The Process of Division of Decimal Numbers

The number systems are very interesting to study. Various arithmetic operations can be carried out on them. Multiplication is one of them. The reverse of this operation is division. Division is bit more different from addition and subtraction. The division of natural and whole numbers are almost similar. Any number divided by ‘0’ gives infinity. Infinity is a number which is very large and not defined. Zero on dividing by zero also gives infinity. This question was first raised by the great Indian mathematician Ramanujam.  Zero divided by any number gives zero. Any number divided by the number one gives the same number. This is the same with multiplication. Any number multiplied with one gives the same number. But any number multiplied by zero gives zero.

The process of dividing decimals with whole numbers is similar to the division of two whole numbers. The simple difference is in the placement of the decimal point. This is very important as the placing of decimal point can change the whole number. So, one has to be very careful about this. There can be dividing decimals problems in mathematics and have to be solved carefully. The decimals division is very much similar to the division of natural numbers or the whole numbers. The placement of the decimal point in the final answer is very crucial. Many dividing decimals examples can be used to explain the concept. The following steps to dividing decimals have to be followed to arrive at the final answer.

These are very easy steps.
If a decimal number is divided by a whole number, the division is carried out as usual. The digits present after the decimal point are noted and the decimal point is placed at the same point in the final answer. If the division is carried out between two decimal numbers then the decimal numbers are first converted into whole numbers by multiplying by 10, 100 or thousand and so on depending on how many digits are present after decimal point in the denominator. If the numbers to be divided are 8.68 and 1.2, then the numbers are multiplied by 10. The numbers now become 86.8 and 12. Now the division process is carried out. First ‘868’ is divided by 12. In the final answer decimal point is placed one point from the right. This is how the final answer is got.

Set Theory Rules are as follows

(A∪A) = A,  (A∩A) = A (idempotent law)
(A∪B)’ = A’∩B’, (A ∩ B) ′ = A ′ ∪ B ′ (De Morgan’s law)
A∪ B = B ∪ A, A∩B = B∩A (commutative law)
(A ∩ B) ∩ C = A ∩ (B ∩ C), (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
A ∪ φ = A, A ∩ U = A, A ∪ U = U, A ∩ φ = φ (identity law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)   (distributive law)
(A’)’ = A (involution law)

Basic Set Theory Properties are:
A st is inherently unordered. This means that the order of elements in a st does not make a new st. Change in order of elements inside a st is feasible. For example if st A = {a, x, c} then A = {a, c, x} = {x, c, a} = {x, a, c} = {c, a, x} = {c, x, a}.
Each element in a st is distinct. Multiple repetition of an element makes no difference. For example if st A = {2, 3} then A = {2, 2, 3, 3} = {2, 3, 3} = {2, 2, 3} = {2, 2, 2, 3, 3, 3, 3} = {2, 3} and so on.
Above rules and properties of Set Theory Help you to solve Set Theory Questions easily.
Let us see some Set Theory Problems and Solutions now:

Q.1) prove that (A ′ ∩ B) ′ ∩ (A ∪ B) = A
Sol.) lets begin with LHS of the above problem:
LHS = ((A ′) ′ ∪ B ′) ∩ (A ∪ B) (by De Morgan’s law)
= (A∪ B’) ∩ (A ∪ B) (by involution law)
= A∪ (B∩ B’) (distributive law/property)
= A ∪ φ (complement law)
= A (identity law) = RHS
Hence, proved.

Q.2) in a community if 70 % people can speak English and 60% people can speak French then find the number of people who can speak both languages.
Sol.) Number of people who can speak English = n(E) = 70 %
Number of people who can speak French = n(F) = 60%
Number of people who can speak both languages will be n(E∩F) = ?
Total number of people in the community n (E∪F)= 100%
Now n (E∪F)= n(E) + n(F) - n(E∩F)
100=70 + 60 - n(E∩F)
100=130 - n(E∩F)
n(E∩F)=30%

Articles – a, an, the

An article is a major concept of language learning. In English an article is considered as an adjective that provides some valuable information about a noun in a sentence. In many other different languages an article is considered as a part of speech. An article can be classified based on definite and indefinite characteristics. These are called definite article and indefinite article.

Definite Article

A definite article is that which defines that a noun is particular and has something definitely addressed. Definite article is used for nouns that have been already used in a speech or for nouns that are uniquely specified. In the English language, there is only one definite article ‘the’. Definite article ‘the’ is used for both singulars and plural nouns. For example: Give me a Barbie doll. Give me the Barbie doll. In the second sentence, a specific Barbie doll is referred to while in the first sentence not any particular Barbie doll is referred.
1) Ram said to his wife, “Please pack the gifts for kids that I bought yesterday”. (Here Ram is speaking about those particular gifts for kids that he bought the previous day.)
2) The Taj Mahal is made by Mughal Emperor Shah Jahan. (Here, Taj Mahal is a uniquely specified noun.)

Indefinite Article

An indefinite article is used for a noun that is not particular and uniquely specified. It can be a noun that the speaker is mentioning for the first time. For example: Give me a Chota Bheem DVD. Give me the Chota Bheem DVD. Here, in the first sentence the speaker is talking about no particular Chota Bheem DVD while in the second sentence the reference is definite. In the English language, there are two indefinite articles ‘a’ and ‘an’. The indefinite article ‘an’ is used before words that begin with a vowel sound. At times even a word that starts with a consonant but sounds like a vowel. For example: an hour, an MBA, etc. Also the indefinite article ‘a’ is used before words that begin with a consonant sound even if the word begins with a vowel. For example: a European, etc. These were the basics about article in English language.

Introduction for Solving Number Patterns

Solving number patterns is the main topic under the solving number systems. In this lesson we have to study about the concepts of divisors, factors, multiples and prime numbers. Also in this topic we have to discuss about properties of number patterns. Here we have to discuss about Divisors, Factors ,Multiples,Even,Odd numbers,Prime numbers and Composite numbers. These are the main topics in the solving number patterns.

Solving Divisors, Factors and Multiples Number Patterns:

Divisors:

Step 1: Consider the number 96. Divide 96 by 4. When we divide 96 by 4, the remainder is 0.

Step 2: 4 is the divisor of 96. Similarly if we divide 96 by 2, we will get 0 as the remainder.

Step 3: Again 2 is the divisor 96. By applying the same process, 1,2,3,4,6,8,12,16,24,32,48,96 are also divisors.

Step 4: Thus the divisors of 96 are 1,2,3,4,6,8,12,16,24,32,48,96.

Note:

The divisor in the division algorithm is different from this divisor. In division algorithm the divisor indicates the denominator only.

Example:

Find the divisors of (a) 28, (b) 35

Solution:

(a) Solving The divisors of 28 are 1, 2, 4,7,14, and 28

(b) Solving The divisors of 35 are 1, 5, 7 and 35

Solving even and Odd Number Patterns and Prime and Composite Patterns:

The numbers that are divisible by 2 is called as even numbers.
If we divide the even number by 2, the remainder is always zero.

The numbers that are not divisible by 2 is called as odd number.
If we divide an odd number by 2, the remainder is always 1

Now observe the following:

(a) 2 + 2 = 4 ; 2 + 6 = 8 ; 4 + 8 = 12 ; 10 + 20 = 30

(b) 2 × 2 = 4 ; 2 × 6 = 12 ; 4 × 8 = 32 ; 10 × 20 = 200

From the above examples the following are true.

Even number + even number = even number
Even number × even number = even number
Odd number + odd number = even number
Odd number × odd number = odd number
These are the main rules used for solving number patterns.

Prime numbers and Composite number patterns

From the above, find the numbers that have only 2 divisors?

2, 3, 5 and 7 have only two divisors.

What are the numbers which have more than 2 divisors?

4, 6, 8, 9 and 10 have more than 2 divisors.

The natural numbers that have only two divisors are known as the prime numbers.
The natural numbers that have more than two divisors are known as the composite numbers.

