continuity functions

A function is said to be continuous if and only if it is continuous at every point of its domain. A function is said to be continuous on an interval, or subset of its domain, if and only if it is continuous at  each point of the interval. The sum, difference, and product of continuous function with the same  domain  are additionally continuous, as is the quotient, except at points at which the denominator is zero. Continuity can  be defined in terms of limits by saying that f(x) is continuous at x0 of its domain if and only if, for  values of x in its domain

Consider the graph of f(x) = x3 − 6x2 − x + 30:

x3 

We can see that there are no "gaps" in the curve. Any value of x will give us corresponding  value of y. We can continue  the graph in the negative and positive directions.Such functions are known as continuous  functions.

A function ƒ : X → Y is a uniform continuity if  for every real number ε > 0 there exists δ > 0 such that d₂(ƒ(x), ƒ(y)) < ε for every x, y ∈ X with d₁(x, y) < δ.

Complex Numbers

Numbers that has both real and imaginary parts are called complex numbers.It can be written as a + bi where a and b are real numbers and i is the standard imaginary unit.The standard imaginary number has the property i^2 = -1.

Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions.Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities.

To assemble a complex number, a second number is associated with each real number.

A complex cardinal is again an ordered pair of absolute numbers (a,b).

We address that fresh cardinal as

a + bi

The '+' and the i are just symbols for now.

We call 'a' the absolute part and 'bi' the abstract part of the complex number.

Ex :

(2 , 4.6) or 2 + 4.6i ;

(0 , 5) or 0 + 5i ;

(-5 , 36/7) or -5 + (36/7)i 

Instead of 0 + bi, we address 5i.

Instead of a + 0i, we address a.

Instead of 0 + 1i, we address i.

The set of all complex numbers is C. 

Next time we will learn how to solve complex numbers problem like sum, product, multiplication, etc.

coordinate plane

When one horizontal line intersect a vertical line, we get a coordinate plane.Generally, we call the horizontal line as x-axis and the vertical line as y-axis.The two lines intersect at 0 point.This point is known as the origin and is written as (0,0).

The following terms are used in the coordinate plane.
1) x-axis
2)y-axis
3)ordered pairs
4)plane                                                   
5)coordinates
6)origin
7)intersection
8)number line.
The following diagram is an example of coordinate plane.


You can also see the given link for coordinate geometry help
A coordinate plane can be divided into 1st quadrant, second quadrant, third quadrant and fourth quadrants.
They are denoted by the Roman letters I,II,III and IV respectively.

Derivatives

Derivative is an important part of calculus.Derivative is defined as the small rate of change of a function with respect to the variables of the function.

There are many application of derivatives.See the following link for application of derivatives help.

The process of finding the derivatives is known as differentiation.Many formula are used in finding the derivatives.Following are the formula used in differentiation.
The sum rule - u(x) + v(x) = du/dx +dv/dx
The product rule - u(x)v(x) = u(x) dv/dx + v(x) du/dx
The quotient rule - u(x)/v(x) = (v(x)du/dx - u(x)dv/dx)/(v(x))^2
The chain rule - y(u(x)) = dy/du du/dx

 Next time we will learn about problems on application of derivatives.
You may look up for the power rule as well in the book or in the website.


 

Quadratic equation

What is a quadratic  equation?

A quadratic equation  is a polynomial expression of the additional degree.

The accepted form is

ax^2+bx+c=0,

where x represents a variable and a, b and c are constants with a ≠ 0. 

Examples of quadratic equations:

(a)5x2 − 3x − 1 = 0 is a quadratic equation in quadratic form where

a = 5, b = -3, c = -1

(b) 5 + 3t − 4.9t2 = 0 is a quadratic equation  in quadratic form.

Here, a = -4.9, b = 3, c = 5 

Quadratic functions:

A quadratic function  is a polynomial function of the form

f(x)=ax^2+bx+c, where a is not equal to 0

The graph of a quadratic function  is a parabola whose major axis is parallel to the y-axis .

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Probability

Probability is the possibility of some events to happen or the belief that something will happen.

What is probability in Math?
In Mathematics, when you do an experiment the possibility of getting some results is called probability.
For example, there is probability of getting 6 in tossing  a dice.

Calculating probability formula: If there is a set of N elements , we can define a sub-set of n favorable event where n is less than or equal to N.Then probability is calculated as P = n/N. All the other formula of probability are derived from this formula.

Next time we will learn how to solve probability problems using the above formula with many examples.

Elementary Math help

In elementary Math, basic arithmetics such as addition, subtraction, division and multiplication are included.Elementary Math is taught to students of lower grades.

Learning elementary Math is very important as all the basics that are needed for higher Maths is given here.You can avail elementary Math help where you will also get elementary algebra help.

In elementary algebra, you will be provided with lots of solved problems on algebra.Online help on elementary Math that covers all the basic concepts are also available in many websites.

Solving linear equations

Linear equation is an algebraic expression in which each terms has degree not more than 1. Linear equation can have one or more variables.

A linear equation can be written as y = m + bx, where m and b are constants.

Let's solve linear equations by taking some examples.
Find x if 4x - 4y = 8
       Soln.4x = 8 + 4y
               4x/4 = 8/4 + 4y/4
               x = 2 + y
      substituting the value of x in the initial equation, we get
          8 + 4y - 4y = 8
                        8 = 8



             

Least Common Multiples

What are common multiples?
Multiples that are common to two or more numbers are called common multiples.

Least common multiple: The least common multiples are the smallest numbers that are both the common multiples of the numbers.

How to find least common multiple?
Let us find the least common multiple of 4 and 6
Multiples of 4 are 4, 8, 12, 16, 20, 24,......
Multiples of 6 are, 6, 12, 18, 24, 30,......
The common multiples of 4 and 6 are 12, 24,..
Since 12 is the smallest multiple for both 12 is the least common multiple of 4 and 6.

Learn sequence and series

A sequence is a set of numbers that are in order.For example, 1,2,3,4,5,6,7,8,9.....is a sequence.The dots indicate that the sequence is infinite.If there are no dots in a sequence , it is a finite sequence.

We get a series when we sum up the terms of a sequence.For example, 1+2+3+4+5+.....is an infinite series.
There are websites where you can avail basic information on sequence and series.

Many sequence and series help are available online from different tutoring company.You can get contact for
live tutor as well at your local area .

Polygon

Polygon is a closed figure made up of several line segments that are joined together.The sides meet at vertices, they do not cross each other.

There are five types of polygon.They are regular polygon, equiangular polygon, equilateral polygon, concave polygon and convex polygon.

The area of a polygon is calculated by the formula, A = (1/2)N sine(360degree/N)S^2
where N is the number of sides and S is the length from center to a corner.

Next time we will learn the different types of polygon in detail.

Prime numbers

Let's learn what is a prime number.
A prime number is a number that can be divisible only by 1 and itself.

For example, 3 and 5 are prime numbers because they can only be divisible by 1 and themselves.

Let's list prime numbers between 1 to 10.
The prime numbers between 1 to 10 are 3, 5 and 7.

Similarly, you can find out prime numbers from number between 1 to 100.