Countably infinite

Countably infinite on the face of it appears to be a contradiction, which it is not.  In everyday language we may be using the term countable to signify a countable (and hence finite) number. But in mathematical terms countably infinite has a very specific and well defined meaning. In order to understand what countably infinite means,  we first need to understand certain concepts and a little about the history of counting.

Countably Infinite :history of Counting

Primitive man had no use for counting large numbers. While hunting or fighting other tribes he found it enough to indicate one, two or more. This sense of one, two or many is also found among the animals. You will almost never find a lonely animal picking a fight with a large group, while it will willingly fight a one to one battle with another animal. If you observe carefully, you will find that at times a lonely animal does fight two after an initial hesitation. This tendency shows that animals too do have a primitive sense of counting and numbers.

With the development of civilization various systems for counting developed, which have ultimately evolved as numbers the way we know them. The concept of infinity also came into existence with this evolution and the meaning of " infinite " was taken as something that cannot be counted.

Finite and Infinite

Concepts of finite and infinite became known and initially none imagined that there could be different degrees of  " infinite"  too. Researches on number theory, set theory and analysis have proved otherwise. Thus we know today that there are finite and infinite numbers but among the infinite numbers there too are various degrees .  Thus, in present day mathematics infinite cannot be just treated as a number that cannot be counted.

Definition of Countably Infinite

Countably infinite in mathematics means a set of elements which can be mapped one to one on to the set of natural numbers. In other words for which a one to one correspondence can be found between all its elements and natural numbers without skipping any natural number and without assigning either two elements to the same natural number, or assigning two natural numbers to the same element..

Not all infinite sets are countably infinite. It requires a very high degree of skills in mathematics and logic to understand how. But it has been established logically and mathematically that not only a higher degree of infinite exists beyond countably infinite, but for any degree of infinite, there exists one which is beyond it and hence larger.

Dividing Rational Numbers

To review, rational number is any number that can be written as a fraction of integers. We shall understand by using example, 3 and 4 are the integers so 3 divided by 4 would be considered as a rational number. Fractions as we know are rational numbers and so are whole numbers. Let us take 3 divided by 1 is 3 and is the same thing so dividing rational numbers using the multiplicative inverse.

Dividing Rational Numbers or How to Divide Rational Numbers – Rational numbers are divided using the multiplicative inverse. When we divide fractions, we find the multiplicative inverse of the divisor; often it is also known as reciprocal. Reciprocal is when we replace the numerator by denominator and denominator by numerator then the resultant fraction. For example: - For a fraction 7/8, its reciprocal will be 8/7.

To Divide Rational Numbers, we find the multiplicative inverse or reciprocal of divisor. For example, if we have to divide 2/5 by ¾ then ¾ is considered as dividend and 2/5 as the divisor. So in this case reciprocal of divisor will be 5/2. Now instead of dividing we will directly multiply 5/2 by 3/4. Therefore in simple words 3/4 is multiplied by the multiplicative inverse of 2/5. When we multiply fractions we multiply straight across the top, 3 times 5 is 15 and 4 times 2 is 8. Thus 15/8 is the solution.

We also understand that in Division of Rational Numbers, we have to turn the fraction upside down and then multiply the first fraction by the resulted reciprocal. If we have two rational numbers 2 /3 and 3/4 and we need to divide ¾ by 2/3 so ¾ will be called as divisor, its reciprocal will be 4/3, which is nothing but done upside down. Then next step will be multiplying the first fraction with the reciprocated fraction. That is 2/3 multiplied by 4/3. Here we shall multiply 2 by 4 and divided by 3 multiplied by 3 and get 8/9 as the solution. We also will understand how we simplify the fractions, 48/108 we simplify by 2 the whole fraction as it is divisible by 2 we get 24/54. Another simplification by 2 gives us 12/27. Here we understand that 12 and 27 will not be simplified by 2 anymore as it’s not divisible by 2, thus which could be another number, 3 is the other number which is divisible so we get 4/9. Thus we simplify the fractions.

Binary Numbers

Binary Numbers Tutorial – Binary numbers are used in computer programming and are used in all modern computer based devices. In Binary numeral system, there are two numeric values which are used to represent the numbers and those two values are 0 and 1. The numerical value represented in every case depends upon the value assigned to each symbol. For example: - 0 is denoted as 0, 1 is denoted as 1 and 2 is denoted as 10. Binary numeral system is also termed as base – 2 system because in this the base 2 is used to give a numeric value to a number.

Binary Representation of Numbers – Binary numbers are represented using two numerical values only that are 0 and 1. We know that in decimal system we use base 10 to represent numbers, the binary works on the same principle but the only difference is that it uses base as 2. For Understanding Binary Numbers, let me show you an example.
For example: - if we have a three digit number 432 then we know that number 4 holds the ones place which means 4 multiplied by 10^0, 3 holds the tens place which means 3 multiplied by 10^1 and 4 holds the hundreds place which means 4 multiplied by 10^2. In Binary system, we apply the same method but the only difference is we use base 2 instead of base 10. For this number we use 2^0, 2^1, 2^2 and so on.

To convert a Binary number to a decimal number, we follow the same pattern. For example: - if we have 100111 then it can be expanded as 1 X 2^5 + 0 X 2^4 + 0 X 2^3 + 1 X 2^2 + 1 X 2^1 + 1 X 2^0 which can be simplified as 32 + 0 + 0 + 4 + 2 + 1 = 39. Hence 39 can be written as 100111.

Multiplying Binary Numbers – Binary numbers are multiplied the same way as we multiply numbers in the decimal system. For example: - 1 X 1 = 1, 1 X 0 = 0 and 0 X 1 = 0. Therefore if we have to multiply 101 and 11 then it will be 1111

Dividing Binary Numbers – Binary numbers are divided the same way as we divide the numbers in decimal system. For example: - 11011 divided by 101 gives 101 as a quotient and 10 as a remainder.

All about Box and Whisker Plot

Definition of Box and Whisker Plot
Statistics tells us that any group of data given for statistical analysis is clustered around a middle or central value.  We can define Box and Whisker Plot as a box drawn graphically by marking the given data points on the graph in such a way that the box so obtained shows the middle part of the data values. The box and whisker plot is represented in box and whisker graph. Definition of Box and Whisker plot also states that it is the representation of data distribution along a graphical number line using quartiles and median. The Box and Whisker plot can be drawn either horizontally or vertically as needed.

How to Create a Box and Whisker Plot?
Once the given data is arranged in ascending order, we can create a Box and Whisker plot by finding the median of the data (Q2) which divides the given data into two halves. The median of the two halves, Q1 and Q3 is found out and it divides the data into quarters or quartiles. Now a box is drawn from Q1 to Q3. The whiskers are drawn at the first and last data value and then they are attached to the box using a line. As the Box and whisker plot visually represents median, the 25th quartile, the 75th quartile, the smallest data value and the highest data value of the given data values, it is also called as five-number summary. These values represent the middle or the centre of the data distribution, the spread of the data and the overall distribution range.

Example of Box and Whisker Plot 
We can see a wide range of applications of box and whisker plot in real life.  Some of the real life examples of the box and whisker plot are comparison of the marks, weight or height of the students in a class, comparison of the rate at which the different commodities are sold, comparison of the weight and the operating time of the phones, comparison of the height climbed by skyscrapers, comparison of cow’s daily milk yield etc.  The box and whisker plot can also be applied on data showing the rainy days, weight of the dogs, weekly earning etc.

Box and Whisker Plot Online
Box and whisker plot can be drawn using free online tools available in the internet. Box and Whisker Plot Online software helps us to draw the box and whisker plot easily for the given data. It is very useful for students to learn for the data required. The data can be user specified or the students can even use the inbuilt data available.