Example:

Solving prime numbers between 20 and 30.

Solution:

The numbers between 20 and 30 are

21 22 23 24 25 26 27 28 29

∴ the prime numbers are 23 and 29

Definition of parabola

Definition of parabola:
A curve in a plane having two ends and one vertex is called a parabola. It is a conic section. That means it is a section of a double cone intercepted by a plane. When a double cone is cut by a plane such that the plane is parallel to any one slant length of the cone, the cross section thus obtained is called as such.

However in co-ordinate geometry we do not study parabola problems as sections of double cone. We use the characteristic property of these to define it and use that to find its equation.

It so happens that in the plane of every parabola there is a line (called the directrix) and a point (called the focus) such that for every point on this the distance from the directrix and the distance from the focus are found to be equal. This is the definition most commonly used in co-ordinate geometry to define these and to derive its equation.

Types of parabola:
Primarily these are of two types and they are horizontal and vertical. In simple terms, the horizontal form of this is x as a function of y. Therefore the equation of a horizontal parabola would be x = f(y) type. On the other hand in a vertical form of this the equation is y as a function of x. Therefore it would look like this: y = f(x).

Another way of classifying these would be: positive and negative parabola. For the horizontal form, if it opens towards the positive side of the x axis, it is called the positive para-bola; where as if it opens on the negative side of the x axis, it is called the negative para-bola. Similarly, for the vertical form, if it opens towards the positive y axis it is called a positive para-bola and if it opens towards the negative side of the y axis, it is called a negative form of same. Below are the pictures of the various types of these:

1. Horizontal positive form of the same:

As you can see in the above picture, this one opens to the right.
X = f(y) = p(y-k)^2 + h
2. Horizontal negative form:

The above form opens to the left.
X = f(y) = -p(y-k)^2 + h

3. Vertical positive form of the same:

A vertical positive form opens up (as can be seen in the picture above).
Y = f(x) = a(x-h)^2 + k

4. Vertical negative form:

Lastly the vertical negative form would open down.
Y = f(x) = -a(x-h)^2 + k

How to calculate Area of Ellipse?

An ellipse is nothing but the planar curve which is resulted by the intersection by a plane of a cone in a way such that it will produce a closed curve. Circles can be said as a special case of ellipse which can be obtained by cutting in orthogonal plane of the cone’s axis. Ellipse can also be said as the locus of all the points in the plane in which the sum of the distances of two fixed points will be a constant. Before going into the formula for area of an ellipse, it is essential to know the elements of ellipse which forms the structure of it. An ellipse is said to be a smooth and closed curve symmetric about its vertical and horizontal axis. The longest diameter of an ellipse is called as major axis and the shortest diameter of an ellipse is called as the minor axis. These both axes are the lines that pass through the centre of an ellipse.

With the measurement of both these axis only, the area of a ellipse will vary. Each of the above said axes bisects perpendicularly with the other. Also, the sum of the distance from the two focus of the ellipse to any point P on the ellipse will be equal to the major axis. It can also be said that each axis cut the other equally into two parts and they cross at right angles to each other. Also, if these both axes are equal in length, then it is called as circle.

Area of an Ellipse Formula
The standard equation for representing the ellipse equation is given as,
X^2 / a^2 + Y^2 /b^2 = 1.

This equation is a standard equation in which when the ellipse is centred with origin.
But algebraically the ellipse area formula is given in other terms with the help of semi major axis (half of major axis) and semi minor axis (half of minor axis). Thus the formula for area of ellipse is given as Pi*a*b, where ‘a’ is the semi major axis and ‘b’ is the semi minor axis. This formula is actually arrived from the formula of circle Pi multiplied by radius square. Here the radius is split into semi major and semi minor axis.
Also at some special cases, when an ellipse is given by the implicit equation which is given as,
AX2 + Bxy + CY2 = 1, then the area of the ellipse would be 2*Pi whole divided by square root of 4AC subtracted with B square.

Understanding Fractions Better

Numbers have always fascinated humankind. Indians are credited with the invention of zero. Indians have always been good with numbers and the world looks at us with awe. We have great strides and come a long way. From natural numbers, whole numbers, integers, real numbers and complex numbers. We have also studied fractions. Let us what are fractions to understand the concepts of denominator numerator. A number can written in the form of A/B where A is an integer and B is an integer. A can be called the numerator and B can be called the denominator. So we see numerator v/s denominator to extend the concept further. The numerator is written on top of the horizontal line and the denominator is written below the horizontal line. So we understand what is a numerator and denominator if we didn’t know it earlier.

We must understand numerators and denominators to understand fractions better. Operations of addition, subtraction, multiplication and division between the fractions can be performed. Here both numerator and also the denominator come into picture. If the denominators are dissimilar they must be made similar to proceed further with the operations. For this we need to take the LCM of the denominators. Taking the LCM is a simple operation. Unless the denominators are equal we cannot proceed further. Once the denominators are equal, we can just add the numerators if the operation to be performed is addition. If subtraction is to be performed we just need to subtract the numerators after making the denominators equal. So we must know the definition of numerator and denominator to perform these operations. Only when we know it, we can do something.

After making the denominators equal if they are not, we proceed to perform addition, subtraction or any operation with fractions. Sometimes the fractions can be in the form of mixed fractions. We need to be careful in such cases. We must first convert them into improper fractions without proceeding. They cannot be directly used. For the process of finding LCM we need improper fractions in our hand. Once we get the improper fractions we first check whether their denominators are same or not. If they are same, our problem becomes very simple and easy. The numerators can be just added or subtracted as the case may be. If they are not equal only then the process of taking LCM is to be performed and addition or subtraction is to be done.

Distributive Property of Addition

The distributive property is expressed in the mathematics expressions as following equation: a (b + c) = ab + ac. You can understand this as the sum of a (b + c) is corresponding to the sum of a times b and a times c. Distributive property of addition would be inaccurate to multiply ab and just add c, or to multiply ac and add b.

Order of Operations:

The distributive property of addition that remind that all contained by the parenthesis needs to be multiplied by the outside number. Distributive property, when they are knowledge the order of operations.

Concept of that the problems anywhere present are different mathematical operations, such as multiple, addition, subtraction, parenthesis, you have to work in a certain order to get the right answer. This arrange is the parenthesis, exponent, multiplication,division , addition and subtraction, that may be abbreviate to the  PEMDAS.

Example Problems (distributive Property of Addition):

Example problems:

When you include a mathematics problem that use parenthesis you need to solve what’s in the parenthesis first, by you can move about on to solve further problems. If the mathematics problem largely has known numbers, it is somewhat easy to solve. 2(10+5) becomes 2(15) or is also equal under the distributive property of addition is 2(10) + 2(5). What obtain additional difficult is when you are functioning with variables (such as a, b, x, y, and so on) in algebra, and when these variables cannot be joint together.

Consider the equation 5(12a + 2), but we don’t know what the variable a stand for, we can’t add 12a + 2, but using the distributive property still allow us to just this expression because we identify this equation is equal to 5(12a) + 5(2). In order to simply the expression we can take each part separately and multiply it to 5, and we get 60a + 10.

Addition, Subtraction, Multiplication and Division

As the learning session for numbers is completed, the next session starts with addition, subtraction, multiplication and division. Understanding these calculations is the basics of learning mathematics. It is quite challenging to make kids’ understand the concepts of addition, subtraction, multiplication and division followed by order of operations, simplification and so on. In that case providing fun examples related to kids can be helpful. Let’s give it a try.

Addition: Addition denoted by the plus sign, “+”, is a mathematical operation that combines two or more collection forming larger collections.

For example: It is Tina’s birthday and she has received many beautiful dresses for girls – one from Annie aunty, another from grandpa and another one from her mother. Now adding all three we get, 1+1+1 = 3 dresses for girls. This mathematical operation is addition.

