Friday, May 31, 2013

Gradient In Math

Introduction of gradient in math:

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.A generalization of the gradient for functions on a Euclidean space which have values in another Euclidean space is the Jacobian.


Example problem for gradient in math:-


Problem :-

The three variables function is x axis, y axis and z axis the region of three dimensional space it is normally show a f(x,y,z) these is called as scalar field and P is a point and specified direction is no need for the x axis, y axis and z axis. You need to calculate and any change in f is called as directional derivative.

Solution:-

Well, let's start by letting R=x0i+y0j+z0k be the position vector for P. Let the specified direction that we want to move from P be given by the unit vector u = u1i + u2j + u3k. Let Q=(x + x, y + y, z + z) be a point along with the vector in a specified direction. Let Deltas be the scalar value such that vecPQ = Deltashatu, that is s is the length of  vecPQ. In following formula for gradient in math

`vecPQ = Deltaxi+ Deltayj+ Deltazk = Deltasu_1i+ Deltasu_2j+ Deltasu_3k`

Let `deltaf ` = f(Q) - f(P). By linear approximation,

`deltaf = fx(P)delta x + fy(P)delta y + fz(P)delta z + erfx(Q)delta x + erfy(Q)delta y + erfz(Q)delta z`
`deltaf = fx(P)(delta s)u1 + fy(P)(delta s)u2 + fz(P)(delta s)u3 + erfx(Q)(delta s)u1 + erfy(Q)(delta s)u2 + erfz(Q)(delta s)u3`
Dividing by deltas, we have,

`(Deltaf)/(deltas) = (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3+ erf_x(Q)u_1+ erf_y(Q)u_2+ erf_z(Q)u_3`

Two points Q and P are approaches to line. It contains three functions error and finally we got a zero and will get directional derivative.

` (Deltaf)/(deltas) (p)= (delf)/(delx)(p)u_1+ (delf)/(dely)(p)u_2+ (delf)/(delz)(p)u_3`

Or, the dot product,

`gradf(p).hatu`

where gradf is a special function defined as follows,

`(Deltaf) = (delf)/(delx) i+ (delf)/(dely) j+ (delf)/(delz) k`

More texts are skips the vector arrow in given arrow of `gradf ` and `Delta ` symbol is called the Del operator. To writing a symbol as vector it is more helpful for gradient of function produces a vector.


Gradient in math Exercises:



In following math gradient to demonstrate the similarity to differentiation

`grad(f+g) = grad f + grad g`
`grad(f+g) = f grad g + g grad`
`grad(f/g) = f grad g - g grad f /g^2`
`gradf^n = nf^(n-1) grad f `

Thursday, May 30, 2013

Solve What is Histograms

INTRODUCTION ABOUT SOLVE WHAT IS HISTOGRAMS:

Let us see about the introduction for solve what is Histograms. The construction of histograms is used  to from a frequency of table. The rectangle height is located at the interval and it is represented by the number of scores. The shapes of the histograms are varying depend upon the choice of the size of the intervals. Let us see the explanation about solve what is Histograms.


Explanation about Solve what is Histograms:



The Solve Histogram is showing as the total distribution in the image. It is a bar chart that counts the pixels of every tone of gray.  The bar chart occurs in the image. It is the basic scheme for solving the permutation of problems in the framework of probabilistic model-building genetic algorithms that uses the edge of histogram is based upon the sampling techniques was reported.


Example for Histogram about Solve what is Histograms:



Example 1:

Solve the frequency distribution question, produce table, and histogram.

The following data represents the rain fall measured annually (in millimeters) by the meteorological office since records began in 1951 up to and including 1995.

Years

1951-57 = 1256  1586  1340  1340  1277  1419  1103
1958-64 = 1263  1231  1763  1509  1292  1227  1288
1965-71 = 1126  1342  1437  1184  1359  1461  1128
1972-78 = 1359  1459  1540  1235  1384  1485  1498
1979-85 = 1232  1670  1505  1572  1517  1417  1485
1986-92 = 1231  1312  1255  1132  1336  1362  1305

1993-95 = 1262  1396  1426

Example 2:

Draw the Histogram for the following data’s:

MonthScales (in millions)
January60
February59
March70
April75
May80
June84
July56
August89
September90
October95
November20
December37



Solution:






 
The months are represented by x-axis and the scales are represented in y-axis.

