side of
a) but not equal to a.
.
.
Notes
Basic Limits
Theorem (Basic Limits)
lim x = a
x→a
lim c = c
x→a
.
.
Notes
Limit Laws for arithmetic
Theorem (Limit Laws)
Let f and g be func ons with limits at a point a. Then
lim (f(x) + g(x)) = lim f(x) + lim g(x)
x→a x→a x→a
lim (f(x) − g(x)) = lim f(x) − lim g(x)
x→a x→a x→a
lim (f(x) · g(x)) = lim f(x) · lim g(x)
x→a x→a x→a
f(x) limx→a f(x)
lim = if lim g(x) ̸= 0
x→a g(x) limx→a g(x) x→a
.
.
. 2
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3. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Hatsumon
Here are some discussion ques ons 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.
.
.
Notes
Outline
Con nuity
The Intermediate Value Theorem
Back to the Ques ons
.
.
Recall: Direct Substitution Notes
Property
Theorem (The Direct Subs tu on Property)
If f is a polynomial or a ra onal func on 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.
.
.
. 3
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4. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Definition of Continuity
Defini on y
Let f be a func on defined near
a. We say that f is con nuous at f(a)
a if
lim f(x) = f(a).
x→a
A func on f is con nuous if it is
con nuous at every point in its
domain. . x
a
.
.
Notes
Scholium
Defini on
Let f be a func on defined near a. We say that f is con nuous at a if
lim f(x) = f(a).
x→a
There are three important parts to this defini on.
The func on has to have a limit at a,
the func on has to have a value at a,
and these values have to agree.
.
.
Notes
Free Theorems
Theorem
(a) Any polynomial is con nuous everywhere; that is, it is
con nuous on R = (−∞, ∞).
(b) Any ra onal func on is con nuous wherever it is defined; that is,
it is con nuous on its domain.
.
.
. 4
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5. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Showing a function is continuous
.
Example
√
Let f(x) = 4x + 1. Show that f is con nuous at 2.
Solu on
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.
.
.
Notes
At which other points?
Ques on
√
As before, let f(x) = 4x + 1. At which points is f con nuous?
Solu on
.
.
Notes
Limit Laws give Continuity Laws
Theorem
If f(x) and g(x) are con nuous at a and c is a constant, then the
following func ons are also con nuous at a:
(f + g)(x) (fg)(x)
(f − g)(x) f
(x) (if g(a) ̸= 0)
(cf)(x) g
.
.
. 5
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6. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Sum of continuous is continuous
We want to show that
lim (f + g)(x) = (f + g)(a).
x→a
We just follow our nose.
(def of f + g) lim (f + g)(x) = lim [f(x) + g(x)]
x→a x→a
(if these limits exist) = lim f(x) + lim g(x)
x→a x→a
(they do; f and g are cts.) = f(a) + g(a)
(def of f + g again) = (f + g)(a)
.
.
Notes
Trig functions are continuous
tan sec
sin and cos are con nuous
on R.
sin 1
tan = and sec = are
cos cos cos
con nuous on their domain,
{π
which is } .
sin
R + kπ k ∈ Z .
2
cos 1
cot = and csc = are
sin sin
con nuous on their domain,
which is R { kπ | k ∈ Z }. cot csc
.
.
Notes
Exp and Log are continuous
For any base a 1,
the func on x → ax is ax
loga x
con nuous on R
the func on loga is
con nuous on its .
domain: (0, ∞)
In par cular ex and
ln = loge are con nuous
on their domains
.
.
. 6
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7. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Inverse trigonometric functions Notes
are mostly continuous
sin−1 and cos−1 are con nuous on (−1, 1), le con nuous at 1,
and right con nuous at −1.
sec−1 and csc−1 are con nuous on (−∞, −1) ∪ (1, ∞), le
con nuous at −1, and right con nuous at 1.
tan−1 and cot−1 are con nuous on R.
π
cot−1 cos−1 sec−1
π/2 tan−1
csc−1
sin−1 .
. −π/2
−π .
Notes
What could go wrong?
In what ways could a func on f fail to be con nuous at a point a?
Look again at the equa on from the defini on:
lim f(x) = f(a)
x→a
.
.
Notes
Continuity FAIL: no limit
.
