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