Educating Through Online

Idea of  Education through Online build on much of the fast development in the field of communication. Practical learning is a relatively new idea in teaching. Online tutoring is an interactive process in which students work in complete coordination with experienced tutors. The interaction between the student and the tutor is one-on-one, live and in real time. All relevant study material is sent via email, and students can engage in live chat with their tutor. For example student has a doubt with a math problem then online math tutor will send step by step solution via  email and via live chat.

Students are taught a mixture of common subjects like English, chemistry or Math .Here math is a subject many students develop a phobia about it and online math tutoring services are ones that are now being in demand. The online Math tutors can do away with the math phobia of many students.

This will because of the learning process for the student. Some of the math tutor will use the animation videos and graphics. The online Math Tutors often use whiteboards. Which gives the tutor and the student a shared screen space. In other words, it serves the purpose of a conventional blackboard that is normally used in a traditional classroom. Math helper try to understand the problems given by each student one-on-one and develop a teaching program as per their convenience.

For example if the student is in grade 3 then math tutor will develop the program as per their understanding label.

Example Which is the same as 10–3
a)4-3
b)4+3                                
c)4/3
d)4*3

Advantages of online tutoring is 24/7  availability and learning at your speed. Online tutoring sessions are customized to match the needs of each individual student. Tutor refer student friendly training methodology that makes the most difficult problems seem like an easy problem for the students.

The dedicated online math tutor can help any student who is weak in maths. With the expert guidance of the tutors a student can attempt any professional exam .The best part about online tutoring is that it operates across all over the continents and students from all over the globe.

Summary : Online tutoring is use full for all grade students and parents, tutors are available 24/7.Education through online is convenient for students, being at home student can clarify their doubts. The online Math tutors can do away with the math phobia of many students.

Online Math Tutoring

In modern life internet is one of the most powerful and far-reaching communication tool and more over online math tutor is one of the most powerful tools. Math Online is a high quality, independent online math tutoring program. For some children, math will be the most difficult thing,When a teacher imparts math knowledge to a student over the Internet, the process is known as online math tutoring.

Now a day free online math tutor is available,these math tutors will teach you maths problem step by step 

For Example Solve 2x^2+7x+6

Here we can write 7x as 4x+3x

=>2x^2+4x+3x+6

Take 2x common

And 3 common 

=>2x(x+2)+3(x+2)

Here (x+2) is common

=>(x+2)(2x+3)

=>x=-2

Or x=-3/2

Math tutor helps parents and children, some are interactive math games, some are tips and videos for studying, some online tutoring, and some others are to help parents learn to teach their kids math concepts and become children math helper.

Online math tutor, you get 24/7 availability of classrooms with highly qualified instructors. You do not need to step out of your house for math tutor and can study at a time of your own choice, with the help of  math tutors you can even schedule your class according to your convenience and interact with the subject matter maths expert as and when required. 

With the help of free online math tutor you can clarify your doubt at any time and any where.

Parallelogram Properties

Properties of a parallelogram

A quadrilateral is a plane geometric figure that has four sides and four vertices. If in such a quadrilateral, both the pairs of the two opposite sides are parallel and they are also congruent to each other, then such a quadrilateral is called a parallelogram. (Abbreviated as ||gm). The figure below shows a sample parallelogram.



In the above picture, ABCD is a parallelogram. The two sides AB and DC are parallel to each other. This is indicated by the single arrows. The other two sides AD and BC are also parallel to each other. This is indicated by double arrows on both the sides. Also the length of the side AB is same as that of DC and the length of the side AD is same as that of BC. The four angles of the parallelogram are: Properties of parallelogram:

1. The first property we already stated in the definition of the parallelogram, that the opposite sides are parallel to each other.

2. The opposite sides are also congruent to each other. This we saw in the figure above and its description.

3. The opposite angles are congruent. Thus in the above figure,
4. The adjacent angles are supplementary. Thus in the above figure:
mmmm
5. The diagonal of a || gm divide the || gm into two congruent triangles. This can be shown with the help of the following figure:

In the above figure, ABCD is a || gm. AC is its diagonal. Now if we consider the triangles DAC and BAC, we see that one of the sides AC is common to both the triangles. We already established as the property number 2 of the || gm that the opposite sides are congruent. Therefore the side DC is congruent to AB and DA is congruent to CB. Therefore by the SSS congruency theorem, the two triangles DAC and BAC are congruent. Hence the property 5 of the || gm stands proved.

6. Area of a parallelogram: Since we just established that the diagonal of the parallelogram divides the || gm into two congruent triangles, if we know the area of one of these triangles we can find the area of the || gm by doubling the area of the triangle. Let us now see how to find the area of the || gm.

Consider the || gm ABCD shown in the figure below:

The triangle BDC has the length of base = DC = b and the altitude = h. The area of this triangle is therefore given by the formula:
∆ =(1/2)*base*height
∆ =(1/2)* b*h

Now we already established that the area of the || gm is twice the area of this triangle. Thus  the area of the | | gm ABCD would be:
A=2* ∆
A=2*(1/2)* b*  h
A=b*h

 Properties of normal distribution is a topic of statistics and therefore shall be tackled under a separate article.

The Stem And Leaf Plots

Stem and Leaf Plot Definition:- The stem and leaf plot is used for the presentation of the quantitative data in the graphical formats. It is similar to the histogram by which the shape of the distribution of the data can be found. It is the useful tool which can be used in exploratory data analysis. This plot was popular during the type writer time. In modern computer the machine language is zero and one.

So this technique of layout of the data is obsolete in modern computers. The stem and leaf display is also called as the stem plot. The stem and leaf displays retain the data to at least two significant digits. This display contains two columns which are separated by a vertical line. The left column contains the stem and the right column contains the leaf of the data.

For example in the nub thirty two the stem is left most digits which are three and the leaf is the rightmost digit which is two is the leaf. In numb ten one is the stem and zero is the leaf of the number. In number twenty nine the leftmost digit is two and rightmost digit is nine. The number two is called the stem and the number nine is called the leaf.

To construct the stem-leaf displays the data numbs are arranged in the ascending orders.  The data value may be rounded to a particular place value that can be used as the leaf. The remaining digits to the left of the rounding digit will be used as the stem. Stem and leaf plots can be used to find the range, median, mean and media. Other statistics parameters can also be calculated with this available data.

Irregular Polygon Definition:- Polygon is the plane figure which is bounded by the finite chain of straight line segments and closed in a loop to form a closed chain or circuit. These straight line segments are known as edges or sides. The junction of the two sides is called the vertex or corner. Polygon means a shape which has many sides and angles.