These are the examples for solve what is Histograms.

Tuesday, May 28, 2013

Easy Multiplication Tables

Introduction to easy multiplication tables:

Multiplication:

Multiplication symbol is "×". There are four basic operations in elementary arithmetic multiplication, addition, subtraction and  division. Easy multiplication tables are the mathematical operation of scale one number by another number.

Since the result of scaling by whole numbers could be consideration of as consisting of many copies of the original, whole-number provides greater than 1 could be calculated by repeated addition.

For example, 5 multiplied by 6 (regularly said as "5 times 6") can be calculating by adding 5 copies of 6 together:

5 x 6 = 5 + 5 + 5 + 5 + 5 + 5 = 30.

2 x 4 = 2 + 2 + 2 + 2 = 12.


Tables on easy multiplication tables


This is 0 to 12th method of easy multiplication tables.

x0123456789101112
00000000000000
10123456789101112
2024681012141618202224
30369121518212427303336
404812162024283236404448
5051015202530354045505560
6061218243036424854606672
7071421283542495663707784
8081624324048566472808896
90918273645546372819099108
100102030405060708090100110120
110112233445566778899110121132
1201224364860728496108120132144



Identification of 9th multiplication tables:
find 9 x 12.

Make use of the 8-method. Multiplying 9 by 12 times means, and then we get 108, as the answer.

And mention the easy multiplication tables with answer (bold number) above.

Identification of 7th multiplication tables:
find 7 x 4.

Make to use of the 7-method. The multiplying 7 by 4 times means, we get 28, as the answer.

And mention the easy multiplication tables with answer (bold number) above.

Identification of 2nd multiplication tables:
find 2 x 8.

Make to use of the 8-method. The multiplying 2 by 8 times means, and then we get 16, as the answer.

And mention the easy multiplication tables with answer (bold number) above.

Identification of 11th multiplication tables:
find 11 x 3.

Make to use of the 8-method. Multiplying 11 by 3 times means, and then we get 33, as the answer.

And mention the easy multiplication tables with answer (bold number) above.


Practice on easy multiplication tables:


1)   Find number from the  multiplication table. Multiplying 2 into 4 times.

2)   Find number from the multiplication table. Multiplying 4 into 5 times.

3)   Find number from the multiplication table. Multiplying 6 into 3 times.

4)   Find number from the multiplication table. Multiplying 7 into 8 times.

5)   Find number from the multiplication table. Multiplying 12 into 5 times.

6)   Find number from the multiplication table. Multiplying 6 into 3 times.


Monday, May 27, 2013

Solve Graphing Test

Introduction to solve graphing test:

Solve graphing test is one of interesting topics in mathematics. Without doing any algebraic manipulations, we can solve two simultaneous equations in x and y by drawing the graphs corresponding to the equations together. An equation in x and y is of the form a x + b y + c = 0. The equation represents a straight line, so, the problem of solving two simultaneous equations in x and y reduces to the problem of finding the common point between the two corresponding lines.
Steps to solve the graph:


The following steps are necessary to solve the graphing test:


Step 1:

 Two different values are substitute for x in the equation y = mx + b, we get two values for y. Thus we get two points (x1, y1) and (x2, y2) on the line.
Step 2:

   Draw the x-axis and y-axis on the graph and choose a suitable scale on the co- ordinate axes. Both the axes is chosen based on the scale values of the co-ordinates obtained in step 1. If the co-ordinate values are large in given data then 1 cm along the axes may be taken to represents large number of units.
Step 3:

Plot the two points (x1, y1) and (x2, y2) in Cartesian plane of the paper.

Step 4:

 Two points are joined by a line segment and extend it in both directions of the segment. Then it is the required graph.


Examples to solve graphing test:


Examples to solve graphing test are as follows:

Example 1:

   Draw the graph y = 3x −1.

Solution:

   Substituting x = −1, 0, 1 in the equation of the line, we get y = −4, −1, 2 correspondingly. In a graph, plot the

   Points (−1, −4), (0, −1) and (1, 2).