Subtraction: Subtraction denoted by the minus sign, “-”, is a mathematical operation that is inverse of addition. In this operation, a part is taken away from a collection and thus forming a smaller collection.

For example: Father brought 50 return gifts for kids on Tina’s birthday. Only 45 of her friends came for the birthday party. Now after subtracting the two, we can find, 50-45 = 5 return gifts for kids remaining. This mathematical operation is subtraction.

Multiplication: Multiplication denoted by the sign “x” is a mathematical operation that scales one number by another.

For example: Mother is planning to organize a fancy dress costumes competition on next Sunday. There are 5 groups of two members and therefore each group requires two costumes. Now by multiplying, we get 5 X 2 = 10 fancy dress costumes. This mathematical operation is multiplication.

Division: Division denoted by the sign “/” is a mathematical operation that includes a dividend, quotient, divisor and remainder.

For example: Tina has 5 best friends and she has 20 chocolates. She wanted to distribute the chocolates among them. Now diving the numbers we get, 20/5 = 4. Each of her friends will get 4 chocolates. This mathematical operation is division.

These are the basic operations of mathematics that form the first steps in learning mathematics for kids.

Problems related to consecutive integers

Many a times in arithmetic we face problems related to finding consecutive integers the sum of which is given to us. The said integers may be odd or even or neither. In this article we’ll try to understand how to solve such type of problems.

Formula for sum of consecutive integers:
As such there is no specific formula for finding sum of consecutive integers. Such problems are solved using algebra and arithmetic approach. The methods may however, vary from problem to problem. Let us look at the following example to understand this better.

Example 1: Three consecutive integers are there whose sum is 66. Find consecutive integers.
Solution: Suppose we assume that the middle or the second integer is n, then the first integer would be n-1 and the third integer would be n+1, because the integers are said to be consecutive, and we know that consecutive integers differ by 1. The three integers are therefore now: n-1, n and n+1. The sum of these integers is given to be 66. Therefore,
n-1 + n + n+1 = 66
3n = 66
n = 66/3
n = 22.
Since n was our middle integer, the other two would be n-1 = 21 and n+1 = 23. So the three consecutive integers are 21, 22 and 23.
Odd consecutive integers:
Odd consecutive integers are such that each of the integer in the list is an odd number. It may be negative or positive, but has to be an odd number. For example, …. -11, -9, -7, -5, -3, -1,  1, 3,  5,  7,  9…These are odd consecutive integers. They go up to infinity in both directions.

Example 2: The sum of three consecutive odd integers is -39. Find the integers.
Solution: Here again, assume that the middle integer is n. Then the odd integer before n would be n-2 and the odd integer after n would be n+2. So the sum
= n-2 + n + n+2
= 3n = -39
n = -39/3
n = -13.
Therefore the other two odd integers would be -15 and -11. So solution: -15, -13, -11 are the required three odd integers.
Even consecutive integers:
The method of solving problems of even consecutive integers is exactly same as that for odd integers.

Example 3: Find the three even integers whose sum is 0.
Solution: If the middle integer is n, then the other two would be n-2 and n+2. Therefore the sum
= n-2 + n + n+2
= 3n = 0
n = 0/3
n = 0.
So the integers would be -2, 0, and 2

Integers: Positive and Negative

Integers are the numbers consisting of positive numbers, zero and also the negative numbers.  The positive numbers and negative numbers are called as positive negative integers. Positive integers are the integers with positive sign for example, 2, 3, 1, 5 etc. and negative integers are the integers with negative sign. Examples of Negative Integers are -21, -3, -1, -5 etc. On a number line the integers are shown as given below

The following are the Positive and Negative Integers Rules:
Addition rules of Integers: 
When two positive integers are added the sum would be a positive integer; (3+5=8)
When two negative integers are added the sum would be a negative integer; [(-3) + (-5)= -(3+5)= -8]
When a positive integer and a negative integer are added, the numbers are subtracted and given the sign of the larger number;[ (3)+(-5)= -2]

Subtracting Positive and Negative Integers:
A positive integer subtracted from a negative integer would give a negative integer; (-7)-(4)=-7+4= -3
A negative integer subtracted from a positive integer would give a positive integer, here the numbers are added and given the positive sign; (5)- (-7)=(5+7) = 12
When two negative numbers are subtracted it gives a negative and a positive integer. The numbers are subtracted and given the sign of the larger number; (-5) – (-7)=-5+7= 2; -7 – (-5)= -7+5= -2

Multiplication Rules of Integers:
A  Positive Integer multiplied with a positive integer, the product would be a positive integer
3x5=15
A positive integer multiplied with a negative integer, the product would be a negative integer
3x (-5)=-15
A negative integer multiplied with a positive integer, the product would be a negative integer
(-7)x3=-21
A negative integer multiplied with a negative integer, the product would be a positive integer
(-7)x(-5)= 35

Dividing Positive and Negative Integers:
A  Positive Integer divided by a positive integer gives a positive integer, 15/5=3
A positive integer divided by a negative integer gives a negative integer, 15/(-3) = -5
A negative integer divided by a positive integer gives a negative integer, -21/3 = -7
A negative integer divided by a negative integer gives a positive integer, (-21)/(-7) = 3

Positive and Negative Integers Word Problems
The temperatures recorded in the Sahara Desert and Thar Desert are 138 degrees Fahrenheit and -55 degrees Fahrenheit respectively. Calculate the difference in the temperatures.
Sahara Desert Temperature= 138 degrees Fahrenheit
Thar Desert Temperature= (-55) degrees Fahrenheit
Difference in temperatures = Temperature in Sahara Desert – Temperature in Thar Desert
      = 138 – (-55) = 138 +55 = 193 degrees Fahrenheit (Answer)

Standard Deviation

Standard deviation is denoted by a symbol of Greek letter (s) and is shows how much variation exists from the mean or average or expected value. If the standard deviation is low, that will indicate that the point tents to very close to the mean, whereas standard deviation is high, the data points are spread out over a large range of values. Standard deviation has very useful important properties such as unlike variance and is expressed in the same units as the data. Mean standard deviation is mainly used in the statistic conclusion to measure confidence. Also used to find how set of data spread out.
Standard deviation (s) is the square root of its variance (s2), which is the average of the squared difference from the average of the mean. Mean (µ) is simple average value of given set of data.
For example just take different height of dogs, find out mean, variance, and standard deviation.
Height of dogs = 700mm, 570mm, 180mm, 530mm, 400mm
To find the mean
Mean = 700+570+180+530+400/5
= 2380/5 = 476
The mean or average height of the dog is 476mm.
Then to calculate the variance first subtracts the value of mean from the every height of the dog and then square the resulting values. Add the sum of squared values and divide with total number of dogs, resulting value gives the variance.
Variance = (700-476)2 + (570-476)2+ (180-476)2 + (530-476)2 + (400-476)2 / 5
  = 50176+8836+87616+2916+5776 / 5
= 155320/5 = 31064
The square root of the variance gives the standard deviation for the height of the dogs.
Standard deviation (s) = vvariance
= v31064
= 176.24 mm


Finding the standard deviation for population and sample, following formulas are used,
Standard deviation for population (s) = v(1/n ?_(i=1)^n¦(xi- µ)2)
Standard deviation for sample (s) = v(1/(n-1) ?_(i=1)^n¦(xi- " " )2)
Where,
µ,   - mean
When we have n value of data, if we are calculating variance, we should divide by n for the population and divided by n-1 for a sample. Standard deviation is used to measure the investment volatility, in finance. It is also called as historical volatility.

Mean and Standard Deviation
Mean and standard deviation is mainly used to find the center of the data set. Mean is defined as; it is the simple average value of the given data set and is represented by a symbol of Greek letter (µ).
Mean for population (µ) = 1/n ?_(i=0)^(n-1)¦xi
Mean for sample ( ) = 1/n ?_(i=1)^n¦x

Where,
n - Size of the sample or number of item in the sample
x, xi - Observed value or set of value

Finding Standard Deviation
Finding standard deviation the following steps should be followed.
First calculate the mean of given set of data by sum of given data divided by total number of data.
Then subtract the mean from each observed value.
Square the each difference and then add all the squared values to get their total sum. The resulting value divided by one less then the number of data in the data set.
The resulting value gives the variance.
Finally standard deviation can get from square root of the variance.

