Means | Calculus

Mean of a function

In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by Recall that a defining property of the average value of finitely many numbers is that . In other words, is the constant value which whenadded to itself times equals the result of adding the terms . By analogy, adefining property of the average value of a function over the interval is that In other words, is the constant value which when integrated over equals the result ofintegrating over . But the integral of a constant is just See also the first mean value theorem for integration, which guaranteesthat if is continuous then there exists a point such that The point is called the mean value of on . So we write and rearrange the preceding equation to get the above definition. In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function. There is also a harmonic average of functions and a quadratic average (or root mean square) of functions. (Wikipedia).

Video thumbnail

Determine if a Relation is a Function

http://mathispower4u.wordpress.com/

From playlist Intro to Functions

Video thumbnail

Working with Functions (1 of 2: Notation & Terminology)

More resources available at www.misterwootube.com

From playlist Working with Functions

Video thumbnail

What is a function?

This video explains what a mathematical function is and how it defines a relationship between two sets, the domain and the range. It also introduces three important categories of function: injective, surjective and bijective.

From playlist Foundational Math

Video thumbnail

What is a Function?

In this video, I talk about the definition of a function and properties of functions. I also go over some examples of how to determine whether a relation is a function or not and how to evaluate functions. Enjoy! Facebook: https://www.facebook.com/braingainzofficial Instagram: https://

From playlist College Algebra

Video thumbnail

Functions of equations - IS IT A FUNCTION

👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r

From playlist What is the Domain and Range of the Function

Video thumbnail

Determine if the equation represents a function

👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r

From playlist What is the Domain and Range of the Function

Video thumbnail

Pre-Calculus - Vocabulary of functions

This video describes some of the vocabulary used with functions. Specifically it covers what a function is as well as the basic idea behind its domain and range. For more videos visit http://www.mysecretmathtutor.com

From playlist Pre-Calculus - Functions

Video thumbnail

How to determine if an ordered pair is a function or not

👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r

From playlist What is the Domain and Range of the Function

Video thumbnail

What is a Function in Math and Physics? (A more intuitive explanation of Function Definition)

0:00 Introduction 3:35 Examples of functions and why we care 14:55 Make own function 19:12 Definition of a function 21:48 Outro

From playlist Summer of Math Exposition Youtube Videos

Video thumbnail

Live CEOing Ep 678: Language Design Review of Foreign Function Interface Functionality continued

In this episode of Live CEOing, Stephen Wolfram discusses upcoming improvements and features to the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch channel of Stephen Wolfram

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Основной доклад

Олег Маричев Wolfram Research, Inc.

From playlist Russian Virtual Conference 2019

Video thumbnail

Live CEOing Ep 23: NeuralNet Framework in the Wolfram Language

Watch Stephen Wolfram and teams of developers in a live, working, language design meeting. This episode is about NeuralNet Framework in the Wolfram Language.

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Live CEOing Ep 630: Language Design in Wolfram Language [Break, Supplied, & Held Expressions]

In this episode of Live CEOing, Stephen Wolfram discusses upcoming improvements and features to the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch channel of Stephen Wolfram

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Live CEOing Ep 386: NetFunction Design Review for Wolfram Language

In this episode of Live CEOing, Stephen Wolfram reviews the design of NetFunction and other upcoming machine learning functionality for the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the of

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Live CEOing Ep 561: Language Design in Wolfram Language [Compiler Review]

In this episode of Live CEOing, Stephen Wolfram discusses upcoming improvements and features to the Wolfram Language. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twitch channel of Stephen Wolfram

From playlist Behind the Scenes in Real-Life Software Design

Video thumbnail

Jia-Kun Liu (7/26/22): Some applications of optimal transportation

Abstract: In this talk, we will introduce some interesting applications of optimal transportation in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric inequality in geometry; and an application in image recognition relating to a transport betwee

From playlist Applied Geometry for Data Sciences 2022

Video thumbnail

Lecture 5 Lp Spaces on the real line

Lecture with Ole Christensen. Kapitler: 00:00 - Repetition; 05:00 - Introduction; 15:00 - Inequalities For Integrals; 26:00 - Support Of A Function; 31:30 - Special Continuous Functions; 34:15 - Examples;

From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math

Video thumbnail

Jianfeng Lu - Mathematical Models of Electronic Structure - IPAM at UCLA

Recorded 08 March 2022. Jianfeng Lu of Duke University presents "Mathematical Models of Electronic Structure" at IPAM's Advancing Quantum Mechanics with Mathematics and Statistics Tutorials. Learn more online at: http://www.ipam.ucla.edu/programs/workshops/advancing-quantum-mechanics-with-

From playlist Tutorials: Advancing Quantum Mechanics with Mathematics and Statistics - March 8-11, 2022

Video thumbnail

5.3 Functions - Evaluating Expressions with Function Notation

More resources available at www.misterwootube.com

From playlist Working with Functions

Related pages

Integral | Calculus | Domain of a function | Interval (mathematics) | Probability theory | Function (mathematics) | Mean value theorem | Mean | Domain (mathematical analysis) | Multivariable calculus | Euclidean space | Continuous function | Jensen's inequality