A regular polygon means the shapes which have many sides of equal lengths and many angles which are equal in measurements. Irregular polygons are those in which length of each side is not equal and the measurement of angles is also not equal.  The polygon in which one or more interior angles are greater than one hundred and eighty degrees is called as concave polygon.

The polygon which has only three sides cannot be concave. The convex polygon has opposite properties to the concave. It means one or more angles of the polygon are less than one hundred and eighty degrees.

A line which is drawn through the concave polygon can intersect it more than two places. It is also possible that some of the diagonals lie outside of the polygons.  In convex polygon all diagonal lie inside the polygon.  The area of the concave polygon can be found by assuming it as other irregular polygon.

Spherical Geometry

Spherical Geometry :- The geometrical symmetry is associated with the two dimensional surface of the sphere. It is not Euclidean. The main application of the spherical geometry is in navigation and astronomy. The plane geometry is associated with the point and lines. On the sphere, points are defined as usual sense. The straight lines are not defined as the usual sense. In spherical geometry angles can be defined between great circles, resulting a spherical trigonometry which is differ from the ordinary trigonometry in many respect; the sum of the interior angles of a triangle exceeds one hundred and eighty degrees.

Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter.  To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².

Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.

Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.

In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line. 
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by  {(x₁+x₂)/2  ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment  joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by  {(x₁+kx₂)/(k+1)  ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the  replacing l/m the by k.

Obtuse Angle

                                                                                                          
In this article we shall learn about What is an Obtuse Angle? And also about Obtuse Scalene Triangle. Before studying about obtuse angle let us learn the definition of ‘angle’. In terms of geometry, an angle is defined as a measure between the circular arc and its radius and it’s shape is formed by two rays called as the sides (arms) of the angle and it shares an end point which is common to the two rays i.e., vertex of the angle. The units to measure an angle are degrees in sexagesimal system, radians in circular system and grades in centismal system.

We have various types of angles such like acute angles, right angles, obtuse angles, supplementary angles, straight angle, reflex angles etc. There are few properties which are listed below:

•    The measure of it can be positive or negative.
•    The measure of it can exceed 360°.
•    When we have two angles such as 30° and 390° where 390° = 360° + 30° where it means that the two terminal sides 30° and 390° belongs to same plane. Hence these two angles are called Coterminal angles.
•    It is expressed in terms of degrees and radians.

So far, we have learnt about definition of angle and its properties, now let us define obtuse angle?
It is defined as an angle whose measure will be greater than 90 degrees but less than 180 degrees. If the measure of this is not between 90 degrees to 180 degrees is not considered as an obtuse angle. Even if the measure is exactly 90 degrees is called as a right angle but not an obtuse. In other words, it is defined as the swipe between the quarter and the half rotation of a circle whose measure varies from 90 degrees to 180 degrees.

Now let us see about Obtuse Scalene Triangle. Firstly, Scalene Triangle is defined as a triangle, whose all sides are unequal and all angles are unequal. Where as an obtuse scalene triangle definition is similar to a scalene triangle where one of its angle is greater than 90°. An obtuse scalene triangle has one obtuse angle and two acute angles, where the two acute angles may be equal or unequal. If the two acute angles are equal then it is known as an obtuse isosceles triangle. There are few facts about scalene triangle such as

•    All interior angles are different.
•    The side which is opposite to the smallest angle will be the shortest side.
•    Similarly, the side which is opposite to the largest angle will be the longest side.

In order to find the area of an obtuse triangle the best formula to use is “heron’s formula”.

According to heron’s formula, the area A of a triangle whose sides are a, b, c is as follows:

A = √s(s-a)(s-b)(s-c), where a, b, c are sides of triangle and ‘s’ is the semi perimeter of the triangle. i.e., S = (a + b + c) / 2.

Polynomials

An expression with a single term is called as monomial, with two terms as binomial and with three terms as trinomial. If the number of terms is more, then such expressions are given a general name as polynomials, the word ‘poly’ means ‘many’. So it can also be written as poly-nomial to emphasize the meaning. Therefore, a poly-nomial is an expression that contains a number of terms consisting variables with constant coefficients. The terms of expression are usually arranged in descending order of the variable powers. Since a constant can also be expressed with the variable power as 0, a constant term can also be a part of a poly-nomial.
But as per convention in algebra, the definition of a polynomial includes certain restrictions. A polynomial can be built up with variables using all operations except division. For example, x³ – (2x + 3)/(x) + 7 cannot be called as a polynomial. However, this restriction applies only division by a variable and not for division by any constant, because such divisions can be considered as equivalent to fractional coefficients. The other restriction is that the exponents of the variables can only be non-negative integers.

A polynomial is generally an expression but acts as part both in equations and functions and such equations and functions are named with prefix ‘poly’ in general. In fact we convert a polynomial function to an equation while attempting to find its zeroes. Therefore, it is imperative to know how to factor a polynomial so that the zero product property can be used to determine the solutions. Using the zero product way is the easiest way to find the solutions of the variables.

Polynomial equations with a single variable with degrees 1 and 2 can easily be solved and the latter type is more popularly known as ‘quadratic’ equation. Mostly quadratic equations are possible to solve by factoring but even otherwise it can be done by using the quadratic formula. But equations with higher degrees are not all that easy to solve. But thanks to the great work by the mathematicians, there are ways to do that. Let us see some of the helpful concepts enunciated by the mathematicians.

As per fundamental theorem of algebra, the number of roots (the number of solutions of variables when equated to 0) of a polynomial is same as the degree of the same. This concept guides us to do the complete solution. We must also be aware that in some cases the solutions or some of the solutions may be imaginary. To get an idea on this, Descartes’s rule of sign changes helps. As per this rules we can figure out the number of real solutions, both positive and negative. Subsequently we can figure out the imaginary solutions with the help of fundamental theorem of algebra.

The rational zero theorem helps us to know what are the possible zeroes of a function. By a few trials, we can figure out a few zeroes and can reduce the polynomial to the level of a quadratic. Thereafter, the remaining zeroes can easily be figured out.

In addition, these days there are many websites advertising as ‘Factor Completely Calculator’ to help us in finding the solutions of a polynomial.

Correlations

Zero Order Correlation :- Zero order correlation means there is no correlation between the two quantities. They vary independently.  If classes in one variable are associated by the classes in the other, then the variables are called correlated. The correlations is said to be perfect if the ratio of two variable deviations is constant. Numerical measure of correlation is called co- efficient of correlations. A group of n individuals may be arranged in the order of merit with respect to some characteristics.

The same group would give different order for different characteristics. Consider order corresponding to two characteristics A and B. The correlations between these n pairs of ranks is called rank correlation in characteristics A and B for the group of individuals.  When the variation of the value of one variable becomes the cause of the variation of the value of other variable then it is called the correlation between the two variables. When the variation between two quantities is directly proportional then it is called positive correlations. It means when one variable increases other also increases or when one variable decreases other variable also decreases.

