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

Lecture

Convexity and Concavity: Inflection Points, Taylor Expansion, and Darboux Sums

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 inflection points, convexity, concavity, and asymptotes in functions. It also explores higher derivatives, Taylor expansion, and Darboux sums, providing examples and applications. The lecture delves into the concepts of local maxima and minima, convex and concave effects, and the properties of functions in terms of convexity and concavity.

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)

Derivative test

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.

Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name). For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient is zero).

Second derivative

In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Inflection point

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

Related lectures (50)

Taylor's Formula: Developments and Extrema

Covers Taylor's formula, developments, and extrema of functions, discussing convexity and concavity.

Taylor Series and Function Analysis

Explores Taylor series, function properties, inflection points, and critical points in graphical and mathematical contexts.

Convexity Study and Limiting Developments

Explores convexity, concavity, and limiting developments for functions, emphasizing extrema and derivative properties.

Chapter 5: Function Studies

Covers the study of functions, including limits, derivatives, and sign variations.

General Case MOOC: Analysis I MOOC: Analysis I (part 6) : Study of functions, limited developments