In mathematics, calculus is a branch that is all about the assessment of numbers that varies in a one-liner way. This branch of mathematics deals with continuous change. Therefore, in this topic, we will teach about the definition of calculus, differential and integral calculus.
Meaning of Calculus
It is a division of mathematics that allocates with discovering the properties of derivatives and integrals of functions, by methods initially based on the summation of insignificant alterations.
Definition of Calculus
It is a branch of mathematics, which was developed by Newton and Leibniz that deals with the study of the rate of change. Moreover, we use it in mathematical models to get optimal solutions. In addition, it helps us to understand the changes between the values that relate by a function. However, it mainly focuses on some important topics such as integration, differentiation, functions, limits, and so on.
Besides, on a broad perspective the calculus is classified into two different types:
- Differential Calculus
- Integral Calculus
The differential and Integral calculus deals with the impact on the function of a slight change in the independent variable as it leads to zeros. Furthermore, both these (differential and integral) calculus serves as a foundation for the higher branch of Mathematics that we know as “Analysis.” Besides, mathematical calculus plays a very important role in modern physics as well as in science and technology.
Basic Calculus
Basically, it is the study of both differentiation and integration. And both these concepts are established on the idea of limits and functions. Besides, some concepts like exponents, continuity are the basis of the advanced calculus. Moreover, it explains the two types that are differential and integral.
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Differential Calculus
It relates to the problems of finding the rate of change of a function with respect to the other variables. Moreover, to get the ideal solution, we use derivatives to find the maxima and minima values of a function. In addition, it arises from the study of the limit of a quotient. Furthermore, it deals with the variables such as x and y, functions f(x), and the consequent changes in the variables x and y.
Besides, the symbols dx and dy are known as differentials. Moreover, the process of finding the derivatives is what we call differentiation. Also, the derivatives of the function are represented by dy/dx or f'(x). This means that the function is the derivative of y with respect to the variable x.
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Integral Calculus
We mostly use it for the following two purposes:
- For calculating f from f’ (that is from its derivative). Moreover, if the function f is differentiable in the interval of consideration, then we can define f’ in that interval.
- For calculating the area under a curve.
Integration
It is just opposite (reciprocal) of differentiation. We can understand differentiation as dividing part into many small parts. In addition, we can say that integration as a collection of small parts in order o form a whole. Generally, we use it for calculating the area.
Definite Integral
It has a definite boundary within which we need to calculate the function. Furthermore, the lower and upper limit of the independent variable of a function is definite. So, we describe this integration using definite integrals, which is:
\(\int_{b}^{a} f(x). dx = F(x)\)
Indefinite Integral
These don’t have a definite border that is no upper or lower pre-defined limits. Hence, the integration value is always accompanied by a constant value (C). Such as:
\(\int f(x). dx = F(x) + C\)
Solved Example for You
Q1: What is the solution to the differential equation?
A1: The solution to a differential equation is a function because when we plug it (and its various derivatives) into the equation, then we can find the equation to be satisfactory.
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