Line Equation

A line equation is nothing but an algebraic equation where every term in that equation will either be a product of a constant and a single variable or simply a constant. These equations will have one or more than one variables. They play an important role in applied mathematics. Equation of a line may simply be said as the linear equation which will not include exponents. In modeling many phenomenons, these equations are used in reducing non linear equations to linear equations from the assumption that the interested quantities will vary from background state only to a small extent.

The most common form used in order to find an equation of a line is given by, Y= mx + b.

The above given equation is used to describe the straight line on a particular coordinate plane. There are many equations representing the equation of a straight line, but this is the common form. This representation is known as Slope and Intercept form. In above said equation, ‘x’ and ‘y’ represents the coordinates of any points lying on the line, ‘m’ is known as the slope of the line(steepness) and ‘b’ is known as the Intercept which is a point formed when the line is crossing the y axis.

Standard Form Line Equation
There are many ways in order to express the equation of a line which are having their own pros and cons. There is a standard form in order to represent the equation of the line, which is given as Ax + By = C, where both A and B are not equal to zero. The equation finally is represented in this form giving out the standard form to represent. This standard form is essential when one needs to graph out the line or in order to find the line y intercept or x intercept. Also, some of the other forms for finding equation of a line are given as follows:

• Point-slope form: y – y1 = m(x-x1), where x1 and y1 are any points present on the line and ‘m’ is the slope, which is nothing but the proportionality constant. 
• Two point form: When there are two points lying on the line, then the formula to find the equation of the line is (x2–x1) (y-y1) = (y2-y1) (x-x1). Here x1, y1 and x2, y2 are two points present on the line, where x1 and x2 are not equal. Here the value y2-y1 divided by x2-x1 is nothing but the slope formula, which when simplified will give the point slope form. 
• Intercept form: This equation is a modification of standard form, where A is 1 divided by ‘a’ and B is 1 divided by ‘b’. This equation states that the sum of x divided by ‘a’ and y divided by ‘b’ will be equal to one.

Algebra Problem

Algebra problems present a challenge to many students who continue to be mystified by the subject throughout high school. The key to understanding algebra is a combination of understanding the concepts, practice, memorizing the equations, and practice.
Since algebra is a new branch of math with concepts and methodology students are completely unfamiliar with, it’s quite important to start students off on a positive note. It’s highly likely that they have already formed a negative impression based on reports from older siblings and friends. The most effective thing that tutors can do is to present algebra as a simple subject which any student can learn with ease.

Of course, presenting the course material in a way that makes it comprehendible to every student is another story altogether. Some students take to algebra like a duck to water, after only a few classes while others tend to take longer. Many students go through high school math with a hazy overview of algebra concepts and many unanswered doubts.  Algebra problems in particular, take some time and guided instruction before students can start solving them on their own, which students may or may not receive in class.

With the number of students in each class, longer and more complicated curriculum to finish, and teachers pressed for time, individual attention for each student is largely an unmet criterion. On the bright side however, there are plenty of learning aids available in the market today which are designed keeping students’ learning issues and hurdles in mind. Many of the learning aids, products and services are focused on providing help with math, particularly algebra theory and problems.

Students who are keen on getting extra help with the subject should seriously consider using one of these services to learn better, understand concepts, solve algebra problems easily, and score better grades. Most of them are available on the internet so all you really need is a computer with an internet connection and you’re all set to access math help, anytime you need it. Students who need personalized instruction can make use of written and video tutorials or live tutoring services. If you’re looking at sharpening your problem solving skills, peruse hundreds of worksheets and practice questions which have different types of questions of varying difficulty. Algebra geniuses can make use of these services to keep challenging themselves, test their knowledge and learn advanced concepts, which may not be covered in class.

Introduction to Statistics Examples

The study of data is called Statistics.  Collections of observation of an individual or a number of individuals is called data.

Collection of data:  There are two types of data namely Primary data and Secondary data.

Primary Data:  The data which is collected by the investigator with a definite object for his own purpose is called Primary Data.

Secondary Data:  The data which is collected by someone other than the investigator is called Secondary Data.

Statistics Examples: Measures of Central Tendency

Measures of Central Tendency:

A numerical value which represents approximately the entire statistical data is called Measures of Central Tendency of the given data.

The different ways of measuring central tendency of a statistical data are

Mean,  Median  and  Mode.

Statistics Examples: Mean

Mean :

The mean of a set of data is the same as finding average.

Mean = `(Sum of all observations )/(Total Number of Observations)`

`Mean of ungrouped data:`

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                ------------------------------
                   `sum_(i = 1)^n` `f_(i)`

Ex :

Find the mean of the following data:

x f
25 25
35 20
45 15
55 15
75 10

Solution:

Construct another tabe:

x f fx
25 25 625
35 20 700
45 15 675
55 15 825
75 10 750
85 3575

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                 ---------------------------------
                    `sum_(i = 1)^n` `f_(i)`

Mean =  `sum`fx /  `sum`f
=3575 / 85
=42.06

Statistics Examples: Median

Median for Raw data:

Arrange the set of datas in ascending or descending order.  The middle most value is the Median.

Rule 1:  If n is odd, the median = `(n + 1)/(2)` th term

Rule 2 :  If n is even, there are two middle terms ie `(n)/(2)`  th term and  `(n)/(2)` + 1 th term.

In this case , the arithmetic mean of these two terms is the median.

Median =     `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                        ----------------------------------------
                                                  2

Ex 1:
Find the median of 6, 7, 2, 5 and 10

Sol:
Arrange the given datas in ascending or descending order:
2, 5, 6, 7, 10
Here n= 5 ( odd number)
Median =  `(n + 1)/(2)` th term =  `(5 + 1)/(2)` th term
=   `(6)/(2)` th term
=  3 rd term
=   6

Ex 2:
Find the median of : 6, 11, 15, 7, 19, 8, 4, 10

Sol :
Arrange the given datas in ascending or descending order:
4, 6, 7, 8, 10, 11, 15, 19
Here n = 8 ( even)
Median =    `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                     ----------------------------------------
                                              2

Median  =   `(8)/(2)`  th term  +  `(8)/(2)` + 1 th term
                     ----------------------------------------
                                             2

Median  =   `(4th term + 5th term)/(2)`
 = ``(8 + 10)/(2)`
= 18 / 2
= 9

Statistics Examples : Mode

Mode:  Mode is the repeated value of the given data

Ex: Find the mode for the given data:  34, 56, 21, 56, 71, 98, 22, 56

Sol: In the given data 56 is repeated thrice.  So the mode is 56.
Mode for tabulated data:

Number  7 8 9 10 11 12 13 14 15
Frequency 3 7 11 14 13 17 12 8 6

Sol:  Since the frequency of number 12 is maximum
Mode = 12

Continuity of a function

Limit of a function can be found from the graph of that function, besides other methods. Some of the graphs are continuous. So what is continuity?  That means they can be drawn without lifting pencil from the paper. See some examples below:

The functions that all the above graphs represent are continuous. Now look at the following graphs:

 All the above graphs are not continuous or discontinuous as they cannot be drawn without lifting the pencil from the paper. With this understanding now let us try to define continuity.

Definition of continuity:
If the domain of a real function f contains an interval containing a and if lim (x->a) f(x) exists and lim (x->a) f(x) = f(a), then we say that f is continuous at x = a.

Thus, if lim (x->a+) f(x), lim (x->a-) f(x) and f(a) all exist and are equal, the f is said to be continuous at x = a.

If f is not continuous at x = a, we say that it is discontinuous at x = a.

(1) In the following picture, x is not defined at x = 2. Therefore f is discontinuous at x = 2.