  X    -1    0    1

  Y    -4   -1    1

  Join the points by a line segment and extend it in both directions. Thus we get the required linear graph



    Example 2:

    Draw the graph of the line  2x  + 3y  = 12.

Solution:

  The given equation is rewritten as 3y = −2x + 12 or y = (−3 / 2)x  + 4.

  Substituting  x = −3 then y = 6

 x = 0, then y = 4
x = 3 then y = 2.

  Plot x and y values in the graph sheet. [(−3, 6), (0, 4) and (3, 2)]

  X    -1      0        1

  Y   5.5     4       2.5

  Join the points by a line segment and extend it in both the directions. Then it is the required linear graph.

Practice problems to solve graphing test:


Some practice problems to solve graphing test

1. Draw the graph of the following : y  = −2x

 Answer:     x    -1   0   1

  y     2   0  -2

 2. Draw the graph of the following equations:  y + 2x −5 = 0.
Answer:    x    -1   0   1
y     7   5   3

Friday, May 24, 2013

Altitude Term In Math

Introduction about altitude term in math:

            Altitude or height term is defined based on the context in which it is used. As a general definition, the term altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. In this article we shall discus about altitude term based problems.




Triangle:


The total space inside the triangle is called as area of that triangle.




Formula to find Area:

            Area of right angle triangle (A) =1/2 (length x height) square unit

                                                               = 1/2 l x h square unit.

                        Here, the term height refers the altitude of the triangle.

Example problem:



A right angle triangle has length 5cm and altitude 13 cm. Find the area of that triangle.
Solution:

Given:

            Length (l) =5cm

            Altitude (h) =13cm

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (5 x 13)

                           = 1/2 (65)

                           =65/2

                           =32.5

Area of triangle = 32.5 cm2



A right angle triangle has length 7.5m and height 10 m. Find the area of that triangle.
Solution:

Given:

            Length (l) =7.5m

            Altitude (h) =10m

Formula:

Area of triangle = 1/2 (l x h) square unit.

                           = 1/2 (7.5 x 10)

                           = 1/2 (75)

                           =75/2

                           =37.5

Area of triangle = 37.5 cm2


Rhombus:


The altitude of rhombus is the distance between base and opposite side of the base.




Formulas:

Area of the rhombus (A) = b x a

                                    b – Base of rhombus.

                                    a – altitude of rhombus

If two diagonal lengths are given:

Area of the rhombus (A) = (d1 x d2)/2

Example problems:

1.      The altitude and base of rhombus are 11 cm and 6cm respectively. Find are of rhombus.

Solution:

      Given:

                  Altitude of rhombus (a) = 11 cm

                         Base of rhombus (b) = 6 cm

            Area of the rhombus (A) = b x a square units.

                                                      = 11 x 6

                                                      = 66

      Area of the rhombus (A) = 66 cm2

2.      The altitude and base of rhombus are 14 cm and 10cm respectively. Find are of rhombus.

Solution:

      Given:

                  Altitude of rhombus (a) = 14 cm

                         Base of rhombus (b) = 10 cm

            Area of the rhombus (A) = b x a square units.

                                                      = 14 x 10

                                                      = 140

      Area of the rhombus (A) = 140 cm2

Tuesday, May 14, 2013

How To Cross Multiply

Introduction of How to Cross Multiply:

Cross multiplication is the easiest method that can be done in the equation form of two fraction form. For Example: `(4)/(16)` = `(3)/(12)`. By using the cross multiply method, we can check this given example as 4 * 12 = 48 and 16 * 3 = 48. Now it be proved that  `(4)/(16)` = `(3)/(12)`. Let see how to do the cross multiply and example problem of cross multiply.


Rules – How to Cross Multiply:


Let take

`(P)/(Q)` ….. Fraction 1

`(R)/(S)` ………. Fraction 2

Now equate the two fractions as in the form of `(P)/(Q)` … = `(R)/(S)` ……fraction equation

Rule 1: How to cross multiply means, multiply denominator of the fraction 2 with numerator of the fraction 1 as well as multiply denominator of the fraction 1 with numerator of the fraction 2

P * S = Q * R

Rule 2: Multiply the term on both sides of the fraction equation, and then the value of the term remains unchanged.

