Vertical Tangent

Vertical Tangent:

In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point (in the sense explained more precisely below). As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.


In mathematics, a vertical tangent is tangent line to be vertical. Since a vertical line contain infinite slope, a function whose graph contain a vertical tangent be not differentiable on the point of tangency. During the definition of the slope, vertical lines are excluded. Although from a simply geometric point of view, a curve might contain a vertical tangent. Imagine of a circle (with two vertical tangent lines). We still contain an equation, namely x=c, except it be not of the form y = ax+b.In fact, such tangent lines contain an infinite slope.
Limit Definition of Vertical Tangent

A function ƒ have a vertical tangent on x = a. Condition the difference quotient use to identify the derivative have infinite limit:

[lim_(h->0)(f(a+h)-f(a))/(h)=+oo] (or [lim_(h->0)(f(a+h)-f(a))/(h)=-oo]

The first case corresponds toward an upward-sloping vertical tangent, with the second case toward a downward-sloping vertical tangent. Easily speaking, the graph of ƒ have a vertical tangent at x = a condition the derivative of ƒ on a be either positive or negative infinity.



Used for a continuous function, it is often possible toward detect a vertical tangent through taking the limit of the derivative.

if [lim_(x->a)f'(x)=+oo]

After that ƒ have to contain an upward-sloping vertical tangent at x = a. In the same way, if

if [lim_(x->a)f'(x)=-oo]

then ƒ have to contain an downward-sloping vertical tangent at x = a. Within these situations, the vertical tangent toward ƒ appears since a vertical asymptote on the graph of the derivative.Let us now look at few examples of Vertical Tangents.

Examples of Vertical Tangent

Example 1 for vertical tangent

The function

[f(x) = root(4)(x)]

have a vertical tangent at x = 0, while it is continuous also

[lim_(x->0)f'(x)=lim_(x->0)(1)/root(4)(x^2)=oo]

[g(x)=root(4)(x^2)]

Have a vertical cusp on x = 0, given that it be continuous,

[lim_(x->0)g'(x)=lim_(x->0)(1)/root(4)(x)=+oo]

[lim_(x->0^-)g'(x)=lim_(x->0^-)(1)/root(4)(x)=-oo]

Example 2 for vertical tangent

What value of x does the function (you mean the graph of the equation) 5x^2+2xy+y^2=36 have a vertical tangent line?

Solution

5x^2 + 2xy + y^2 = 36
and put y = -x
5x^2 - 2x^2 + x^2 = 36
4x^2 = 36
x^2 = 9
x = +3,-3


Hope you like the above example of Vertical Tangent.Please leave your comments, if you have any doubts.

Angles

Angles:


In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).

The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek (ankylοs), meaning "crooked, curved," and the English word "ankle." Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".

Let OA and OB two half lines with common end point O. The half lines OA and OB are the sides of an angle and the point O is the vertex of the angle. An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position.We usually come across the questions such as how to measure an angle,what are the various methods to measure and angle etc,the explanation to these questions is given below.



Measurement of angle

The amount of rotation from initial side to terminal is called the measure of an angle.
Positive and Negative angles

Angles that are formed by counter clockwise (anti clockwise) rotation, such as the one shown in fig (ii) are said to be positive or to have positive measure.
Angles that are formed by a clockwise rotation, like the one in fig(iii) are said to be negative or to have negative measure.
Lines at right angles

The lines are said to be at right angles if the rotating half line (or ray) from starting from initial position to the final position describes one quarter of a circle.


Hope you like the above example of Angles.Please leave your comments, if you have any doubts.

