Interjection and its types

Interjection is one of the eight parts of speech in English grammar along with noun, pronoun, verb, adverb, adjective, conjunction and preposition. Interjection is a part of speech that conveys emotions or expresses a meaning or feelings. Interjection does not add meaning to a sentence but express the feeling of the speaker. An interjection is sometimes followed by the exclamation sign i.e. “!”. For example: The kid exclaimed, “Hurrah! I got Barbie coloring pack of pens”. Here, the interjection “hurrah!” is expressing the excitement of the kid after getting Barbie coloring pack of pens. There are five different types of interjections classified based on expressions such as greetings, joy, approval, surprise and grief. Lets’ have a closer look at each of the types of interjections.

Interjections to express Greetings:
Interjections to express greetings are type of interjections that are used to wish someone. Popularly used interjections of expressing greetings are: hello, hi and so on. For example: Hello, how have you been? Here, the speaker is wishing hello and how the other person has been by using “hello”.

Interjections to express Joy:
As the term suggests, these types of interjections convey the feeling of joy and excitement of the speaker. Commonly used interjections are: hurrah, yippee, hey and more. For example: Hey! Look Farlin India collection has real good stuffs for kids. Here, the speaker is expressing his excitement at Farlin India brand’s grand collection.

Interjections to express Approval:
Interjections to express approval are used to express or congratulate on someone’s effort. Commonly used interjections of expressing approval are: bravo, wow etc. For example: Bravo! You won the match. Here, the speaker is congratulating some for winning the match.

Interjections to express Surprise:
Interjections to express surprise are used to express the emotion of surprise by the speaker. Commonly used interjections of expressing surprise are: oh, eh and so on. For example: Oh! Nuby baby brand is giving away annual sale this month. Here, the speaker is expressing his surprise at Nuby baby brand’s annual discount.

Interjections to express Grief:
Interjections to express grief as the term describes is used to express sorrow or grief by the speaker. Commonly used interjections expressing grief are: alas, ouch and more. For example: Alas! The man is dead. Here, the speaker is using “Alas” to express his grief on the man’s death.

Set Theory and Various Types of Sets

Set theory is one of the important theories in mathematics. It can help in solving of various mathematical problems. The problems can be represented in the graphical form with the help of set theory. A set is nothing but a collection of objects. There can be intersection of sets and union of sets. On performing the intersection operation the common elements of both the sets are got. The union of two sets gives all the elements present in both the sets. Basically a set can consist of various objects in it. But usually in mathematics sets usually deal with the objects which are related to mathematics.

The concept of set theory is very ancient. In the 1870’s itself good research was done on this topic and considerable progress was made. A Venn diagram is best used to represent the operations on sets. The process of intersection and union can be easily represented on the Venn diagram. There can be bigger set and a smaller which is part of it is called its subset. This can be explained with the help of an example. If a set contains the elements {1, 2, 3, 4} and there is another set which contains the elements {1, 2} then the latter set is subset of the former set. It simply means the all the elements present in the second set are also present in the first set. First set covers the whole of the second set in it.

The compliment of a set is nothing but a set containing elements which are present in the universal set but are not present in the given set. The set complement can be explained with the help of an example. If there are elements like {a, b, c, d, e} in the universal set and elements in the given set whose compliment is to be found out, are {a, c, e} then the complement set is given by { b, d}. So, the elements b, d is contained in the universal set but is not present in the set whose compliment has to be found. So, these elements are part of the required set. The compliment of a set X can be represented by the notation X’. This is a simple notation and there is another method of representing the same. Instead of the apostrophe symbol the letter ‘c’ can also be used to represent compliment.

Introduction to polygon basics

The polygon is a basically called plane figure and that is surrounded with closed path, collected of a fixed series in straight line segments with a nearer polygonal chain. And these segments are called its edges or sides. Let us discuss  the topic called polygon basics. (Source – Wikipedia)

Types of Polygon Basics

Names of polygons basics with different number of sides:

• Triangle : Triangle consists of three sides
• Quadrilateral :Quadrilateral consists of four sides
• Pentagon: Pentagon consists of five sides
• Hexagon:Hexagon consists of six sides
• Octagon:Octagon consist of eight sides.
• Nanogon : Nonagon consists of nine sides
• Decagon : Decagon consists of ten sides
• Heptagon:Heptagon consists of sevensides.
• Triskaidecagon or Tridecagon : Trikaidecagon or tridecagon consists of  thirteen sides. 

• Tetrakaidecagon or Tetradecagon : Tetrakaidecagon or tetradecagon consists of fourteen sides.
• Pendedecagon : Pendedecagon consists of fifteen sides.
• Hexdecagon: Texdecagon consists of  sixteen sides.
• Heptdecagon: Heptdecagon seventeen sides.
• Octdecagon: Octdecagon eighteen sides.
• Enneadecagon: Enneadecagon consists of nineteen sides
• Icosagon: Icosagon consists of twenty sides.
• Triacontagon : Triacontagon consists of thirty sides.
• Teracontagon: Teracontagon consists of forty sides.
• Tetracontagon: Tetracontagon consists of fifty sides.
• Pentacontagon: Tentacontagon consists of sixty sides.

 The triangle is the basic method in polygon and it has three angles. This involves three sides and three vertices.

 Triangle

Square

The octagon

The drawn diagramatic representation are the  basics form in polygon.

Problems Based on Polygon Basics

Example 1 :

Determine the side distance end to end of hexagon is 12 cm. Mention the area of the hexagon.

Solution:

            Given:

                        Side distances  (t) = 12 cm

            Formula:

                        Area of the hexagon (A) = t2 2.6

                                                = 122 x 2.6

                                                = 144 x 2.6

                                                = 374.4

Example 2:

The side distances of hexagon is 13cm. Find the area of the hexagon.

Solution:

            Given:

                        Side length (t) = 13 cm

            Formula:

                        Area of the hexagon (A) = t2 2.6

                                                = 132 x 2.6

                                                = 169 x 2.6

                                                = 439.4

              Area of the hexagon = 439.4 cm2    

Real numbers

Real Number Definition– As the name says “Real”, so these numbers are actually numbers that really exists. Any number that we think of is considered as a real num, be it positive or negative, fraction or decimal. Real num. are numbers those includes both rational and irrational number. A real-number has to have a value. If there is no value to any number then we can call that number as an imaginary number. All integers like -75, 89, and 84 etc. are considered as real num. All fractions like 3/5, 7/2, -9/7 are considered as real no. too.

Decimals along with repeating decimals are also considered as real num.
These can be any positive or negative number. We can plot All Real Numbers on the number line too.  Therefore we can order these numbers and that we cannot do in case of imaginary numbers. The name of imaginary numbers itself says that they are imaginary so we can just imagine them; they do not have a specified value. These numbers can be plotted the same way we plot the integers that is smaller numbers on left and larger numbers on right. So greater the number, more it will be towards right side of number line.
So we can call real nos. as all those numbers which are present on number line are termed as such.
Some Real Numbers Examples are pi, 34/7, 5.676767, -1034, 45.87 etc.
Some examples of imaginary numbers are square root of -34 or square root of -2, as there is a negative under the root, so the value of this number cannot be found.