(2) In the following picture, f(-1) is defined, but the left hand limit and the right hand limit at x = -1 are not equal. So the function is discontinuous at x = -1

(3) The above picture, at x = 1, both left and right hand limits exist and are equal but the limit of the function is not equal to f(1) so the function is again discontinuous at the point x = 1.
In simple words we can state continuity as follows:
A function is said to be continuous at any point x = a if the following three conditions are met:
(a) f(a) exists
(b) lim (x->a-) f(x) = lim (x->a+) f(x) = lim(x->a) f(x)
(c) quantities in (a) and (b) are equal.
If any of the above conditions is not met, we say that the function is discontinuous at the point x = a.

Mean median and mode

What is mean, median and mode?
In many statistical situations, like the distribution of weight, height, marks, profit, wages and so on, it has been noted that starting with rather low frequency, the class frequency gradually increases till it reaches its maximum somewhere near the central part of the distribution and after which the class frequency steadily falls to its minimum value towards the end. Thus, the central tendency may be defined as the tendency of a given set of observations to cluster around a single central or middle value and the single value that best represents the given set of observations is called the measure of central tendency.  Mean, median and mode are all measures of central tendency.

Define mean, median and mode:
Mean: The average value of a set of data is called the mean. If x1,x2, x3, …. Xn are n values of a given variable then the mean value, represented by μ, would be sum of these x values divided by n.
Median: The middle value of a data set is called the median. It is represented by ‘Me’.
Mode: In a data set of various values of a variable, the number that occurs maximum number of times is called the mode. In other words the value with maximum frequency is called the mode.

How do you do mean median and mode?
Mean median and mode problems usually involve calculating mean median and mode. That can be done using the following formulas:
Mean = μ = [∑xi]/n
Median = Me = middle value obtained after arranging the values in ascending order.
Mode = Mo = the value that occurs most number of times or the value with maximum frequency.

Solved example: Find the mean, median and mode of the following numbers: 5, 4, 5, 5, 6, 7, 8, 9, 6, 8
Solution:
Mean = μ = [∑xi]/n = [5+4+5+5+6+7+8+9+6+8]/10 = 63/10 = 6.3
Median: Firs arrange the data in ascending order. So we have:
4, 5, 5, 5, 6, 6, 7, 8, 8, 9. The two middle numbers are 6 and 6. The average of these numbers is (6+6)/2 = 6. Therefore,
Me = 6
Mode: The number that occurs most number of times is 5. Therefore,
Mo = 5

Identities of Pythagorean Theorem

Pythagorean Theorem is derived from the right angle triangle. This theorem is very important and useful for all calculus part and geometrical part. In this article we discuss about Pythagorean identities. It is also known as Pythagorean trigonometric identities mean expressing the Pythagorean Theorem in terms of trigonometric functions. It also includes formula of sum of angles and it shows the basic relations between sine and cosine functions and from this basis other identities are derived. For deriving the identities we also have to know the theorem and the theorem states that in a right angle triangle the square of hypotenuse is equal to sum of square of base and square of height.

There are 3 Pythagorean identities. We discuss about all three identities in theory and mathematical forms. Suppose we have given a unit circle, we mark a point on the circle the point is vertex of the right angle and it is represented by the coordinates. Suppose the coordinates are (sinθ, cosθ).the given circle is unit circle so hypotenuse is 1. And the two legs of right angle triangle in a unit circle are sinθ and cosθ. By using Pythagorean Theorem we can write as (sin^2θ+cos^2θ=1). This basic equation is known as first Pythagorean identity.

For the second Pythagorean identity we start from the first Pythagorean identity. First write the identity (sin^2θ+cos^2θ=1). Now we divide each term by (cos^2θ). While dividing we have to remember that (sinθ/ cosθ= tanθ) and (1/ cosθ= secθ). After dividing we get (tan^2θ+1=sec^2θ). This equation is known as second Pythagorean identity.

Now we derive the third Pythagorean identity. For this again we have to use the first Pythagorean identity. First we write the equation (sin^2θ+cos^2θ=1). Now we divide each term by (sin^2θ). While dividing the equation we have to remember some trigonometric formula such as (cosθ/ sinθ= cotθ) and (1/ sinθ= cosecθ). After dividing each term we get (1+ cot^2θ=cosec^2θ). This equation is the third Pythagorean identity.
Pythagorean identities list are...
1. (sin^2θ+cos^2θ=1)
2. (tan^2θ+1=sec^2θ)
3. (1+cot^2θ=cosec^2θ)
Some Pythagorean identities problems. First problem is, suppose we have given secx= (-2/3) and tanx>0 then we have to find values of other trigonometric functions.  Solution of this problem is first find (cosx=1/secx=-3/2), then by using first identity find (sinx). After this we can easily determined the remaining functions.
Second problem is suppose we have to solve (sinθcos^2θ-sinθ). A Solution of this problem is that first take out the common term and then we use the first Pythagorean identity. Finally we get the result as (-sin^3θ).

What is an ogive?

What is an ogive?
Ogive in mathematics it comes under the part of statistics. It is a way to show the data in a single line. Representation of all the data with help of single line diagram. We can show an olive by graph.  An ogive can be used to show the result at any time. What changes are occurring in curve point by point we can understand by an ogive? Either any particular value curve increases or decreases we clearly understand by ogive curves. It also helps to describe the slopes of curve.

We know the frequency distribution in statistics. An ogive is the graph of all frequencies of a particular frequency distribution. All these frequency make a continue series. In any graph we have X-axis and y-axis. X-axis denotes boundaries and Y-axis denotes frequencies.

There are two types of ogives. First type is less than ogive; it means that we plot the graph between less than cumulative frequencies and upper limits of boundaries. It is an increasing curve. Second type is more than ogive, it means that we plot graph between higher values from cumulative frequencies and lower limits of boundaries. It is a decreasing curve. Ogives are very useful in several areas such as median, quartiles, and deciles, percentiles etc. ogives also to differentiate from a given set that which value is above and which is below from a particular value. We also compare between frequency distributions.

Definition of ogive
Ogive is a curve in frequency distribution. It shows the relationship between cumulative frequencies and boundaries.
In architecture ogive means a diagonal rib of a vault. Similar to conic shape, in staring it is in pointed and then it gradually become wider. For example we can say, in astronautics conical head of any missile or any rocket.

Ogive in statistics
Statistics ogive shows a single line curve. All data are mentioned in that single line graph. Graph may be increasing or decreasing depends upon the data values. If we want that all the values individually categorized then an ogive gives ideal curve. We can understand statistics ogive by some examples.
Suppose we have give data in two columns. First shows range of class(15-20, 20-25, 25-30, 30-35, 35-40, 40-45, 45-50) and second column shows frequency(4,6,10,12,15,3,5). We have to find median quartile and decile. First we calculate cumulative frequency then separate higher and lower values. After this make a table and finally we get result.

More about Quartiles

Quartiles are the values that divide the given data arranged in ascending order into subdivisions of twenty five percent, fifty percent and seventy five percent. First quartile is the twenty fifth percentile also known as the lower quartile. Second quartile is the 50th percentile also known as the median and the third quartile is the seventy fifth percentile also known as the upper quartile. The lower quartile or the first quartile is the middle value or the median of the first half of the data values arranged in the numerical order. It is denoted as Q1. Q1 =( ¼). (n+1)th value of the data set, here n is the total number of data values. The second quartile denoted by Q2 is the median of the data set arranged in the numerical order. Q2 = median = (1/2)(n+1)th value of the data set. The third quartile or the upper quartile is denoted by Q3. Q3 is the (3/4)(n+1)th value which is the middle value or median of the upper half of the data set.