Coefficients

Coefficients:

Introduction to coefficient:

In simple terms the meaning of a coefficient is,a factor that contributes to produce a result.In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in

7x2 − 3xy + 1.5 + y

the first three terms respectively have coefficients 7, −3, and 1.5 (in the third term there are no variables, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression). The final term does not have any explicitly written coefficient, but is usually considered to have coefficient 1, since multiplying by that factor would not change the term. Often coefficients are numbers as in this example, although they could be parameters of the problem, as a, b, and c in

ax2 + bx + c

when it is understood that these are not considered as variables.Also it is defined as the number in front of the variable is called as coefficient. For example 2x is the given expression here the coefficient is 2. The coefficient is usually in numeral with the variable. The expression contains variable and coefficient of the variable.The other way we can learn about a coefficient is by solving few example problems related to coefficient.
Example Problem for the Coefficient:

Example 1:

Find the coefficient of the variables in the given expression:

X + 2y

Solution:

Given that X + 2y

Here x and 2y is the expression with the coefficient

The coefficient of x is 1

The coefficient of y is 2.

Example 2:

Find the coefficient of variables in the given expression:

Y2 + 2y + 3xy +1

Solution:

Given that Y2 + 2y + 3xy +1

Here y2 , 2y, 3xy, is the expression with the coefficient and 1 is with out variable

The 3xy having two variables that two variable consider as a one variable

1 is the constant term of the given expression

The coefficient of y2 is 1

The coefficient of 2y is 2

The coefficient of 3xy is 3

Example 3:

Find the coefficient of variables in the given expression:

Z5 + z + y +6y

Solution:

Given that Z5 + z + y +6y

Here z5, z, y, 6y is the expression with the coefficient

The coefficient of z5 is 1

The coefficient of z is 1

The coefficient of y is 1

The coefficient of 6y is 6.
Example Problem for the Coefficient:

Example 4:

Find the coefficient of variables in the given expression:

X3 + y 5 +2xy + 5yx

Solution:

Given that X3 + y 5 +2xy + 5yx

Here X3, y 5, 2xy, 5yx the expression with the coefficient

The coefficient of x3 is 1

The coefficient of y5 is 1

The coefficient of 2xy is 2

The coefficient of 5yx is 5.

Example 5:

Find the coefficient of variables in the given expression:

100x2 + 2z3 + 102z

Solution:

Given that 100x2 + 2z3 + 102z

Here 100x2, 2z3, 102z the expression with the coefficient

The coefficient of 100x2 is 100

The coefficient of 2z3 is 2

The coefficient of 102z is 102.

Rectangular Coordinates

Rectangular Coordinate:

The rectangular coordinate system otherwise called as Cartesian coordinate system. The rectangular coordinate system is always depends on the grid. In rectangular coordinate every point on the plane can be recognized by the distinctive x and the y coordinates. Now any point on the Earth can be recognized by giving its latitude and the longitude.

Here, o-origin of coordinates, right side value from 0 is point as positive numbers i.e., 1,2,3,etc., left side value called negative numbers -1,-2,-3,....

Problems on Rectangular Coordinates:

Problem 1:

Convert the following polar coordinate to rectangular coordinates R = 5

Angle q =30 degree

Solution:

R = 5 and Angle q = 30

For the given polar coordinates we have to find the equal rectangular coordinate

Where the rectangular coordinates are (x , y)

Here x = R Cos q

And y = R Sin q

So

X = 5 Cos 30

Y= 5 Sin 30

So the rectangular coordinate (4.3301, 2.5)

Problem 2:

Change the given polar coordinate to rectangular coordinates

(R, q) = (5, 53.1)

Solution:

R = 5 and Angle q = 53.1

For the given polar coordinates we have to find the equal rectangular cordinate

Where the rectangular coordinates are (x , y)

Here x = R Cos q

And y = R Sin q

So

X = 5 Cos 53.1

Y = 5 Sin 53.1

So x=3

And y = 4

So the rectangular coordinates are (3, 4)

Problems on Rectangular Coordinates:

Problem 3:

Change the rectangular coordinates to polar coordinates (-3, -7)

Solution:

We know x2 + y2 = r2

So

(-3)2+ (-7)2 = 9 +49 =58

Where q = Tan(y / x )

R2 = 58

R =7.6

Tan q = (-7/-3) =66.8 degrees

So the polar coordinates are (7.6, 66.8)

We can use the same for three dimensional. So we will use x, y, z coordinates.

Let us look at a more wider explanation about the Rectangular Coordinates.
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes.