Find Standard Deviation
Find standard deviation for the list of numbers, 1, 3, 4, 6, 9, 8
Mean (µ) = 1+3+4+6+9+8/6
= 5.16
Variance (s2) = (1-5.16)2 + (3-5.16)2 + (4-5.16)2 + (6-5.16)2 + (9-5.16)2 + (8-5.16)2  / ( 6 – 1)
= 17.31+4.66+1.35+0.71+8.07+14.75 / 5
= 46.85 / 5 = 9.37
Variance (s2) is 9.37
We know that standard deviation is the square root of the variance
So the standard deviation (s) = vs2
= v9.37
s = 7.81

Introduction To Parabola

Graphing of a Parabola  : Let us understand  how to Graph Parabola . For graphing parabola we will first draw rough sketches of parabola and various terms associated to them are given below:
1. Equation: y^2 = 4ax, vertex = (0, 0), focus = (a, 0), Latustrectum = 4a, Directrix: x = -a.
2. Equation: y^2 = -4ax, vertex = (0, 0), focus = (-a, 0), Latustrectum = 4a, Directrix: x = a.
3. Equation: x^2 = 4ay, vertex = (0, 0), focus = (0, a), Latustrectum = 4a, Directrix: y = -a.
4. Equation: x^2 = -4ay, vertex = (0, 0), focus = (0, -a), Latustrectum = 4a, Directrix: y = a.

Sketching of curves represented by y = ax^2 + bx + c. For graphing a parabola equation y = ax^2 + bx + c always represents Vertex of Parabola (-b/2a, -D/4a) and axis x = -b/2a.

The parabola graph opens upwards or downward according as a > 0 or < 0. It meets x-axis at (alpha, 0) and (beta, 0), where alpha and beta are the roots of the equation ax^2 + bx + c = 0.

If the roots of this equation are not real, then the parabola does not cross x-axis. In order to draw rough sketch of the parabolas given by the equations of the form y = ax^2 + bx + c, we may follow the following algorithm.

Algorithm: 
Step 1: Obtain the equation and observe the sign of the coefficient of x^2 in it.
Step 2: Put y = 0 in the given equation and get the values of x. Let the values be alpha and beta.
Step 3: Mark the points A(alpha, 0) and B(beta, 0) on x-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on x = -b/2a = (alpha + beta)/2 and opening upward and downward according as the coefficient of x^2 in the given equation is positive or negative.

In the above algorithm, if the values of alpha and beta are imaginary, then the equation y = ax^2 + bx + c represents a parabola having vertex at (-b/2a, -D/4a) and opens upward or downward according as a > 0 or a < 0.

Sketching of curves represented by x = ay^2 + by + c. For a graphing a parabola the equation x = ay^2 + by + c also represents a Parabola Vertex at (-D/4a, -b/2a) axis y = -b/2a and so the parabola graph opens leftward or rightward according as a < 0 or > 0.

It crosses y –axis at (0, alpha) and (0, beta), where alpha and beta are the roots of the equation ay^2 + by + c = 0.
If alpha and beta are not real, then the parabola does not cross y-axis and it opens rightward if a > 0 and leftward if a < 0.
In order to draw a rough sketch of the parabolas given by the equations of the form x = ay^2 + by + c, we may follow the following algorithm.

Algorithm:
Step 1: Obtain the equation and observe the sign of the coefficient of y^2 in it.
Step 2: Put x = 0 in the given equation and get the values of y. Let the values be alpha and beta.
Step 3: Mark the points A(0, alpha) and B(0, beta) on y-axis.
Step 4: Draw a parabola passing through points A and B having its vertex on y = -b/2a = (alpha + beta)/2 and opening upward and downward according as a > 0 or a < 0.

What are lines?

Lines are more related to geometry than to math and could be considered as under math lines. Therefore if the above title was ‘geometry lines’, also it would have been apt.
What are lines?
A line is a set of points that has only one dimension, length.

A line AB is shown in the given diagram. The arrow heads on the line AB show that it is extending endlessly in both the directions and has no end points. Hence a line has no fixed length. Or we can say that the length of a line is infinite. We used the two points A and B to define the line. Therefore in general we can say that two distinct points in a plane determine a line. In other words, if there is only one point under consideration, then there can be infinitely many lines passing through that point as shown in the figure below:

However if we have two points to describe a line, then there can be only one line that passes through both these points.

In co-ordinate geometry, a line can be defined as a set of ordered pairs with a definite property. For example, if we have an ordered pair (x,y) such that y = 2x + 1, that means that all those values of  x and y that satisfy the above equation, would be a part of the line defined by that equation.

Some special subsets of a line:
1. Line segment: A line segment is simply a part of a line that has a specific length and specific end points as shown in the diagram below:

Here CD and PN are both line segments. Note that the end points are dots and not arrows as was the case with line.

2. Ray: A ray is a part of a line that has only one end point. It extends endlessly in one direction. See figure below:

In the above figure AF is a ray. It has one end point at A and extends endlessly in the other direction. Therefore it has a dot at one end and an arrow at the other end.

Two lines can be related to each other in four different ways:

1. Lines that have just one point in common are called intersecting lines.

2. Lines that lie the same plane but never intersect even if produced endlessly in both directions are called parallel lines.

3. Two intersecting lines that form a right angle are called perpendicular lines.

4. Lines that are not in the same plane and do not intersect are called skewed lines.

What is a line plot?

Define line plot:
A line plot is a method of data representation. Of the many methods to represent statistical data, one is graphical representation. A line plot is a type of graphical representation of data. When frequency of data is plotted along a number line it is called a line plot. A line plot makes sense only when the number of observations is lesser than 25. For small number of observations a line plot gives a very good visual representation of data. Following are some examples of line plots:

Line plot graph:

The first step for any data representation is collection of data. Data can be collected from various sources such as, the internet, person to person observations, questionnaire, survey records etc. After collecting the data we can proceed to make the line plot. For making a line plot, first we need to come to a suitable scale for plotting the data. As we can see in the two examples above, in one the scale 1 unit = 5 marks and each unit is subdivided to 5 marks. That is because, the marks are all lying between 30 to 50. For the second example we see that the scale is 1 unit = one day, since we are interested in number of cars sold each day.

Once we decide on the scale, we can actually make the line and mark the numbers as per our scale. Ensure that the distance between consecutive points on the line is proportional to the number the point represents.

Next take the data and list it. Now for each observation, mark a X over the number on the number line that is equal to the observation value. For example, if we have 4 cars sold on Monday, then we mark four Xs over the Monday mark of the number line. Similarly we go on for each of the observation from the data available with us.
See the example chart below;

Understanding the line plot:

We have seen above how to make a line plot. But now the question is that if such a plot is given to us, how do we read that plot (or understand the line plot)? A line plot easily brings out the outlier. An outlier is an observation that is very large or very small as compared to all of the other observations. A line plot also helps us identify gaps and clusters in observations.

Guided reading answers

The guided reading answers are nothing but getting help from others to reading the subjects.In online only we can get the help of tutor for reading any subjects.Initially the math problems can be solved by using some arithmetic operations  like addition,subtraction,division and multiplications and these can be denoted by (+ ,`xx` ,`-` ,÷ ).The following article shows some guided reading answers.

Solved Math Problems with some Guided Reading Answers

Problem 1:

Solve the 56x + 47y = 2632 given equation on the x and y intercepts.

Given:

56x + 47y = 2632

Solution:

56x + 47y = 2632

To find the x intercept of y = 0 and solve for x.

