Lesson 27: Integration by Substitution (Section 041 slides)

This talk was presented as part of the Trends and Advances in Monte Carlo Sampling Algorithms Workshop.

This document outlines lecture material on evaluating definite integrals from a Calculus I course at New York University. The lecture covers using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Properties of the definite integral such as additivity and comparison properties are discussed. The Second Fundamental Theorem of Calculus is proved, relating the definite integral of a function to the antiderivative of that function. Examples are provided of using this theorem to compute areas under curves defined by functions.

The first report of Machine Learning Seminar organized by Computational Linguistics Laboratory at Kazan Federal University. See http://cll.niimm.ksu.ru/cms/lang/en_US/main/seminars/mlseminar

This document discusses using moment conditions to perform Bayesian inference when the likelihood function is intractable or unknown. It outlines some approaches that have been proposed, including approximating the likelihood using empirical likelihood or pseudo-likelihoods. However, these approaches do not guarantee the same consistency as a true likelihood. Alternative approximative Bayesian methods are also discussed, such as Approximate Bayesian Computation, Integrated Nested Laplace Approximation, and variational Bayes. The empirical likelihood method constructs a likelihood from generalized moment conditions, but its use in Bayesian inference requires further analysis of consistency in each application.

This document introduces tensors through examples. It defines a vector as a rank 1 tensor and a matrix as a rank 2 tensor. It then provides an example of a rank 3 tensor. The document discusses how to define an inner product between tensors and provides examples using vectors and matrices. It also gives an example of how derivatives of a function can produce tensors of different ranks. Finally, it introduces the concept of decomposing matrices into their symmetric and antisymmetric parts.

The document discusses achieving higher-order convergence for integration on RN using quasi-Monte Carlo (QMC) rules. It describes the problem that when using tensor product QMC rules on truncated domains, the convergence rate scales with the dimension s as (α log N)sN-α. The goal is to obtain a convergence rate independent of the dimension s. The document proposes using a multivariate decomposition method (MDM) to decompose an infinite-dimensional integral into a sum of finite-dimensional integrals, then applying QMC rules to each integral to achieve the desired higher-order convergence rate.

1. The document discusses universal prediction without assuming data is either discrete or continuous. It presents a method to estimate generalized density functions to achieve universal prediction for any unknown probabilistic model. 2. A key insight is that universal prediction can be achieved by estimating the ratio between the true density function and a reference measure, without needing to directly estimate the density function. This allows universal prediction for data that is neither discrete nor continuous. 3. The method involves recursively refining partitions of the sample space to estimate the density ratio. It is shown that this ratio can be estimated universally for any density function, achieving the goal of prediction without assumptions about the data type.

This document provides an outline and introduction to the topics of pattern recognition and machine learning. It begins with an overview of key concepts like probability theory, decision theory, and the curse of dimensionality. It then covers specific techniques like polynomial curve fitting, the Gaussian distribution, and Bayesian curve fitting. The document also includes an appendix on properties of matrices such as determinants, matrix derivatives, and the eigenvector equation.

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. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.

The document discusses numerical methods for solving nonlinear equations, including root finding and systems of nonlinear equations. It covers the basics of nonlinear solvers like bisection, Newton's method, and fixed-point iteration. For one-dimensional root finding, it analyzes the convergence properties and order of convergence for these methods. It then extends the discussion to systems of nonlinear equations and shows how Newton's method can be applied by taking derivatives to form the Jacobian matrix.

This document outlines a calculus lecture on evaluating definite integrals. The lecture will: 1) Use the Evaluation Theorem to evaluate definite integrals and write antiderivatives as indefinite integrals. 2) Interpret definite integrals as the "net change" of a function over an interval. 3) Provide examples of evaluating definite integrals and estimating integrals using the midpoint rule. 4) Discuss properties of integrals such as additivity and illustrate how a definite integral from a to c can be broken into integrals from a to b and b to c.

This document presents two tensor factorization methods: Exponential Family Tensor Factorization (ETF) and Full-Rank Tensor Completion (FTC). ETF generalizes Tucker decomposition by allowing for different noise distributions in the tensor and handles mixed discrete and continuous values. FTC completes missing tensor values without reducing dimensionality by kernelizing Tucker decomposition. The document outlines these methods and their motivations, discusses Tucker decomposition, and provides an example applying ETF to anomaly detection in time series sensor data.

This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.

This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.

f is decreasing on (−∞, −4/5] and increasing on [−4/5, ∞). . . . . . . V63.0121.021, Calculus I (NYU) Section 4.2 The Shapes of Curves November 16, 2010 13 / 32

The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.