Similarly value of infinity cannot be determined too .
Hence these numbers are not considered as real and are considered as an imaginary numbers. So these numbers are all integers, fractions decimals and repeating decimals numbers. We can add, subtract, multiply and divide real nos. just like another numbers. We can perform the operations on real-numbers same way as we do them on other numbers.

Now the question arises that Is 0 a Real Number– Zero is considered as an integer and all integers are real-numbers. Therefore zero is considered to be as a real-number.
 All Real Numbers Symbol is R which is used by many mathematicians. The symbol R is used to represent the set of real nos. All such numbers can be seen on the number line but we cannot find imaginary numbers on that.

Construct angles tutoring

In geometry, we have many constructions of different geometrical shapes, angles,angle bisectors etc, these constructions are really challenging and mind blowing,now let see and the steps of  few constructions of angles.

Constructions of angles can be done by two methods,one by using ruler and compass and another by using protractor. Now we see and understand the steps of angles constructions by using both the methods.

Construct Angle 60' by Two Methods

Construct angle 60' by a ruler and compass.

Construction method I: Draw a convenient line segment AB,keep the compass at A ,draw a semi circle.the semi circle should intersect the line segment AB,then mark the intersect the point as C, then again keep the compass at C,without changing the measurement of the compass draw an arc , which cut the semicircle at the point D.Now join A and D and extend the line.Hence we get angle 60'.

Construction method II:  

In this method we draw a line segment AB and keep the protractor at A ,mark 60' from right to left .Mark the new point as ,C Join AC.Now we get angle 60' using protractor.

Construct Angle 90' by Two Methods

Construct angle 60' by a ruler and compass.

Construction method I:

Draw a line segment AB,draw a semi circle at A,which intersect C,then keep the compass at C draw an arc which intersect the semi circle at D,again with the same measurement keep the compass  at D ,draw one more arc ,which intersect the semi circle at E,do not change the measurement of compass, keep the compass at D and E, cut two more arc's which intersect each other exactly at F, join A and F.The line AF exactly perpendicular to the line segment AB.Hence angle 90' constructed.

Construction method II :

Now in this method, draw a line segment AB keep the protractor at A and mark 90', name as C , join A and C , we get line AC is perpendicular to the line AB. Hence  angle 90' constructed using protractor.

Exercises on Construct Angles

1) Construct angle 120' using ruler and compass.

2) Construct angle 120' using protractor.

3) Construct angles 15' , 30', 45' , 75', 105', 135' using ruler and compass.

4) Construct angles 15' , 30', 45' , 75', 105', 135' using protractor.

Learning to Simplify Mixed Numerals

We have studied the number line. We know about the natural numbers, whole numbers, integers, real numbers and also imaginary numbers. When these numbers are combined we get interesting combinations. We need to do a study on fractions. Fractions can be numbers in the form of P/Q. P and Q are integers. These fractions now are of three types namely proper, improper and mixed fractions. Mixed fractions are bit complicated compared to the other two. So we need to study them in detail. First we need to learn the process of simplifying mixed fractions as a first step in this direction. It is a relatively simple process.  To simplify mixed fractions we convert them to improper fractions. Improper fractions are easy to handle and can be used for the purpose of calculations. After converting the mixed fraction into an improper one, we can further simplified if need be. For this we need to convert mixed fraction to decimal and solve this problem.

After getting the improper fraction, we take the numerator of the fraction as the dividend and the denominator as the divisor. We divide the dividend with the divisor till we get the remainder as zero. When the digits in the dividend get over we continue the process of division by adding zeros to it and a decimal point to the quotient. Sometimes we don’t get the remainder as zero. In that case the decimal part might contain digits that are repeating. It can be ended there. So the question arises how to solve mixed fractions after we have converted them into decimals. Solving can involve addition of two mixed fractions or it can also be subtraction of two mixed fractions. Both the processes can be relatively simple if the denominators in both the cases are the same. But if the denominators are different the process is bit different than when they are same. Now we will learn subtracting mixed fractions with different denominators to get the final answer. When the denominators are not same first convert into similar ones and then do the process of addition or subtraction. This can be done by taking the LCM of the denominators. After taking the LCM the denominators become common and the numerators can be added or subtracted to get the answer. This helped us to know what is a mixed fraction better. It contains a whole number part and a proper fraction.

Line Plot Examples

One of the important educational tools is graphs. Through graphs we learn how to organize and interpret information. Graphs help not only to analyze data in math but also help convey information in business. Many types of graphs exist, but we use line plots to show frequency of data.

A line plot shows frequency of data along a number line with “x” mark or any other marks to show frequency. It is best to use a line plot while comparing fewer than 25 frequency data. It is a quickest and simplest way to organize data.

Steps to Follow to Plot the Examples on Line Plot

Here are few steps to show how to sketch a line plot;

Step 1 : Gather the given  information which is called data for which a line plot has to be drawn. Look for those data sets that need to show frequency.

Step 2 :Group the data items that are the same and then create and label a chart to help you organize the list from the data.

Step 3 : Determine an approximate scale to draw the line plot. If the scale consists of numbers, then break it into even parts.

Step 4 : Draw a horizontal line and label it according to the scale chosen. This looks  similar to a number line.

Step 5 : Put a mark, say X that corresponds to the number on the scale according to the data that is organized. This is the line plot.

For example,

We use the above steps to give the line plot examples

Sketch the line plot for the following given data of heights of students of a class in centimeters.

120,110,100,120,105,110,105,100,110,105.

Solution : Let the labels be the heights of the students (in centimeters) in a class. If two students are of 120cms , you'd place two X's above 120. If three students are of 110cms , you'd place two X's above 110. If three students were 105cms , you'd put three X's above 105. If two students are of 100cms , you'd place two X's above 100. The total would be ten students, thus ten X's altogether.

Some more Line Plot Examples

Use all the steps to draw the line plot examples:

(1) Sketch the line plot to show the test scores of 17 students that are given below:

10,  30,  10,  10,  20,  30,  50,  40,  45,  50,  10,  30,  35,  30,  20,  40,  50

 (2) Represent the data in a line plot. The following data is obtained from a survey made in an area for number of pets in each household.

4, 0, 1, 3, 2, 6, 1, 4, 2, 8, 10, 2, 1, 3, 0, 13 ,4, 2, 1, 14, 0, 3, 0, 2, 1, 0, 3, 1

Multiplying mixed fractions

Mixed fractions are just the mixture or a combination of two things of whole number and a fraction. Mixed fractions are specially used when we have to show how much a whole thing we have and how much of part of something we have. The form for mixed fraction looks like this, a and b/c, where ‘a’ is our whole number and b over c is b/c is our fraction.

Multiplying Mixed Number Fractions - For Multiplication of Mixed Fractions, we first convert the mixed fractions into improper fractions and the multiply them and then convert the solution back into mixed fraction. How do I Multiply Mixed Fractions – let us understand by taking an example, if we have to multiply 1 times 3/4 and 5 times 7/9 then we can convert 1 times 3/4 to improper fraction by multiplying 4 with 1 and adding 3 to it. We will get 7, which will be our numerator for the improper fraction. The denominator will remain same that is 4. So our improper fraction will be 7/4. Similarly we can convert 5 times 7/9 as (9*5) +7/9 which will be 52/9.