56x + 47(0) = 2632

Solve the value of x.

  x = `2632/56`

  x = 47

  To find the y intercept of x=0 and solve for y.

56(0) + 47y = 2632

Solve the value of y

47y = 2632

  y = `2632/47`

  y = 56.

The equations of x intercept on (47,0) and y intercept on (0,56).

Problem 2:

Solve the problem 185 + 35 ( 40 + 27 ) ÷ 67 – 60 in method of order of operation
Solution:

Given:

 `=>`  185 + 35 ( 40 + 27 ) ÷ 67 – 60

Step 1: we need to simplify the parentheses

 `=>` 185 + 35 `xx` 67 ÷ 67 – 60

Step 2: We need to simplify the multiplication

 `=>` 185 + 2345 ÷ 67 – 60

Step 3: We need to simplify the division

 `=>` 185 + 35 – 60

Step 4: We need to simplify the addition

 `=>`220 –  60

Step 5: We need to simplify the subtraction

 `=>` 160

Answer: 185 + 35 ( 40 + 27 ) ÷ 67 – 60 = 160

Solved more Math Problems with some Guided Reading Answers

Problem 3:

Solve the given problem 11(s – 9) – 6s ` - ` 26 = 13(s + 33)
The Solutions follows below:

Step 1: Given expression is,

11(s – 9) – 6s ` - ` 26 = 13(s + 33)

Step 2: Multiplying the integer terms

11s – 99 – 6s – 26 = 13s + 429.

Step 3: Grouping the above terms

5s –125 = 13s + 429

Step 4:  Add 125 on both sides

5s –125 + 125 = 13s + 429 + 125

Step 5: Grouping the above terms

5s = 13s + 554

Step 6: Subtract 13s by on both sides

5s `-` 13s = 13s `-` 13s + 554

Step 7: Grouping the above terms

–8s = 554

S = `- 554/8`

The required answers is

S = `- 554/8`

Problem 4:

Solve given problem (156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)
 Solution:

The problem can be solved in simplifying method .

Step 1: (156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)

Step 2: 156x2 – 141x – 91 + 213x2 – 181x – 144 + 916x2 `-`    41x `-` 20

Step 3:  1285x2 – 363x – 255      

The required answer is

(156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20) = 1285x2 – 363x – 255

Interior Angles of Different Shapes

An interior angle is the angle formed when the endpoints of a polygon’s two sides are shared. In simple terms, when a new line intersects two straight lines that are parallel, then four angles will be formed inside them. These angles are known as Interior Angles. This angle will be inside a shape always. Also the angles formed outside the shape are known as exterior angles. Interior angles are sometimes referred as “Internal Angles”.

Interior Angles of a Triangle
The interior angles of a triangle will always be 180 degrees. For example if A, B and C are three sides of a triangle, then A+B+C = 180 degrees. Due to this reason, only one of the angles formed inside a triangle can be obtuse (i.e. greater than 90 degree). Eventually in a right angle triangle, since one angle is 90 degree, the other two angles will always add up to 90 degree.

Interior Angles of Square and Quadrilaterals
A square can be made up of two triangles. Hence, the sum of interior angles of square will add up to 360 degrees.  Quadrilateral also lies in the same case in which the sum of interior angles gives 360 degrees. Meanwhile, in this case maximum of two angles can be greater than 90 degrees.
Interior Angles of Polygons

The interior angle of polygon is nothing but the angle formed at every vertex inside it. Hence for a polygon of N vertices and N sides, there will be N interior angles. The sum of these interior angles will sum up to a constant value always. The formula that relates the sum of interior angles and number of sides of polygon is as follows:
Sum of Interior Angles = 180(n-2) degrees, where n is the number of sides. For example, sum of square that has four sides will be 180*2= 360 degrees, sum of pentagon which has five sides will be 540 degrees. Similarly Interior Angles of a hexagon, which has six sides, will be 720 degrees.
Also, the above said formula will not be applied to regular polygons, whose all sides and interior angles are always equal. Therefore, Interior Angles formula for a regular polygon will be 180(n-2) divided by n, where ‘n’ is the number of sides. This formula applies to its children such as regular pentagon whose angle is 108 degree, regular hexagon whose interior angle is 120 degree etc.
Also, two interior Angles at the same side of a polygon located at both sides are referred as “Adjacent Interior Angles”.

Modal value in statistics

Mode Statistics
In statistics, the four basic measures of central tendency are Mean, Median and Mode. We know that Mean is the Average of the total data values in a given data set and Median is the middle value of the given data set when values are arranged in the numerical order. What is Mode, at times we come across a data set in which a number repeats most number of times that is it occurs most often such a data value is called the Mode. For example, consider the data, 1, 1, 3, 3, 3, 3, 5, 5, 7. As you can see the number that occurs most number of times is 3 as it is appearing four times which is the most number of times in the given list and hence 3 is the Mode

Mode Definition
Let us now define Mode, in simple words mode can be defined as the data value that occur most number of times in the data list. There can be more than one mode in a given data set, if there are two modes in a given set it is called bi-modal and if more than two modes in the data set is called multimodal. If there is no number that occurs most frequently then we can say that the data set has no mode. For example the data, 11, 12, 13, 8, 9, 11, 14, 11, 9, 15, 11, 12. First let us arrange the given data set in the numerical order. 8, 9, 9, 11, 11, 11, 11,12, 12, 13, 14, 15. Now let us list the data values that occur more than once. The data value 9 occurs 2 times, the data value 11 occurs 4 times, the data value 12 occurs two times. So, the data value that occurs most number of times is 11 which is four times and hence the mode of the data set is 11.

Mode Formula: the value that occurs most frequently in a data set when arranged in numerical order. The typing speed (in words per minute) of a typist working in a firm are 46, 52, 40, 38, 52, 36, 46, 52, 37, 46  find the modal value or the mode using the mode formula.

First we arrange the typing speeds in the numerical order which gives us 36, 37, 38, 40, 46, 46, 46, 50, 50, 50. The values that are occurring most frequently are 46 and 50, both the values are occurring three times in the list and hence we can call it bimodal with the modes 46 and 50.

Quick Algebra Answers

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics (Source: Wikipedia).

In this article we are going to see some quick algebra problems with answers.

Quick Algebraic Example Problems with Answers:

Problem 1:

Given the equation

5(-3x - 2) - (x - 3) = -4(4x + 5) + 13

Multiply factors.

-15x - 10 - x + 3 = -16x - 20 +13

Group like terms.

-16x - 7 = -16x - 7

Add 16x + 7 to both sides and write the equation as follows

0 = 0

Hence the statement is true for all values of x and therefore all real numbers are solutions to the given equation.

Problem 2:

Given the line

5x - 5y = 7

Rewrite the eqn in slope intercept form y = mx + b and identify the value of m the slope.

-5y = -5x + 7

y = x - 7/5

The slope is given by the coefficient of x which is 1.

Problem 3:

To find the eqn of the line through the points (-1, -1) and (-1, 2), we use the slope m.

m = (y2 - y1) / (x2 - x1) = (2 - -1) / (-1 - -1) = 3 / 0

The slope is defined which means the line is perpendicular to the x axis and its equation has the form x = constant. Since all points have equal x coordinates -1, the equation is given by:

x = -1

Problem 4:

Given the equation

2x - 4y = 10

To find the x intercept we set y = 0 and solve for x.

2x - 0 = 10

Solve for x.

x = 10 / 2

= 5

The x intercept is at the point ( 5 , 0).

Quick Algebra Practice Problems with Answers:

1. Find the statement is equal or not equal: 7(-4x - 3) - (x - 4) = -8(2x + 6) + 11

2. To find the equation of the line through the points (-2, -2) and (-2, 1), we first use the slope m.

3. Solve: 8x - 2y = 3

Answers for quick algebra:

1.Not Equal.

2.Slope m =1/0

3. The x intercept is at the point (3/8 , 0).

Introduction to practice pre algebra answers

The word algebra is defined from the Arabic word al–jabr. In Arabic language, ‘al’ represents ‘the’ and ‘jabr’ represents ‘reunion of broken parts’. We can use letters like a, b, x and y to denote numbers. Using the basic operations add (+), subtract (-), Product(x) and Divide (/)or extraction of roots on these symbols and real numbers, we obtain what are called algebraic expressions. This contains following types of problems like Polynomials, Algebraic Identities, and factorization and Division of a Polynomial by a Polynomial.

Practice Pre Algebra Example Problems with Answers

Practice Pre Algebra Example Problems with Answers are given below:

1. Steven has 18 rupees. She buys fruits for 8 rupees. Find the remaining amount with her.

Solution:

We can write it as 18 – 8 = 10

2. Write 25–x=10 as a Mathematical statement.

Solution:

What number is to be subtracted from 25 to get 10?

3. Pick out the variables and constants: A, – 15, q, l, 22.3, and 73

Solution:

The variables are A, q and l

The constants are – 15, 22.3 and 73.

4. Write the following using powers:

(a) 5×m×m×m (b) a×b×a×a×b

Solution:

(a) 5×m×m×m=5m^3

(b) axb×a×a×b=a×a×a×b×b

= a^3×b^2=a^3b^2

5. Write the following algebraic expression in the product form:

(a) 8xy^2 (b) 10p^2 qr^3

Solution:

(a)8xy^2= 8 × x × y × y

(b) 10p^2 qr^3 = 10 × p × p × q × r × r × r

6. Write the coefficient in the following algebraic terms:

(a) The coefficient a in 10a is     10 .

(b) The coefficient of c in 3c is     3  

7. Identify the like and unlike terms:

(a)    12a and 15a -These are like terms

(b)   – 3b and 6c-These are unlike terms because here the variables are different.

8. Add and subtract the algebraic terms 12x and 5x.

Solution:

The above terms are like terms. Hence we can add and subtract directly.

12x + 5x = (12 + 5) x= 17x

12x - 5x = (12 - 5) x= 7x

Practice Pre Algebra Problems for Answers

Practice Pre Algebra Problems for finding answers are given below:

Practice 1: The symbol Δ, in Δ + 5 is called a ___________ .

Practice 2: A quantity which takes different values is called a __________ .

Practice 3: The coefficient of a b in 15ab is____ .

Practice 4: 2×m×n×n×m×m can be written as ____________ .

Practice 5: Add the following terms: 3xy, 7xy and 18xy.

Introduction to my math answers sheet

Math is a family where we can declare any kind of the numbers like symbols, letters and the different numbers, complex numbers, real numbers, rational numbers etc...Math is a group of sciences (arithmetic, geometry, algebra, trigonometry, calculus etc..) that explains measures,magnitudes,area of the region, statistics. we are going to learn how to solve some of the math problems. In addition, math helps a lot in our daily life. Here we include some of the math problems to help you better.The answer sheet to the problems follows.

Math Answer Sheet to Given Problems

Ex 1: Let A = {1, 2}, B = {a, b}. Find some relations from A → B and B → A.

Sol:
Since relation from A to B is a subset of the Cartesian product A × B = {(1 , a) , (1, b) , (2 , a) , (2 , b)} any subset of A × B is a relation from A → B. Therefore{(1 , a), (1 , b), (2 , a), (2 , b)}, {(1, a), (1, b)}, {(1, b, (2, b)}, {(1 , a)} are some relations from A to B. Similarly any subset of B × A = {(a , 1), (a , 2), (b , 1), (b , 2)} is a relation from B to A. {(a , 1), (a , 2), (b , 1), (b , 2)}, {(a, 1), (b, 1)}, {(a, 2), (b, 1)} are some relations from B to A.

Ex 2: Evaluate the equation     2(-3x - 4) - (-x - 4) = -2(x + 1) + 2

Sol:
Given the equation  2(-3x - 4) - (-x - 4) = -2(x + 1) + 2
Multiply factors.
-6x - 8 + x + 4 = -2x - 2 +2
Grouping the terms.
-5x - 4 = -2x
-3x = 4
X = `(-4)/(3)`  is the answer to the math problem

Answer Sheet to Probelms on Math :

Ex 3: If x <4 -="-" 4="4" 5="5" nbsp="nbsp" p="p" simplify="simplify" x="x">
Sol:
Given the expression  |x - 4| - 5|-4|
If x <; 4 then x - 4 < 4 and if
x - 4 < 4 the |x - 4| = -(x - 4).
Substitute |x - 4| by -(x - 4) and |-4| by 4 .
lx - 4| - 5|-4|
= -(x - 4) -5(4)
= -x -16 is the answer to the math problem

Ex:4 Find the next two numbers in the number pattern.
 2, 6,18 , ___, ___,.
Sol:
Step 1: The given pattern is obtained by multiplying 3 to each number to get the next number in the pattern.
Step 2: The next two numbers in the pattern are 54,162.

Ans: 54,162 is the answer to the math problem

Mean frequency distribution

What is frequency distribution:
Frequency distribution is a term associated with statistics. In statistics, when we study some characteristic of a sample and then group the observations under various groups, a frequency distribution comes to play. A frequency distribution is a table showing the values of one or more variables and their respective frequencies of a given sample.

How to construct a frequency distribution:
In a univariate data, (univariate means that there is only one variable that we are interested in studying, the other factors are assumed to be constants) if x1,x2,x3,….xn are the possible values of the variable and f1,f2,f3,….fn are the frequencies with which each of the variables occur respectively. Then a table drawn as follows:
Variable (xi) Frequency (fi)
x1 f1
x2 f2
x3 f3
: :
: :
: :
xn fn
Is called a frequency distribution table of the data. Here i = 1,2,3,…..n. That means that the variable x can have n possible values and the frequency of each of the values are f. The above is a frequency distribution when the variable in question is discrete. For a continuous variable, the frequency distribution would look like this:
Class interval Frequency (fi)
C1 – C2 f1
C2 – C3 f2
C3 – C4 f3
: :
: :
: :
Cn-1 - Cn fn
Here, C1 – C2, C2 – C3 etc are the class intervals and f1,f2,f3… etc are the frequencies of the respective classes. The above distribution can be converted to a mean frequency distribution by replacing the class intervals with the respective class means.  We would need to do that to be able to easily work with the data. Its not exactly possible to calculate sample mean, sample standard deviations etc from a distribution having only class intervals. Therefore the mean frequency distribution would look like this:
Class interval Class mean Frequency (fi)
C1 – C2 (x1) ̅ f1
C2 – C3 (x2) ̅ f2
C3 – C4 (x3) ̅ f3
: : :
: : :
: : :
Cn-1 - Cn (xn) ̅ fn
Here, (x1,) ̅  (x2,) ̅  (x3,) ̅ …. (xn) ̅ etc are the respective class means.
The type of frequency distribution that we would construct would depend upon the frequency distribution example given to us. As we saw above, for a single variable, the distribution would be different if the variable is discrete and different if the variable is random. Similarly for more than one variables also the distributions would be different.

Statistical Mean a stepwise approach

Mean Statistics
In Statistics a branch of Mathematics, the expression for the mathematical mean of a statistical distribution is the mathematical average of all the terms in the data. Here we add up the values of the terms given and divide the sum by the number of terms in the data. This expression is also called the Arithmetic Mean. For example, let us find the Statistical Mean of the following data 6, 8, 5, 10, 10, 12, 8, 6. Here first we need to find the summation of the data values which would be 6 + 8 + 5 +10 + 10 +12+8 +6 = 65. The Arithmetic Mean is got by dividing this sum with the total number of data values. The total number of data values is 8 and so Mean = 65/8= 8.125

Mean Math of the given data is the average of the total number of given data values. It is very simple to calculate we just add up the data values and divide by the count of data values. When the give data values are positive, we just add them and divide by the count to get the mean. The mean of 5, 8, 10, 12, 15 would be, (5+8+10+12+15)/5 = 50/5= 10. When the given data has negative data values, the method would be the same except that we need to combine the like terms. For example, The mean of 5, -3, 7,       12, - 2 would be (5 +(-3) +7 + 12- 2). Combining the like terms, [5+7+12 +(-3-2)] = [24 – 5] = 19. The Mean would be, sum of the data values/total number of data values = 19/5 = 3.8

Short cut Method
A short cut method of calculating the arithmetic mean is based on the property of arithmetic average. In this method the deviations (D) of the items from an assumed mean are first calculated and then multiplied with their respective frequencies (f). Then, the total of these products [summation(fD)] is divided by the total frequencies [summation(f)]and added to the assumed mean(A). The figure we get is the actual arithmetic average or the Arithmetic Mean.

Formula used in the short-cut method of calculating the arithmetic mean:
X(bar) = A + summation(fD)/summation(f)
Given the following data, calculate the Mean using the short cut Method
Weight(Kg) 68 70 72 74 76
Number of students 3 4 2 1 2
Summation of f(i) = 3+4+2+1+2 = 12

Let us assumed mean A= 72, let us tabulate the deviation
X(i) – A f(i).x(i)
(68 – 72) = -4 -4x 3 = -12
(70-72) = -2 -2x4  = -8
(72 – 72) = 0 0 x 2 = 0
(74 – 72) = 2 2 x 1 = 2
(76 – 72) = 4 4 x 2 = 8

Summation(f(i)x(i) = -10
So, we have, summation f(i) = 12, Summation(f(i)x(i) = -10 and A = 72
X(bar) = 72 + (1/12)(-10) = 72 – 0.833 =  71.17 kg

Derivatives of Inverse Functions made simple

Derivatives of Inverse Functions can be understood easily when we first learn about inverse functions. Consider an example, f(x) = 2x and g(x)=x/2  here, the first function doubles the input values and the second function halves the input values, they are going in opposite direction. These functions f and g are called inverse functions. The inverse is showed by using ‘-1’ in the power. It does not mean x-1=1/x as in exponential functions, it just denotes that the function is the inverse of the original given function.

An Inverse Function Solver helps us to find the inverse function of the given function. We can find the inverse function y in terms of x by first solving for x and then interchanging the x and y.  The function we get by doing so will be the inverse function of the original given function. We can also find inverse function solver online.  For better understanding let us consider Inverse Function Examples as given:

Example: Find the inverse of y= 2x-3
Solution: f(x) = y = 2x-3 [original function]
Solving for x, we get
2x = y+3
x = (y+3)/2
to  get the inverse function we interchange the x, y in the above step
here, y = y-1 =(x+3)/2  is the required inverse function of 2x-3

While Graphing Inverse Functions we use the same method as we use in graphing functions. The only difference being, first we find the inverse of the given function which will be taken as y and then various x values are plugged in, to get the coordinates(x,y). These coordinates are plotted on the graph which when joined give the required graph of inverse function.
Now that we have learnt about inverse functions, let us learn about Derivative of Inverse Function. If f(x) and g(x) are inverse functions then, the derivative of inverse function is given by the formula:
g’(x) = 1/f’[g(x)]

To find the derivative of inverse function of f(x) = 2x-1, first we need to find the inverse function of f(x), which is, f-1(x)=(x+1)/2. In the next step we find the derivative of this inverse function. [f-1(x)]’= d/dx[x/2 +1/2]=1/2 [applying the derivative rule] . So, the derivative of inverse function of f(x)=2x-1 is 1/2

Let us now go through Inverse Sine Function which is given as, y= sin-1(x) which implies sin(y)=x where y lies between – pi/2 and pi/2. Evaluating an inverse sine function is same as asking what angle we need to plug into the sine function to get the value x. Let us evaluate sin-1[1/sqrt(2)]. Here we are asked for what angle ‘ y’ we arrive to the value 1/sqrt(2), written as sin(y) = 1/sqrt(2). We have already learnt that sin(45) gives us the value 1/sqrt(2) and hence we get, y = pi/4.
Learn more about how to solve Calculus Problems.

Trigonometric function integrals tables

Trigonometric functions Integrals: If u is a differentiable function of x , then sin u is a differentiable  of x . The chain rule gives the derivative of sin u as d/dx sin u = cos u du/dx . From another  point of view , however , this same equation says that sin u is one of the  antiderivatives  of the product  cos u .(du/dx) .
Therefore , integration (cos u du/dx ) dx = sin u + C .

A formal cancellation of the dx’s in the integral on the left leads to the following rule . If u is a differentiable function , then  integration  cos u du = Sin u + c .

Let us take an example of Trigonometric Integral  top understand the concept . suppose vwe have given to integrate cos ( 7 (theta) +5) d(theta) so , to solve this let us take u = 7 (theta) + 5 , du = 7 d(theta) => 1/7 du = d(theta)  so plugging in the problem we have  integration  cos u . 1/7 du = 1/7 integration cos u du = 1/7 sin u + c  .

The chain rule formulae for the derivatives of the tangent , cotangent , secant , and cosecant of a differentiable function  will give us the following Trigonometric Integrals Table:

Table of Trigonometric Integrals
(i) Integration of sin  x dx = cos x + c  ,  Extension sin( ax +b) dx = -cos (ax+b) / a+ c
(ii) Integration of cos x dx = sin x + c  , extension integration cos (ax + b) = sin (ax +b) / a+  c
(iii) Integration tan x dx = -log mod (cos x ) + c , extension integration tan ( ax+ b) = -log mod ( cos (ax +b)/ a+ c
(iv) Integration  cot x dx = - cosec ^2 x  + c , extension  integration cot (ax +b) = -cosec^2(ax +b) / a+ c
(v) Integration cosec x dx = log mod (cosec x  - cot x ) + c , extension integration cosec (ax +b) dx = - log mod( cosec (ax +b) – cot (ax +b))/ a+ c
(vi) Integration sec x dx = log mod (sec x + tanx ) + c  , extension sec  (ax + b) dx = log mod sec (ax +b) + tan (ax +b)/a + c
(vii) Integration sec x . tan x = sec x + c , extension  integration sec (ax + b).tan (ax+b) dx = sec (ax +b)/ a+ c
(viii) Integration cosec x . cot x dx –cosec x + c , extension cosec (ax + b) . cot (ax + b) x = -cosec (ax +b) / a+ c

Integration formula

The following is a non comprehensive list of integration formulas:

Understanding logarithmic functions

The function f(x) = a^x is a one to one function provided that a > 0 and a =! 1. Therefore f would have an inverse which we will call the logarithmic function.

What are logarithmic functions?
If a > 0 and a =! 1, then the function  log_a(x), called the logarithm of x to the base a, is the inverse of the one to one function a^x:
Y = log_a(x)  x = a^y, (a > 0, a =! 1)
Since a^x has the domain (-inf, inf), log_a(x) has the range (-inf, inf). Since a^x has the range (0,inf), log_a(x) has the domain (0, inf). Since a^x and log_a(x) are inverse functions, the following cancellation identities hold:
log_a (a^x) = x for all real x and a^(log_a x) = x for all x > 0.
The graphs of some examples of logarithmic functions are shown in the picture (a) below. They all pass through the point (0,1). Each graph is the reflection in the line y = x of the corresponding exponential graph in the picture (b).

From the laws of exponents we can derive the following laws of logarithmic functions.
Laws of logarithms: If x > 0, y > 0, a > 0, b > 0, a =! 1, b =! 1, then
(i) log_a(1) = 0
Proof: We know that a^0 = 1 for any real number a. Therefore by the definition of log we can say that log_a(1) = 0. Hence proved.
(ii) log_a(xy) = log_a (x) + log_a(y)
Proof: Let u = log_a(x) and v = log_a(y). By defining property of inverse of log function we know that x = a^u and y = a^v. Thus xy = a^u * a^v = a^(u+v). Inverting again we have, log_a(xy) = u + v = log_a(x) + log_a(y). Hence proved!
(iii) log_a(1/x) = - log_a(x)
(iv) log_a(x/y) = log_a(x) - log_a(y)
Proof: Let u = log_a(x) and v = log_a(y). Then again by defining property of inverse of log function we know that x = a^u and y = a^y. Thus a^u/a^v = a^(u-v). Inverting again we have, log_a(x/y) = u - v = log_a(x) - log_a(y). Hence proved. The rule number (iii) can be proved in the same manner.
(v) log_a(x^y) = y log_a(x)
(vi) log_a(x) = (log_b x)/(  log_b a)
The logarithm law (vi) presented above shows that if you know the logarithms to a particular base b, you can calculate the logarithms to any other base a.

Some standard derivatives: Derivative of csc function

CSC is the acronym for cosecant function. Csc is the reciprocal of sine function. Symbolically it can be written as : csc (x) = 1/sin(x).

Derivative of csc function:
The derivative of csc function is the slope of tangent to the curve of the equation y = csc (x) at any point x. It can also be called the gradient of the csc function. To find the derivative of the function csc (x) we use the quotient rule which is as follows:
If a function f is such that it is a quotient of two functions g and h, symbolically,
f(x) = h(x)/g(x), then the derivative of f, represented by f’(x) is given by the formula,
f’(x) = [g(x)*h’(x) – h(x)*g’(x)]/(g(x))^2
Using the above rule we can find the derivative of csc function as follows:
(d/dx) csc (x) = (d/dx) (1/sin x) = [sin x * (d/dx)(1) – 1* (d/dx) (sin x)]/sin ^2 (x)
= -cos x/sin^2(x)
= -csc x cot x
Therefore derivative of csc is –csc x cot x.

Derivative of csc^-1 (x):
Inverse of the cosecant function is written as csc^(-1) (x). The derivative of the inverse of csc function can be found as follows:
Let y = csc^-1 (x), |x| >1
Thus, x = csc y, y belongs to (0,pi) – {pi/2}
So, dx/dy = - csc y cot y ? 0 because csc y ? 0 and since y belongs to (0,pi) – {pi/2}, cot y ? 0
Therefore, dy/dx = -1/(csc y cot y)
Since y belongs to (0,pi) – {pi/2} that means, either y belongs to (0,pi/2) or y belongs to (pi/2 , pi)
In both cases the following holds.
Then x = csc y > 0 and so |x| = x
Also cot y > 0 and so cot y = sqrt(csc^2 (x) – 1) = sqrt(x^2 - 1)
Therefore dy/dx = -1/x*sqrt(x^2 – 1)
Thus derivative of csc^-1 (x) = -1/x*sqrt(x^2 – 1)

Derivative of csc^squared (x):
Derivative of csc^2 (x) can be found using the chain rule.
Let y = csc^2 (x) and let csc x = u
Then y = u^2
Therefore dy/du = 2u
And since u  = csc x
So, du/dx = -csc x  cot x
Thus dy/dx = dy/du * du/dx
= 2u * (-csc x cot x)
= 2*csc x * (-csc x cot x)
= -2 csc^2 (x) cot x
Thus we see that derivative of csc^2 (x) = -2 csc^2 (x) cot x

Vertical Asymptotes: The Vertical Lines that never intersect the function

The vertical lines that correspond to the zeroes of the denominator of a rational function are called the Vertical Asymptotes. They occur only for those values of ‘x’ that produce zero in the denominator but not in the numerator. If  0/0 occurs, then we simply say we have a ‘hole’ in the graph. The next thing that comes to the mind is, How to find Vertical Asymptote of a given function. It involves a simple method where we just need to follow few steps to find all the values of x for which the denominator equals zero.

Following are the steps involved in finding the Vertical Asymptotes:
1. Vertical asymptote is in the form of an equation in x, x=a where f(x) is a function and f(a) does not exist.  A vertical asymptote is a vertical line which never intersects the function f(x).
2. We know that a fraction is undefined if its denominator is equal to zero. So, to find the vertical asymptote of a rational function, we need to  find the value of x such that the denominator is equal to zero.
3. In the next step, we need to equate the denominator to zero.
4. Then, we need to solve the equation in x to get the value of a.
5. Finally we need to plug in the value of a in the equation x=a, which is the vertical asymptote of the given function.

Note:  We can get more than one vertical asymptote depending on the given function

Finding Vertical Asymptote of a rational function, f(x) =(x^2+2x+3)/(x^2-5x+6). Let us first find all the x values by setting the denominator (x^2-5x+6) equal to zero and solving for ‘x’. Factorizing (x^2-5x+6) we get (x-6)(x+1)=0 and hence x = 6 and x=-1 which will make the denominator zero. Hence the vertical asymptotes of the given function are, x = 6 and x=-1.

Let us now Find Vertical and Horizontal Asymptotes of f(x)= (2x^2-5x+3)/x^2-1. The vertical asymptote is got by equating the denominator to zero, x^2-1=0. On factorization, we get (x+1)(x-1)=0. This gives us x=1 and x=-1 as the vertical asymptotes. To find the horizontal asymptote we need to first find the degree of the numerator and the denominator. Here the degree of the numerator and the denominator is the same, so, the horizontal asymptote is given by y= the coefficient of the highest degree in the numerator divided by the coefficient of the highest degree in the denominator. That gives us y=2/1=2, which is the horizontal asymptote.

Know more about the Asymptote Calculator. This article gives basic information about Asymptotes.

Introduction to Statistics

Introduction Statistics: The word statistics seems to have been derived from the Latin word ‘status’ or the Italian word ‘statista’ or the German word ‘statistik’ or the French word ‘statistique’, each of which means a political state.  In a short statistic summary we can understand the Importance of statistics in some different disciplines like

Statistics in planning: Statistics in indispensable planning – may it be in business, economics or government level.The modern age is termed as ‘the age of planning’ and almost all the organisations in the government or business or management are resorting to planning for efficient working and for formulating policy decisions. 

Statistics in states: As has already been pointed out, in the old days Statistics was the science of State-craft and its objective was to collect data relating to manpower, crimes, income and wealth etc. for formulating suitable military and fiscal policies.

Statistics in Mathematics: Statistics is intimately related to and essentially dependent upon mathematics.

Statistics in economics: The interaction between Statistics and Economics was first observed by William Petty in his book but it took fairly long time for effective use of Statistics in formulation of economic theories and economic policies.

Statistics in business and management: Prior to industry revolution, when the production was at the handicraft stage, the business activities were very much limited and were confined only to small units operating in their own areas.

Statistics in accountancy and Auditing: Today, the science of Statistics has assumed such unprecedented dimensions that even subjects like Accountancy and Auditing have not escaped its domain.

Statistics in industry: In industry, statistics is extensively used in Quality Control. The main objective in any production processes it to control the quality of the manufactured product so that it conforms to specifications. Statistical tools are widely used by business enterprises for the promotion of new business. Before embarking upon any production process, the business house must have an idea about the quantum of the product to be manufactures. 


Statistics in Insurance: Probability theory on which modern theory of statistics is based is the backbone of the Insurance.


Statistics in Astronomy: Even in the ancient past the astronomers made recordings about the movements of heavenly bodies like stars and planets for the study of eclipses.


Statistics in Physical Sciences: The application of Statistics in Astronomy, which is a physical science, has already been discussed. In physical sciences, a large number of measurements are taken on the same item. 

Statistics Summary:
In short summary statistics, we can more learn that In the ancient times the scope of Statistics was primarily limited to the collection of the following data by the governments for framing military and fiscal policies:
(i) Age and sex wise population of the country
(ii) Property and wealth of the country. The former enabling the government to have an idea of the manpower of the country (in order to safeguard itself against any outside aggression) and the latter providing it with information for the introduction of new taxes and levies.