Complex Number

Complex number

Introduction:

Consider a simple quadratic equation x2 + 1 = 0. There is no real number which satisfies this equation. So there was a need to find a system which could answer to this problem. Euler used the symbol 'i' to denote to solve the above equation.

Complex number system consists of the set of all ordered pairs of real numbers (a, b) denoted by a + ib, where i =
Complex number

Square root of a negative number is known as an imaginary number.

If x and y are real numbers, then x + iy is called a complex number. x is called the real part and y is called the imaginary part.

The following are the types of complex numbers: Equality of Complex numbers, Sum of two Complex numbers, Negative of a Complex number, Additive identity of the Complex number, Additive inverse of a Complex number, Product of two Complex numbers, Multiplicative identity of Complex numbers, Conjugate complex numbers, Quotient of two non-zero Complex numbers, Reciprocal of a non-zero complex number or multiplicative inverse of a non-zero complex number.
Properties of Complex numbers

The Properties of Complex numbers are:Commutative Law for Addition, Commutative Law for multiplication, Additive Identity Exists, Multiplicative Identity Exist, Reciprocals (Multiplicative Inverses) Exist for nonzero complex numbers, Negatives (Additive Inverses) Exist for all complex numbers, Non Zero Product Law.
Graphical representation of Complex numbers

The complex number Z = x + iy may be represented graphically by the point P whose rectangular co-ordinates are (x, y)। Thus each point in the plane is associated with a complex number.

Mathematical Induction

Mathematical Induction:

Let us first understand the meaning of mathematical induction,the word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a principle by which one can arrive at a conclusion about a statement for all positive integers, after proving certain related related proposition.

Statements:

Some sentences depend on a variable for its truth value (i.e., true or false).
e.g., "2+4+6+…2n=2n" is true for n=1 but false for n=2, n=3 etc.
As the above sentence is definitely true or definitely false for a particular positive integral value of n, the sentence is a statement and it depends on nÎN for its truth-value. Such statements are called predicates and are symbolised as P(n).

Principle of Mathematical Induction (PMI)

A statement P(n) is true for all nÎN if
(i) P(1) is true (ii) P(r) is true implies P(r+1) is true.

Illustrative Examples

The following are the Illustrative Examples:
If P(n) is the statement n²-n+41 is prime, prove that P(1), P(2) are true but P(41) is not true.

Summary

1 A sentence is called a statement if it can be adjudged as true

or false

2. Every statement is a sentence, but a sentence may or may not be a statement.
3. A statement involving natural number n is generally denoted by P(n).

Conclusion:


Let n N and P(n) denote a certain statement or formula or theorem. Then P(n) holds good for every natural number n if
(i) it holds for n = 1 and
(ii) it holds for n = k+1 whenever it holds for n = k.

Geometric Series

Geometric Series:

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series

is geometric, because each term except the first can be obtained by multiplying the previous term by \frac{1}{2} \ .

Geometric series are one of the simplest examples of infinite series with finite sums. Let us understand the role of geometric series,historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
A sequence of non negative numbers is called a geometric sequence where the ratio of of each term to its preceding term is the same except for the first term. A geometric series is a series whose terms are in geometric sequence.
There is an interesting story related with geometric series. It is said that the King of Persia was very impressed with the inventor of the board game chess ( believed to be from India ) that he offered to give him any reward. The inventor wanted one grain of wheat to be placed on the first square of the chessboard , two grains on the second square , four on the third , eight on the fourth and so on. This demand of the inventor seemed very small to the King.
We can see that the grains placed in the various squares are the terms of a geometric sequence 1, 2 ¹, 2², 2³ , 2⁴, ... 2 ⁶³
Sum of grains = S ₆₄ = 1 + 2 ¹ + 2 ² + 2 ³ + ...... + 2 ⁶³
We take the reverse order S ₆₄ = 2 ⁶³+ 2 ⁶² +.....+ 2 ³ + 2 ² + 2 + 1 ------------> 1
2 * S ₆₄ = 2 ⁶⁴ + 2 ⁶³ +.....2 ⁴ + 2 ³ + 2 ² +2 + 1 ----------->2
Subtract 1 from 2 S ₆₄ = 2 ⁶⁴ - 1
A huge amount of grain.