Hence we got two fractions as 7/4 and 52/9. We can multiply both the numerators and the denominators. We will get 7*52/4*9 which will be 364/36. We can reduce this fraction by dividing both the numerator and the denominator by 4 which makes the fraction as 91/9.

Now we can convert it into a mixed fraction by dividing. So 91/9 can be written as 10 times 1/9. Multiplying Mixed Fraction always gives a mixed fraction as the solution. The same we witnessed in the above example. We multiplied two mixed fractions and we got a mixed fraction as an answer. So to multiply mixed fractions we need to follow the following steps: -
1. Convert the mixed fractions that we have to multiply into improper fractions.
2. Multiply the improper fractions by multiplying the numerators and the denominators separately that is if we have two fractions, the numerator of both the fractions should be multiplied together and the denominator of both the fractions together. For example if we have 11/2 times 8/5. Then we can multiply 11 with 8 which will be 88 and 2 with 5 which will be 10. Therefore 11/2 times 8/5 gives 88/10
3. Reduce the fraction as much as we can.
4. Convert the answer back into the mixed fraction form.

Discovery of algebra

The discovery of the algebra starts from the place Egypt and Babylon. In Egypt and Babylon the people were very interested to learning the linear equations such as cx=d and the quadratic equations such as c2+dx = e and in intermediate equations. The Babylonians basic steps are used to solve the quadratic equations. Nowadays we also use these basic steps only. Now in this article we just simply see about the discovery of algebra.

Explanation for Discovery of Algebra:

The Diaphanous who is called the father of algebra. The book named Arithmetic book of Diaphanous gives an advanced level and many unexpected solutions to the intermediate equations for discovery of algebra. The first Arabic algebras that is a systematic expose of the basic theory of equations, is written by AL-Khwarizmi. The basic laws and identities of algebra are stated and solved by the Egyptian mathematician Abu Kamil in nineteenth century.

Discovery of Algebra in Persian Mathematics:

The Persian mathematician, who called omar kayyam, mentioned in his book how to express roots of cubic equations by line segment which is obtained by intersecting conic equations. But he did not find the formula for the roots. But in 13th century, the great Italian mathematician named Leonardo Fibonacci achieved the close approximation to the solution of the cubic equation.

After the introduction of the symbols for unknown and for algebraic powers and operations there was a development in algebra. It was happened in 16th centuries.

Algebra entered in to the modern phase in the gauss time. After the discovery the Hamilton, the German mathematician Hermann Grassmann examining the vector. Because of its abstract character, American physicist named Gibbs acknowledged the vector algebra for system of great utility for physicists. In that time period, algebra of modern that means abstract algebra has continued to develop. After that it is used in all branches of mathematics and in other science.

Mathematical Sentence

In Algebra, Mathematical sentence is a term which is also known as an expression. An expression is usually defined as a sentence that has a number, an operation and a letter in it.

When a mathematical sentence is not in an algebraic form, it will just have two numbers and an operation. In other words, an expression is the mathematical analogue of an English noun. It is a correct arrangement of mathematical symbols which is used to represent a mathematical object of interest.

Example for Mathematical Sentence:

In general, a mathematical sentence is a formula that is right or wrong, true or false. If a mathematical sentence has an equal sign, it is referred as an equation.

Let us consider the following simple examples to show what a Mathematical Sentence means:

3 + 2 = 5. We know this to be true, it is a mathematical sentence.

3 + 4 = 5

we know this to be false, however, since we know definitely that it is false, But still it is a mathematical sentence.

5x + 8 = 13, for all values of x

This statement is neither true nor false: For some values of x, the statement is true and for some other values, it is false. Hence as the statement is neither true nor false, this is not a mathematical sentence.

Mathematical Sentence as an Open Sentence:

A mathematical sentence that contains one or more variables is referred as an open sentence

Some examples for an open sentence are listed below:

2a = 5+ b, 4x = b + 2a

Solved example for an open sentence:

To show ‘3(2b) = 3' an open sentence?

Solution:

Step 1: In the equation given, the number of variable is One

Step 2: Open Sentence is a mathematical sentence with one or more variables

Step 3: So, '3(2b) = 3' is an open sentence.

Subtracting Fractions

Subtraction is one of the basic operations in math. It is a process of finding the difference of two numbers. In case of whole numbers the process is simple but in case of fractions, the process may involve a number of steps.

How to Subtract Fractions
As mentioned earlier, subtraction of fractions require a number of steps. Of course, in the simplest cases, where the denominators of the fractions are same, the process is just like subtracting whole numbers. All that need to be done is just subtract the numerators over the given common denominator.

But in case of fractions with different denominators, first task is to find the equivalents of the given fractions with same denominator.  Such a equivalent common denominator is also called as Lowest Common Divisor. It is same as the lowest common multiple of the denominator.

How do you Subtract Fractions
Let us concentrate on the case of subtractions of fractions with different denominators, as the case of subtraction of fractions with same denominators is as simple as subtraction of whole numbers.
The easiest method for students to understand is the method of equivalent fractions. The first step here is, determine the lowest common multiple of the given denominators.

Then find ‘equivalent fractions’ of the given ones with the lowest common multiple as the denominator. That is rewrite the given fractions as if their denominators are changed to the lowest common multiple found. For example, 1/2 can be rewritten in the equivalent form as 2/4, 4/8,8/16 etc. depending on the need.
The subtraction of given fractions is same as the subtraction of their equivalent fractions and now the process is simple because the denominators are made same.

Subtract Fractions
Let us discuss a specific example to illustrate how to subtract fractions with different denominators. Let is consider that 2/5 is subtracted from 3/7.

The denominators are 5 and 7 and the lowest common multiple is 35. The equivalent fraction of 2/5 with denominator is 14/35 and that of 3/7 with the same denominator is 15/35.

Now subtraction of 2/5 from 3/7 is same as subtraction of 14/35 from 15/35 which is (15 – 14)/35 = 1/35. Thus the answer is 1/35.

Subtracting Fractions with Whole Numbers
For subtracting fractions with whole numbers, first subtract the fraction parts and then the whole number parts and combine them For example, 5(1/2) – 3 (1/4) is done by (1/2) – (1/4) 1/4 and 5 – 3 = 2, which gives the final answer as 2(1/4).

However trouble arises when the fraction part of the first mixed number is less than that of the second mixed number. In such cases, convert the first fraction part to an improper fraction by borrowing 1 from the whole number. The rest of the procedure is same as explained but remember the whole number part of the first one is reduced by 1!

How to Multiply Two Digit Numbers

If we have practiced and hopefully memorized the multiplication tables and Multiplication Rules, then we can solve any multiplication problem. We just have to understand the system to how to do it. We know that multiplication is just repeated addition but if we have large numbers, we cannot add them repeatedly to get the solution. To multiply one digit number and two digit number with a one digit number is an easy process but multiplying two double digits numbers uses a different process that needs to be followed.