We know that range of a given data set is the value got by calculating the difference between the highest and lowest values in the data set. So, range = highest score – lowest score.  Now Quartile Range is a bit similar to range, it is the difference between the upper quartile (Q3) and lower quartile (Q1). Quartile Range is given as (Q3 – Q1) for a particular data set which is also called the Inter Quartile Range denotes as IQR. So, IQR = (Q3 – Q1). Let us learn about Quartile Deviation which is the absolute measure of dispersion. It is also called the semi Inter Quartile Range and is half of the Inter Quartile Range. It is written as Q.D in short, Q.D. = (1/2) (Q3 – Q1)

For example let us find the lower quartile, median, upper quartile, inter quartile range and quartile deviation of the data set 15, 18, 14, 20, 26, 16, 18.
First arrange the data values in the numerical order:
14, 15, 16, 18, 18, 20, 26
The number of values, n = 7
Lower Quartile = (1/4)(n+1) = (1/4)(7+1) = 8/4 = 2nd value which is 15(Q1) in the data set
Second Quartile = Median = (1/2)(n+1)= (1/2)(7+1)=8/2= 4th value which is 18(Q2)in the data set
Third Quartile= (3/4)(7+1)=3. 8/4 = 3.2 = 6th value which is 20(Q3) in the data set
Inter Quartile Range = IQR = (Q3 – Q1)= (20 – 15) = 5
Semi Inter Quartile Range = Quartile Deviation= Q.D. = (1/2)(IQR)= (1/2) (Q3 – Q1) = 5/2 = 2.5

Step by step math solution

Line Plot Graph made simple
Line Plot Definition
A data represented on a number line with marks like ‘x’ or any other mark which shows the frequency of a value in the data is defined as the line plot. For example:
The line plot below shows the marks of 20 pupils in a class.

The ‘x’ marks show the frequency of the marks obtained by the pupils
Let us now take a quick look at how to make a line plot
• First we need to gather the information. Once the information is ready we look for the data sets which occur often that is the data which is frequently shown. Something like the favorite flavor of certain people or the number of pets a group of people have.
• The data is to be sorted and then a chart is created so as to organize the list. We then name the chart for convenience.
• This is an important step which involves determining the scale. The scale might not have the labels that are not the data values as per the given information and hence we need  to decide the scale depending upon the frequency of the data items for which a numerical scale is used which begins with the least number and ends in the highest number in the data set.
• Now we draw a horizontal line which is similar to  a number line according to the chosen scale.
• Finally we start marking ‘x’ above the line corresponding to the number on the scale as per the data we have. Once the markings are done, the line plot for the given data is ready for further analysis.

Let us now make line plot graphs using a line plot example
Given are the costs of 15 books sold in a book store, represent a data as a line plot graph.
$20 $15 $9 $15 $9 $20 $9 $20 $20 $35 $25 $20 $9 $30 $15
We need to decide on the scale, the lowest value is 9 and the highest value is 35. So, the scale should start from 5 and end in 35

Solve by the addition method

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign (+). For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5. (Source: Wikipedia)

Example Problems for Solve by the Addition Method:-

Problem 1:-

Solve 453 + 213 by the Addition method.

Solution:-

In the following step by step process of addition method

Step 1:-

               453
            + 213
           -----------
           -----------
The above equation 453 is adding to 213 in between (+) plus operation. In basic addition process start with right side value to left side value

Step 2:-

               453
            + 213
           -----------
                   6
           -----------
Adding the right side values 3 and 3. 3 is equal with 3 in 3+3 =6. Then move to next value.

Step 3:-

               453
            + 213
           -----------
                 66
           -----------
Adding the next two values 5 and 1. 5 adding with value 1 in 5+1 =6. Then move to next value.

Step 4:-

               453
            + 213
           -----------
               666
           -----------

Adding the last two values 4 and 2. 4 adding with value 2 in 4+2=6. We get the final answer is 666.

Problem 2:-

Solve 654 + 323 by the Addition method.

Solution:-

In the following step by step process of addition method

Step 1:-

               654
            + 323
           -----------
           -----------

The above equation 654 is adding to 323 in between (+) plus operation. In basic addition process start with right side value to left side value

Step 2:-

               654
            + 323
           -----------
                   7
           -----------

Adding the right side values 4 and 3. 4 adding with value 3 in 4+3 =7. Then move to next value.

Step 3:-

               654
            + 323
           -----------
                  77
           -----------

Adding the next two values 5 and 2. 5 adding with value 2 in 5+2 =7. Then move to next value.

Step 4:-

               654
            + 323
           -----------
               977
           -----------

Adding the last two values 6 and 3. 6 adding with value 3 in 6+3=9. We get the final answer is 977.


Practice Problems for Solve by the Addition Method:-

Problem 1:-

solve 421 + 167 by the addition method.

Answer:- 588

Problem 2:-

solve 217 +171 by the addition method.

Answer:- 388

Problem 3:-

solve 383 + 71 by the addition method.

Answer:- 454

Problem 4:-

solve 152 + 38 by the addition method

Answer:- 190

Problem 5:-

solve 43 + 24 by the addition method.

Answer:- 67

Frequency Distribution in Statistics

In mathematics frequency distribution is used in statistics. Mean of a frequency distribution is that the arrangement in which sets of value occurs and in the values one or more variable takes place. Frequency distribution is in the form of either graphical or tabular. Each value in the table contains frequency or count of values, how many times they occur. The values of frequency in group or interval forms.  After summarizing the entire values frequency distribution table is formed. Mean of frequency distribution is also that it shows the total number of observations within a given interval. The interval is either exclusive or exhaustive. The size of intervals generally depends on the data which we have to analyze and calculate. One thing we have to remind that the intervals must not be overlapped to each other.

Now we discuss that how to construct frequency distribution tables. We use some steps to make a frequency distribution table. In step one; we determine the range of given data. Range of given data means the difference between the higher value and the lower value. In step two, we decide that which data can be grouped means formulation of approximate number of groups. There are no particular rules for step two. It can be 5 groups to 15 groups. But there is one formula for this (K=1+3.322logN), where K is the no of groups, logN is the total number of observations.

In step third, we decide the size of intervals.  The size of interval is denoted by (h). To determine the size we can use a formula (h= range/number of groups). If result is in fraction then we choose next higher value. In step fourth, we decide start point means starting from the lowest value and in the ascending order. In step fifth, we determine the remaining groups. It is determined by adding the interval size corresponding to all values. In step sixth, we distribute all the data into their groups. For this we use tally marks method because it is suitable for tabulating the observations into their respective groups. By using these six steps we can construct a frequency distribution table.

Now we come to standard deviation for frequency distribution. It is a measure of variation or measure of dispersion amongst the data. In place of taking absolute deviation we may square each deviation and obtained the variance. The square root value of variance is known as standard deviation for given values of frequency.

Derivatives of inverse trigonometric functions

Like many functions, the trigonometric functions also have inverse.  Just like how we can find the derivative of trigonometric functions, we can also find the derivatives of inverse trigonometric functions. In this article we shall take a quick look at inverse trig functions derivatives. To be able to find the standard formulae for derivatives of inverse trig functions, we would need the formula for derivative of any general inverse function.

