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.