How to Multiply Two Digit Numbers - We will learn here how Double Digit Multiplication works. Let us start with an example of the multiplication problem of two double digits numbers. Let us do 16 times 19; we can break down the Two Digit Multiplication into series of steps which are given as follows: -

• In 2 Digit Multiplication, firstly we take the numbers present in ones place and multiply them together; in this case we multiply 9 times 6 which equal 54.
• With the product of the digits at ones that is 54, we only write the 4 down while multiplication and 5 is carried forward on the tenth place just like when we add two numbers.
• Then we multiply 9 times 1 which is 9 and then add 5, and which is 14 so the solution to 9 times 16 is 144. Thus 16 times 9 is 144.
• Then we take the digit on the tens place on one number that is one in the number 19 and multiply it with 16, which gives us 16.
Now we have two solutions, one is 144 and other is 16, to find the final answer we add a zero to the second solution and then add the solutions together. That means 16 will become 160 and then we add it to 144 which equals 304. So the solution of 16 times 19 is 304.

 Only by using these breaking down method we have solved this big problem.  To multiply two digit numbers we can either break the one double digit number into two digits. For example, in the previous example, we can break 19 into 1 and 9. 9 is at ones place so we can write it as 9 but 1 is at tens place so we can use its face value which is 10. Then we can multiply 9 by 16 which give us 144 and then we multiply 10 by 16 which equals 160. We can add both the solutions to get the solution 304.

Definition of an Acute Angle

The different types of angle have different types of names. Where the angle is less than 90 radiant is called as acute angle. The acute angle is smaller angle compared with other type of angle such as right angle, obtuse angle, straight angle, reflex angle, and full angle.  The angle which is larger is called as reflex angle.

Definition of a Acute Angle
Definition acute angle, an angle which is less then 90 radiant is called as acute angle. Compared to the other types of angle, an acute angle is very smaller that is in between 0-90 radiant. Also that acute triangle are those where all the interior angle are acute.

Definition of an Acute Angle
An angle of less then 90˚ is called as acute angle and it has three angles. An angle which is exactly 90˚ is a right angle but it is not an acute angle, since it is less than 90˚ or between 0˚-90˚.

Acute Angle Definition Math
An angle which measures less than 90˚ or in between 0˚-90˚ is called as acute angle. In other words, acute angle is a positive angle that measures less than 90˚. Simply it is defined as, an angle of less than 90˚.

Geometry Acute Angle
In geometry, an angle is line segments that intersect in the same plane or it is the amount of bend between two lines. The angles are formed by two intersecting rays at the point of intersection or sharing of same end point. The angles are used to measure the turns in between the two arms. The unit of an angle is degree or radiant. In order to measure an angle from a circle that is a two dimensional plane, if the angle measures greater than 0˚ and less than 90˚ is called as acute angle. The acute angle is referred by the alphabets like ABC or DEF.

In above figure angle A is acute which measures greater than 0˚ and less than 90˚. It is formed by the intersection of two rays like AB and AC, which are called as sides of angle or arms. Where the angle is formed is called as vertex. In figure, the angle 1 which is, marked by square sign is called acute angle. Acute angle is smaller than right angle which is exactly equal to 90˚. Similarly the angle 2 is complementary of an angle 1. Complementary angle of an acute angle is also acute as it measures greater than 0˚ and less than 90˚.

Exterior Angles in Polygon and Triangle

Exterior Angle
Consider a shape such as square ABCD. Extend the horizontal bottom of the square at the point D up to F. Now we have a vertical line BD and a horizontal line DF joining at the common vertex D. The angle BDF formed between the original side BD of the mathematical shape i.e. square and the extended side DF is termed to be the exterior angle. The angle formed on the inner side of BD is the interior angle BDC.  The sum of the interior angle and the exterior angle formed using the side BD is 180 degrees. This forms a horizontal straight line CF measuring 180 degrees.

Polygon
A mathematical shape which has straight sides and flat shape is termed to be polygon.

Exterior Angles Polygon
Exterior Angles Polygon definition states that: With one of the angles of the polygon, a linear pair is formed by an angle which is termed as an exterior angle of the polygon.

At every vertex of a polygon, two exterior angles can be formed at the maximum.  Each exterior angle of a polygon is formed in between a side of the polygon and the line extended from its adjacent side.

Finding Exterior Angles of Polygons
We can find the exterior angles formula for polygons. The important point to be noted here is that at each vertex of the polygon, two equal exterior angles can be drawn but the formula for finding the exterior angles of polygon uses only one exterior angle per vertex.

Polygon Exterior Angles Formula:
The polygon exterior angles formula states that the sum of Polygon Exterior Angles is 360 degrees, irrespective of the type of the polygon. In other words, we can say that the sum of all the exterior angles of a polygon is equal to one full revolution.

Sum of one exterior angle of all vertex of any polygon = 360 degrees.

In case of a regular polygon, the formula to find any exterior angle is obtained by dividing the sum of the exterior angles i.e. 360 by the number of angles, say “n”.

The value of an angle of a regular polygon = 360/n

Here are few examples of Hexagon:

Example 1:
Hexagon has 6 sides. Therefore every exterior angle in hexagon = 360/6 = 60 degrees.

Example 2:
The value of an angle of a regular polygon is 30 degrees. How many sides are there in the polygon?
360/n = 30 which can be rewritten as 360 = 30n.
Thus, n is given by dividing 360 by 30 which results in 12.

Exterior Angles Triangle
In case of triangle, exterior angle lies in between one side of the given triangle and the extension of the other side of the same triangle. Exterior Angles of Triangle can be obtained by adding the measures of the two of the non-adjacent interior angles.

Math Homework Help

As many parents will know, math homework is something kids avoid like the plague. Math may not be everyone's favorite subject but there is no denying it's importance in day to day life. Math skills are necessary and students who take an interest in the subject early on, will have a much easier time studying it in high school.

Much of the lack of interest in math boils down to the way it is delivered in class. To ensure that every student understands what is being taught, teachers need to employ different techniques to explain the concepts. This does not happen very often and students vary between having a vague idea and being completely clueless. When they have to finish their homework, in most cases students simply don't know enough to do their work. Some parents help out but in many households parents are either too pressed for time or not well-versed enough in math themselves to help their kids.

Getting help with math is the best solutions for students and parents who find themselves in this situation. Math tutoring has become very popular over the past decade with several students signing up for them, starting as early as elementary school. Math tutors wok with students on an individual basis, giving them ample time to learn at their own pace and clear every single doubt. Hundreds of students in schools across the country use math tutoring and as a result, have aced their tests and exams, getting As and Bs where they were previously failing.

Many math helpers feature help with math homework as a regular part of their services that students can make use of everyday. Unlike online calculators and programs which calculate the answer for the questions students input, getting professional help ensures that students really learn the concept or theory and how to put it into practice. Students can also practice with math tutors which allows them to really explore the topic as there is someone who can correct them, if need be. Homework help has helped students keep up with their schedules and submit assignments on time, all the while learning more about the topic.

All about acute angles

A very common question for a 5th grader who has just started to learn geometry would be: what is an acute angle? Or What is a acute angle? (Considering that they are not really sure what article to use before the word ‘acute’). Let us now describe an acute angle in simple terms.

A acute angle:
Look at the hands of a clock on your bedroom wall when the time is say 2:00 pm. The hour and the minute hand of the clock make an angle with each other at the centre of the clock. That angle would be an acute angle. From 12:00 noon to 3:00 pm, the angles made by the hands of the clock are acute angles. At 3:00 pm the hands of the clock are at right angles to each other. After 3:00 pm the angle between the hands of the clock are obtuse angles.

If you have a pine tree around your home, look at the top of the tree. The angle made by the two lines in the form of an inverted V at the top is an acute angle.