When the variation between two quantities is indirectly proportional then it is called negative correlations. It means when one variable increases other decreases or when one variable decreases other variable increases. For example the production of any quantity is indirectly proportional to the cost of that quantity. When the temperature increases the charge of the electricity bill also increases. This is the example of positive correlations.

On the other hand during the summer season the length of night hours decreases. It means the day hours are increasing and night hours are decreasing. During the cold season the day hours are decreasing whereas the night hours are increasing. There is a negative correlation between the day and the night hours. The reception of the radio signals is indirectly proportional to the distance from the transmitter. The reception near the transmitter is better than the reception at a distance. As the distance between the transmitter and the receiver increases the signal strength in the receiver decreases. It means the signal strength is indirectly proportional to the distance from the transmitter.

This is an example of the negative correlation. Zero order correlation means there is no correlation and all the quantities have their own graph.

Correlation Equation: - Correlation equations are written to find the order of correlations. We can find the perfect positive correlation, negative correlation or zero order correlations.  The numerical measure of correlation is called co- efficient of correlation and is defined as
r = (∑XY)/(n σx σy ) =(∑XY)/√(∑X² ∑Y²), where X and Y are the deviation from the mean positions.

Deviations are used to find the lower and upper bound of that quantity. Let a resistance value is written as 200 ± 5%.

The lower value of the resistance is equal to 190 and upper value is 210 ohm.
σ²x=(∑x²)/n  ,   σ²y=(∑y²)/n

Where, X = deviation from mean, ¯x=x-¯x

Y = deviation from mean, ¯y=y-¯y
σ_(x =  Standard deviation of x series )
σ_(y =  Standard deviation of y series )
n = number of values in two variables

We can also use the direct Method by substituting the value of σ_(x ) and  σ_(y ) in the above mentioned formula.

r = (∑XY)/(n σx σy ) = (∑XY)/√(∑X² ∑Y²)          or (n∑xy- ∑x∑y)/√(n∑x²-(∑x )²4 ×{n∑y²-∑x²}

Where r is the coefficient of correlation which can be used to find the rank of two quantities.

Vectors

In this section we are going to understand the basic concept of vectors and we will see a very important operation that can be done on vectors which is cross product. 

Let us see what are vectors first.Vectors are used to represent quantities where magnitude and direction both are essential to define a quantity properly. A vector hence represents magnitude and direction of a quantity. For example quantities like velocity and acceleration needs to be defined with magnitude and direction as well. A velocity of 5 km/hr due east represents that the object is moving with a speed of 5 km/hr in east direction. This means that velocity is a vector quantity. 

A vector is denoted graphically by an arrow where the head of the arrow represents the direction of the vector and its length represents the magnitude of the vector. The length of a vector is used for comparing magnitude of 2 or more vectors. A vector with shorter length will have smaller magnitude than other vectors. 

A vector a can be represented in 3 dimensional space as:
 a= a₁i+a₂j+a₃k

Here i, j and k are unit vectors in the x, y and z direction respectively. 

Let us see how cross product can be performed on the vector  Cross product is also known as vector product. This is due to the fact that the result after the product is also a vector. It is a binary operation on 2 vectors in 3 D space. The resulting vector is perpendicular to both the vectors and thus it is perpendicular to the plane in which initial vectors lie. 

If the direction of the two vectors is same or they have zero magnitude or length, then the cross product of the two vectors results zero. The magnitude of the cross product vector is equal to the area of the parallelogram which has the two vectors as its sides. If the two vectors are perpendicular and form a rectangle, then the resulting magnitude will be the area of rectangle i.e. product of lengths of the vectors. 

The operation of cross product is denoted by ×. If the two vectors are ‘a’ and ‘b’ then the cross product of the two vectors is denoted as a×b. Note that a×b is not equal to b×a. The resulting vector of this cross product denoted by c is a vector that is perpendicular to both the vectors. Its direction can be given by using right hand rule. Formula for Cross Product of Two Vectors is given as:
a×b=|a||b|sinӨ n

Here Ө is the smaller angle between the vectors a and b. |a| and |b| denotes the magnitude of the vectors a and b respectively. n here represents the unit vector perpendicular to the plane of a and b. Its direction can be given by right hand rule. The right hand rule says that if a×b is the cross product that you are calculating then role your fingers of right hand from vector a to vector b, the direction of your thumb will give the direction of resulting vector as shown below:

If the vectors a and b are parallel to each other, then the cross product of then is a zero vector as sin0 = 0
Cross product is considered as anti commutative. This means that a×b = -(b×a) 

Algebra Word Problems

In the field of mathematics where many forms are present algebra is one such form that deals with the basic concepts of Mathematics. We all know that Maths is a game of mathematical operations, so to deal with addition, subtraction, multiplication, division with the known and unknown values of the variables is known as algebra. In a simple way to say is that when we play computer games then we deal with jumping, running or finding secret things and in Maths we play with letters, numbers and symbols to extract the secret things. Now let’s see what is included in algebra.

Get easy steps to solve Algebra Problems (click here)


WHAT IS THERE IN ALGEBRA?
1. Algebraic terms – Algebraic terms are those terms which are infused with the values of variables. For example 3a.
In this example 3 is the numerical coefficient and a is the variable. The main work of numerical coefficient is also to deal with + 3 or – 3 of the variable.

2. Algebraic expressions – Algebraic expressions deal with a logical collection of numbers, variables, positive or negative numbers in the mathematical operations. For example we can write an expression – 3a + 2b.

3. Algebraic equations – Algebraic equations refer to the equivalency of expressions considering both the left hand side and the right hand side. For example an equation is x – 35 = 56k2 + 3. Over here both left hand side and right hand side are dealing with expression. If both the expressions are equal then that is said to be an equation.

ALGEBRA WORD PROBLEMS

The main part in Algebra is to solve the word problems, which we also did in Pre Algebra Word Problems.  It can be harder if you don’t think and do it but it can be easy if you think and do it. If we say in English how are you and the simple answer to the question is I am fine which can be simply understood but to convert the English language in to the language of maths requires certain techniques. You need to practice a lot with proper understanding of the translations. Let’s go step by step.

1. First of all read the entire word problem carefully. Make it a point that we should not start solving the problem by just reading half of a sentence. Read it properly so that you can gather the information that what you have and what you still need to have.

2.  Then, pick the information so that it can be changed into certain variables. Some might be known and some might be unknown. If required then draw pictures and symbols in the rough work so that you can deeply understand the problem without leaving any loopholes.

