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.

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.

SYMBOL		REPRESENTS
Symbol	=	Represents	Is equal to
Symbol	+	Represents	Plus or Addition
Symbol	-	Represents	Minus or subtraction
Symbol	/	Represents	Division
Symbol	X or *	Represents	Multiplication
Symbol	≠	Represents the	Inequality
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”
Symbol	є	Represents the	Is an element of
Symbol	∑	Represents the	summation
Symbol	α	Represents the	Alpha
Symbol	β	Represents the	Beta
Symbol	γ	Represents the	Gamma
Symbol	δ	Represents the	Delta
Symbol	ζ	Represents the	Zeta
Symbol	η	Represents the	Eta
Symbol	θ	Represents the	Theta
Symbol	λ	Represents the	Lamda or Lambda
Symbol	μ	Represents the	Mu
Symbol	π	Represents the	Pi
Symbol	σ	Represents the	Sigma
Symbol	φ	Represents the	Phi
Symbol	χ	Represents the	Chi
Symbol	ω	Represents the	Omega
Symbol	#	Represents the	Number  Sign
Symbol	±	Represents the	Plus Minus
Symbol	Ω	Represents the	Omega
Symbol	ι	Represents the	Iota
Symbol	℮	Represents the	Estimate sign
Symbol	√	Represents the	Square Root
Symbol	∞	Represents the	Infinity
Symbol	∫	Represents the	Integral
Symbol	≈	Represents the	Almost equal to
Symbol	∂	Represents the	Partial differential
Symbol	∆	Represents the	Increment
Symbol	w.r.t	Represents the	With respect to
Symbol	log	Represents the	Logarithm
Symbol	!	Represents the	Factorial
Symbol	%	Represents the	Percentage

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...

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

Geometrical Meaning of the Zeroes of a Polynomial

We know that  a real number k is a zero of the polynomial p(x) if p(k) = 0 in a geometrical way.

The x-coordinate of the point, where the graph of a polynomial intersect the x-axis is called the zero of the polynomial.

An nth - degree polynomial intersects the x-axis of n points and therefore, has a maximum of n zeros in geometrical graph

In a quadratic polynomial ax2 + bx +c,

If a>0, then the graph is a parabola that open upwards.

If a<0, then the graph is a parabola that open downwards.

Special case to the zeroes of a polynomial in geometrical meaning


In a geometrical way the evaluation of zeroes with polynomial evaluation of the equation in geometrical meaning is follows

Case (i) :

Here, the graph cuts x-axis at two distinct points A and A′.
The x-coordinates of A and A′ are the two zeroes of the quadratic polynomial ax2 + bx + c in this case is shown in the following figure



Case (ii) :

The graph given here cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (i) coincide here to become one point A is shown in the following figure.



The x-coordinates  of the  A is the only zero for the quadratic polynomial ax2 + bx + c in this case.

No zero case

The graph given here is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point

So, the quadratic polynomial ax2 + bx + c has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also refers that the polynomial of degree 2 has at-most two zeroes is shown in the following figure.

Concept of the Derivative Online Help

In online, concept of derivative is clearly explained with the help of solved example problems. In online so many websites explain the concepts with the help of math sites. The derivative concept is clearly explained in calculus whereas derivative concept helps to find the rate of change for the given function with respect to change in the input function. The following are the example problems with detailed solution helps to explain the concept of derivative in online.


Concept of derivative online help example problems:


The following example problems explain the concept of derivative in online.

Example 1:

Determine the derivative by differentiating the polynomial function.

f(t) = 3t 3 +4 t 4  + 5t

Solution:

The given function is

f(t) = 3t 3 +4 t 4  + 5t

The above function is differentiated with respect to t to find the derivative

f '(t) = 3(3t 2 )+4(4t 3  ) + 5

By solving above terms

f '(t) = 9t 2 +  8t 3 + 5

Example 2:

Determine the derivative by differentiating the polynomial function.

f(t) = 6t6 + 5 t5 + 4 t4 + t

Solution:

The given equation is

f(t) = 6t6 + 5 t5 + 4 t4 + t

The above function is differentiated with respect to t to find the derivative

f '(t) =  6(6t 5)  +5 (5 t4 ) +4(4 t3) + 1

By solving above terms

f '(t) =  36t 5  + 25 t4  + 16 t3 + 1

Example 3:

Determine the derivative by differentiating the polynomial function.

f(t) = 2t 2 +4 t 4  + 15

Solution:

The given function is

f(t) = 2t 2 +4t 4  + 15

The above function is differentiated with respect to t to find the derivative

f '(t) = 2(2t  )+4(4 t 3 ) + 0

By solving above terms

f '(t) = 4t +16t3

Example 4:

Determine the derivative by differentiating the polynomial function.

f(t) = 5t5 +4t 4 +3t 3  + 2

Solution:

The given function is

f(t) = 5t5 +4t 4 +3t 3  + 2

The above function is differentiated with respect to t to find the derivative

f '(t) = 5(5t 4 )+4(4t 3 ) +3( 3t 2) +0

By solving above terms

f '(t) = 25t 4 +16t 3  +9 t 2


Concept of derivative online help practice problems:


1) Determine the derivative by differentiating the polynomial function.

f(t) = 2t 3 +3 t 4  + 4 t 5

Answer: f '(t) = 6t 2 +12 t3 + 20 t 4

2) Determine the derivative by differentiating the polynomial function.

f(t) = t 3+t5 + 4 t 6

Answer: f '(t) = 3t2 + 5t4 + 24 t 5

Solve Online Scale Factor

Scale is a thing which is used to measure the dimensions of the object. The terms scale factor is used to mention the dimensions of the geometric shapes. In geometry, the term scale factor is usually used to change the dimensions of the image. Using the scale factor, any shapes dimension can be increased or decreased. In online, students can learn about various topics. In online, students can learn about scale factor clearly. Online learning is very interesting and interactive. Moreover online learning is different from class room learning in different ways. In online, students have one to one learning.


More about scale factor:


A scale factor is a figure used as a figure by which another figure is multiplied in scaling.

A scale factor is new to scale shape in 1 to 3 proportions.
Scale factor can be established by the following scenario
Size Transformation:     In size change, the scale factor is the ratio of express the amount of reduction.

Scale Drawing:    In scale diagram a diagram that is like but either superior or lesser than the real object, for example, a road map or a building blueprint...

Comparing Two Similar Geometric Images:      The scale factor when compare two parallel geometric imagery is the ratio of lengths of the equivalent sides.






Solve Scale factor Problem:


Q 1:   The area of a rectangle is 5 and it is enlarged a scale factor by 5. Solve for the area of an enlarged rectangle?

Sol :  The Area of a rectangle(R) is 5 and its Enlarged by a scale factor of 5.

that means the length L and the width W are equally multiply by 5.
As well note that the area of Rectangle is the length L times the width W, so Rectangle = LW.

Because the area of a rectangle is given by Rectangle(R) = LW, and L turn out to be 4L and W turn out to be 4W,

Solve for new triangle,

the new area of a enlarged rectangle  = (5L)(5W) = 25LW = 25(R).
Now we know that Rectangle(R) = 5, so 25(R) = 125.

Q 2:  The side of the square is 6 cm and it is enlarged a scale factor by 4. Solve for the area of an enlarged square?

Sol :  The Area of a square = a2

=  6 * 6

= 36 cm2.

Scale factor is 4 cm.

New side of square  = 4 * 6

= 24 cm.

Area of enlarged square = 242.

= 576 cm2.

Answer is 576 cm2.

Study Online Interest

To study online interest is useful to calculate the interest easily. We have two types of interest Simple interest and compound

interest. Formula for simple interest is,    Interest = Principle `xx` Rate `xx` Time

Where  I = Total amount of interest paid

P = Amount lent

R = Percentage of principle charged as interest each year.

Example: rate = 2% means then 2/100 =0.02. Use this value in formula.

Formula for compound interest is,       A = p`(1+r)^(n)`


Examples of study online interest:


Ex:1

To find simple interest if principle= 3000, rate= 3%, time= 5

Sol:

To study online interest we have to write the formula first

That is Simple interest = Principle `xx` Rate `xx` Time

Apply the given values in the formula,

Interest = 3000 `xx` 3%`xx` 5               R = 3% = 3/100 = 0.03

= 3000`xx` .03`xx` 5

= 450

The answer is 450

Ex:2

Shan invested Rs.6000 for 5 years. He received the simple interest of Rs.1,500.Find the rate of interest.

Sol:

To study online interest we have to write the interest formula first

That is Simple interest = Principle `xx` Rate `xx` Time

Here principal, P= Rs.6000

Number of years, n= 5

Simple interest, I = Rs.1500

Rate of interest, r =?

We know that rate of interest, r =`(100 xx 1500)/(6000 xx 5)`

=5

Therefore Rate of interest = 5%

Ex:3

If the principal amount is $7000, the periodic rate of interest is 3% and the number of compounding periods is 3, what is the value at maturity?

Sol:

To study online interest we have to write the formula first

A = p`(1+r)^n`

Here principal, P= $7000

Rate of interest, r= 3%

= 3/100 = .03

n = 3

A = P (1 + r) n

A = 7000 (1 + .03) 3

A = 7649.089

Ex:4

If the principal amount is $2000, the periodic rate of interest is 2% and the number of compounding periods is 5, what is the value at maturity?

Sol:

To study online interest we have to write the formula first

A = p`(1+r)^n`

Here principal, P= $2000

Rate of interest, r= 3%

= 2/100 = .02

n = 5

A = P (1 + r) n

A= 2000 (1 + .02) 5

A= 2208.1616

These are the some examples to study online interest.