That would be like this:
f’(x) = 1/g’(f(x)), where f and g are the inverse functions of each other.
Let us first try finding the inverse trig function derivatives of the sine function. The sine function inverse is written as arc sin (x). Therefore the function would look like this: y = arcsin(x),
Therefore, x= sin (y) for –pi/2 ≤ y ≤ pi/2. We fix this domain for the sine function to ensure that our inverse exists. If we don’t restrict the domain, then y could have multiple values for same value of x. For example, sin (pi/4) = 1/sqrt(2) and sin (3pi/4) is also 1/sqrt(2). With that in mind, we can write the relationship between sin and arc sin as follows:
Sin(arc sin (x)) = x and arc sin(sin(x))  = x
Thus here our f(x) = arc sin(x) and g(x) = sin (x), then using the derivative of inverse formula that we stated above we have:
f’(x) = 1/f’(g(x)) = 1/cos (arc sin (x))
This formula may to be applicable in practice. So let us make a few changes in there. We had earlier x = sin y so y = arc sin (x). Using that, the denominator of our derivative would become
Cos(arc sin (x)) = cos y
Next we use our primary trigonometric identity which was:
Cos^2 (x) + sin^2 (x) = 1
Thus, Cos^2 (y) + sin^2 (y) = 1 from this we have
Cos (y) =√( 1- sin^2 (y)) = √(1 – (sin y)^2). But we have already established that sin y = x. So replacing that we get,
Cos (y) = √(1 – x^2). So now plugging all that back to our equation of derivative of inverse trig functions for sine f’(x), we have:
f’(x) = 1/cos(arc sin(x)) = 1/cos y = 1/√(1-x^2)
Therefore we see that derivative of arc sin (x) = 1/√(1-x^2).
The derivatives of other trigonometric inverse functions of arc cos, arc tan etc can be derived in a similar way. They are:
d/dx arc cos (x) = -1/√(1-x^2) and d/dx arc tan (x) = 1/(1+x^2)

All about the hypotenuse of a right triangle

Like every triangle, a right angled triangle would also have three sides. However, in a right triangle one of the angles is a right angle. That means one of the angle measures 90 degrees (or 𝛑/2 radians). Since the sum of angles in any triangle has to be 180 degrees, in a right triangle as one angle is already 90 degrees, the sum of the other two angles have to be 90 degrees. That means that the other two angles are compliments of each other. It also means that the other two angles have to be acute angles. A typical right triangle would look as follows:

The longest side is called the hypotenuse. The side that is adjacent to the know angle is called the adjacent side and the side opposite to the known angle is called the opposite side. By default the hypotenuse will always be the side opposite the right angle.

The adjacent and opposite sides together are also called the legs of the right triangle. The length of the hypotenuse of a right angled triangle can be found using different methods, depending on what part of the triangle is given to us.

To find the hypotenuse of a right triangle given the length of the legs:
If the hypotenuse = c and the legs are ‘a’ and b. If a’ and b’ are known, then we can calculate the length of the hypotenuse using the Pythagorean rule as follows:
C^2 = a^2 + b^2
Finding hypotenuse of a right triangle given one of the angles and the adjacent side:
In the picture below,

Suppose the angle marked in red is x and the adjacent side = a, then the length of the hypotenuse H can be given by the formula:
H = a/Cos (x)
Formula for the hypotenuse of a right triangle given one of the angles and its opposite side:
Again from the picture above, if we are given the opposite = b instead of the adjacent side. Then the formula for the hypotenuse can be written as follows:
H = b/sin (x), where x is again the angle marked in red.
Thus as we saw above there are more than one ways to find the length of the hypotenuse of a right triangle.

Trapezoidal Numerical Integration

Introduction to numerical integration:  As we have seen, the ideal way to evaluate a definite integral a to b f(x) dx is to find a formula F(x) for one of the antiderivatives of f(x) and calculate the number F(b) – F(a). But some anti derivatives are hard to find, and still others, like the antiderivatives of (sin x)/x and sqrt(1 + x^4), have no elementary formulas. We do not mean merely that no one has yet succeeded in finding elementary formulas for the anti-derivatives of (sin x)/x and sqrt(1 + x^4). We mean it has been proved that no such formulas exist. Whatever, the reason, when we cannot evaluate a definite integral with an anti-derivative, we turn to numerical methods such as the trapezoidal rule and Simpson’s rule.
Numerical Integration Methods are Trapezoidal rule and Simpson’s rule. Let us now describe Trapezoidal numerical integration.

Trapezoidal Numerical Integration: When we cannot find a workable anti-derivative for a function f that we have to integrate, we partition the interval of integration, replace f by a closely fitting polynomial on each sub interval  integrate the polynomials and add the result to approximate the integral of f. The higher the degrees of the polynomials for a given partition, the better the results are.  For a given degree, the finer the partition, the better the results, until we reach the limits imposed by round-off and truncation errors.

The polynomials do not need to be of high degree to be effective. Even line segments (graphs of polynomials of degree 1) give good approximations if we use enough of them  .To see why, suppose we partition the domain [a, b] of f into n subintervals of length delta x = h = (b – a)/n and join the corresponding points on the curve with line segments.

The vertical lines from the ends of the segments to the partition points create a collection of trapezoids that approximate the region between the curve and the x-axis. We add the areas of the trapezoids, counting area above the x-axis as positive and area below the axis as negative.
T = ½ (y0 + y1) h + ½ (y1 + y2)h + ….. + ½ (yn-2 + yn-1) h + ½ (yn-1 + yn)h= h (1/2y0 + y1 + y2 + ….. + yn-1 + ½ yn) = h/2 (y0 + 2y1 + 2y2 + …. + 2yn-1 + yn).Where, Y0 = f (a), y1 = f(x1), Yn-1 = f(x n-1), yn = f(b).

The trapezoidal rule says: use T to estimate the integral of f from a to b. Numerical Integration in R As we have seen , the definition of the Riemann integral is not very efficient way to prove that a function is Riemann integral . However once it is known that a function f is Riemann integral on some interval [a, b] a modification of the definition makes it possible to evaluate the integral of simple limit.

Volume of sphere

Introduction to Sphere : A tennis ball and a fully blown football  are some familiar objects which bring to our mind the concept of a sphere .A sphere is a three dimensional geometrical object  which can be defined as follow The set of all points in space which are equidistant from a fixed point , is called a sphere.

The fixed point is called the center of the sphere and the constant distance is called its radius. A line segment through the center of a sphere, and with the end points on the sphere . All diameters of a sphere  are of constant length , being equal to twice the radius of the sphere .Thus , if d is the length of a diameter of a sphere   of radius r  then d= 2r . The length of diameter is also called the diameter of sphere .The solid sphere is the region in sphere, bounded by sphere .

Also every point whose distance from the center is less than or equal to the radius is a point of the solid sphere .A sphere can also be considered as a solid obtained on rotating a circle about its diameter.

Volume of a sphere: The volume of a sphere (v) of radius r is given by v = 4/3 ∏r3 cubic units .Let us take an example how to volume of sphere.

Find the sphere volume  of radius 7 cm We know that formula for volume of a sphere of radius r is given by v = 4/3 ∏r3  cubic units here r = 7 cm  therefore v = 4/3 x 22/7 x 7 x 7 x 7 cm3 another example of volume of a sphere.

Calculate volume of a sphere whose surface area is 154 square cm. Let the radius of the sphere be r cm .then ,
Surface area = 154 cm2 => 4∏r2  = 154 => 4 x 22/7 x r2 = 154 => r2 = 154 x 7 / 4 x 22 = 49/4 => r = 7/2 cm , so let v be the volume of sphere .
We will use volume of a sphere equation

That is , v= 4/3∏r3   = 4/3 x 22/7 x 7/2 x 7/2 x 7/2 cm3 = 179.66cm3.