3. Note the key words as for different mathematical operations there are different key words –

a) Addition – increased by, total of , sum, more than, plus
b) Subtraction – decreased by, minus, less than, fewer, difference between
c) Multiplication – of, times. Multiplied by, product of, increased or decreased by the factor of
d) Division – per, out of, quotient
e) Equals -  is, gives, yields, will be


4. After going though these expressions carefully you can solve various word problems logically. Limit Problems are part of Calculus.

Rational Inequalities

Rational Inequalities : - The rational expressions are written in the P(x)/q(x) form where p(x) and q (x) are the polynomials i.e p(x), q(x)∈z(x), and q(x)≠0 are known as rational functions. the set of rational function is denoted by Q (x).

Therefore Q(x) = (p(x))/(q(x)) (where p(x), q(x) ∈z(x) and q(x) ≠0. for, if any a(x)∈z(x), then we can write a(x) as  ( a(x))/1, so that a(x)∈z(x)  and therefore,(a(x))/1  ∈Q(x);i.e  a(x)∈Q(x). The rational function example is given below.   

(2x+4)/(x²+5x+8)>0
(ax+b)/(cx+d)<0 p="">
Rational inequalities means the left hand side is not equal to the right hand side of the equation. The inequalities are associated with the linear programming.  The step by step procedure to solve the inequalities is given here. 

The very first step to solve the inequality problems is to write the equation in the correct form. There are two sides in the equation, left hand side and the right hand side. All the variables are written in the left hand side and zero on the right hand side. The second step is to find the critical values or key. To find this we have to keep the denominator and the numerator equal to zero.

The sign analysis chart is prepared by using the critical values. In this step the number line is divided into the sections. The sign analysis is carried out by assuming left hand side values of x and plugs them in the equation and marks the signs. Now using the sign analysis chart find the section which satisfies the inequality equations.

To write the answer the interval notation is used. Let us solve an example (x + 2) / (x² – 9) < 0. Now put the numerator equal to zero, we get x = -2. By putting the denominator equal to zero we get x = +3 and x = -3.  Mark these points on the number line. We get minus three, minus two and plus three on the number line.
Now try minus four, plug the value of x equal to minus four we get – 0.286.

Plug x equal to minus 2.5 we get + 0.1818. Now plug 2 in the equation we get minus zero point eight.
Now we try x equal to plus four which is at the right hand side of the number line. We get plus 0.86. The sign is minus, plus and plus sign. The equation is less than zero. To get less than zero we need all the negative values of the equation.

For x equals to plus three and minus three the equation is undefined. For x equals to minus two the equation is zero. The equation is satisfied by the value of x which is less than minus three. Therefore the answer which is the solution of the equation is minus infinite to minus three.

Rational Expressions Applications : - The equations which do not have equal sign are called the expression. The expressions are used to simplify the equations. The expressions are written to form the conditions of the problems.

According to the direction of the question the expressions are formed to solve a problem.

Set Operations

Set Operations: - The collection of the distinct objects is called the sets. For example the number 5, 7 and 11 are the distinct object when they are considered separately. Collectively they can form a set which can be written as [5, 7, 11]. The most fundamental concept of the mathematics is the sets.  The objects which are used to make the sets are called the elements. The elements of the sets may be anything, the letters of the alphabet, people or number. The set are denoted by the capital letters.  The two sets are equal when they have equal elements. The set can be represented by two methods, roster or tabular form and the sets builder form. In the roster form the elements are separated by commas. For example a sets of positive and odd integers less than or equal to seven is represented by {1, 3, 5, 7}.

The set of all the vowels of the alphabet is {a, e, I, o, and u}. The sets of the even numbers are {2, 4, 6,}. The dots indicate that the positive even integers are going to infinite. While writing the set in the roster form the element must be distinct. For example the set of all the letters in the word “SCHOOL” is {s, c, h, o, and l}. In the set builder form all the elements of the sets have the common property which is not possessed by any element which is outside of the set. For example vowels have the same property and can be written as {a, e, I, o, and u}. The set in the set builder form can be written as V = {x; x is a vowel in the English alphabet}.  A = {x; x is the natural number and 5 < x < 11}. C = {z; z is an odd natural number}.

Operations on Sets: - Like the mathematical operations, there are number of operations which are carried out by the sets. We will study them one by one. Union of sets is the union of the all the elements of a set and all the elements of the other  taking one element only once. Let A = {2, 3, 4, 5} and B = (4, 5, 6}. Therefore the union  of the two sets is written as A ∪B  = {2, 3, 4, 5, and 6}. Intersection of two sets A and B is the set of all elements which are common in both the sets. If the intersections of the two sets are equal to null then the set is called the disjoint set.

The difference of sets A and B can be defined as the A – B. It is equal to the elements which belong to set a and the element which does not belong to the set B. The operation on the sets such as union and intersection satisfy the various laws of the algebra such as the associative laws, the commutative laws, idempotent laws, identity laws, distributive laws, De Morgan’s laws, complement laws and involution law. These all the laws can be verified by the Venn diagram. 

Dispersion Statistics

Measures of Dispersion : - The dispersion is also known as the variability is the set of constant which would in a concise way explain variability or spread in a data. The four measures of dispersion or variability are the range, quartile deviations, average deviation and the standard deviation. The difference between two extreme observations in the given data is known as the range. It is denoted by R. In frequency distribution, R = (largest value –smallest value). It is used in statistical quality control studies rather widely. Median bisects the distribution. If we divide the distribution into four parts, we get what are called quartiles, Q(1 ),Q2 (median) and Q(3.)

The first quartile Q(1,) would have 25 % of the value below it and the rest above it; the third quartile would have 75% of values below it. Quartile deviation is defined as, Q. D.  = 1/2  ( Q3-Q1). If the average is chosen a, then the average deviation about A is defined as A.D. A.D. (A) = 1/n ∑|(xi- A)|  for discrete data. The Standard deviation is also called as the Root mean square deviation. The formula for the standard deviation is given as Standard deviation,σ=√(1/n ∑(xi- x ̅)^2 ) for discrete data.

The Square of the standard deviation is known as the variance. It is denoted by the square of sigma. Out of these measures, the last σ is widely used as a companion to x ̅ on who is based, when dealing with dispersion or scatter. Measure of dispersion is calculated for the data scattering. Deviation means how a value is deviated from it mean or average value. The mean of the two groups of the data may be same but their deviation may be high.

Central Tendency Measures : - The central tendency measures are also called the statistics central tendency. The clustering of data about some central value is known as the frequency distribution. The measure of central tendency is the averages or mean. The commonly used measures of central values are mean, Mode and median. The mean is the most important for it can be computed easily. The median, though more easily calculated, cannot be applied with case to theoretical analysis. Median is of advantage when there are exceptionally large and small values at the end of the distribution. The mode though easily calculated, has the least significance. It is particularly misleading in distributions which are small in numbers or highly unsymmetrical. In symmetrical distribution, the mean, median and mode coincide.

For other distributions, they are different and are known to be connected by empirical relationship. Mean – Mode = 3 (mean – median). The sum of the values of all the observations divided by the total number of observations is called the mean or average of a number of observations.  The value of the middle most observations is called the median. Therefore to calculate the median of the data, it is arranged in ascending (or descending) order. The observation which is found most frequently is known as mode.

The central tendency measures and the variability or dispersion are used in the statistical analysis of the data.

Interpolation

Interpolation is a concept that is used in numerical analysis. It means finding an intermediate value from the given data. That is, for a set of function values, sometimes a situation arises to know what the value of the function is, for some intermediate value of the variable.

Let us explain the concept with a simple example. Let the ordered pairs of the data of a function be (0, -1), (1, 7), (2, 22), (3, 40), (4, 69), (5, 98) and we need to find the function value of 3.8 of the variable. The attempt to find that value is called interpolation and there are several methods. Let us describe those one by one.

The first method is to round the given value of the variable to the nearest value in the data and take the corresponding value of the function. So for the given data, the value 3.8 of the variable can be rounded to 4 and assume the value of the function approximately as 40. But as one can easily see it is a crude method and far from accurate. This method can only be used just for guidance.

The next method is called as linear interpolation. That is, the function is assumed to be linear in the interval that contains the required value of the variable and accordingly the value of the function is determined. In the given example 3.8 falls in the interval [3, 4]. Considering the function to be linear in this interval, the slope of the function in this interval is (69 – 40)/(1) = 29. So the value of the function at x = 3.8 is, 29(0.8) + 40 = 63.2. Though this method also is not very accurate still the accuracy is much better than that in method 1. This method, even at the cost of sacrificing some accuracy, is preferred because it is easy to work with. In general the linear interpolation y at a point x  is given by the formula y = [(y_b – y_a)/(x_b – x_a)](x – x_a) + y_a, where [a, b] is the interval in which the required point occurs. This is the algorithm used in any linear interpolation calculator.

It is always possible a curve rather than a straight line could better cover the points plotted from a data. In other words a polynomial function can give a better interpolation. Finding a suitable polynomial function for a given data is called as polynomial regression.  Suppose we consider the same data, the three degree polynomial function f(x) = 3x² + 5x – 1 will be very close to the desired results. In such a case evaluating the function for x = 3.8, the value of the function is will be a very accurate interpolation. The accuracy can further be improved when a polynomial function of degree same as the number of data points is determined.

A high level interpolation polynomial can be derived by extensive methods and such an interpolation is called as Lagrange interpolation introduces by the famous Italian mathematician.

Analytical Geometry

Analytical Geometry:

Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.

The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.

All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.

Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).

Also check out the video streaming of Analytical Geometry

Variables

                                                                                                               
What are Variables?

A variable is something which always varies. That is it does not have a fixed value. In any situation if we are not sure of a value for a quantity then we represent such things using alphabets like x, y, z, a, b, c etc known as variables.

Let’s assume that Peter is working in a hotel and earns $12 per hour and on top of his hourly wage he also get tips for each hour. This expression $(12 + x) would give how much peter earn in a given hour. Here x denotes the tips earned. Now, you might also realize that number of tips or the amount of tips Peter make per might change dramatically from hour to hour. It varies consistently. Since, we are not sure of the tips Peter make each hour exactly, this value is called a variable.

Random Variables

In this section i will introduce you to the concept of a random variable. For me this is something I had lot of trouble for getting my head around. That’s really because it called variable. Generally, variable are unknowns used in algebraic equations/ expressions.

For example: x + 2 = 15. We can find the value of x, by subtracting 2 from both sides.
Or if we have an equation in two unknowns that is y = 2x + 5, here x is an independent quantity and y is a dependent quantity. So, we can assume any x value and find respective value for y. In this case we will have infinite combinations of x and y satisfying the equation.

A random variable is kind of same thing that it can take multiple values but it is not something which you really ever solve for. A random variable is usually denoted by a capital letters say X. It can take bunch of different values but we are never solving for it. In fact it a function that maps from you from the world of random processes to the actual numbers. Let us see one example.

Let the random process be it is going to rain tomorrow or not. Now, how are we going to quantify this, let say X = 1 if it rains tomorrow and X = 0 if does not rain. It is not compulsory that we have to use 1 and 0. Here we can assign to any number. So, we need to keep in mind is a random variable is not a traditional variable.

It is defined as a numerical value to each outcome of a particular experiment. For every element of an experiment’s sample space, it can take only one value. When a random variable can take finite (countable) finite set of outcomes then it is known as discrete random variable. Individual outcomes for a random variable are denoted by lower case letters. A continuous random variable would take on any of the countless number of values in the given line interval.

Using Matrices

In this article we shall study about one of the methods used to solve system of linear equations using matrices. Before we study about the method let us first see few definitions

Consider the following system of simultaneous non – homogeneous linear equations
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Expressing the above equations in matrices, we get

These equations can be represented as a matrix equation as AX = D, where







Here A is called the coefficient matrix.X is called the variable matrix.
D is called the constant matrix.

Augmented Matrices

The coefficient matrix augmented with constant column matrix, is called the augmented matrix, generally denoted by [AD]. Hence the augmented matrix of the above system of simultaneous linear equations is 

Sub Matrices

A matrix obtained by deleting some rows or columns (or both) of a matrix is called a sub matrix.

Definition (Rank of a matrix)

Let A be a non – zero matrices. The rank of A is defined as the maximum of the orders of the non – singular square matrices of A. The rank of a null matrix is defined as zero. The rank of A is denoted by rank (A).

It is to be noted that :-

If A is a non zero matrix of order 3 then rank of A is
(i)    1 if every 2 x 2 sub matrices is singular
(ii)    2 if A is singular and atleast one of its 2 x 2 sub matrices is non – singular.
(iii)    3 if A is non – singular.

Consistent and Inconsistent systems

A system of linear equations is said to be
(i)    Consistent if it has a solution
(ii)    Inconsistent if it has no solution.

A system of three simultaneous equations in three unknowns whose matrix form is AX = D has
(i)    A unique solution if rank (A) = rank ([AD]) = 3
(ii)    Infinitely many solutions if rank (A) = rank([AD]) < 3
(iii)    No solution if rank (A) is not equal to rank ([AD])

It is to be noted that the system is consistent if and only if rank (A) = rank ([AD])

The different ways of solving non homogenous systems of equations are
(i)    Cramer’s Rule
(ii)    Matrix inversion method
(iii)    Gauss – Jordan method

In Gauss – Jordan method we try to transform the augmented matrix by using elementary row transformations. So that the solution is completely visible that is x = α, y = β and z = γ. We may get infinitely many solutions or no solution also.

For solving a system of three equations in three unknowns by Gauss – Jordan method, elementary row operations are performed on the augmented matrix as indicated below.

(i)    Transform the first element of 1^st row and 1^st column to 1 and transform the other non zero elements if any in of 1^st row and 1^st column to zero.
(ii)    Transform the second element of 2^nd row and 2^nd column to 1 and transform the other non zero elements if any in of 2^nd row and 2^nd column to zero.
(iii)    Transform the third element of 3^rd row and 3^rd column to 1 and transform the other non zero elements if any in of 3^rd row and 3^rd column to zero. 

Matrices Calculator
We shall study about Matrices Calculator in some other article.

Right Circular Cylinder

Right Circular Cylinder :- If the number of circular sheets is arranged in a stack, if the stack is kept vertically up then it is called the right circular cyl. If the base is circular and the other sheets are kept at ninety degrees at the base, then it is called the right circular cyl.  The right circular cyl can be made by a rectangular sheet by folding it in round shape. The area of the sheet gives us the curved surface area of the cyl because the length of the sheet is equal to the circumference of the circular base which is equal to 2 π r . Therefore the curved surface area of the cyl is equal to the area of the rectangular sheet. The area of the rectangular sheet is equal to the product of the length and the breadth. The curved surface area of the cylinders is also equal to the product of the perimeter of the base and the height of the cylinders.  If the top and bottom of the cylinders is also consider because it is needed to make the circular tin, If the area of the bottom and top of the cylinders is included in the curved surface area we get the total surface area. It means total surface area of the right circular cylinder is equal to the area of the base plus the area of the top of the cylinder plus the curved surface area of the cylinder. The cylinder has two circular base at the bottom and at the top.The diameter of the base of the cylinder can be measured directly with the help of the scale. 

What is the Formula for Volume of a Cylinder :- As we know that the volume of cuboids is equal to the product of the length, breadth and the height. The volume is three dimensional. The cuboids are built up with the rectangle of the same size sheets. In the same way the right circular cylinders can be made by using the same principle. So, by using the same argument as for cuboids, we can see that the volume of a cylinders can be obtained by the product of the base and height of the cylinders.  Therefore, the volume of a cylinder = the base area of the cylinder X height of the cylinder. As we know that the base of the cylinder is the circle. The area of the circle is equal toπ r². The height of the cylinder can be assumed as h. Therefore the volume of the cylinder = π r²h, where r is the base radius and h is the height of the cylinder.

Example :- Let us find the volume of the cylinder which has the radius of the base is twenty one centimeter and its height is thirty five centimeter.

Solution :- The formula for the volume of the cylinder, V = π r²h
Radius r = 21 centimeter, height h = 35 centimeter
Therefore the volume of the cylinder V = π ×21² ×35 = 48490.48 cubic centimeters.

Trend Line

The trend lines are drawn for the prediction of the data. These lines are associated with the data series. The trend lines are used for any type of improvements from the available data. The trends lines can be drawn to follow the equation of a line, i.e y = mx + c, where m is the slope of a lines. The value of c is constant which decides the equation of a lines. The trends line is used for exponential or logarithmic formulae. We can choose the right  type of it for our data as per our requirements. Linear trend lines, logarithmic trend lines, Polynomial trend lines, power trend lines, exponential and moving average trend lines can be drawn as per our requirements. We can add, remove or modify a trend lines as required.

Now a days, online tools or calculators are available to draw a trend line. We can display the equation of any trend line on the chart. The R squared value can be displayed. We can format the present structure to draw a fresher one. The online tools are available to draw any type or desired line quickly to take the faster decisions. Prediction of whether forecasting, draught, can be made by the data available. These  are the key of the success of our business, trade, education and research.

Equation of a Line Calculator: - Line calculators are the online tool which can be used to find the slope of a lines. Slope of a line is the tangent of the line. Slope is the ratio of the rise to run.  A line can be written in the standard form, slope intercept form, intercept form, general form, point slope form, two point form.
The equation of line calculators can also be used to find the equations of the lines when the coordinates are given. This calculator has four inputs and one output. The input to the calculators is the value or the coordinates of two points. The output terminal shows the value of the slope which is ratio of the difference between the y coordinates to the difference of x coordinates. The four inputs to the line calculators are x- coordinate one, x_ coordinate two, y- coordinate one and y- coordinate two. After giving all the inputs we have to press the output button. In the output, we can get a fraction or the degree of the tangent. If we want to check the equation of a line then press the output button for equation of the line. The slope of the lines can be used to find the height and the base when we making the bridge or when a road is planned in the hill area. The tangent angle is required to hit a moving or stationary target.

In the hanging bridge technology the line calculators are used to find the slope of the tension wire to hold the bridges. All the computers which are used to find the moving target and to fire guided missiles use the line calculators to calculate the tangent angles.

Mixed Fractions

Mixed Fraction: - The mixed numbers are like 2 1/2 which is two and one half or 35 3/32 which is equal to thirty five and three thirty two seconds.  To express the mixed fractions we have to keep exactly one blank between the whole numbers and the fractions. The mixed fraction are in the form of 3 2/3 ( three and two third) , 7 1/5 ( seven and one fifth) , 13 1/7 ( thirteen and one seventh ) , 113 7/100 ( one hundred thirteen and seven hundredth) and so on.

The rules for adding the mixed numbers: - To add the mixed number, convert them into the fraction. The algebraic formulae can be used for the addition of these, for example a/b + c/d = (ad + bc) / bd.

Example :- Let us add the two mixed fractions a b/c and d b/c

Solution :- The given fraction are a b/c and d b/c. The first step which is to be used is to convert the mixed fraction to the fraction. The fraction of the mixed term a b/c is equal to (ac + b)/c. The procedure is to convert the mixed fraction to the fraction is to multiply the whole number to the denominator of the fraction and then add the fraction. The total value is divided by the denominator of the fraction. The fraction of the mixed term d b/c is equal to (dc + b)/c. Now we have to add the two fractions as below.

(ac + b )/c + (dc + b)/c. As the denominators of both the terms are same, the numerators can be added directly as below
(ac + b + dc + b ) /c =   (ac +2 b + dc) /c

The rules for subtracting the mixed numbers: - To subtract the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the subtraction of the fractions, for example a/b - c/d = (ad - bc) / bd.

Example :- Let us subtract the two mixed fractions a b/c and d b/c
(ac + b )/c - (dc + b)/c

As the denominators of both the terms are same, the numerators can be subtracted directly as below
{(ac + b) – (dc + b)} /c = (ac + b – dc - b) /c = (ac – dc)/ c = (a – d) [by canceling the common term in the numerator and denominator]. 

The rules for multiplying the mixed numbers: - To multiply the mixed numbers or the fractions we have to convert them into the fractions. The algebraic formulae can be used for multiplication of the fractions, for example a/b *c/d = a c /bd

The rules for dividing the mixed numbers :- To divide the mixed numbers or the fractions, convert them into the fractions. The algebraic formulae can be used for the division of the fractions, for example a/b ÷ c/d = a/b ×d/c = ad/bc.

Fraction Calculator Online: - The fraction calculator online is a tool which can be used to add, subtract, to multiply and to divide the fractions. We have to enter the fractions to be calculated, enter the function is to be carried out (i.e add, subtract, multiply, divide). The output in the form of the fraction will be displayed in the output window.

Direct and Inversely Proportional

Proportionality:

A quantity is said to be proportional to another quantity if change of one of the quantities is always accompanied by the change of the other. This property is known as proportionality.

Proportionality is of two types:

(i)                  Direct proportionality

(ii)                Inverse proportionality

Direct Proportionality:

A quantity is said to be directly proportional to another quantity if the change in both of them is in the same direction. This means that if one of the quantities increases then the other also increases. If one of the quantities decreases then the other also decreases.

It is denotes by the symbol a. If ‘a’ is directly proportional to ‘b’ then:

We write as a a b ==> a = k * b where k is the proportionality constant.

i.e.  a / b = k = constant à a1 / b1 = a2 / b2

Graph:

Let us consider that x a y. If we plot the values of x and y on a graph sheet we obtain the graph showing the relation between these two quantities. Generally, the graph of directly proportional quantities is a straight line. Thus by seeing the graph we can conclude the relation and proportionality between two quantities.

Inversely proportionality:

A quantity is said to be inversely proportional to another quantity if the change in both of them is in the opposite direction. This means that if one of the quantities increases then the other quantity decreases. If one decreases then the other increases.

Inverse proportionality is also uses the symbol a but the reciprocal of the second quantity is written.

If ‘a’ is inversely proportional to b then:

We write as a a 1 / b

You can see that the reciprocal of b is written to indicate inverse proportionality. We can also say that ‘a’ is directly proportional to the reciprocal of ‘b’.

If a a 1 / b à a = k / b where k is the proportionality constant.

i.e. a * b = k = constant à a1 * b1 = a2 * b2

Graph:

Let us consider x is inversely proportional to y i.e. x a 1 / y

Now plot the values of x and y on the on a graph sheet we obtain the graph showing the relation between the two quantities. Generally the graph is not a straight line but a curve. On seeing the graph we can analyze the relation and proportionality between the two quantities.

Problem:

If the volume of a gas at a given temperature is 2 liters when its pressure is 1 bar, then what will its volume when the pressure increases to 3 bars? (Volume in inversely proportional to pressure)

Sol: Given, initial volume v1 = 2 l

Initial pressure p1 = 1 bar

Final pressure p2 = 3 bars

As volume is inversely proportional to pressure we have, p1 v1 = p2 v2

Now v2 = p1 v1 / p2 = (2 * 1) / 3 = 0.66 liters (approx.)

Thus final volume the gas is 0.66l

Properties and Area of a Rectangle

Rectangles
For a normal 4th grader, a rectangle would mean a plane figure that has four sides. However, more precisely in geometry, a branch of math, a rectangle is a special type of a quadrilateral that has 4 right angles. It would look as shown in the picture below:

Properties of a rectangle:
1. It has four sides.
2. It has four angle and all the angles are right angles.
3. It has four vertices.
4. Each pair of opposite sides are congruent.
5. Opposite sides are parallel.
Examples of rectangles:
1. Top of a book.
2. Face of a cuboid.
3. Top of a table.
4. Front of a cupboard.
5. Etc.

The Area of the Rectangle:
The Formula for the Area of a Rectangle is as follows:
A = l * w
Where,
A = area of the rectangle,
L = length of the rectangle
W = width of the rectangle.

It is customary to denote the longer side of the rectangle as length and the shorter side as width. Another custom is to denote the horizontal sides as the length and the vertical sides as the width of the rectangle. It is shown in the picture below.

Let us now try to understand how to calculate the area of a rectangle with the help of a sample problem question.

Example 1: Find area rectangle from the figure shown below:

Solution:
From our formula for area of a rectangle we know that,
Area = A = L * W
For this problem,
L = length = 5 units and
W = width = 3 units
Therefore substituting these values of L and W into the above formula for area of the rectangle we have,
A = 5 * 3 = 15 sq units <- answer="" p="">
If instead of being given the measures of length and width, we are given the co ordinates of the vertices of the rectangle then its area can be found out as follows:

Consider a rectangle with the vertices at A (x1,y1),B  (x2,y2), C (x3,y3) and D (x4,y4) taken in clock wise direction. Therefore we know that if AB is the length of the rectangle then BC would be the width of the rectangle. The distance AB can be found using the distance formula as follows:

Length = L = AB = √[(x2-x1)^2 + (y2-y1)^2]

Similarly the distance BC can also be found using the distance formula as follows:

Width = W = BC = √[(x3-x2)^2 + (y3-y2)^2]

Both the above can be now used to find the area of the rectangle as follows:

A = L * W.

Sample problem:
Find the area of a rectangle having vertices at (3,7), (0,7), (3,-2) and (0,-2)

Solution:
First let us sketch a graph of the said rectangle.

From the picture we see that
Length = L = 3-0 = 3 and
Width = W = 7 – (-2) = 7+2 = 9

Therefore,
Area of rectangle = 3 * 9 = 27 sq units.

Define Absolute maximum

Optimization is one of the most vital applications of differential calculus, which guides the business and the industry to do something in the best way possible. Business enterprises ever need to maximize revenue and profit. Mathematical methods are employed to maximize or minimize quantities of interest. Absolute maximum value is when an object has a maximum value.

In mathematics, the maximum and minimum of a function, identified collectively as extrema , is the largest and smallest value that the function obtains at a point either within a given local or relative extremum (neighborhood) or on the function domain in its entirety.

A function f has an absolute maximum at point x1 , when f(x1) =  f(x) for all x. The number f(x1) is called the maximum value of ‘f on its domain. The maximum and minimum values of the function are called the extreme values of the function. If a function has an absolute maximum at x = a , then f (a) is the largest value that f that can be attained.

A function f has a local maximum at x = a if f (a) is the largest value that f can attain "near a ." Simultaneously, the local maxima and local minima are acknowledged as the local extrema. A local minimum or local maximum may also be termed as relative minimum or relative maximum.
Both the absolute and local (or relative) extrema have significant theorems linked with them Extreme Value Theorem is one of it.

To find global maxima and minima is an objective of mathematical optimization. If a function is found to be continuous on a given closed interval, then maxima and minima would exist by the extreme value theorem.
Moreover, a global maximum either have to be a local maximum within the domain interior or must lie on the domain boundary. So basically the method of finding a global maximum would be to look at all the local maxima in the interior, and also look at the maxima of the points on the boundary; and take the biggest one.
For any function that is defined piecewise, one finds a maximum by finding the maximum of each piece separately; and then seeing which one is biggest

In mathematics, the extreme value theorem signifies that if a real valued function f is continuous in the closed and bounded intermission [x,y], at that moment f should attain its maximum and minimum value, each of it at least once. That is, there prevail numbers a and b in [x,y] in such a way that:
F(a) = f(c) = f(b) for all c summation [x,y].
A related theorem is also known as the boundedness theorem which signifies that a continuous function f in the closed interval [x,y] is bordered on that interval. That is, there always exist real numbers m and M in such a way that:
m = f(c) = M for all c summation [x,y].
The extreme value theorem thus enhances the boundedness theorem by demonstrating that the function is not only bounded, but also accomplish its least upper bound as its maximum as well as its greatest lower bound as its minimum.

Algebra variable

Algebra is a subject which helps us to find unknown quantities with minimal information. But how do we carry all mathematical operations when something is not known? We assign letters from alphabet (mostly small case letters) for the unknown.

Such letters are only called as variables or changeable. Why the name is selected as such? Because the value assigned to these could be varied according to our wish but gives the correct information at a required condition. For example I earn $100 per day.

How do I calculate the total earnings after a certain number of days? Here the word certain is really uncertain! In other words it is an unknown. So we assign ‘x’ as the number of days and I can formulate that my total earnings in dollars will be 100x. Now my job is simple. I just to need to replace ‘x’ by the actual number of days I decide and I get the required information correctly.

Thus basically the changeable are employed as algebra variables or simply math variables. But why we had done away with the prefixes ‘algebra’ or ‘math’ and we simply refer as such?
Because, such uses of letters even in other subjects are subjected to mathematical or algebraic operations.
The types of variables are many. In most cases more than one forms are used. For example one may represent an input and another may be the corresponding output. Obviously the former can be assigned any value and hence it is described either as independent forms and the latter, because of the dependence on the input is referred as dependent forms.

In most of the cases they are represented by the letters ‘x’ and ‘y’ respectively. In such cases they are related by equations or functions.
Further, different types of variables are used depending on the context. For example when you study about distance versus time for a moving object, lettert is used for the unknown time and ‘s’ is used for the corresponding distance.

The same letter tis also used for denoting temperatures in normal scale. (whereas, T is used for temperature in absolute scale!). Letters p and v are used for denoting pressure and volume respectively.
We mostly see that the letters at the second half of English alphabet are mostly used for the unknown quantities. It is just a convention and the first of half is generally reserves for constants. However use of letters like ‘a’, ‘b’ ‘c’ is common in geometry.
In addition to English alphabetical letters Greek letters are also extensively used especially in the topics of trigonometry.

Linear Transformation

Let us try to give a simple introduction and explanation about linear transformations. Let us not scare the readers with hi-fi terms like ‘vector spaces’ ‘matrices’ and symbols like ‘e’ ‘Rn’ etc. Yes, let us involve those at a higher level after getting acquainted with what basically a linear-transformation means.

In a data different scores, that is, items described by numbers are exhibited. There is a possibility to express the items of the data in the form of a pattern. If such a pattern is in the linear form that the transformation of the data set to a pattern is called as linear transformation. It may be noted that such a transformation can be fairly accurate for a limited interval meaning limited number of items in the data set.

This type of transformation is obviously results as a linear function in the form a + bx (similar to mx + b in analytical geometry). Since linear functions are always ‘one to one’, this type of transformation is also referred as one to one linear transformation. Recording the scores of student in a class can be cited as one of the linear transformation examples.

Let us discuss about the basic concept of this type of transformation. Suppose Xi represents the item in general, of the given data, and if X’I is the same after the transformation of the data, then the linear relation is X’ = a + b Xi, where a and b are constants for the particular transformation. The letter a is called additive component and b is the multiplicative component of transformation.

These are analogous to y-intercept and slope of linear algebraic functions. One must know what should be mean and standard deviation of the transformed data and accordingly the values of constants a and b are determined. Because the condition of a linear-transformation is X’m =   a + b Xm and X’s = b Xs, where the subscripts m and s refer the respective mean and standard deviation.

Let us illustrate a linear transformation example. Suppose a data is describes the scores as 13, 16, 21, 21, 24. This has to be linearly transformed with a mean of 95 and standard deviation of 15. What is the formula of transformation?

The mean and the standard deviation of the given data are Xm = 19 and Xs = 4.42, rounded to nearest hundredth. The set of desired figures in the transformation is X’m = 95 and X’s = 15. Since, X’s = b Xs,
b = 3.39 and a = 95 - 3.39*19 = 30.59, rounded to two decimal place.
Thus the transform relation is X’ = 30.59 + 3.39X.

Linear Programming with the Help Of Simplex Algorithm

The concept of programming is very important. It is being used in the field of mathematics as well. Linear programming is a very important concept and is now very widely used in the field of mathematics. The Simplex method tutorial is a part of the linear programming model. This method is also called an algorithm. This algorithm is used as part of linear programming. This is used in finding a optimal solution.

The Simplex method examples can be very helpful in understanding and knowing more about this algorithm. For understanding this method a geometric figure called the polytope has to be studied. Basically a polygon is a geometric figure which has many sides. So, hexagon is a geometric figure which has six sides.

A pentagon is a geometric figure which has five sides. Similarly there are other geometric figures which have different number of figures and they are given various names. In Simplex methods the polytope plays a very important role as this gives the area which is under consideration for finding the optimal solution. So, this concept has to be learnt properly.

There is different number of vertices present in a polytope. To find the optimal solution, the process begins from any one of the vertices of the polytope and moves towards the vertex which shows the optimal solution. This can be represented in a standard form.

Another form can be used in this case, namely the canonical form. There are two methods that can be used. The two methods are called the M-method and the other one is called two-phase method. As the name suggests in the two-phase method there are two phases that are to be considered to arrive at the final solution.

The final solution is nothing but the optimal solution. The ultimate purpose is to arrive at the optimal solution. An example can be used to explain the concept. An equation will be given for simplification. There will also be some constraints given. The simplification has to be done keeping these constraints in mind.

The constraints can also be in the form of equations. These equations must be taken into account while performing the simplification procedure. Then they can be represented in the canonical form and a feasible solution is found for the variables present in the equation, keeping in mind the constraints given. Once this is done the optimal solution is found out.