Calculus

Calculus

Calculus is all about the comparison of quantities which vary in a 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.

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”.

Derivatives & its Properties:

Whereas a derivative refers to the instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount. The Derivative of a function is represented as:
\(\lim\limits_{x \to h}\frac{ f(x+h)- f(x)}{h} = A\)

Continuity:

A function is said to be continuous at a particular point if the following three conditions are satisfied-

  1. f(a) is defined
  2. \(\lim\limits_{x \to a}f(x)\) exists
  3. \(\lim\limits_{x \to a^{-}}f(x) = \lim\limits_{x \to a^{+}}f(x) = f(a)\)

Continuity and Differentiability:

A Function is always continuous if it is differentiable at any point, whereas the vice-versa condition is not always true.

Derivative of a function:

Derivative of a function f(x) signifies the rate of change of the function f(x) with respect to x at a point lying in its domain. For a function to be differentiable at any point x = a, in its domain, it must be continuous at that particular point but vice-versa is necessarily not always true.

Differentiation:

Differentiation can be defined as a derivative of a function regarding the independent variable and can be applied to measure the function per unit change in the independent variable.

Quotient Rule:

The Quotient rule is a method for determining the derivative (differentiation) of a function which is in a fractional form.

Chain Rule:-The rule applied for finding the derivative of the composition of a function is basically known as the chain rule.

Differential Formula:-It is important to be aware of the differentiation of various function to calculate the differential value of a various composite function. Thus here are the important formulas that will help in solving complex differential problems.

Derivative:-The instantaneous rate of change of a quantity with respect to the other. It helps to investigate the moment by moment nature of an amount.

Derivative of function of parametric form:-The representation of a function y(x) when represented as a third variable is known a parametric form. A relation between x and y expressible in the form x = f(t) and y = g(t) is a parametric form representation with parameter as t.

Differential calculus and Approximation:-We are already aware of the differential calculus. This technique is also helpful in calculating the approximate value of root values.

Derivative of Inverse Trigonometric Function:-These arcus functions are used to obtain angle for a given trigonometric value. Inverse trigonometric functions have various application in engineering, geometry, navigation etc.

Integral Calculus:-It is the branch of calculus where we study about integrals and their properties. It is mostly useful for following two purposes:

  1. To calculate f from f’(i.e from its derivative). If a function f is differentiable in the interval of consideration, then f’ is defined in that interval.
  2. To calculate area under a curve.

Integration:-Integration is reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as collection of small parts in order to form a whole. It is generally used for calculating area.

Antiderivative Formula:-Anything that is an opposite of a function and has been differentiated in trigonometric terms is known as an antiderivative. It is important to know the antiderivative of various functions in order to calculate the integration value of functions.

Methods of Integration:-It can be a challenging task to calculate the integration value of various composite functions. Thus we use various methods of integration for calculating the antiderivative of functions.

Definite Integral:-A definite integral has a specific boundary within which function needs to be calculated. The lower limit and upper limit of the independent variable of a function is specified, its integration is described using definite integrals. A definite integral is denoted as:
\(\int_{a}^{b}f(x).dx = F(x)\)

Indefinite Integral:-An Indefinite Integral do not have specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied with a constant value (c). It is denoted as:
\(\int f(x).dx = F(x)+ c\)


Practise This Question

Two players A and B throw a die alternately for a prize of Rs 11, which is to be won by a player who first throws a six.  If A starts the game, their respective expectations are