Calculus is all about the comparison of quantities which vary in one-liner way. It has significant applications in Science and Engineering. Many of the topics that we study like velocity, acceleration or current in a circuit do not behave in a linear fashion. If quantities are changing continually, we need calculus to study about it. Calculus is the branch of mathematics that deals with continuous change. It has two main branches.
- Differential Calculus- Concerned with the study of the rates at which quantities change.
- Integral Calculus- Concerned with the accumulation of quantities and areas within the curves.
Making the perspective simple, we can say that differential calculus cuts something into very tiny pieces to find the rate of change while Integral calculus accumulates the tiny pieces to find the quantity. Just like division and multiplication are inverse to each other, Integral and Differential are inverse to each other.
Application
Aryabhatta used differential calculus to find the motion of the moon. Similarly, modern mathematics uses differential equations and functions to find out maxima and minima of curves which are in common usage.
Few other applications are as follows
- Automatic air conditioners- temperature control.
- Cruise control in cars
- Water mixers
- Industrial control systems rockets, ships etc.
Limits
It is the degree of closeness to any value or the approaching term.
Limits are all about approaching.
A limit is normally expressed as-
\(\lim\limits_{x \to c}f(x) = A\)
It is read as “the limit of f of x as x approaches c equals A”.The “lim” shows limit, and fact that function f(x) approaches the limit A as x approaches c is described by the right arrow as
\(f(x) = A\)
Derivatives
Differential calculus is the study of the definition, properties, and applications of the derivative of a function.The process of finding the derivative is called differentiation.
The expression for derivative is defined as-
\(f'(x)= \lim\limits_{\bigtriangleup x \to 0}\frac{f(x+\bigtriangleup x) – f(x)}{\bigtriangleup x}\) |
Integration
Integration is the process of finding the area under the curve.It can be used to find the areas, volumes etc.
It can be of two types
- Definite Integral- where the limits are defined.
- Indefinite Integral- where limits are not defined.
Differential Equation
The basic parts of a differential equation are functions and their derivatives.The function represent physical quantities, derivative represents the rate of change and their relationship is represented by the differential equation.
Isaac Newton gave three kinds of differential equation-
\(\frac{\mathrm{d} y}{\mathrm{d} x}= f(x)\) \(\frac{\mathrm{d} y}{\mathrm{d} x}= f(x,y)\) \(x_{1}\frac{\partial y}{\partial x_{1}} + x_{2}\frac{\partial y}{\partial x_{2}} = y\) |
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