The document discusses optimizing the area of a rectangular field using 320 yards of fencing. It poses the problem of determining the shape of the field that maximizes the enclosed area given the fixed amount of fencing. It also provides a link to an online quiz about derivatives and graphing that contains problems requiring deep thought about concepts learned.

This document is from a Calculus I course at New York University. It outlines the objectives and content for Sections 2.1-2.2 on the derivative and rates of change. The sections will define the derivative, discuss how it models rates of change, and how to find the derivative of various functions graphically and numerically. It will also cover how to find higher-order derivatives and sketch the derivative graph from a function graph.

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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.

This lecture covers various solution methods for unconstrained optimization problems, including: 1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems. 2) Newton's method and the false position method for curve fitting to minimize functions in one dimension. 3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.

- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals. - It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.

This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.

Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.

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 reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.

This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.

This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.

This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.

The derivative of a sum of functions is the sum of the derivatives of those functions, but the derivative of a product or a quotient of functions is not so simple. We'll derive and use the product and quotient rule for these purposes. It will allow us to find the derivatives of other trigonometric functions, and derivatives of power functions with negative whole number exponents.

This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.

This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.

basic limits, limit laws, limits of algebraic and rational functions, limits of piecewise functions, limits of trigonometric quotients.

The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.

This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.

This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.

This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.

The document is a summary of lecture notes for a Calculus I class. It discusses integration by substitution, providing theory, examples, and objectives. Key points covered include the substitution rule for indefinite integrals, working through examples like finding the integral of √x2+1 dx, and noting substitution can transform integrals into simpler forms. Definite integrals using substitution are also briefly mentioned.

The document discusses the fundamental theorem of calculus. It begins by outlining the topics to be covered, including a review of the second fundamental theorem of calculus, an explanation of the first fundamental theorem of calculus, and examples of differentiating functions defined by integrals. It then provides more details on these topics, such as defining the integral as a limit, stating the second fundamental theorem, using integrals to represent concepts like distance traveled and mass, and working through an example of finding the derivative of a function defined by an integral.

The document outlines a calculus class lecture on the fundamental theorem of calculus, including recalling the second fundamental theorem, stating the first fundamental theorem, and providing examples of differentiating functions defined by integrals. It gives announcements for upcoming class sections and exam dates, lists the objectives of the current section, and provides an outline of topics to be covered including area as a function, statements and proofs of the theorems, and applications to differentiation.

The document provides an overview of Section 5.4 on the Fundamental Theorem of Calculus from a Calculus I course at New York University. It outlines topics to be covered, including recalling the Second Fundamental Theorem, stating the First Fundamental Theorem, and differentiating functions defined by integrals. Examples are provided to illustrate using the theorems to find derivatives and integrals.

Here are the key points about g given f: - g represents the area under the curve of f over successive intervals of the x-axis - As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative - The concavity (convexity or concavity) of g will match the concavity of f over each interval In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.

The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.

The document provides an example of using the substitution method to evaluate the indefinite integral ∫(x2 + 3)3 4x dx. It introduces the substitution u = x2 + 3, which allows the integral to be rewritten as ∫u3 2 du and then evaluated as (1/2)u4 = (1/2)(x2 + 3)4. The solution is compared to directly integrating the expanded polynomial. The document outlines the theory and notation of substitution for indefinite integrals.

The document provides notes on integration by substitution. It begins with objectives of being able to transform integrals using substitutions, evaluate indefinite integrals using substitution, and evaluate definite integrals using substitution. It then gives examples of using substitution to evaluate indefinite integrals of expressions like tan(x) dx and definite integrals like ∫01 cos2(x)sin(x) dx. The document emphasizes that substitution allows integrals to be transformed into simpler forms.

The document discusses partitions, Riemann sums, and the definite integral. It begins by defining partitions of an interval [a,b] and Riemann sums with respect to those partitions. Examples are given of partitions and calculating Riemann sums. The definite integral is then defined as the limit of Riemann sums as the partition size approaches zero. Several properties of definite integrals are stated, including linearity and the Fundamental Theorems of Calculus. Examples are provided of evaluating definite integrals using these properties.

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 method of substitution is the chain rule in reverse. At first it looks magical, then logical, and then you realize there's an art to choosing the right substitution. We try to demystify with many worked-out examples.

This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.

This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.

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.

We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties

The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.

The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.

This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.

The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.

This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.

This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of

This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.

The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.

This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.

This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.

This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.

No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output. V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33

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.

linear function, quadratic functions, cubic functions, power functions, transformations, and compositions.

linear function, quadratic functions, cubic functions, power functions, transformations, and compositions.

This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.