Graph of exponential function

Exponential function : if a is positive real number , then the function which associate every real number x to a^x i.e f(x) = a^x is called the exponential function                                                                
Graph exponential function: Here let a >1 , then the exponential function for +ve base is defined as y= f(x) = a^x

Graph of exponential function of y=a^x, a>1

Graph of exponential function y = f(x) = a^x, here  0

Here in both the above the exponential Function Graph, we can see clearly the domain of the exponential function is R and the range is the set of all positive real number .
Some more graphing exponential functions . 

Let us check the graph of y = 3^x

From the graph of y=3^x , we can see  that graph is increasing as , x increasing , y is also increasing . Graph of y = 3^x  passes through (0,1 ).Domain and range of the graph is set of all real number.The graph of the function is continuos and smooth.

Graph of y=3^x

Graph of y =3^x is decreasing as x decreases the graph of the function decreases. Here the graph passes through (0, 1) and domain of the function is all real number .the graph is smooth and continuos. . The graph is asymptotic to the x-axis as x approaches positive infinity . The graph of the function y= 3^x increases without bound as x approaches negative infinity
****   Followings are the result  about exponential functions :-
1. Domain of exponential function is the set of all real number.
2. Range of exponential function is the ser of all +ve number
3. Exponential functions graph  always passes through (0,1)
4. Exponentia l function graph is an increasing  function as we move from left to right  when x increases function  also increases.
5. For every large negative values of x , the value of   the function approaches 0 
Common exponential function
If the base of an exponential function is taken as 10 ,then it is called common exponential function.Thus f(x) = 10^x is common exponential function
Natural exponential function
If the base of an exponential function is taken as e , then it is called natural  exponential function.thus f(X) = ex is natural exponential function where.


e = 1+(1/1!)+(2/2!)+(3/3!)+(4/4!)+...........  = 2.718(approximately)

Graph of y=e^x

Derivatives in Calculus

Derivative or Derivatives:
 In calculus, a branch of mathematics, derivative is the measure of how a given function changes according to the inputs.
 A derivative can be taken as how much one quantity is changing according to the changes in some other quantity; for instance, the derivative of the position of a moving object with respect to time is its instantaneous velocity
In other words, for a real valued function of a single real variable, the derivative of a point is the slope of the tangent line to the graph of the function at that given point.
What is Derivative?
Derivative is the measure of the rate of change at any given point on a curve. The rate of change of a function is the slope of a line.
The derivative of f(x) when the limit extends from h to 0 is given by the formula:

Definition of Derivative

Prime numbers

Numbers are classified into two categories, prime numbers and composite number. Let us learn about prime numbers. What is a prime number? Or what is prime number? Prime number is a positive whole number which can only be divisible by 1 and itself. For example: - 2, 3, 5, 7 etc.

Prime number only has two factors.  A list of prime number up to 100 is given as follows:-
• 2 = 1 X 2
• 3 = 1 X 3
• 5 = 1 X 5
• 7 = 1 X 7
• 11 = 1 x 11
• 13 = 1 X 13
• 17 = 1 X 17
• 19 = 1 X 19
• 23 = 1 X 23
• 29 = 1 X 29
• 31 = 1 X 31
• 37 = 1 X 37
• 41 = 1 X 41
• 43 = 1 X 43
• 47 = 1 X 47
• 53 = 1 X 53
• 59 = 1 X 59
• 61 = 1 X 61
• 67 = 1 X 67
• 71 = 1 X 71
• 73 = 1 X 73
• 79 = 1 X 79
• 83 = 1 X 83
• 89 = 1 X 89
• 97 = 1 X 97
All these numbers listed above makes a list of prime numbers upto 100. We can list prime numbers in such a way. All these numbers are divisible by number one and themselves. Rest all other numbers are known as composite numbers. Following is the prime numbers chart of numbers up to 100:-

Prime number

 So this makes a prime number chart. All those numbers marked in blue are prime numbers. Now the question arises, is 1 a prime number? 1 is not a prime number, by definition a prime number is a number that is divided by one and itself so 1 does not fit under the definition of a prime number. Hence, 1 is not a prime number. It is the building block of the positive integers. A prime number should have two factors but number just has one factor that is 1, therefore it is not considered as a prime number.