Acute angle definition:
An angle whose measure is more than 0 degrees and less than 90 degrees is called an acute angle. If the angle measure is in radians, then the angle whose measure is more than 0 radians and less than pi/2 radians is called an acute angle.
Mathematically it is written like this:
If angle A is such that measure of angle A = m0 < = (mIn geometry an acute angle can be generally sketched as follows:

Acute angles in geometry:

An equilateral triangle has all three angles as acute angles. Each of the angles in an equilateral triangle is 60 degrees or pi/3 radians. See picture below:


In a right triangle, one of the angles is 90 degrees (or pi/2 radians), but the other two angles are acute angles. For example see the picture below:

The angle subtended by the chord of a circle at any point in the major segment of the circle would always be a right angle, except if the chord is a diameter of the circle. In such a case, the angle subtended would be a right angle. See the following figure to understand that better:

In the above figure, BC is the chord of a circle and the angle subtended by BC at A = α. Α would be an acute angle.

Basics about Circles

Definition: A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). Properties of a circle encompass use of terms such as chord, segment, sector, diameter etc of a circle. Now let us try to understand some other terms related to properties of circles.

Properties of circle:
A straight line that intersects a circle in two distinct points is called a secant to that circle. In the picture below, we have a circle with centre at C. A line l intersects this circle in two points, A and B. This line is a secant to the circle.

A straight line that intersects (or touches) a circle in just one point is called a tangent to the circle at that point. For a circle at a given point, there can be only one tangent. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide. See picture below.

Circle theorems:

1. The tangent at any point of a circle is perpendicular to the radius through the point of contact. This we can see in the above picture. The tangent is perpendicular to the radius that joins the point of contact with the centre of the circle.

2. The lengths of tangents drawn from an external point to a circle are equal.

Circle formulas geometry:

Area of a circle: Area of a circle is given by the formula:
A = pi r^2
Area of semi circle: Area of a semi circle of radius r is given by the formula:
A = (pi/2)r^2

Segment of a circle:

The portion (or part) of the circular region enclosed between a chord and the corresponding arc of the circle is called a segment of the circle.

In the picture above, the orange portion is called the minor segment of the circle and the yellow portion is the major segment of the circle. The minor segment corresponds to minor arc and the major segment corresponds to the major arc of a circle.

Area of a segment of a circle is found using the formula below:

Where, theta is the angle subtended by the chord at the centre of the circle and r is the radius of the circle.

Calculate Time - Fourth Grade Math

In grade four, new concepts are introduced in math. Some new concepts are time, multiples and factors, addition and subtraction of three digit numbers, unitary method, measures of length, mass and capacity, fractional numbers, addition and subtraction of fractions, decimals, addition and subtraction of decimals, Introduction to angles.

Introduction to grade four math:

In grade four, the topic time contains the following sub units.

- Measurement of time

-Calender

- Time in second

- Addition and subtraction of time

In this article let us learn about 24-hour clock time.

In the present day world business houses, airlines, railways are busy round the clock. Hence it is convenient to use 24 - hour time representation instead of a.m. and p.m.

1. 12 O' clock midnight is expressed as 00 00 or 24 00

2. 12 O' clock is expressed as 12 00

3. The time between 12 O' clock noon and 12 O' clock midnight is expressed by adding 12 hours to the given hours period.

For example:

25 minutes past 6 in the evening is expressed as 18 25

45 minutes past 11 midnight is written as 23 45

Rules for Writing 24-hour Clock Time:

A day begins at 12 midnight (00:00 hours) and hence at 12 midnight the following day.

Thus 1 day = 24 hours

Rule 1: For any time in a.m. we simply put down the time by writing hours and minutes in two digits numbers.

Rule 2: For any time written in p.m. we simply add 12 hours to the number of hours period and write minutes without separating them.

6:25 a.m is written as 06 25 hours

10:45 a.m is written as 10 45 hours

3:10 p.m. is written as 15 10 hours (3 + 12 = 15)

10:50 p.m. is written as 22 50 hours (10 + 12 = 22)

Example Problems on Grade Fourth Math:

Ex 1: Express 11:25 p.m. in the 24 hours system.

Sol:

Step 1: See which rule can be used.

Step 2: Since the time given is in p.m., add 12 to 11

Step 3: So, 11:25 p.m. = (11 + 12 hours) : 25 min

= 23 25 hours

Ex 2: Express 18 30 hours in terms of a.m. or p.m.

Sol:

18 30 hours means (12 + 6 hours) 30 minutes

= 6:30 p.m.

Ex 3: Express 07:45 p.m. in the 24 hours system.

Sol:

Step 1: See which rule can be used.

Step 2: Since the time given is in p.m., add 12 to 7

Step 3: So, 07:45 p.m. = (7 + 12 hours) : 45 min

= 19 : 45 hours

Trigonometric Integrals

Trigonometry is a fundamental concept of mathematics. It is used in calculus functions and vectors. In this topic we have to use trigonometry as integral function. That means how to integrate trigonometric functions. For this we also have to know what is integration?  Integration means to calculate area of a given curve, and the curve is a closed curve made by x axis and y axis.

Trigonometric integrals mean integration of trigonometric functions. As we know these trigonometric functions are basic formulas for solving trigonometric integral. To more simplify this term, let’s take an example like sin2X. This is a trigonometric function. And we integrate this function for this first we have to expand this term by using formula of trigonometry. After expanding we carry out the constant term then by using product rule of integral, we can integrate this trigonometric function.

Above example is simple it has only one trigonometric function but trigonometric function may be combine with other function also. It can be algebraic function with trigonometry, logarithmic function with trigonometry and exponential function with trigonometry. These are also called integrals of trigonometric functions. To solve this type of problem either we can use integration by substitution method or integration by parts method.

Inverse trigonometric integrals such as sin^-1X and cos ^-1X etc. now to integrate this type of functions we have to use basics of calculus. We need  to take this function equal to any constant like Y. means we have to write Y= sin ^-1X. now we transfer sin function to other site the we get. X=sin Y. Now we can simply integrate this term.

Trigonometric substitution integrals, here we also integrate trigonometric functions and calculus functions, but procedure is different. To integrate this type of function we have to substitute and equal trigonometric term in place of other trigonometric term. The first from of integrals is integration of [f’(x)/f(x)] dx=logf(x) . In this form integral of a function whose numerator is the exact derivative of its denominator and equal to the logarithmic of its denominator? The second form is, in the integrand consist of the product of a constant power of a function f(x) and the derivative of f(x), to obtain the integral we increase the index by unity and then divide by increase index. This procedure is known as power formula. Lets take an example suppose we have to integrate (4x^3/1+x^4) dx= ln (1+x^4). By using this method we substitute 1+x^4 = any constant term like (t), and after that we integrate this function.

