# Introduction Calculus

Calculus is a branch of Mathematics that deals with the study of limits, functions, derivatives , integrals and infinite series . The subject comes under the most important branches of applied Mathematics, and it serves as the basis for all the advanced mathematics calculations and engineering applications.

## Categories of Calculus

There are two major categories of Calculus:

•  Differential Calculus
•  Integral Calculus

In this content, we will focus majorly on different solving techniques of Calculus and will also throw some light on a wide range of concepts associated with the subject.

## Pre-Calculus

Before we jump into the detailed study of the subject, we must be familiar with some basic terms that are associated with the course. A good understanding of Calculus requires you to have a basic knowledge of:

## Functions

These functions are further characterized as

•  Polynomials
•  Rational Functions
•  Logarithms
•  Exponentials
•  Trigonometric

Throughout this course, we will be making use of these terms frequently, so it is better if you have a good understanding of the terms listed above. These are not very difficult-to-understand concepts. You may study them on your own before you proceed further into learning concepts of Calculus. Next we move to the core concepts and examples of Calculus.

## Polynomials

A polynomial function has the form f(x)=a_n x^n=a_(n-1) x^(n-1)+...+a_1 x+a_0, where a_n ,a_(n-1),...,a_0 are real numbers and n is a nonnegative integer. In other words, a polynomial is the sum of one or more monomials with real coefficients and non-negative integer exponents. The degree of the polynomial function is the highest value for n where n is not equal to 0.

Polynomial functions of only one term are called monomials or power functions. A power function has the form f(x)=ax^n.

For a polynomial function f, any number r for which f(r)=0 is called a root of the function f. When a polynomial function is completely factored, each of the factors helps identify zeros of the function.

## Rational Functions

Rational function" is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers. Rational functions supply important examples and occur naturally in many contexts. All polynomials are rational functions.

## Logarithms

Logarithmic functions are used to simplify complex calculations in many fields, including statistics, engineering, chemistry, physics, and music. For example,log(xy)=logx+logy and log(x/y)=log x - log y are logarithmic functions that essentially simplify multiplication to addition and division to subtraction. Logarithmic functions are the inverse of their exponential counterparts.

## Exponentials

An exponential function is a mathematical function of the following form: f ( x ) = a x where x is a variable, and a is a constant called the base of the function. The most commonly encountered exponential-function base is the transcendental number e , which is equal to approximately 2.71828. Thus, the above expression becomes: f ( x ) = e x` When the exponent in this function increases by 1, the value of the function increases by a factor of e . When the exponent decreases by 1, the value of the function decreases by this same factor (it is divided by e ).

## Trigonometric

A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, and cosecant. Also called circular function.

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