Let us take more example volume of a sphere. A sphere of diameter 6 cm is draped in a right circular cylinder vessel partly filled with water .The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

We have radius of sphere = 3cm, volume of a sphere formula = 4/3 ∏ r3  cm3 = 4/3 ∏ (3)³ cm³ = 36cm³, radius of cylindrical vessel = 6cm.
Suppose of water level rises by h cm and radius 6 cm = (∏x 62 x h) cm 3 = 36∏h cm3 , clearly volume of the water displaced by the sphere is equal to the volume of  the  sphere =36∏h = > h=1 cm , hence water level rises by 1 cm

Difference (Newton) Quotient Made Simple

To set up a difference quotients for a given function requires an understanding of a function notation. Given the function f(x)=4x^2-3x-7. This notation is read as “f of x equals..” This implies that the value of the function, that is the y-value depends upon the replacement for x. We get the numerical value for the function by substituting a number for ‘x’. If a non-numerical quantity is substituted for ‘x’, we get an expression rather than a numerical value. One important point to be remembered is careful use of parenthesis which is essential. For instance, f(x)=4x^2-3x-7; f(3)=4(3)^2 – 3(3)-7=36-9-7=20

Derivative Quotients  at x for a function f is given by, [f(x+h)-f(x)]/h. Sometimes it is written using delta(x) for the change in x and delta(y) for change in y; delta(x)=h and delta(y)=f(x+delta(x))- f(x). The Difference Quotient is so called as it involves the operations subtraction and division. The common forms of Difference Quotient are as follows:
1. [f(x+h) – f(x)]/h
2. [f(a+h) – f(a)]/h
3. [f(x+delta(x)) – f(x)]/delta(x)

Simplifying Difference Quotient
The difference quotient is simplified to get a h or delta(x) in the denominator which can be canceled to get the final value. Let us consider a difference quotient example to understand the step involved in simplifying difference quotient ; f(x) = 4x^2-3x-7. First we need to find the function f(x+h),which we can get by substituting (x+h) in all x in the given function.  f(x+h)= 4(x+h)^2 – 3(x+h) – 7 = 4(x^2+2xh+h^2) – 3x – 3h – 7= 4x^2+8xh + 4h^2-3x – 3h – 7. Subsituting f(x+h) in the difference quotient, we get,
Difference Quotient = [f(x+h) – f(x)]/h
       = {[4x^2+8xh+4h^2-3x-3h-7] – [4x^2-3x-7]}/h
        = 4x^2+8xh+4h^2-3x-3h-7 -4x^2+3x+7]/h  (opening the parenthesis)
On simplification, we get
        = [8xh+4h^2-3h]/h  
        = h[8x +4h-3]/h            (taking h common)
        = (8x +4h-3)     (canceling h)

Difference Quotient Example
Given function, f(x) = 2x^2-1
First we need to calculate f(x+h) which is got by substituting (x+h) in all x of the function
f(x+h) = 2(x+h)^2 -1 = 2(x^2+2xh+h^2) -1 = 2x^2+ 4xh+2h^2 -1
Next we substitute f(x+h) and f(x) in difference quotient
 [f(x+h) – f(x)]/h ={[2x^2 +4xh +2h^2-1]- [2x^2 -1]}/h
= [2x^2+4xh +2h^2-2x^2+1]/h
= [4xh+2h^2]/h  (combining like terms)
= h[4x +2h]/h      (taking h common)
= [4x +2h] (canceling h)

Newton Quotient
The difference quotient is attributed to Sir Isaac Newton and hence given the name Newton Quotient. The slope of a line through the points [(x+h),f(x+h)] and [x, f(x)] is given by [f(x+h) – f(x)]/h. This expression is the Newton Quotient or Newton’s difference quotient.
Newton Quotient  =[f(x+h) – f(x)]/h

Anti Derivatives and their Rules

What are Anti derivatives?
Anti derivative is nothing but indefinite integral or primitive integral in calculus. If there is a function h, then the anti-derivative of this function will be a differential function, say H. The derivative of H will be equal to h.
H’ = h

The anti derivates are solved by a process called indefinite integration or anti differentiation, which is the opposite process of differentiation that finds the derivative.

Rules of Anti derivatives
The rules of anti derivatives are generally the reverse of the rules of the derivatives. We can say that the anti derivative of a function is equal to the sum of the derivative of the function and a constant, so it is just one step and simple.

Constant Rule
Consider a constant “a”, which has to be multiplied with the function g(y). The value obtained by multiplying the constant with the anti derivative of the function g(y) for all values of y will be equal to the anti derivative of the function g(y), which is calculated after multiplying the constant with the function g(y) for all values of y.

Sum Rule
If there are two functions f(y) and g(y), the anti derivative of the sum of the two functions will be equal to the sum of the anti derivative of the function f(y) and the anti derivative of the function g(y).

Difference Rule
If there are two functions f(y) and g(y), the anti derivative of the difference of the two functions will be equal to the difference of the anti derivative of the function g(y) from the anti derivative of the function f(y).

Reverse Rules
Listed below are the anti derivative rules that form the reverse of derivative rules:
• The anti derivative of cos y is given by sin y + a, where “a” is a constant. Thus we can say, the anti derivative of the derivative of y will result in y + a.
• The anti derivative of sin y is the negative of the sum of cos y and a.
• The anti derivative of the square of sec y is the sum of tan y and a.
• The anti derivative of the square of cosec y is the negative of the sum of cot y and a.
• The anti derivative of the product of sec y and tan y is given by the sum of sec y and a.
• The anti derivative of the product of cosec y and cot y is given by the negative of the sum of cosec y and a.

Power Rule
The power rule of the anti derivative is the reverse of the power rule of the derivative. The power rule of the derivative usually comprises of two steps as the power is brought in the front to be multiplied with the derivative and then the power is reduced by 1 and then it is simplified.

In case of power rule for anti derivative, the power rule comprises of the following two steps.
Step 1: The value of the power in the function is increased by 1. Say for example, g(y) = 6y^2. Then by increasing the power value by 1, the function will become g(y) = 6y^3.

Step 2: Divide the function g(y) obtained from step 1 by the new power value. In this example, it will be: (6 divided by 3) (y^3), which will result in g(y) = 2y^3.

Thus anti derivative of this example is given by 2y^3 + a, where “a” is a constant, as any anti derivative is the sum of the derivative and a constant “a”.

Calculus: Rules of Integration

Calculus Integration Rules
Following are the Rules of Integration where a, b, c, n are some constants and u=f(x);v=g(x) and w=h(x)
General Integration Rules
1.Integral [a dx]= ax +c
2. Integral [a f(x) dx]= a Integral [f(x) dx]
3.Integral [x^n dx]= x^(n+1)/(n+1) +c
4.Integral [f(x)+g(x)+h(x)]dx = Integral[f(x)dx]+ Integral[g(x)dx]+ Integral[h(x)dx]
5. Integral[f(x)-g(x)-h(x)]dx = Integral[f(x)dx] – Integral[g(x)dx] – Integral[h(x)dx]
6. Integration by parts: Integral [u dv] = uv – Integral [v du]
7.Integral[F(u)dx] = Integral [F(u)/u’] du
8. Integral[1/x dx] =ln|x| +c
9. Integral [1/(x^2+a^2)]dx =1/a tan^-1[x/a] +c
10.Integral [1/(x^2-a^2)]dx = [1/2a ]ln |x-a/x+a| +c

Integral Rules of Exponential Functions
1.Integral [e^x dx]= e^x +c
2. Integral [a^x dx]= a^x/ln a +c
3.Integral [ln x dx]= x(ln x -1) +c
4.Integral[log base a of x]dx= (x/ln a)(ln x -1) +c
5. Integral[x e^(ax)]dx= [e^(ax)/a^2](ax-1) +c
6. Integral[e^(ax)/x] dx = ln|x| + summation(i=1 to infinity) [(ax)^i/i.i!] +c
7.Integral[x^2 e^(ax)]dx= e^(ax)[(x^2/a – 2x/a^2 + 2/a^3)] +c
8. Integral[x^n e^(ax)]dx = (1/a)x^n e^(ax) – (n/a)Integral [x^(n-1) e^(ax)]dx
9.Integral[e^(ax)/x^n] dx = [1/(n-1)][- e^(ax)/x^(n-1) + a Integral e^(ax)/x^(n-1)]dx
10.Integral[x^n ln x] dx= [x^(n+1)]/(n+1)^2 [(n+1)lnx – 1] + c