Exponential Function an Introduction

An Exponential Function is a function which involves exponent which is the variable part rather than the base as in any normal function. For instance f(x)= x^3 is a function and an exponential function is something like g(x)= 3^x, here the exponent or the power is a variable (x) and the fixed value is the base (3). So, the definition of Exponential functions can be given as a function whose base is a fixed value and the exponent a variable. Example: f(x) = 5^x, here the base 5 is fixed value and the exponent ‘x’ is the variable.
In general, we can define Exponential Functions as a function which is written in the form ‘a^x’ in which ‘a’ is the base which is a fixed value or constant (‘a ‘not equal to 1) and ‘x’ the variable which is any real number. The most common exponential function we come across in math is e^x which is known as the Euler’s number.
Let us now take a quick look at the Exponential Function Properties. Consider the Exponential function f(x) = b^x for which the properties are as follows:
• The domain of the exponential function consist of all real numbers
• The range is the collection of all positive real numbers
• When b is greater than 1 then the function is an increasing function also called exponential growth function and when b is less than 1 then the function is a decreasing function also called exponential decay function
• The other properties that an exponential function satisfy are,
1. b^x.b^y = b^(x+y) [when bases are same and a multiplication operation then we can add the powers]
2. b^x/b^y = b^(x-y)[when bases are same and a division operation then we can subtract the powers]
3. (b^x)^y = b^(xy) [when a base is raised to a power x and raised to whole power y then we can multiply the powers]
4. a^x.b^x= (a.b)^x [when bases are different with the same power and a multiplication operation then we can multiply the bases whole raised to power]

We come across a function called an Inverse Exponential Function; this is nothing but a logarithm function.  We know that the exponential function is written in the form f(x) = b^x, to find the inverse of a given function we need to interchange x and y and solve for y. By interchanging we get x = b^y  and then solving for y gives us y = log x (base b) which is a logarithm function.

Points and lines tutoring

Tutor is the person who teaches the kids and this teaching section is the tutoring. Tutoring is an open source for the students to gain knowledge that is in online a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.

Points and Lines Tutoring:

Tutoring about the geometric points and lines we have to know the classification of a points and lines. points and lines classification are as follows.

Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.

Problems in Points and Lines Tutoring:

Example 1:
Find the distance between the points A(6,3) and B (2,1).

Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (6,3), (x2,y2)= (2,1).

Then d (A, B) =`sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

= `sqrt((2-6)^2 +(1-3))^2)`

= `sqrt((-4)^2+(-2)^2)`

= `sqrt(16+4)`

=`sqrt20`
=2`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-5,3) and B(2,1)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-5,3),(x2, y2) = (2,1)

=  `((-5+2)/(2))``((3+1)/(2)) `

=   `(-3/2) ` , ` (4/2)`

=    `(-3/2, 2)`

Example 3:
Find the slope of the lines given (8,-5) and (4,2)

Solution:
(x1,y1)= (8,-5), (x2,y2)= (4,2).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(2+5)/(4-8)`

m =`7/-4`

Example 4:
Find the equation of the line having slope  3 and y-intercept 5.

Solution:
Applying the equation of the line is y = mx + c
Given,       m =3 ,c = 5
y =  3x +5

or  y = 3x+5
or  -3x+y-5 = 0
3x-y+5 = 0.

Introduction for division math facts

Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than one; otherwise it is less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.

Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it obeys all the properties of multiplication. (Source: Wikipedia)

Definition and Steps for Division Math Facts:

Definition for division:

Division is defined as an arithmetic function, which is the opposed process of multiplication. From the process of division, the proportion or ratio of two numbers be capable of be calculated.

Otherwise, the process of decision how many periods of one number is included in a further one. Symbol of division is ‘/’ or ‘÷’.

If we divide a number by another number, then

Dividend = (Divisor * Quotient) + Remainder

Steps for division math facts:

Step1. Division of two integers by the related signs resolve be positive sign

a) Positive integer ÷ positive integer = positive integer

b) Negative integer ÷ negative integer = positive integer

Step2. Division of two integers by the unlike signs will be negative

a) Positive integer ÷ negative integer = negative integer

b) Negative integer ÷ positive integer = negative integer.              

Division Math Facts Example Problems and Practice Problem:

Example problem for division math facts:

1. Solve the following division

36 ÷ 6

Solution:

36 ÷ 6

= (6 * 6) ÷ 6

Answer: 6

2. Solve the following division,

32 ÷ 4

Solution:

32 ÷ 4

= (4 * 8) ÷ 4

Answer: 8

3. Solve the following division,

48 ÷ 8

Solution:

48 ÷ 8

= (6 * 8) ÷ 8

Answer: 6

Practice problem for division math facts:

1.      - 49 ÷ 7 = -7 (unlike sign)

2.      56 ÷ 7 = 8 (like sign)

3.      48 ÷ 12 = 4 (like sign)

4.      81 ÷ 9 = 9(like sign).

Learning subtraction of square roots

A square of a number a is a number x. therefore x2=a .A number x whose square is a. Every positive real number a has a unique positive square root, called the principal square root. Square root denoted by a radical sign as sqrt of a. For positive ax, the principal square root can also be written in exponent notation, as a1/2. We can undo a exponent with a radical, and a radical can undo a power. The “`sqrt(a)` “symbol is called the "radical “symbol..The line across the top is called the vinculum.

Subtracting Square Root Terms

Subtracting square roots

Subtracting square roots is combining like terms when we need to do that with algebraic expressions. The induces (a square roots index is 2 `root(2)(a)` , a cube roots index is 3  `root(3)(a)` , a 4th roots index is 4 `root(4)(a)` ,a 5th roots index is 5 `root(5)(a)` etc.) or the radicands (enclosed by parentheses after SQRT or the expression under the root sign) are the same.

Just as with "regular" numbers, square roots can be subtracted together. But you might not be able to simplify the subtraction all the way down to one number. Just as "you can't subtract apples and oranges", so also you cannot combine "unlike" radicals. To subtract radical terms together, they have to have the same radical part.

Simplifying Square-Root Terms

Simplify a square root, we take out anything that is a perfect square; that is, we take out front anything that has two copies of the same factor.

We can raise numbers to powers other than just 2; we can cube things, raise them to the fourth power, raise them to the 100th power, and so forth.

(ab)^2=a^2b^2  and`sqrt(ab)`   = `sqrt(a)``sqrt(b)` but we can’t write this subtraction of square root  `sqrt(a-b)`  = `sqrt(a)` - `sqrt(b)`

Example Problems on Subtracting Square Root

Example Problems

1. (4 * `sqrt(2)` ) - (5 * `sqrt(2)` ) + (12 * `sqrt(2)` )
Solution: Combine like
= (4 - 5 + 12) * `sqrt(2)`
Answer is 11 * `sqrt(2)`

2. (53 * `sqrt(5)` ) - (5 * `sqrt(5)` )
Solution: Combine like terms
(53 - 5) * `sqrt(5)`
= 48 * `sqrt(5)`

3. (40 * `sqrt(5)` ) - (48* `sqrt(5)` )
Solution: Combine like terms by subtracting the numerical coefficients.
(40 - 48) * `sqrt(5)`
= -8 *`sqrt(5)`
`sqrt(3x+1)`-( -`sqrt(2x-1)` )= 1    subtracting 2 square roots with variables
`sqrt(3x+1)` = 1 - `sqrt(2x-1)`
Take square both sides
3x + 1 = 1 - 2 `sqrt(2x-1)` + 2x-1
3x + 1 - 1 -2x + 1 = -2 `sqrt(2x-1)`
x+1 = -2 `sqrt(2x-1)`
take Square both sides again
x^2 + 2x + 1 = 4(2x -1)
x^2 + 2x + 1 = 8x -4
x^2 -6x + 5 = 0
(x-5)(x-1) = 0
x1 = 5, x^2 = 1