PS = QR

Multiply term A on both sides then its value is not changed. PSA = QRA.

Let we see example problem of how to cross multiply.


Example Problem – How to Cross Multiply:



Example 1:

How to find the value of x in given equation `(3)/(x)` = `(4)/(8)`?

Solution:

Step 1: Multiply 8 with the numerator of the fraction `(3)/(x)`.

3 * 8 = 24

Step 2: Multiply the denominator of the fraction 3/x with numerator of the fraction 4/8.

x * 4 = 4 x

Step 3: Equate above two steps, 4 x = 24

Step 5: Divide 4 by both sides, we get 4 * (`(x)/(4)`) = `(24)/(4)`

Then the value of x = 6

Answer: x= 6

Example 2:

How to solve this equation `(X - 3)/(4)` = `(5)/(2)`?

Solution:

Step 1: Multiply 2 with the numerator of the fraction `(X - 3)/(4)`.

2 * (X - 3) = 2X - 6

Step 2: Multiply the denominator of the fraction `(X - 3)/(4)` with numerator of the fraction `(5)/(2)`.

4 * 5 = 20

Step 3: Equate above two steps, 2X – 6 = 20

2X = 14

Step 5: Divide 2 by both sides, we get 2 * (`(X )/(2)`) = `(14)/(2)`

Then the value of X = 7

Answer: x= 7

Monday, May 13, 2013

Math h Functions

Introduction:

The hyperbolic functions are defined as the analogs of common trigonometric functions. The fundamental hyperbolic functions include sin h called as the hyperbolic sine, cos h called the hyperbolic cosine and the tan h called the hyperbolic tangent function.

The hyperbolic functions are just the rational functions of the exponentials. The mathematical hyperbolic functions are also called as the math h functions. In this article we are going to see various math h functions.


Expressions of math h functions:


There are various mathematical algebraic expressions given to the math h functions.

The hyperbolic sine of x is given by sinh x = ½ (ex-e-x).

The hyperbolic cosine of x is given by cosh x = ½ (ex+e-x).

The hyperbolic tangent of x is given by tanh x = sinh x/cosh x

= (ex-e-x) / (ex+e-x)

= e2x-1 / e2x+1

The hyperbolic cosecant of x is given by csch x = (sinh x)-1

= 2 / ex-e-x

The hyperbolic secant of x is given by sech x = (cosh x)-1

= 2 / ex+e-x

The hyperbolic cotangent of x is given by coth x = cosh x / sinh x

= (ex+e-x) / (ex- e-x)

= e2x+1 / e2x-1 .


Math h functions with respect to circular functions:


The hyperbolic functions or math h function with respect to the circular functions is given by

x = a cos t and y = a sin t

Here the circle is given as a rectangular hyperbola.

The math h functions exist in many applications of mathematics which involve the integrals with v (1+x2) and the circular functions with v (1-x2)

The math h functions also include many identities similar to that of the trigonometric identites. The identites of the math h functions includes

Cosh2 x – sinh2 x = 1

Cosh x + sinh x = ex

Cosh x – sinh x = e-x

The identities for the complex arguments includes

Sinh (x+iy) = sinh x cos y + i cosh x sin y

Cosh (x+iy) = cosh x cosy + i sinh x sin y.

Saturday, May 11, 2013

Symbols That Represent Me







Introduction to symbols that represent me:
There are numerous mathematical symbols that can be used in mathematics mode. This is a listing of common symbols found within all branches of Mathematics.
symbols that represent me

