FUNCTIONAND

LIMIT

2

2.1 FUNCTION

• BACKGROUND

The term of “function” was first used by Leibniz in 1673 to denote the dependence of one quantity on another

Example :

The area of a circle depends on its radius r by the equation A = r2;

We say that “ A is a function of r “

Notation

• Leonard Euler introduced the using of a letter of alphabet such as f to denote a function or relationship.

Example :

y = f(x)

is read “y equals f of x”, that is the value of y depends on the value of x

DEFINITION

• A function is a rule that assigns to each element of set A one and only one element of set B

• The set A is called domain of the function

• The set of all possible value of f(x) as x varies over the domain is called the range of f

DEFINITION

y = f(x)

• y is called dependent variable

• x is called independent variable

• The graph of a function f is the graph

of the equation y = f(x)

Example

Example

• f(x)=2x-1

• g(x)=x^2

Which is a function?

2.2 OPERATION ON FUNCTION;

CLASSIFYING FUNCTIONS

• Given function f and g, their sum f+g, difference f-g, product f.g and quotient f/g are defined by

(f+g)(x)=f(x)+g(x)

(f-g)(x)=f(x)-g(x)

(f.g)(x)=f(x).g(x)

(f/g)(x)=f(x)/g(x)

• For the function f+g, f-g, and f.g the

domain is defined to be the

intersection of the domains of f and g

and for f/g the domain is this

intersection with the points where

g(x) = 0 excluded

• If f is a function and k is a real

number, then the function kf is

defined by

(kf)(x)=k.f(x)

and the domain of kf is the same as

the domain of f

• Given two function f and g, the composition of f with g, denoted by f o g, is the function defined by

(fog)(x)=f(g(x))

where the domain of f o g consists of all x in the domain of g for which g(x) is in the domain of f.

Classification of Functions

• Constant function, f(x)=c, c is a constant value

• Monomial in x, f(x)=cxn, c is a constant value, n is a nonnegative

• Polynomial in x, f(x)=a0+a1x+a2x2+…+anx

n

• Linear, f(x)=a0+a1x

• Quadratic, f(x)=a0+a1x+a2x2

• Cubic f(x)=a0+a1x+a2x2+a3x

3

• Rational function, ratio of two polynomial

2.3 INTRODUCTION TO CALCULUS :

TANGENTS AND VELOCITY

• TWO FUNDAMENTAL PROBLEM OF CALCULUS :

1. The tangent problem (differential calculus)

2. The area problem

(integral calculus)

The tangent problem

• Given a function f

and a point P(x0,y0)

on its graph, find

the equation of the

tangent to the

graph at P (figure

2.3.1)

The area problem

Given a function f,

find the area

between the graph

of f and an interval

[a,b] on the x-axis

(figure 2.3.2)

• Secant line is the line

through P and Q where

Q is any point on the

curve different from P.

• If we move the point Q along the curve toward P, the secant line will rotate toward “limiting” position. The line T occupying this limiting position is called the tangent line.

• If P(x0,y0) and Q(x1,y1) lie on the graph f so

that f(x0)=y0 and f(x1)=y1, then the slope of the

secant line through P and Q is :

msec =slope of PQ =y1-y0 =

f(x1)-f(x0)

x1-x0 x1-x0

• If x1-x0=h so that x1=x0+h then we can write :

• As Q approaches P along the graph of f, or

equivalently as h=x1-x0 gets closer and

closer to zero, the secant line through P and

Q approaches the tangent line at P.

• Thus the slope of the secant line msec

approaches the slope of the tangent line

mtan.

msec=f(x0+h)-f(x0)

h

mtan=limiting value as h approaches zero of

f(x0+h)-f(x0)

h

Velocity

• The average velocity of an object moving in one direction along a line is :

Average velocity =Distance traveled

Time elapsed

• If over the time interval from t0 to t1 the

distance traveled is

s1-s0

and the time elapsed is

t1-t0

so the average velocity during the interval is

given by

Average velocity =s1-s0

t1-t0

Geometric Interpretation of

Average Velocity

• For a particle moving in one direction on a

straight line, the average velocity between

time t0 and t1 is represented geometrically

by the slope of the secant line connecting

(t0, s0) and (t1,s1 ) on the position versus

time curve.

Geometric Interpretation of

Instantaneous Velocity

• For a particle moving in one direction on a

straight line, the instantaneous velocity at

time t0 is represented geometrically by the

slope of the tangent line at (t0,s0) on the

position versus time curve.

Geometric Interpretation of Average

and Instantaneous Velocity

2.4 LIMIT (AN INTUITIVE

INTRODUCTION)

• In the last section we saw that the concepts of tangent and instantaneous velocity ultimately rest on the notion of a "limit" or "value approached by" a function. In this section as well as the next few we will investigate the notion of limit in more detail. Our development of limits in this text proceeds in three stages:

1. First we discuss limits intuitively.

2. Then we discuss methods for computing limits.

3. finally, we give a precise mathematical discussion of limits

• Limits are used to describe how a function

behaves as the independent variable moves

toward a certain value. To illustrate, consider

the function

• If the value of f(x) approaches the

number L1 as x approaches x0 from the

right side, we write

1)(lim0

Lxfxx

• If the value of f(x) approaches the

number L1 as x approaches x0 from the

left side, we write

2)(lim0

Lxfxx

Lxfxfxxxx

)(lim)(lim

00

• If limit from the left side is the same as

the limit from the right side, say

Then we write

Lxfxx

)(lim0

2.5 LIMITS (COMPUTATIONAL

TECHNIQUES)

THEOREM

EXERCISE