Calculus is a branch of mathematics that deals with the study of change. The concept of antiderivative is an essential part of calculus. It is a fundamental concept that is used in various fields such as physics, engineering, economics, and more. In this article, we will explore the definition, meaning, and origins of antiderivative.
Definitions
An antiderivative is a function that, when differentiated, gives the original function. In other words, it is the reverse process of differentiation. The antiderivative of a function f(x) is denoted by F(x). Mathematically, we can write:
F'(x) = f(x).
Where F'(x) denotes the derivative of F(x) with respect to x.
Origin
The concept of antiderivative has been around for centuries. The ancient Greeks were aware of the concept of differentiation, but it was not until the 17th century that the concept of antiderivative was developed. The concept was first introduced by Isaac Barrow, a mathematician, and theologian. Later, it was further developed by other mathematicians such as Newton and Leibniz.
Meaning in different dictionaries
According to the Oxford English Dictionary, an antiderivative is “a function whose derivative is a given function.” Merriam-Webster defines it as “a function whose derivative is a given function.” The Cambridge Dictionary defines it as “a function that, when differentiated, gives a given function.”.
Associations
Antiderivatives are closely related to integrals. In fact, the antiderivative of a function is also known as the indefinite integral of the function. Antiderivatives are used to find the area under a curve, which is the definite integral of a function.
Synonyms
The synonyms of antiderivative include indefinite integral, primitive function, and anti-derivative.
Antonyms
The antonyms of antiderivative include derivative, differential, and slope.
The same root words
The root words of antiderivative are anti-, which means opposite or against, and derivative, which means a function that is obtained from another function through differentiation.
Example Sentences
- The antiderivative of x^2 is (1/3)x^3 + C, where C is a constant of integration.
- To find the area under a curve, we need to find the antiderivative of the function and evaluate it at the limits of integration.
- The concept of antiderivative is used in various fields such as physics, engineering, and economics.
- The antiderivative of sin(x) is -cos(x) + C, where C is a constant of integration.
- The antiderivative of e^x is e^x + C, where C is a constant of integration.