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**Introduction to Calculus**

Calculus is a subfield of mathematics, which mainly concerned with the area of math focused on limits, derivatives, functions and infinite series. Calculus math is concerned with a main part of modern mathematics education. Two major categories of Calculus are given below

- Differential calculus
- Integral calculus, which are correlated by the fundamental theorem of calculus.

Geometry is concerned with the study of shape, Calculus is the study of change and algebra is the study of many operations and their application to solving given equations. The calculus theory provides gateway to other, progression in mathematics devoted to the study of limits and functions. In a broad way this is called a mathematical analysis. Calculus has many more applications in the field of science, economics and engineering and these can be solved so many problems for which algebra alone is not sufficient.

**Differential calculus:** Differential calculus is a sub-discipline or sub-category of calculus, which is mainly concerned with the study of the rates where quantities change. The principal objects of study in differential calculus are the derivative of a function, related notions similar to the differential and the applications of differential. The derivative of a function at a selected input value describes the rate of change of the function near that input value. The method of finding a derivative is called differentiation. As per theory of geometry, the derivative at a point equals the slope of the tangent line to the graph of the function at that point, for a real-valued function of a single real variable. The derivative of a function at a point typically defines the best linear approximation to the function at that point.

The two sub fields of calculus which are Differential calculus and integral calculus are connected by the basic theorem of calculus, which states that differentiation is the reverse process to integration.

**Integral calculus: ** Integration is an necessary concept in mathematics and together with it is inverse; differentiation is one of the two main operations in calculus. Here an interval [a, b] of the real line and a function f is given of a real variable x

The definite integral is not described formally to be the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the vertical lines x = a, and x = b, such that area below the x-axis subtracts from the total, and that above the x-axis added to the total.

The word integral may also refer to the notion of the ant derivative F whose derivative is the given function f. In this situation, it is called an indefinite integral and is written:

The rules of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. From the fundamental theorem of calculus, which they independently developed the integration is connected with differentiation: if f is a continuous real-valued function defined on a closed interval [a, b], then, once an ant derivative F of f is known, the definite integral of function f over that interval is given by

**Pre-calculus**

In American mathematics education, pre-calculus, also recognized as algebra 3, is a superior form of secondary school algebra, is a foundational mathematical subject. It is also known as Introduction to Analysis. Pre-calculus math are in fact two separate courses in some schools: Algebra & Trigonometry. Pre-calculus trains students for calculus the same way as pre-algebra trains students for Algebra I. Whereas pre-algebra explains students several different essential algebra topics, pre-calculus does not include calculus, but elaborates topics which will be applied in calculus. It often covers algebraic topics which might not have been given attention in earlier algebra courses. Various pre-calculus courses might be dissimilar with others in terms of content. Such as, an honors level course might spend more time on conic sections, vectors and additional topics required for calculus. A college introductory class might spotlight on topics used in industry-related careers, like matrices.

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