Math Homework Help
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# Math Homework Help

## Graph of Functions (Constant Function)

March 15th, 2008 . by Vanaja

A function of the type y=f(x)=k where k is a fixed real number.

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The graph of the constant function is a straight line parallel to x axis, which is above or below according to k is positive or negative. That is if k>0 the graph will be above x axis and at a distance k units above it. If k<0, then it will be k units below it. If k=0, then the graph will coincide with the x axis.

The domain of the constant function f(x)=k is the set R of all real numbers and range of the function is the singleton set {k}

So, we can see a constant function is a many-one into function.

## Limit of a function at a point.

February 16th, 2008 . by Vanaja

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

It is very important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus.

Definition

Let f(x) be a function of x. Let a and l be constants such that as $x\rightarrow a$, we have $f\left(x \right)\rightarrow l$. In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

$\lim_{x\rightarrow a}f\left(x \right)=l$

In case , no such number l exist, then we say that $\lim_{x\rightarrow a}f\left(x \right)$does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have $\lim_{n\rightarrow \infty }$ (Area of the polygon of n sides)=Area of the circle.

## The Fundamental Principle of Counting

January 13th, 2008 . by Vanaja

If an event can happen in exactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.

Illustration.

Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?

Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.

There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.

Generalisation

If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp….

## Some defnitions of average

January 10th, 2008 . by Vanaja

One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called central value or an average or the expected value of the variable.

The term ” average ” has been defined by various authors. Some important definitions are given below.

• “Average is an attempt to find one single figure to describe whole of figures” -Clark
• ” An average is a single value selected from a group of values to represent them in some way-a value which is supposed to stand for whole group, of which it is a part, as typical of all the values in he group” -A.E Waugh
• ” An average is a typical value in the sense that it is sometimes employed to represent all the individual values in a series or of a variable” -Ya-Lun-Chou
• “An average is a single value within the range of the data that is used to represent all the values in the series.Since an average is somewhere within the range of the data. It is also called a measure of central value” - Croxton &Cowden

## Algorithm & Lemma

April 3rd, 2007 . by Vanaja

An algorithm is a series of well defined steps
which gives a procedure for solving a type of
problem?

The word algorithm comes from the name
of the 9th century Persian mathematician
al-Khwarizmi. In fact, even the word ‘algebra’
is derived from a book, he wrote, called Hisab
al-jabr w’al-muqabala.

A lemma is a proven statement used for
proving another statement.

## Applied Mathematics

February 26th, 2007 . by Vanaja

Applied mathematics is the application of pure mathematics in the service of a given purpose. It has some direct or practical application to objects and happenings in the material world. It plays a great role in the development of various subjects. Every discovery in science owes much to applied mathematics. Principles of applied mathematics have been useful in the investigation of such phenomenon as heat , sound, light, optics,navigation and astronomy. applied mathematics is a part of mathematics definitely related to or suggested by some tangible situations, though not always intended for practical use. It is the connecting link between pure mathematics on one side, physical, biological, social sciences and technology on the other.

## Pure Mathematics

February 20th, 2007 . by Vanaja

Pure mathematics involves systematic and deductive reasoning. It treats only theories and principles without regard to their application to concrete things. It is developed on an abstract, self contained basis without any regard to any possible kind of practical applications that may follow. It consists of all those assertions as that if such and such proposition is true of anything,such and such another proportion is true of that thing.

## Angles

December 12th, 2006 . by Vanaja

An angle is the union of two non- collinear rays with a common initial point.

Two rays forming an angle are called the ‘arms’ of the angle and the common initial point is called the ‘vertex’ of the angle. sometimes, it will be convenient to refer to angle BAC, simply angle A. However this can not be done if there are more than one angle, with the same vertex A.

Interior of an angle: The interior of an angle BAC, is the set of all points P in its plane, which lie on the same side of line AB as C, and also on the same side of line AC as B.

Exterior of an angle: The exterior of an angle BAC is the set of all points Q in its plane, which do not lie on the angle or in its interior.

Types of Angles
An angle whose measure is 90 degrees is called a right angle.
An angle whose measure is less than 90 degree is called an acute angle.
An angle whose measure is more than 90 degrees is called an obtuse angle.

## Ratio

December 5th, 2006 . by Vanaja

The ratio of two quantities of the same kind and in the same units is a comparison by division of the measure of two quantities.
In other words ,the ratio of two quantities of the same kind is the relation between their measures and determines how many times one quantity is greater than or less than the other quantity.
The ratio of a to b is the fraction a/b, and is generally written as a:b.

• Example 1: The ratio of $25 to$50 is 25:50 or25/50 or 1:2
• Example 2: The ratio of 2m to 80 cm is 200:80 or 200/80 or 5:2
• Example 3: There is no ratio between \$10 and 5 meter.

Since the ratio of two quantities of the same kind determines how many times one quantity contains other, is an abstract quantity. In other words, ratio has no unit or it is independent of the units used in the quantities compared.

For the ratio a:b, a and b are called terms of the ratio. The former a is called the first term or antecedent and the later b is known as the second term or consequent.

## Periodic Functions

November 16th, 2006 . by Vanaja

Periodic functions are functions that repeat its values over and over, after some definite period or cycle on a specific period. This can be expressed mathematically that A function f is said to be periodic if there exists a real T>0 such that f (x+T) = f(x) for all x.

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it’s the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.

If a function is periodic, then the smallest t>0 ,if it exists such that f (x+t) = f(x) for all x, is called the fundamental period of the function.

The trigonometric functions sine and cosine are common periodic functions, with period 2?.
ie. sin (x+2?)= sin x , cos(x+2?)=cos x

But tan and cot remain unchanged when x is increased by pi.
ie. tan(x+?)=tan x, cot(x+?)= cot x
So, they are periodic functions with period ? .

An aperiodic function (non-periodic function) is one that has no such period