SYMBOLREPRESENTS
Symbol=Represents Is equal to
Symbol+Represents Plus or Addition
Symbol-Represents Minus or subtraction
Symbol/RepresentsDivision
SymbolX or *Represents Multiplication
SymbolRepresents theInequality
Symbol< and >Represents the“Is less than” and “is greater than”
Symbol≤ and ≥Represents the“Is less than or equal to” and “is greater than or equal to”
SymbolєRepresents theIs an element of
SymbolRepresents thesummation
SymbolαRepresents theAlpha
SymbolβRepresents theBeta
SymbolγRepresents theGamma
SymbolδRepresents theDelta
SymbolζRepresents theZeta
SymbolηRepresents theEta
SymbolθRepresents theTheta
SymbolλRepresents theLamda or Lambda
SymbolμRepresents theMu
SymbolπRepresents thePi
SymbolσRepresents theSigma
SymbolφRepresents thePhi
SymbolχRepresents theChi
SymbolωRepresents theOmega
Symbol#Represents theNumber  Sign
Symbol±Represents thePlus Minus
SymbolΩRepresents theOmega
SymbolιRepresents theIota
SymbolRepresents theEstimate sign
SymbolRepresents theSquare Root
SymbolRepresents theInfinity
SymbolRepresents theIntegral
SymbolRepresents theAlmost equal to
SymbolRepresents thePartial differential
SymbolRepresents theIncrement
Symbolw.r.tRepresents theWith respect to
SymbollogRepresents theLogarithm
Symbol!Represents theFactorial
Symbol%Represents thePercentage


History for symbols that represent me


A very elongated form of the modern equality symbol (=) was first introduced in print in The Whetstone of Witte (1557) by Robert Recorde (1510-1558) the man who first introduced algebra into England.  He justified the symbol by stating that no two things can be more equal than a pair of parallel lines...
The infinity symbol was first given its current mathematical meaning in "Arithmetica Infinitorum" (1655) by the British mathematician John Wallis (1616-1703).

Symbols that represent me : Further history


Gottfried Wilhelm Leibniz (1646-1716) viewed integration as a generalized summation, and he was partial to the name "calculus summatorius" for what we now call [integral] calculus.  He eventually settled on the familiar elongated ‘s’  for the sign of integration, after discussing the matter with Jacob Bernoulli (1654-1705) who favored the name "calculus integralis" and the symbol  I  for integrals...  Eventually, what prevailed was the symbol of Leibniz, with the name advocated by Bernoulli...

Friday, May 10, 2013

In Math What Does Range Mean


Introduction to range in math:

  • In math, Data set is a collection of data, which is usually presented, in tabular form. Each column represents a variable. In math, Range is generally defined as the value we obtained as a result of difference between a greater value and a smaller value.
  • In other words, range is defined as the difference between a maximum and minimum value.

Steps to learn the range in math:


     In order to learn the range of data set the following steps are necessary.
     Step 1: Arrange the numbers from ascending to descending order.
     Step 2: Identify the greater value in the given set
     Step 3: Identify the smaller value in the given set
     Step 4: Find the difference between the greater value and the smaller value to identify the range of the given data set.

Worked Examples for range in math:


Example 1:
Find the range of the data set given below:
                    8 , 15 , 13 , 7 , 24 , 37 , 6.
    Step 1: Arranging the numbers given in data set from least to greatest.
                    6 , 7 , 8 , 13 , 15 , 24 , 37.
    Step 2: Identify the greater value in the given set
                  From the given set, we can identify 37 as the greater value.
    Step 3: Identify the smaller value in the given set
                  From the given set, we can identify  6 as the smaller value.
    Step 4: Find the Range.
                          Range     =   Greater value – Smaller value
                                           =   37 - 6
                                           =  31
                   Hence, the range of the given data set is 31.
Example 2:
Find the range of the data set shown below:
                    33 , 12 , 79 , 24 , 34 , 53 , 27 , 61
     Step 1: Arranging the numbers given in data set from least to greatest.
                    12 , 24 , 27 , 33 , 34 , 53 , 61 , 79.
     Step 2: Identify the greater value in the given set
                    From the given set, we can identify 79 as the greater value.
     Step 3: Identify the smaller value in the given set
                    From the given set, we can identify  12 as the smaller value.
    Step 4: Find the Range.
                              Range    =   Greater value – Smaller value
                                               =   79 - 12
                                               =   67
                        Hence, the range of the given data set is 67.

Practice problems for range in math:


1) Find the range of the data set shown below:
                    7 , 9 , 19 , 31 , 37 , 43 , 6 , 37
     Answer: 37
2) Find the range of the data set shown below:
                    14 , 11 , 32 , 10 , 32 , 77 , 27 , 43
     Answer: 67