Ogive – The cumulative line graph

In statistics, a frequency chart displays the given data, in which the frequency of each data item is found.  What does frequency mean? Frequency as we use in case of the frequency of the flight from one place to another means the number of times the particular flight travels from one place to another.  In statistics, frequency is used to display the number of times the data item occurs in a data set.  Tally marks or tallies are used to record and show the frequency of an item in a data. Now, let us learn about cumulative frequency. It is the total of the frequency and all the frequencies below it in a frequency distribution.  In simple words, it is the running total of frequencies. Given frequency of a set of data, the Ogive chart looks something similar to the chart given below:

Age Frequency      cumulative frequency
8      4 4
9      6 4+6 =10
10    15 10+15=25
11     9 25+9=34
12    18 34+18 = 52
13    10 52+10 =62

The Ogive Definition can be given as a distribution curve in which the frequencies are cumulative
Now that we have the cumulative frequencies, we shall now plot the graph. To plot the graph we take the ages on the x-axis and the cumulative frequencies on the y-axis as we plot a normal line graph. Once all the points are plotted, we now join the points. The curve we get is the cumulative frequency curve, also called the Ogive

We can define Ogive as a cumulative frequency graph which is a curve or graph showing the cumulative frequency for a given set of data. When the given data is an un-grouped data, to get Ogive, we find the cumulative frequency of the data and plot that on the y-axis and the given data to which cumulative frequency is calculated is taken on the x-axis. The graph we get is the Ogive of ungrouped data. When the data given is a grouped data, we divide the group into classes with upper and lower boundary which is taken on the x-axis and the cumulative frequency of the data on the y-axis. The graph we get here is the Ogive of a grouped data.

Ogive Example
For example, let us assume the amount of savings for the months of January and March as $200 and the savings of $125 for the months February, April and May.  For the given data, the ogive displays a running total of the savings with the amount saved in dollars on the y-axis and the months on the x-axis.

Trigonometric Identities | Theorems Based on Trigonometric Identities

Trigonometric Identies are some identies used in Trigonometry in order to make the calculations easier.
Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Ratios
In any right-angled triangle ABC,
let  angle B = 90 o  and angle C = T.                                                      

Line segment AC is the hypotenuse.
With reference to angle C, we can say that,
Line segment AB is the opposite side of Line segment BC is the adjacent side of Therefore, trigonometric ratios are given as,


Trigonometric Identities
Basic trigonometric identities are:
sin^2 T + cos^2 T = 1.
tan^2 T + 1 = sec^2 T
1 + cot 2T  = cosec^2 T
Theorems Based on Trigonometric Identities

Theorem 1: sin^2 T + cos^2 T =1
In right-angled triangle ABC, let angle< B = 90, angle< C = T.
Let AB = a, BC = b, and AC = c
By Pythagorean theorem we can say,
(hypotenuse)^2 = ( side)^2 + (side)^2
From figure we can say,
(AC)^2 =  (AB)^2 + (BC)^2
c^2 =  a^2 + b^2
divide throughout by c^2, we get,
(c^2 ) / c^2 =  ( a^2 + b^2 ) / c^2
1  =  a^2 / c^2 + b^2 / c^2
=  ( AB )^2 / ( AC)^2 + ( BC)^2 / (AC)^2
=   (AB / AC)^2 + ( BC / AC)^2
=   ( sin T )^2 +  ( cos T )^2
Therefore,
1 = sin^2 T + cos^2 T

Theorem 2: tan2 T + 1 = sec2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by cos^2 T,
( sin^2 T + cos^2 T ) / cos^2 T  =  1 / cos^2 T
(sin^2 T / cos^2 T) + (cos^2 T / cos^2 T)  =  1 / cos^2 T
By using trigonometric ratios,
sin T/ cos T  =  tan T
1 / cos T  =  sec T
substitute the values we get,
( sin T / cos T )^2 + 1  =  ( 1 / cos T)^2
(tan T)^2 + 1  =  ( sec T )^2
tan^2 T + 1  =  sec^2 T
Theorem 3: 1 + cot2 T  = cosec^2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by sin^2 T,
( sin^2 T + cos^2 T ) / sin^2 T  =  1 / sin^2 T
(sin^2 T / sin^2 T) + (cos^2 T / sin^2 T)  =  1 / sin^2 T
By using trigonometric ratios,
cos T/ sin T  =  cot T
1 / sin T  =  cosec T
substitute the values we get,
1 +  ( cos T / sin T )^2   =  ( 1 / sin T)^2
1 +  (cot T)^2  =  ( cosec T )^2
1 +  cot 2T  =  cosec^2 T

Standard Deviation of Mean in a nutshell

Standard deviation of Mean is the measure of the spread of the data about the mean value. If the standard deviation is low it shows that the values of the data are not spread out much and if the standard deviation is high it shows that the values of the data are spread out. At times we come across data which has the same mean but different range; to compare the sets of data standard deviation is very useful.  The average squared deviation from the mean is called the Variance. The square root of variance is the Standard Deviation of Mean. It is a statistical measure to know how the data is spread in the distribution, in simple words statistical measure of dispersion. Standard Deviation Means is also called the Mean of the Means.

In a population Variance is given by the formula: sigma^2 =summation[x – mu]^2/n
Where, x is each value in the data, mu is the mean of the data, n is the total number of values in the data.  Usually variance is estimated from a sample in a population. Variance calculated from a sample is given by the formula: sigma^2 = summation[x – x bar] ^2/ (n-1), here, x is each value from the sample, x bar is the mean of the values in the sample; n-1 is one less than the total number of values in the sample.  One Standard Deviation of the Mean is given by sigma= sqrt [summation[x – x bar] ^2/ (n-1)]

Standard Deviation of the Mean Equation
The equation or the formula to be used to calculate the standard deviation depends on whether the data is grouped or non-grouped. For example, given data, 42, 35, 48, 53, 47 is a non-grouped data.

In such a case, the standard deviation of the mean is calculated using the equation:
sigma = sqrt [summation (x- x bar) ^2/ (n-1)] where sigma is the standard deviation, (x-x bar) ^2 is the square of the deviations of the data values and n is the total number of values. Let us consider the data given below
Hours of components Frequency
300-400                   13
400-500                   25
500-600                   66
600-700                            58
700-800                   38
Understanding statistics problems is always challenging for me but thanks to all math help websites to help me out.
The data is a grouped data, here the standard deviation of the mean is estimated using the equation given by, sigma = sqrt [summation f(x-x bar) ^2/summation (f)] where sigma is the standard deviation, f is the frequency, x is each value of the data, x bar is the mean of the data values, summation is the sum of.

Introduction to indefinite integrals

In derivatives we learn about the differentiability of a function on some interval I and if it is differentiable, how to find its unique derivative f’ at each point of I. In application of derivatives we learn that using derivative we can find the slope of the tangent at any point on the curve, we can find the rate of change of one variable with respect to the other. Now let us look at an operation that is inverse to differentiation. For example we know that the derivative of x^5 with respect to x is 5x^4. Suppose the question is like this: derivative of which function is 5x^4∫ Then it may not be that easy to find the answer. It is a question of inverse operation to differentiation.