Integration Rules of Trigonometric Functions
1.Integral [sin x dx]=  - cos x +c
2. Integral [cos x dx]= sin x +c
3. Integral [tan x dx]= ln |sec x|+ c
4. Integral [cot x dx]= ln|sin x| + c
5. Integral [sec^2(x) dx]= tan x +c
6. Integral [csc^2(x) dx]= - cot x +c
7. Integral [tan^2(x) dx]= tan x – x +c
8. Integral [cot^2(x) dx]= cot x – x +c
9.Integral [sec x tan x dx]= sec x +c
10. Integral [csc x cot x dx] = - csc x +c
11. Integral [sec x dx]= ln |sec x +tan x| +c
12.Integral [cos^2(x) dx] = x/2 + ¼(sin 2x) +c
13. Integral[sin ^n(x) dx]= (-1/n)sin^(n-1) x cos x + (n-1)/n .Integral [sin^(n-2) x dx]
14. Integral [cos^n(x)dx]= (1/n)cos^(n-1) x sin x + (n-1)/n. Integral[cos^(n-2) dx]

Integration Rules of Hyperbolic Functions
1.Integral[sinh x dx] = cosh x +c
2.Integral[cosh x dx]= sinh x +c
3. Integral[tanh x dx]= ln cosh x +c
4. Integral[coth x dx]= ln |sinh x| + c
5. Integral[sech x dx] = sin^-1[tanh x] +c
6. Integral[csch x dx] = ln tanh (x/2) +c
7. Integral[sinh^2(x)dx] = (sinh 2x)/4 – (x/2) + c
8. Integral[cosh^2(x)dx]= (sinh 2x)/4 + (x/2) +c
9. Integral[sech^2 (x) dx]= tanh x +c
10. Integral[csh^2(x)dx]= -coth x +c
11. Integral[tanh^2(x) dx]= x – tanh x +c
12. Integral[coth^2(x)dx]= x – coth x +c
13.Integral[sechx tanh x] dx =  - sech x +c
14. Integral[csch x coth x]dx = -csch x +c

Derivatives of Exponential functions of e

Derivative is the rate of change at a point which gives the slope of the curve at that point. When the given equation is y=f(x), the derivative is written as dy/dx or d[f(x)]/dx.  To find the derivatives of exponential functions, let us take a quick look at them. Exponential functions are the functions written in the form y = b^x, where b is a positive number that does not equal 1 and x is any real number.  They have a constant base and the exponent is a variable. The most important exponential function is e as the base, which is an irrational number. The function is written as, e(x) and is called the natural exponential function.  Now that we learnt about the natural exponential function e^x, let us learn more about the Derivatives of E.

The natural exponential function is remarkable and so are its derivatives. Let us first find the derivative of E, where E is f(x)=e^x :
As per the definition of derivatives, we get,
d[f(x)]/dx = lim(delta(x)?0) e^[x+delta(x)-e^x]/delta(x)
=lim(delta(x)?0) [e^xe^delta(x)- e^x]/delta(x)
=lim(delta(x)?0) e^x[e^delta(x)-1]/delta(x)
=lim(delta(x)?0) e^x[1+delta(x)-1]/delta(x)
= lim(delta(x)?0)e^xdelta(x)/delta(x)
=e^x
If f(x) = e^x then f’(x) = e^x. This means that slope is the same as the given function value or value of y for all the points on the graph. The other Derivatives of E or derivatives of e^x are as given below:
If u is a function of x, the derivative of an expression in the form e^u can be obtained and is given by d(e^u)/dx = e^u. du/dx

If an exponential function with base b is given, then the derivative of that expression is given by
d(b^u)/dx = b^u.ln b.du/dx

Let us take an example, derivative of E 2 which is derivative of e^x where x=2.  At this point x=2, the value of y=e^x  is approximately 7.39. We know that the derivative of e^x is e^x. So, the slope of the tangent, that is the derivative of e^2 at x=2 is also 7.39 approximately.

Derivative of E 2x will be the derivative of e^2x. To find the derivative of this exponential function, let us take y= E 2 or y= e^2x where u=2x. Using the chain rule, we get  dy/du = de^u/du , where du/dx equals 2.  So, d/dx of [e^2x] is (e^u). du/dx = 2. e^u , substituting u=2x, the derivative of e^2x is 2e^2x

Derivative of E 3x will be the derivative of e^3x. Using d(e^u)/dx = e^u. du/dx where u=3x, we get, e^3x. du/dx which will be 3e^3x as du/dx = 3

Law of Cosines Explained

Trigonometry Law of cosines:
Trigonometry is a field of study relating the angles and sides of a triangle. However, the fundamental ratios are derived easily from a right-angled triangle and are as identified Pythagorean ratios, yet the definite correspondence between the sides and angles can be established using the law of cosines. It is the relation between the sides and cosine of angle.


Convention:
All the angles are depicts upper case letters and sides are depicted by lower case letters. The side opposite to a vertex is represented by the corresponding lower case letter. In a triangle ABC, the side AB= c, BC = a, and AC = b.


Trigonometry law of cosines:
a^2= b^2+c^2 - 2bc cos (A)
b^2 = a^2 +c^2 – 2ac cos (B)
c^2 = a^2 + b^2 – 2ab cos (C)


Law of cosines Example Problems: 
For example, let us consider a triangle ABC, in which a= 3, b= 4 and c= 5.
To evaluate the angle C,
Substitute the values of a, b and c in
c^2 = a^2 + b^2 – 2ab cos (C)
5^2 = 3^2 + 4^2 – 2(3) (4) cos (C)
25= 9+16 – 24 cos(C)
25 = 25 – 24 cos(C)
Solving for cos (C), we get cos(C) =0. Hence, C = 90?.


Prove Law of Cosines:
We always rely on the principles of geometrical principles to prove the laws in trigonometry. To prove the law of cosines we use Pythagorean principle.

 In the above figure, CP is perpendicular to AC extended to P. Hence, BP = a sin(C) and CP = a cos(C).
Applying Pythagoras theorem in the right-angled triangle APB we get,
(AP)^2 + (BP)^2 = (AB)^2
(b – a sin(c))^2 + (a sin(C))^2 = c^2
Expanding        b^2 -2ab cos(C) + b^2 cos2(C) + b^2sin2(C) = c^2
b^2 +a^2 – 2ab cos(C) = c^2


Law of Cosines Problems
Law of cosines, in Physical sciences and technology, has very wide applications. The law of cosine gives us the magnitude of the difference of two vectors acting at an angle. For example, to evaluate the magnitude of the difference of two vectors of magnitude 100 units each acting at an angle of 120?, we get
c^2= 1002+1002 – 2(100) (100) cos (120)
c^2= 1002+1002 -1002
Solving for c, we get c= 100 units.

Derive Law of Cosines
Though there are so many methods by which one can prove the law of cosines, we stick to the use of Pythagorean principles.