Reciprocals

Reciprocals:

Introduction:

In this Blog Let us see the meaning of Reciprocal .In general definition of reciprocal number is multiplicative inverse of a number. The reciprocal number is commonly specified in following way. The number is n it is commonly denoted the reciprocal is 1/n. Another method for the denoted the reciprocal number is m/n the multiplicative inverse of a fraction is n/m. The example reciprocal of 91 is 1/91.Reciprocal math is nothing but the reciprocal of a number in math. Reciprocal is any number that divides 1.If any number is considered and that number is represented as 1 divided by the considered number then such form of the number is called as reciprocal math.

Example:

If the number is 5 then the reciprocal of the number 5 is [1/5]

Reciprocal Math is mostly used in the division of a fraction number. In the case of dividing a fraction number by another fraction number then we just change the division symbol as multiplication symbol and take reciprocal math of the second number and perform multiplication. So in many cases to solve the math problem of different chapter reciprocal math is being used.

Examples on Reciprocal Math:

1. Find the reciprocal math of the following numbers.

a) 5

b) 21

c) 99

d) 10

e) [5/3]

f) [2/7]

g) [ 1/6]

Solution

The Reciprocal math of the number 5 is [1/5]

The Reciprocal math of the number 21 is [1/21]

The Reciprocal math of the number 99 is [1/99]

The Reciprocal math of the number 10 is [1/10]

The Reciprocal math of the number [5/3] is [3/5]

The Reciprocal math of the number [2/7 ] is [7/2]

The Reciprocal math of the number [1/6 ] is [6/1.] [6/1 ] can be written as 6

2. Divide [2/3] and [8/6]

Solution

[ 2/3] / [8/6]

Keep the first fraction as it is and change the divisible sign as multiplication and find the reciprocal math of second fraction [8/6]

The reciprocal of [8/6 ] is [6/8]

[2/3] * [6/8]

Now by simplifying these we get [1/2]

3. Perform the operation

[7/9] divide by 3

Solution

Here 3 can be written as [3/1]

[7/9] / [3/1]

Reciprocal math of [ 3/1] is [1/3]

[ 7/9] / [1/3]

[ 7/27]

Scatterplots

Scatterplots:
The general meaning of a scatter is to unevenly distribute,something that is not in an order.But here in mathematics the meaning of a scatterplot is a type of mathematical diagram using Cartesian coordinates to display values for two variables for a set of data.

It is very important to learn about the uses of a scatterplot,in Statistics a scatterplot is a graphic tool used to show the relationship between two quantitative variables. The variable that might be measured an expounding variable is plotted on the x-axis, and the response variable is plotted on the y-axis.It provides a graphical expose of the link between the two variables. It is also useful in the early stages of study when exploring records ahead of really devious a parallel coefficient or corrects a regression curve. For example, a scatter plot can aid one to decide whether a linear waning model is proper.

Statistical Scatterplots can explain a variety of patterns and associations in Statistics, such as:

• Data association
• Optimistic or direct relations between variables
• Pessimistic or opposite associations between variables
• Scattered record points
• Non-linear pattern
• Increase of data
• Outliers.

Histogram

Histogram:

A two dimensional frequency density diagram is called a histogram. A histogram is a diagram which represents the class interval and frequency in the form of a rectangle. There will be as many adjoining rectangles as there are class intervals.

In statistics,simple explanation of a histogram is- a histogram is a graphical display of tabular frequencies, shown as adjacent rectangles. Each rectangle is erected over an interval, with an area equal to the frequency of the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e. the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. A histogram may also be based on the relative frequencies instead. It then shows what proportion of cases fall into each of several categories (a form of data binning), and the total area then equals 1. The categories are usually specified as consecutive, non-overlapping intervals of some variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size, but not necessarily so.

Now we will learn about the uses of a Histogram.Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot.