Statistics analysis:- - Statistics analysis is associated with the methods for collection, classification and analysis of numerical data for drawing valid conclusions and making the reasonable decisions. In the field of production engineering and in the analysis of the data used for experiments, It has the meaningful applications. There are the two parts of the statistical analysis, measure of central tendency and probability.  The study of certain numerical representative of the ungrouped data, namely, mean, median, mode is associated with the Central Tendency. There are two main requirements of the statistical analysis, Quality and survey.

We need the population and sample to find the quality and survey. The item name is known as the population and the sample is a portion of the population. The items or the population are scattered here and there covering thousands of kilometers. Secondly one item or population can be produced by the number of people in our country or any other countries. An international organization ISI has been formed to keep the check on all the populations. The quality measures are mean, median, mode, range, percentile, deviations, variance etc.

Introduction to numerical analysis: - The study of the algorithm to find the numerical approximations in the mathematical analysis is called the numerical analysis. It is used to find the most approximate roots of the given equations in the discrete mathematics. It is used to design and analysis the techniques to find the approximate and accurate solutions of the hard problems. It is also beneficial to find the feasibility of numerical whether prediction. The numerical analysis is associated with the numbers. It can be used in the ordinary differential equations to find the accurate numerical solutions to compute the trajectory of a space craft.

Numerical analysis can be used to find the depth of the ocean.  It can be used in the partial differential equations to find the accurate numerical solutions to simulate the car crashes.  Numerical analysis methods can be used to find the value of stocks and the derivatives more accurately in private investment funds. 

To find the optimization algorithms to decide the fuel needs, ticket prices, crew assignments and in airplanes it is used in the overlapping field of the operation research. For actuarial analysis in the insurance companies, the numerical analysis programs are used.  The direct and the iterative methods are used to find the approximate roots of the given equations. There are many methods which are used to find the more appropriate roots of the given equations.  To find the solution in finite number of steps the direct methods are used.

For solving the system of linear equations which includes the simple method of linear programming the direct method is used. To find the solution of the true approximations the finite precision is used.  The iterative methods are not terminated in few steps. They start from the initial guess and then iterative methods are used for more accurate approximation. The Newton’s method, the bisection methods, Jacobi iteration methods are used in the iterative method.

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21
Okt
Permutations

Define Permutation (Read more): - Permutation means the arrangement. Each of the arrangements which can be made by taking some or all of a number of things is called a permutation.  For example, suppose there are three quantities a, b and c. the different orders of arrangements of these three will be six. The permutation of the three quantities can be given as abc, bca, cba, acb, bac and cba. Let four persons enter a train compartment where there are six seats. How these four persons can be arranged on the six seats, this science or calculation is known as the permutation or the arrangement. The required arrangement is 6p4  which is equal to thirty.


Let us to find the number of permutations of n things taken all together, when the things are not all different.  Let the   n  things be represented by  n  letters and suppose   P  of  them to be a’s,  q of them to  be b’s,  r   of them to be c’s, and  the rest  to be unlike. Let N represent the required number of permutations. Then, if in any one of these permutations, the  P  letters,  a  were  changed into P unlike letters different from any of the rest,  then without altering the position of the remaining letter, from this single permutation alone we could form p different permutations. Hence if this change were made in each of N permutations the whole number of permutation would be N fact P. Similarly, if in each of these new N fact P permutations, the q  letters, b, were changed in to q unlike letters, different from any of the rest, then by rearranging q   letters alone amongst themselves, in all possible ways, each would produce factq different permutations. 

Hence the total number of permutations when these changes are made) would be N. fact p. fact q , Likewise,   in each of these N.fact pfactq  permutations, again ,  r  letters C  were changed into  r  unlike letters, different from any of the rest, the whole number of permutations would be N fact p fact q fact r. Thus, when all the letter are different from one another the total number of permutations

N   =        fact n / fact P fact q fact r

Permutation Definition (Read more): - The number of permutations of n same as the number ways in which r places can be filled up by n different things. As many one of the n things can be put on the first place. It can be filled up in b ways, the second place can be filled up in (n – 1) ways. Since each way of filling up the first place can be associated with each way of filling up the second place.

Thus, the first two places can be filled up in (n- 1) ways, the third place can be filled up by any one of the remaining (n- 2) different ways. Hence, the number of different ways in which the first three places can be filled up in n (n- 1) (n – 2) ways.     Proceeding in this way, we find that total number of ways of filling up any number of places in equal to the product of same number of factors.

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In this article we will understand the Integral of E^x, Integral Calculator Step by Step (Read more).The Integral  function is nothing but the inverse of the differentiation and these both play an important role in calculus.

Integral of E^x and Integral Calculator Step by Step

We will see how the integral is evolved with a simple definition.
Let E be a subset of R. let f: E→ R be a function, if there is a function F on E such that F'(x) = f(x) for all x belongs to E, then we call F an anti derivative or integral of f or a primitive of f.

Let us see an example:
Derivative of the trigonometric function (sin x) = cos x, where x belongs to R. hence if f is the function given by f(x) = cos x, x belongs to R, then the function F given by F(x) =sinx, x belongs to R is an integral of f on E, then for any real number k,

we have (F + K)'(x) = f(x) for all x belongs to E
Here, F + K is also an integral of f, thus in the above example, if c is any real constant then the function G is given by
G(x) = sin x +c, x belongs to R is also an integral of cos x.

The standard forms and properties of integrals are :

1.    We know that derivative of (xn+1/n+1) = xn for n not equal to -1. Hence, if n not equal to -1, we have ∫xn dx = (xn+1/n+1) + c , where c is a constant.

2.    We know that derivative of (log x) = 1/x if x > 0. Hence, ∫1/x dx = log ǀxǀ +c where c is a constant .

Some of the properties of integrals are:
1.    since the functions f and g are have integrals on I ,then f + g has an integral on I  and then

∫(f + g)(x) dx = ∫f(x) dx + ∫g(x) dx +c.

∫(kf)(x) dx = k∫f(x) dx +c

Integral of E^x

1.    As we know that  the differentiation of ex  is equal to ex.

2.    Now apply the integral on the d/dx ( ex)

3.    As we all know that the differentiation and integration are inverse to each other so both will cancel .

4.    The remaining ex + c is the answer .where c is the constant of integration for all indefinite integrals .

Therefore integral of  ex is equal to ex +c. Here is an example problem based on the integral of e x:

Solve integral of 5ex dx?

Solution:

Integral Calculator Step by Step

We know that  differentiation of  ex is ex so
d/dx 5 ex = 5 ex. Now applying integration on ∫5 ex dx

As we know that integration and differentiation are inverse and they both cancel, giving us the result as
∫5ex dx = 5∫ex dx  = 5ex +c  where c is the constant of integration.

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A number that can be written as a fraction involving only integers is called a rational number or 'R". Examples of rational numbers (Read more) are ½, -4/3, 7/9, 102/5, -124, 5568 etc. There can be infinitely many rational numbers. It is possible to insert any number of rational (abbreviated as r’al) number (abbreviated as num) between any two rational numbers. In spite of that, the set of rational numbers is not a complete set. That means if they are plotted on a num line, there still would be gaps that need to be filled, but there are no rational numbers available to do so.  In other words, if we  have a continuous line that extends to infinity on both sides, there would be points on this line, for which there is no r’al num. The set of rational number is denoted by the alphabet Q.

Properties of rational numbers:
1. Property of closure: The set of rational numbers is closed under addition, subtraction and multiplication. That means, for any two or more rational numbers, their sum, differences as well as their products are also always rational numbers.
2. Commutative property: The process of adding or multiplying rational numbers is commutative in nature. That means for two r’al num, a and b, a + b is same as b + a and a * b is same as b * a.
3. Associative property (Read more): When multiplying or adding r’al nums, the associative property holds. Thus, for any three numbers a, b and c, a + (b+c) is the same as (a+b) + c and a * (b*c) is the same as (a*b) * c.
4. Additive identity property: This is the property by which we say that the when zero is added to any "R" num, then the value of that num does not change. Thus the additive identity of "R" nums is 0.
5. Multiplicative identity property: According to this property, the multiplicative identity of r’al nums is 1. Any "R" num multiplied by 1 yields the same "R" num, thereby not altering its value.
6. Additive inverse property: For any r’al number p/q, its additive inverse would be –p/q. Thus p/q + (-p/q) = 0.
7. Multiplicative inverse property: For any such num p/q, its multiplicative inverse would be q/p. This holds only if neither of p and q are equal to zero. The q/p is also called the reciprocal of p/q and vice versa. p/q * q/p = 1. Thus product of a "R" num and its reciprocal would always be 1.
8. Distributive property: For any three random r’al nums, a, b and c, the product of a * (b+c) is the same as the sum of the products a*b + a*c. Thus, a * (b+c) = a*b + a*c and conversely, a*b + a*c = a* (b+c). This is called the distributive property.

Note that r'al nums are not commutative under subtraction. That means a - b is not the same as b - a for any two r'al numbers a and b.

Examples of probability (Read more): This is a different topic altogether, thus will be discussed under a separate head.

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A closed figure that is formed by three line segments such a way that no two segments meet each other more than once. This two dimensional figure is known as a triangle. The point where the two segments meet each other is known as the vertex. There are three angles and three sides for a triangle.

Triangle Types (Read more)

They can be classified based on the angles and sides.
There are three types based on sides. They are scalene, Isosceles and equilateral.
There are three types based on angles. They are acute angled, right angled and obtuse angled.
In this article we shall learn more about a right triangle.

What is a Right Triangle? (Read more)

It is a triangle in which one of its angle measures 90 degrees. It cannot have two right angles because the sum of two right angles is 180 degrees whereas the sum of three angles of a triangle is 180 degrees. From this we understand that the other two angles of a right angled triangle are acute angles (measure of the angles is between 0 and 90)

In a right triangle the side that is opposite the angle 90 degrees is known as hypotenuse. The other two sides that contain the right angle are known as the legs of the triangle.

There is a famous rule that connects all these three sides. That is square on the hypotenuse is equal to the sum of the squares of the two legs. This is used to find the unknown side while the two are given.

Example: 1

Find the unknown side of a right triangle whose hypotenuse (Read more) is 25 cm and one of its leg measures 7 cm.

Solution:
We know that (hypotenuse)2 = (Leg 1)2 + (Leg 2)2
Here Hypotenuse = 25 cm and let Leg 1 = 7 cm, Leg 2 = x Substituting these values in the above formula we get
(25)2 = 72 + x2
625 = 49 + x2
625 – 49 = x2
576 = x2
x = 24
Therefore, the unknown side = 24 cm

Special Cases:

When the angles measure 30 degrees, 60 degrees and 90 degrees. That is the angles are in the ratio 1: 2:3. In this the sides of such a triangle are in the ratio 1: √3 : 2.

The other special case is when the angles measure 45 degrees, 45 degrees and 90 degrees. That is the angles are in the ratio 1: 1: 2. In this the sides are in the ratio 1: 1: √2. This is also known as isosceles right triangle.

Let us see few examples based on the special cases

Example:

Find the missing sides of the triangle given below.


Solution:

The given three sided figure a special case. Here we see that the angles are in the ratio 1: 2: 3. Therefore the sides are in the ratio 1: √3 : 2.

To find y, we do 1: 2 = 6: y
That is y = 12 cm

To find x, we do 1: sqrt(3) = 6: x
That is x = 6√3 cm
 

Example:

Find the missing sides of the triangle given below.

 
Solution:

The given three sided figure a special case. Here we see that the angles are in the ratio 1: 1: 2. Therefore the sides are in the ratio 1: 1: √2.

To find a, we do 1: 1 = 3: a
That is a = 3 cm

To find b, we do 1: √2 = 3: b
That is b = 3√2 cm

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26
Aug
Percentages

Percent comes from the Latin word “Per Centum”. Centum means 100, so percent means for every hundred. For example, the passing percentage in an exam is 65% means 65 out of 100 students have passed the exam. Percent can be expressed as ratio and also in decimal. For example 65%=65/100=0.65. If we want to find percentage of 70 books out of 350 books, then we have to find ratio of 70/350=0.2 and then 0.2 is multiplied by 100 to get 20%, so 20% represents 70 books out of 350 books. So decimals, fractions and percentages are inter-related and can be converted to each other. Percentages are used in everyday life. When we go to shops, we see the discount sales put in the form of percent. For example, discount of 25% , discount of 50% and so on. Percentages are used in various fields like in commission, mark up, sales tax, rate of interest, and so on. Let us now see how the percentages, fractions and decimals are inter-related.
1)    Converting a fraction to a percent- To do this divide numerator by denominator and then multiply by 100. The answer with the % sign gives the percentage. For example: ¼ =0.25*100 =25%
2)    Converting a percentage to a fraction- To do this eliminate the % sign and now the given number divided by 100 gives the desired fraction. The fraction can be simplified if possible. For example: 25% = 25/100 = ¼
3)    Converting decimals to percentages- To do this multiply the given decimal by 100 and keep the percent sign. For example: 0.25 = 0.25 * 100 = 25%
4)    Converting percentages to decimals- To do this eliminate the % sign and divide the number by 100 using long division process. The quotient is the desired answer. For example: 35% = 35/100 = 0.35

Examples:

1)Find 14% of 850.
Solution:
14% of 850 = 0.14 times 850 = 119.

2) What percent of 900 is 30?
Solution: 
Let the percent be x.
X% of 900 = 30
(x/100) * 900 = 30
x/100 =30/900
x = ( 30/900) x 100 = 3.33
so 3.33% of 900 is 30.

Let us now understand how to find perimeter of a triangle. A triangle is the smallest polygon which is composed of 3 sides. Perimeter of a triangle is the sum of all 3 sides. So perimeter is found only when all 3 sides are known. In general, if a, b and c are side lengths of a triangle, then perimeter is given by a+ b+ c. For example: the perimeter of a triangle with sides 5cm, 6cm, 7cm is 5+6+7=18cm.
In case of right triangle, perimeter can be found even if we only two sides are known. This is because we can find the third side of the right triangle using Pythagorean Theorem. Pythagorean Theorem states that sum of square of legs is equal to the square of the hypotenuse. The formula is a2+b2=c2, where c is the hypotenuse, a and b are the legs of the right triangle.

Problem 1: Find the perimeter of right triangle whose legs are 3 cms and 4cms.
Solution: we know that a = 3 cms and b= 4 cms. We have to find c using the Pythagorean theorem formula a2+b2=c2 .
32 +42= c2
9+16= c2
25= c2
C= 5.
So perimeter = a+ b+ c= 3+4+5 = 12 cms.

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Application of second derivative

In calculus, we know that if we want to find the slope of a tangent at any given point on the graph of a function, we find the derivative of that function. This derivative that we find is called the first derivative of that function. The first derivative of the function besides giving us the slope of the tangent at a particular point, also gives us the critical point. When we equate the first derivative f’(x) of a function y = f(x), to zero and then solve for x, we get some values of x. These values of x are called critical points. They are denoted by c. These critical points divide the x axis into various intervals. Next we use the first derivative to find the intervals in which the function is increasing and the intervals in which the function is decreasing. At the critical points, f’(x) = 0, therefore at these points the function would have either a local maximum or a local minimum. The point at which the slope of a tangent to the curve is 0 is called the critical point. Thus at the critical point, the tangent would be horizontal. Therefore summarizing we can say that,

Critical point occurs when f’(x) = 0 or f’(x) does not exist.

Function f(x) is increasing in the interval where f’(x) > 0 and it is decreasing in the interval where f’(x) < 0.

That sums up how we use the first derivative for understanding the graph of a function. Now let us take a look at how the second derivative also helps us to understand the graph of a function.

Just like how the first derivative helps us to know where a function is increasing or decreasing, the second derivative helps us to know where a function is concave up and where is it concave down. When f’’(x) > 0, the function in that interval is concave up and if f”(x) < 0, then in that interval the function is concave down. The point at which the concavity of the function changes from concave up to concave down or from concave down to concave up is called the inflection point. As in the case of first derivative, when we equated f’(x) to zero and solved for x we got the critical point. In the same way, in case of the second derivative, when we equate f”(x) to zero and solve for x we get the x co ordinate of the point of inflection. To find the y co ordinate of the point we substitute the x co ordinate thus found into the original function f(x).

Consider the following example.

Find the inflection point for the function f(x) = x^4 + 2x^3 – 12x^2

Solution: First we find f’(x) for the given f(x). Therefore,

f(x) = x^4 + 2x^3 – 12x^2

f’(x) = 4x^3 + 6x^2 – 24x

Next we find the second derivative. Thus we get,

f”(x) = 12x^2 + 12x – 24

Now we set the f”(x) = 0. So,

12x^2 + 12x – 24 = 0

X^2 + x – 2 = 0

(x+2)(x-1) = 0

X = -2, x = 1

(-2, -48), (1, -9)

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Rules of inequalities I

 

 

 

 

 

Solving Inequalities Rules


A mathematical expression which involves symbols >, <, ≥ and ≤ is an inequality. To solve such we need go find a range of values that an unknown variable can take and then satisfy the in-equality. The rules for solving such are that, if the same number is added to both sides of an in-equal form, the form remains true.

If the same number is subtracted from both sides of the inequality, the in-equality remains true. If multiplication is done by the positive number, the in-equality remains true. But if multiplication or division of such an form is done by a negative number, then it is no longer true. In fact, the relation becomes reversed. These are explained briefly with examples below.


 

Rules for Inequalities

While solving an such problem there are inequality rules that needs to be followed. If the form involves addition or subtraction then we add or subtract by the same number both sides in order to have the variable to one side.

For example take x – 5 > 12. Here we have -5 so we add +5 both sides and get x > 17. Take another example x +6 > 15. Here we subtract 6 both sides. We get, x > 9.


If it involves multiplying then we have to divide both sides by the same number. For example take 4x >12.

So divide by 4 both sides we get x > 3. If it involves division then we multiply both sides.

x/4 > 12. Here multiply by 4 both sides. We get x > 48.


 

When Multiplication is done by a negative number the sign of the in-equality should be flipped x/-3 > 12. Here we times both by -3. So x < -36. The sign of the in-equality is flipped.

The > sign is now changed to <. Similarly when we divide by a negative number we flip the sign. Take -5x > 35. Divide by -5 both sides and flip the relation. X < -7. Here also the > sign is flipped t the < sign.



While graphing inequalities if the relation has a < or > sign when we have to draw a dotted line. This indicates that the line is not a part of the solution. If the in-equality is less than, or equal to and or greater than / equal to then we use a solid line for graphing. This indicates that the line contains solutions for the inequality. These are the rules of Solving Inequalities

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15
Jul
Cramers Rule

Cramer's Rule is a usefull tool to solve a set of simultaneous equations. This tool uses Cramer's Rules and formulas to solve these equations by converting them into a matrix form. This matrix form maybe like Ax =B. Each value of the variable is solved by using the determinants. The value of the variable is calculated separately and directly. Let us consider an example. Suppose we have N number of linear equations have n number of unknowns and the value of det(A) is not zero. This system will have a unique solution. The value of unknown x is given by the formula
Xn = det(An)/ det(A).


In this method we use the coofficents of the linear equations and make a matrix A. This way it becomes very easy to solve complex collection of equations. These are many steps involved in solving sets of equations using Cramer's Rules. The 1st step is to read the problem and see if the given matrix is a square matrix or not. Then the determinant of the given matrix is calculated and check if the value of A is not zero. If these 2 points are true then we will be able to apply Cramer's Rules. The next step is to replaces the 1st column by the elements which are on the right hand side of the equation that gives us determinant D1. Then D2 is got by replacing 2nd column with the elements in the right. Then by D1/D we get the variable x and D2/D will give value of variable y. We keep repeating these steps for other unknown variables.


Trapezoidal Rule is a useful method to find the exact value of a given definite integral. This is a numerical method and based on Newton-Cotes formula. The Newton-cotes formula states that the exact value of the integral can be calculated as nth order polynomial. This nth order polynomial integral gives us the exact value of the function. Lets say the n is one, in that case Trapezoidal Rule says that linear polynomial’s area will be given as
ʃba f(x) dx = ( b - a ) [ { f(a) + f(b) } / 2]
Lets takeup an example of Trapezoidal Rule. Now solve ʃ10 xdx using 4 sub-intervals for comparing actual and estimated value. In this we will find the change of x by b-a/n. From the given problem we get, ( 1 – 0 ) / 4 = ¼ = 0.25. Now we find the integral replacing the values into the formula we get 0.3125. Then since the actual value is 0.5, calculated by finding integral, we can find the percentage error. This percentage error is found by substracting the estimated values from the actual, divide that by actual and times 100. This formula is applied and we get 37.5%.
This way both Cramer's Rules and Trapezoidal Rule and helpful tools to calculate or solve problems. This tool is very effective and also helps in find the solutions of typical problems.

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Process of Graphing Rational Functions

There are various number systems in mathematics. The number systems can be like the natural number system, whole number system and so on. The rational number system is one of the important number system in mathematics. The concept of function is very important in mathematics. They can be graphically represented as well. The rational numbers graphing can be very interesting to learn. But to understand this one must be thorough with the concept of functions. The function has both inputs and outputs. The input is transformed to get the output. A function cannot take all values. There will be restriction on the values a function can take. This is a very important concept. The concept of range and domain must be clear in order to understand the concept of function. At some values the function may not be defined. This means that the function will not have definite value at these points in a given particular interval on numbers.

The rational numbers have a numerator and a denominator. This is one of the properties of a rational number but the denominator must not be equal to the number zero. This is because if the number in the denominator is equal to the number zero then the rational number will become infinity and hence cannot be defined. The task of graphing general rational functions can be a easy task if the concepts are clear. So, graphing rational functions practice is required in order to get a hold on the subject. Once the concepts are clear one can easily solve problems related to this field. Graphs help in better understanding of a concept. They are pictorial representations and hence make the understanding clear. Most of the concepts explained with the help of graphs will help in better understanding.

The graph basically contains the horizontal axis and the vertical axis. The horizontal is called the x-axis and the vertical axis is called the y-axis. There are also three dimensional spaces in which the graphs can be drawn. The graph of rational function can be drawn in both the spaces. Once the graph the interpretation of the graph must be done, only then the concept will become clear. The interpretation of a graph is very important and can help in solving many problems. The interpretations can lead to the solutions of many problems. The graph has to be kept in mind that the value of the denominator cannot become zero. At this stage the function cannot be defined. So, one has to be very careful about this fact. The concept of asymptotes and intercepts must be clear in order to understand the process of graphing. So, in order to graph the rational functions one must learn the concepts of asymptotes and intercepts very clearly. If this is not done the process of graphing can become a difficult task and can also lead to many mistakes in the process of drawing the graph. But with the concepts clear the graph can be easily drawn.

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