In the above figure we have, when CP is perpendicular to AB,
c= a cos (B) + b cos (A)
c^2= ac cos (B) +bc cos (A)
Similarly
b^2=bc cos (A) + ba cos(C)
a^2= ab cos(C) + ac cos (B)
Adding the above two equations we have
b^2 + a^2 = ac cos (B) + bc cos (A) + 2 ab cos(C)
Comparing the above equation with (1), we get
b^2 + a^2 = c^2 + 2ab cos(C)
b^2 + a^2 -  2ab cos(C) = c^2

Derivatives and Graphs of Exponential and Logarithmic Functions

Exponential and Logarithmic Functions
The Logarithmic function with base b is a function, y = logb x. Here b is greater than zero and the function x is defined for all x greater than zero. An Exponential function with base b is a function, y=bx, defined for every real number.
Inverse Function: To find an inverse function (f-1), we need to interchange x and y and then solve for y. Example: f -1( x)  of 2x +1 will be, y=2x+1 (interchange x and y and solve for y)
x =2y+1
y = (x-1)/2 = f-1(x)
The Exponential functions and Logarithmic functions are inverse functions, that is, for any base b, the functions f(x) = logb x, g(x) = bx  are inverses.
Example: let f(x) =ln x and g(x) = ex then f and g satisfy the inverse functions. f(g(x) = ln ex=  x and g(f(x) = eln x= x, f(g(x) = g(f(x) and hence the functions f(x) and g(x) are inverses

Derivatives of Logarithmic and Exponential Functions
The most common exponential and logarithmic functions are natural exponent function ex, and the natural logarithm function, ln(x). The derivatives of Exponential and Logarithmic Functions are:
Exponential Functions derivative: d/dx (ex) = ex   d/dx(ax) =ax ln a
Logarithmic Functions derivative: d/dx(ln x) =1/x    d/dx(loga x)= 1/x ln a
Example: Derivative of f(x) = e3x+2  is given by d/dx (e3x+2) = e3x+2. 3 = 3e3x+2
   Derivative of f(x) = ln (3x+2) is given by d/dx[ln (3x+2)] = [1/(3x+2)]. 2 = 2/(3x+2)

Graphing Exponential and Logarithmic Functions
Exponential functions play a large role in real life. From science to money, graphing these exponential functions provide a visual representations to many applications in real life. Graphs of Exponential and Logarithmic functions using examples are as follows,
Let us now graph an exponential function, f(x) = 2x. First we evaluate f(x) using the integers -3, -2, -1, 0, 1,2,3 and tabulate the values.
x   -3          -2        -1        0 1 2        3
f(x)    1/8 1/4 1/2 1 2 4 8


(x,y)   (-3,0.125)      (-2, 0.25)   (-1,0.5)   (0,1)       (1,2)         (2,4)       (3,8)
Once we get the ordered pairs (x,y) plot the points which gives us the graph of the exponential function f(x)=2x.

Graphing Logarithmic Functions: There are several ways to graph logarithmic functions. The easiest way to graph them is to re-write them in exponential form.
Example:  Graph the logarithmic function, f(x) = log5 x. Re-writing f(x) = y =log5 x in exponential form we get x = 5y, choose values for y and then compute corresponding values for x.  Tabulating the values,
y   -1         0 1        2
x = 5^y 1/5         1             5       25
(x,y) (0.2,-1)     (1,0)           (5, 1)      (25, 2)
Plot the (x,y) values. The graph we get is the graph of the logarithmic function, f(x)= log5 x.

Natural Numbers

Number theory : The set of it integers and its properties are at  the  root of all mathematical disciplines. In fact , it is impossible to do mathematics without making use of integers in some form or another. Number theory which involves the study of integers  itself , is a rich and fascinating branch o mathematics .many volume have been written on this subject and some of the best mathematician in history have devoted much of their time to the study of number theory.

We can subdivide the number theory as follow :
(i) Combinatorial Number Theory
(ii) Algebraic Number Theory
(iii) Analytic Number Theory
(iv) Transcendental number theory
(v) Geometric number theory
(vi) Computational number theory

What is a Natural Number?
Introduction To natural number : Since our childhood we are using numbers 1 , 2, 3, 4,………………………..t count and calculate. For example 3 banana , 5 apples , 7 mangoes , 2 books etc. here banana , apples , mangoes are objects whereas three , five , seven , two etc indicates about the quantities of theses objects .

To define natural numbers we might put these in this way, as when we count objects in groups of objects , we start counting from one and then go on to two, three , four etc . there is a  natural way of counting  objects .Hence 1 , 2 , 3 , 4 , …………………are called natural numbers .In fact number from 1 to 1crore are all natural numbers.Let us see what  is  a natural numbers ?We start counting from1  , so 1 is the first natural number , if we add1 to the first natural number , then we get 2  the second natural number/.by adding 1 to any natural number , then we get 2 , the second natural number. In fact adding 1 to  any natural number , we get the next natural number let us take few  examples of natural numbers  , 1000 is the natural number next to 999 , 10001 is the natural number, next to 10000 and so on .Thus if we think of any natural number , there is always a natural number next to it .Consequently there is no last  or greatest natural number consequently there is no last or greatest natural number. now in simple word , let  us define natural numbers: natural numbers are number from 1 onwards, ie , 1 , 2  3  4 , 5 , 6 ……………………..and are used for counting

Properties of natural numbers :Following are some properties of natural number (i) The first and smallest natural number is 1.(ii) Every natural number (except 1) can be obtained by adding to 1 to the previous natural number(iii)For the natural number 1, there is no previous natural number (iv)There is no last or greatest natural number (v)We cannot complete the counting of all natural number .

Ratios and Scale drawings

A ratio is a comparison of one thing with another.  It shows the relative size of two or more quantities. The order of the ratio is very important. a:b is not equal to b:a. Now let us find how to do ratios? A ratio can be expresses in three different ways:-

1. Fractional notation : 2/5
2. Odds notation: 2:5
3. Using the word “to” : 2 to 5

How to find ratios? Ratios can be solved by reducing them.  We can multiply or divide both the terms of ratio by the same number; it makes no change to the ratio. So this clearly answers how to solve ratios?

For example: - In a class of 20 students, 5 are girls and 15 are boys. What is the ratio of girls to boys?
Solution: - There are 5 girls and 15 boys.
The ratio of girls to boys will be 5:15
We can reduce this fraction by dividing both the terms by number 5
The ratio will be 1:3
Let us learn about ratio word problems now. Ratio word problems are problems that require use of ratios to relate the different quantities.
Following points should be kept in mind while solving ratio word problems:-

• Convert all the quantities to same units if required
• Write the quantities in the ratio as a fraction
• Remember to keep same quantities in numerator and denominator

Example:- There are blue and red ball in a bag, the ratio of blue balls to red balls is 5:6. If the bag contains 60 red balls, how many blue balls are there?
Solution: -
Let x be blue balls
Red/Green = 5/6 = x/60
X = 50 balls.
Let us learn about scale drawings. If any ratio is expressed in 1:n  form, then n is called the scale factor. It is not possible to draw on paper the exact size of real- life objects. Therefore we make use of scale drawing to draw such figures like a car, a building or any map.

Example: -A figure has a scale of 1:10 that means anything drawn with the size of “1” would have a size of “10” in real world.

How to Convert Fractions to Decimals

Let's learn about How to Convert Fractions to Decimals.

In order to Convert Fractions to Decimals, the proper or improper fraction has to undergo division. The numerator has to be divided by the denominator and the resultant will be a decimal value. Below is an example:
Example:
5/2
= 2.5

For more help, get it from an expert free math tutor. Next time i will help you with the concept of solving equations with decimals.

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Bias Statistics

Let's learn about bias in the study of statistics.

There are four different types of bias in statistics and these are listed below:

• Spectrum bias
• Omitted Variable bias
• Systematic bias
• Cognitive bias  

 Next time i will help you with some other concept of statistics such as inference statistics. For more help you can also connect with an online tutor and get your help. Not just in statistics but in other topics such as geometry tutoring as well.

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Critical Value

Let's learn about Critical Value in statistics in today's learning.

Critical value is an important concept of statistics which is used in calculus as well. The point where a function has its maxima and minima is called the critical value.

This cutoff value determines the boundary between the samples, on the basis of which , it is determined, whether to reject the null hypothesis or whether to not reject the null hypothesis. If the calculated value from the statistical information is greater than the critical value, then the null hypothesis is rejected in favor of the choice hypothesis and vice versa.

Next time i will share with you about statistics data graphs. For more help connect with a calculus tutor and get your help.

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