CALCULUS INTRODUCTION AND MAIN TYPES

by Eeshan Baheti

Calculus is basically a branch of mathematics which is vastly used in 

physics and chemistry. In physics, it is widely used especially in limits and functions. 

Calculus denotes the courses of elementary mathematical analysis.  

The word calculus (plural:calculi) is a latin term, meaning originally 'small pebble'. 

Because such pebbles were used for calculation and today it means method of computation.

HISTORY OF CALCULUS

The branch of mathematics that is widely used in many fields and subjects was invented 

by Gottfried Wilhelm Leibniz.

Calculus, known in its early history indefinite calculus, is a mathematical discipline 

focused on limits, continuity, derivatives, integrals and infinite series.

Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory

of indefinite calculus in the later seventeenth century.

DEFINITION OF CALCULUS

Calculus is the study of how things change . In this we study about the

relationship between continuously varying functions.

Or we can say that calculus is a branch of mathematics that involves a study of rate of change.

Calculus helps to determine how particles, stars and matter actually move 

and change in real time. Calculus is used in many fields of physics and chemistry.

MAIN TYPES OF CALCULUS

The main types of calculus are differentiation and integration. It also involves 

sub topics such as calculus in limits which is used in physics and to find average value 

of continuous functions in an interval which is applicable in physics and chemistry.

DIFFERENTIATION

The purpose of differential calculus is to study the nature which is increase

or decrease. And it is also used to study the amount of variation 

in a quantity when another quantity (on which first quantity depends) varies independently.

Differentiation is used to calculate the average rate of change. We can do it by

the formula given below of the following average rate of change.

Let a function = f(x) be plotted as shown in figure.

Average rate of change w.r.t. x in the interval [x1,x2] is

Average rate of change = change in y / change in x

=> Average rate of change = △y / △x

=> Average rate of change = y2-y1 / x2-x1

=> Average rate of change = slope of chord AB

Instantaneous rate of change is defined as the rate of change in y with x

at a particular value of x. It is measured graphically by the slope of the tangent  

drawn to the y-x graph at point (x,y) and algebraically by the first derivative of function

y = f(x). The formula to find instantaneous  rate of change is given below of the given figure.

Instantaneous rate of change = dy / dx

=> Instantaneous rate of  change = slope of tangent

=> Instantaneous rate of change = tan Ө

INTEGRATION 

Integration is the reverse process of differentiation. By help of integration

we can find a function whose derivative is known.

Consider a function f(x) whose differentiation w.r.t. x is equal to g(x) then-

                   

               ∫ g(x)dx = f(x) + c

Here c is the constant of integration and this is called indefinite integration.

When a function is integrated between a lower limit and an upper limit, 

it is called definite integration.

Consider a function f(x) whose differentiation w.r.t. x is equal to g(x), 

in an interval a ≤ x ≤ b then-

                 

               ∫ g(x) dx = f(b) - f(a)    (int limits - b,a)

Here int limits b,a will reside in upper part of int and lower part of int.

So, in a conclusion, integration has two types - indefinite integration and definite integration.