Differential Calculus: Introduction

UNIT‒I

DIFFERENTIAL CALCULUS

INTRODUCTION

Calculus is the mathematical tool used to analyze changes in physical quantities. It was developed in the 17th century to study four major classes of scientific and mathematical problems of the time. It is used to find the tangent line to a curve at a point, the length of a curve, the area of a region, and the volume of a solid. Find minima, maxima of quantities, such as the distance of a planet from sun. Given a formula for the distance traveled by a body in any specified amount of time, find the velocity and acceleration or velocity at any instant, and vice versa.

Before we study differential calculus, it is important to understand the concept of functions and their graphs. This is a major pre‒requisite before any Calculus course often dealt with in a separate course called Pre‒Calculus. The fundamental and important aspects of calculus are depending upon functions. The basic concepts of calculus are concerned with function, graphs, and their transformations. A function can be represented by an equation, table, graph or simply in words. Function arises when one quantity depends on another quantity.

Let as consider the following example

The area A of a circle depends on the radius r of the circle. Therefore, the area is related by the equation A = πr², where r (positive number) is associated with one value of A and it is defines as A is a function of r. Thus r is called the independent variables and A is called the dependent variable.

Function

A function is a rule that assigns to each element x in a set D to exactly one element, called f(x), in a set E. It is denoted by f: D→ E.

Consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f(x) is the value of f at x and is read "f of x". The range of f is the set of all possible values of f(x) as x throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable.

A function can be represented using an arrow diagram as shown below:

Each arrow connects an element of D to an element of E. The arrow indicates that f(x) is associated with x, f(a) is associated with a, and so on. The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs

{(x, f(x))|x ϵ D}.

In other words, the graph of ƒ consists of all points (x,y) in the coordinate plane such that y = f(x) and x is in the domain of f.

The graph of a function f gives us a useful picture of the behavior or "life history" of a function.

Since the y‒coordinate of any point (x, y) on the graph is y = f(x), we can read the value of f(x) from the graph as being the height of the graph above the point x. This procedure is demonstrated in the following figure.

Using the graph of ƒ, the domain off on the x‒axis and its range on the y‒axis can be obtained and the same is explained in the following figure.

Thought out this chapter D and E are a set of real numbers. The set D is called domain of the function. The range of f is the set of all possible value of f(x) as x varies thought the domain.

Note:

If y = f(x) is a function, then x is called independent variable and y is called dependent variable.

Vertical Line Test:

A curve in the xy plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

Let us consider the two curves below to represent vertical line test.

If each vertical line x = a intersects a curve only once at (a, b), then exactly one functional value is defined by f(a) = b.

If the line x = a intersects the curve at two different points (a, b) and (a, c), then the curve can't represent a function because a function can assign two values to a.