Example
{
x2 if 0 ≤ x ≤ 1
Let f(x) = . At which points is f con nuous?
2x if 1 x ≤ 2
Solu on
At any point a besides 1, lim f(x) = f(a) because f is represented by a
x→a
polynomial near a, and polynomials have the direct subs tu on property.
lim f(x) = lim− x2 = 12 = 1 and lim+ f(x) = lim+ 2x = 2(1) = 2
x→1− x→1 x→1 x→1
So f has no limit at 1. Therefore f is not con nuous at 1.
.
.
. 7
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8. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Graphical Illustration of Pitfall #1
y
4
3 The func on cannot be
con nuous at a point if the
2
func on has no limit at that
1 point.
. x
−1 1 2
−1
.
.
Notes
Continuity FAIL: no value
Example
Let
x2 + 2x + 1
f(x) =
x+1
At which points is f con nuous?
Solu on
Because f is ra onal, it is con nuous on its whole domain. Note that
−1 is not in the domain of f, so f is not con nuous there.
.
.
Notes
Graphical Illustration of Pitfall #2
y
1 The func on cannot be
con nuous at a point outside
. x its domain (that is, a point
−1 where it has no value).
.
.
. 8
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9. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Continuity FAIL: value ̸= limit
Example
Let {
7 if x ̸= 1
f(x) =
π if x = 1
At which points is f con nuous?
Solu on
f is not con nuous at 1 because f(1) = π but lim f(x) = 7.
x→1
.
.
Notes
Graphical Illustration of Pitfall #3
y
7 If the func on has a limit and
a value at a point the two
π must s ll agree.
. x
1
.
.
Notes
Special types of discontinuities
removable discon nuity 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 con nuous
x→a
at a
jump discon nuity The limits lim− f(x) and lim+ f(x) exist, but are
x→a x→a
different. The func on cannot be made con nuous by
changing a single value.
.
.
. 9
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10. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Special discontinuities graphically
y y
7 2
π 1
. x . x
1 1
removable jump
.
.
Notes
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 . x
2.1 2 −2 −1 1 2 3
−0.5 −1 −1
−0.9 −1
−1.1 −2 −2
This func on has a jump discon nuity at each integer.
.
.
Notes
Outline
Con nuity
The Intermediate Value Theorem
Back to the Ques ons
.
.
. 10
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11. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is con nuous 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.
.
.
Notes
Illustrating the IVT
f(x)
Theorem
Suppose that f is con nuous f(b)
on the closed interval [a, b]
N
and let N be any number
between f(a) and f(b), where f(a)
f(a) ̸= f(b). Then there exists
a number c in (a, b) such that
f(c) = N. . a c b x
.
.
Notes
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.
S ll, it can be used in itera on or in conjunc on with other
theorems to answer these ques ons.
.
.
. 11
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12. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Using the IVT to find zeroes
Example
Let f(x) = x3 − x − 1. Show that there is a zero for f on the interval
[1, 2].
Solu on
f(1) = −1 and f(2) = 5. So there is a zero between 1 and 2.
In fact, we can “narrow in” on the zero by the method of bisec ons.
.
.
Notes
Finding a zero by bisection
y
x f(x)
1 −1
1.25 − 0.296875
1.3125 − 0.0515137
1.375 0.224609
1.5 0.875
2 5
. x
(More careful analysis yields
1.32472.)
.
.
Using the IVT to assert existence Notes
of numbers
Example
Suppose we are unaware of the square root func on and that it’s
con nuous. Prove that the square root of two exists.
Proof.
.
.
. 12
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13. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Notes
Outline
Con nuity
The Intermediate Value Theorem
Back to the Ques ons
.
.
Notes
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.
.
.
Notes
Question 1
To be discussed in class!
.
.
. 13
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15. . V63.0121.001: Calculus I
. Sec on 1.5:. Limits February 7, 2011
Summary Notes
What have we learned today?
Defini on: a func on is con nuous at a point if the limit of the
func on at that point agrees with the value of the func on at
that point.
We o en make a fundamental assump on that func ons we
meet in nature are con nuous.
The Intermediate Value Theorem is a basic property of real
numbers that we need and use a lot.
.
.
Notes
.
.
Notes
.
.
. 15
.