Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Taylor Approximation: Understanding the Basics

In course

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. ...

Description

This lecture covers the concept of Taylor approximation, focusing on developing an understanding of how to approximate functions using simple polynomials around a chosen point. The instructor explains the process of controlling errors in approximations and demonstrates the application of Taylor series in various examples.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Thomas Mountford

D'origine britannique, né en 1961. Il a reçu la bourse présidentielle des jeunes investisseurs (Presidential Young Investigator Grant) en 1990 et le prix de la Fraternité Sloan (Sloan Fellowship) en 1991. Son travail montre que les valeurs critiques d'une classe large de de systèmes de particules proches sont égales à 1, et que, suite à plusieurs travaux sur la trajectoire du mouvement brownien, incluant une simulation numérique, le nombre d'îlots browniens en 2 dimensions tend vers l'infini quand leurs tailles tendent vers 0. Il reçoit le prix Rosenbaum en 1993 lui ouvrant ainsi les portes du Isaac Newton Institute à Cambridge. Il est également décoré par le prix Rollo Davidson en 1995 et nommé membre honoraire de l'Institute of Mathematical Statistics en 2001. Assistant Profeseur de Mathematique a UCLA 1987- 1991 Associate Professeur de Mathematique a UCLA 1991-1993 Professeur de Mathematique a UCLA 1993-2001 Professeur Departement de mathématiques, EPFL dès 2001

Official source

Ontological neighbourhood

Mathematics

Analysis: Calculus

Related concepts (32)

Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

Artificial intelligence

Artificial intelligence (AI) is the intelligence of machines or software, as opposed to the intelligence of human beings or animals. AI applications include advanced web search engines (e.g., Google Search), recommendation systems (used by YouTube, Amazon, and Netflix), understanding human speech (such as Siri and Alexa), self-driving cars (e.g., Waymo), generative or creative tools (ChatGPT and AI art), and competing at the highest level in strategic games (such as chess and Go).

OpenAI

OpenAI is an American artificial intelligence (AI) research laboratory consisting of the non-profit OpenAI, Inc. and its for-profit subsidiary corporation OpenAI, L.P.. OpenAI conducts research on artificial intelligence with the declared intention of developing "safe and beneficial" artificial general intelligence, which it defines as "highly autonomous systems that outperform humans at most economically valuable work".

Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that where is the remainder term.

Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation.

Related lectures (30)

Taylor Series: Approximating Functions with Polynomials

Explores approximating functions with polynomials using Taylor series and discusses the convergence of series representations.

Calculus Foundations: Taylor Series and Integrals

Introduces calculus concepts, focusing on Taylor series and integrals, including their applications and significance in mathematical analysis.

Taylor Series: Basics

Covers Taylor series, polynomial approximations for function behavior, coefficient finding, and error estimation.

Developing Cosine Composition MOOC: Analysis I MOOC: Analysis I (part 6) : Study of functions, limited developments

Explores the composition of Taylor expansions through an example, simplifying complex compositions to identify essential terms.

Approximation of Functions by Taylor Polynomials