Exponentials | Analytic functions | E (mathematical constant) | Special hypergeometric functions | Elementary special functions
The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity which, along with the definition , shows that for positive integers n, and relates the exponential function to the elementary notion of exponentiation. The base of the exponential function, its value at 1, , is a ubiquitous mathematical constant called Euler's number. While other continuous nonzero functions that satisfy the exponentiation identity are also known as exponential functions, the exponential function exp is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1; that is, for all real x, and Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form where the base b is a positive real number. The relation for positive b and real or complex x establishes a strong relationship between these functions, which explains this ambiguous terminology. The real exponential function can also be defined as a power series. This power series definition is readily extended to complex arguments to allow the complex exponential function to be defined. The complex exponential function takes on all complex values except for 0 and is closely related to the complex trigonometric functions, as shown by Euler's formula. Motivated by more abstract properties and characterizations of the exponential function, the exponential can be generalized to and defined for entirely different kinds of mathematical objects (for example, a square matrix or a Lie algebra). In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, computer science, chemistry, engineering, mathematical biology, and economics. The real exponential function is a bijection from to . Its inverse function is the natural logarithm, denoted or because of this, some old texts refer to the exponential function as the antilogarithm. (Wikipedia).
How to determine, domain range, and the asymptote for an exponential graph
👉 Learn all about graphing exponential functions. An exponential function is a function whose value increases rapidly. To graph an exponential function, it is usually useful to first graph the parent function (without transformations). This can be done by choosing 2-3 points of the equatio
From playlist How to Graph Exponential Functions | Learn About
Math tutorial for graphing an exponential equation
👉 Learn how to graph exponential functions involving vertical shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a
From playlist How to Graph Exponential Functions with Vertical Shift
Graphing the parent graph of an exponential equation with base 2
👉 Learn how to graph exponential functions. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a range of values of x and
From playlist How to Graph Exponential Functions with Stretch and Compression
Graphing exponential growth equations
👉 Learn how to graph exponential functions involving horizontal shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing
From playlist How to Graph Exponential Functions
Graphing an exponential function and determine the domain and range
👉 Learn how to graph exponential functions involving vertical shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a
From playlist How to Graph Exponential Functions with Vertical Shift
Math tutorial for graphing an exponential function with transformations
👉 Learn how to graph exponential functions involving horizontal shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing
From playlist How to Graph Exponential Functions
How to graph an exponential equation with multiple transformations
👉 Learn how to graph exponential functions. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a range of values of x and
From playlist How to Graph Exponential Functions with Stretch and Compression
Graphing the exponential function with multiple transformations
👉 Learn how to graph exponential functions involving horizontal shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing
From playlist How to Graph Exponential Functions
Graphing exponential functions
👉 Learn how to graph exponential functions involving vertical shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a
From playlist How to Graph Exponential Functions with Vertical Shift
Math 1A/1B. Pre-Calculus: Intro to Exponential and Logarithmic Functions
UCI Math 1A/1B: Pre-Calculus Pre-Calculus: Intro to Exponential and Logirthmic Functions View the complete course: http://ocw.uci.edu/courses/math_1a1b_precalculus.html Instructor: Sarah Eichhorn, Ph.D and Rachel Lehman, Ph.D License: Creative Commons CC-BY-SA Terms of Use: http://ocw.uci
From playlist Math 1A/1B: Pre-Calculus - Exponentials and Logarithms
The exponential function and the logarithm -- College Algebra
This lecture is on College Algebra. It follows the introductory part of the book Calculus Illustrated by Peter Saveliev. The text of the book can be found at http://calculus123.com.
From playlist College Algebra
Worldwide Calculus: The Exponential Function
Lecture on 'The Exponential Function' from 'Worldwide Differential Calculus' and 'Worldwide AP Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Worldwide Single-Variable Calculus for AP®
The Exponential Distribution and Exponential Random Variables | Probability Theory
What is the exponential distribution? This is one of the most common continuous probability distributions. We'll go over an introduction of the exponential distribution and exponentially distributed random variables in today's probability theory video lesson. The exponential distribution
From playlist Probability Theory
Extremes and Records by Sanjib Sabhapandit ( Lecture - 2 )
PROGRAM BANGALORE SCHOOL ON STATISTICAL PHYSICS - X ORGANIZERS : Abhishek Dhar and Sanjib Sabhapandit DATE : 17 June 2019 to 28 June 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore This advanced level school is the tenth in the series. This is a pedagogical school, aimed at bridgin
From playlist Bangalore School on Statistical Physics - X (2019)
Sebastian Eterović, UC Berkeley
April 12, Sebastian Eterović, UC Berkeley Existential Closedness and Differential Algebra
From playlist Spring 2022 Online Kolchin seminar in Differential Algebra
Lec 16. Exponential Functions College Algebra with Dennis Allison
See full course at: https://cosmolearning.org/courses/college-algebra-pre-calculus-with-dennis-allison/ Video taken from: http://desource.uvu.edu/videos/math1050.php Lecture by Dennis Allison from Utah Valley University.
From playlist UVU: College Algebra with Dennis Allison | CosmoLearning Math
MIT 18.650 Statistics for Applications, Fall 2016 View the complete course: http://ocw.mit.edu/18-650F16 Instructor: Philippe Rigollet In this lecture, Prof. Rigollet talked about linear model, generalization, and examples of disease occurring rate, prey capture rate, Kyphosis data, etc.
From playlist MIT 18.650 Statistics for Applications, Fall 2016
Stanford CS229: Machine Learning | Summer 2019 | Lecture 6 - Exponential Family & GLM
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3Eb7mIi Anand Avati Computer Science, PhD To follow along with the course schedule and syllabus, visit: http://cs229.stanford.edu/syllabus-summer2019.html
From playlist Stanford CS229: Machine Learning Course | Summer 2019 (Anand Avati)
Welcome to Quantitative Risk Management (QRM). There is so much confusion about tails, that it is time to clarify what we are speaking about. Heavy tails, long tails and fat tails are not the same thing from a statistical and probabilistic point of view. In mathematics we need to be preci
From playlist Quantitative Risk Management
How to graph an exponential function
👉 Learn how to graph exponential functions involving vertical shift. An exponential function is a function that increases rapidly as the value of x increases. To graph an exponential function, it is usually very useful to make the table of values of the function. This is done by choosing a
From playlist How to Graph Exponential Functions with Vertical Shift