The answer to the question: " Whose derivative is a given function f ∫ " is provided by an operation called anti derivation. It is possible that we may not get an answer to this question or we may have more than one answer. For example, (d/dx) (x^4) = 4x^3, (d/dx)(x^4 + 3) = 4x^3 and in general (d/dx)(x^4+C) = 4x^3, where c is some constant.

Definition of integration (integrals): If we can find a function g defined on the interval I such that (d/dx)(g(x)) = f(x), for all x belonging to I, then g(x) is called a primitive or anti derivative or indefinite integral of f(x). It is denoted by ∫ f(x) dx and is called indefinite integral of f(x) with respect to x. The process (operation) of finding g(x), given f(x) is called indefinite integration.

Thus the question when can we find the integral of f cannot be easily answered. There are some sufficient conditions such as, continuous functions and monotonic functions have integrals. Sin x/x is continuous, hence ∫(sin x/x)dx is defined but cannot be expressed as any known elementary function. Similarly, ∫ v(x^3+1) dx and ∫v(csc x) are defined but canoe be expressed as known elementary functions. If anti derivative of f exists, then it is called integrable function.
(1) ∫ f(x) dx means, integral of f(x) with respect to x.
(2) In ) ∫ f(x) dx, f(x) is called the integrand.
(3) In ) ∫ f(x) dx, ∫ …. dx indicates the process of integration with respect to x.

For evaluating indefinite integrals we use the following standard table of indefinite integrals:

Calculus - Limits and Continuity

The concept of Limitlays the foundation for the popular branch of Mathematics called Calculus. Calculusinvolves the analysis of functions and their behaviour. To study the behaviour of functions one needs to have a good hold on the fundamental concepts of Limit and Continuity. Calculus Limits and Continuity is the language of science and engineering.Limits and Continuity in Calculus have led to the development of ideas like derivative, integration etc.

Consider a function f(x):R->R and ‘a’ is any point in the domain of the function, limit of the function f(x) exists at x = a, if f(x) = f(a), where x is arbitrarily close to a. The idea is that as we approach the point x = a on the real line, f(x) approaches f (a). The limiting value of the function when x is very close to ‘a’ is ‘f (a)’. In the language of mathematics it is commented as “lim x->a, f(x) =f(a)”.

Continuity is another important concept which is based on the concept of limits. The function f(x) is said to be continuous at x = a, if the limit of the function at x = a on approaching the point x = a from both sides is equal to the value of the function at x = a. Precisely this can be stated as – if  lim h->0, f(a+h) = f(a-h)=f(a), then f(x) is continuous at x = a. Mathematically this can be stated as – a function is said to be continuous at a point if the right hand limit and the left hand limit of the function at that point is equal to the value of the function at that point. Geometrically speaking, a continuousfunction is the one which can be sketched on paper without lifting the pen even once.


 Properties of Continuous Functions:There are two important results for continuous functions which are stated in the form of theorems-
• Intermediate value theorem: The intermediate value theoremstates that if f is a real valued continuousfunctionon the closed interval [a, b] andt is any number between f(a) and f(b), then there exists a numberc in [a, b] such that f(c) = t.Example: The height of a child increases from 1 m to 1.7 m between the ages of eight and sixteen years, then, at some point of timebetween eight and sixteen years of age, the child must have had a height of 1.5 m.
• Extreme value theorem: The extreme value theorem states that if f is continuousreal valued function on [a,b], then f has a maximumvalue in [a,b], i.e. there exists some c in [a,b] such that f(c) >= f(x)  for all x in  [a,b]. The same holds for the minimum of f(x). For example, consider the function sin(x) where x lies between [0,2*pi]. This function attains at maximum at x = pi/2 and a minimum at x = 3*pi/2.

Derivatives of Exponential Functions with Trignometric function power

What are exponential functions? A mathematical function which is in the form, f(x) = a^x is called an exponential function, here x is a variable and a is a constant which is the base of the function (a is greater than zero but does not equal one). The most commonly used exponential function is e which is the natural exponential function which is denoted as e^x. Some definitions of e are
• e=lim(n?inifinity)[1+1/n]^n
• Lim(h?0)[e^h-1]/h=1, where e is a unique positive number
• e=summation(n=0 to infinity)[1/n(factorial)]
We know that derivative is defined as, f’(x) = lim(h?0)[f(x+h) – f(x)]/h
Using the above, we can find the derivative of the natural exponential function f(x) = e^x
d[e^x]/dx = lim(h?0)[e^(h+x) – e^x]/h
    =lim(h?0)[e^h.e^x – e^x]/h  [using rule of exponents]
    =lim(h?0)e^x[e^h-1]/h            [taking e^x common]
    =e^x lim(h?0)[e^h-1]/h
    =e^x .1
    = e^x
As we observe, e^x is its own derivative. Let us find the derivative of the exponential function with trigonometric function as the power.

E-sinx
We have, derivative of e^u is given by e^u.du/dx. So, the derivative of E-sinx written as e^-sinx would be d[e^-sinx]/dx. Taking –sinx = u which gives du/dx = -cosx, we get, e^u. du/dx which equals e^-sinx.(-cosx)
Finally we get the derivative of E-sinx as –cosx.e^-sinx

E –sinx
We have, y= e^-sinx
     y= e^u where u= -sinx, derivative of u= du/dx = -cosx
taking derivative, we get,
dy/dx= dy/du. du/dx
          = e^-sinx. [-cosx]
          = -cosx.e^-sinx

E^-sinx
Let us find the derivative of the exponential function with the trigonometric function –sinx as the power.
Using the chain rule, we get,
d[e^(-sinx)]/dx = d[e^(-sinx)/dx. d[-sinx]/dx
             = e^(-sinx). (-cosx)
             = -cosx.e^(-sinx)

Basics of exponential functions

An exponential function is a function of the form y = a^x where a belongs to positive real numbers and x is any real number. We shall first try to understand such numbers with the help of graphs.

Exponential function graph:
Let us try to graph exponential function y = 2^x. So here we see that a = 2 (which is a positive real number and x is any real number). To obtain certain number of points on the graph we construct the following table:

Note that both the above functions are not inverse of each other. With this understanding, let us now define an exponential function.

Definition: Let a belongs to R+. Then the function f: R ->R+, f(x) = a^x is called an exponential function. a is the base of the function. The corresponding exponential function equation would be: y = a^x
For example:
f(x) = 3^x, g(x) = (1/4)^x (where x belongs to R), h(x) = 1^x, (where x belongs to R), are all exponential functions.

Natural exponential function:
We are familiar with the irrational number pi which we come across in connection with the area and circumference of a circle. There is another important irrational number, which is denoted by e and which lies between 2 and 3. Its approximate value is 2.71828. Exact value of e is given by the sequence: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....An exponential function to the base e, f:R->R+, f(x) = e^x is called the natural exponential function. This function is very often used in study of various branches of science and math.

Inverse of exponential function:
Inverse of an exponential function is a logarithmic function. In other words, the exponential function and the logarithmic function are inverses of each other. Thus, if f:R->R+, f(x) = a^x, a belongs to R+ -{1}, then f^(-1):R+->R, f^(-1)(x) = log(a)x [read that as log of x base a]

Derivative of exponential function:
Based on limit definition of derivatives, the derivative of an exponential function can be shown as follows:
If y= a^x, then dy/dx = (a^x)Ln(a)