Exponential function

 

        The exponential function is a function in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number.
                                 
                                     

          The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point.

           As a function of the real variable x, the graph of y=e^x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. The exponential function is occasionally referred to as the anti-logarithm. However, this terminology seems to have fallen into disuse in recent times.

           Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form cb^x, where b, called the base, is any positive real number, not necessarily e. See exponential growth for this usage.

           In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object

The exponential series

          The function e^x can be expressed as an infinite power series as follows:

 

The exponential function (in blue), and the sum of the first n+1 terms of the power series on the left (in red).

The exponential function e^x can be defined, in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series

You can see the pattern of the coefficients in this case, if the power of x is n, say, then the coefficient is 1/n!. So there is a term x⁶/6! and a term x²⁰/20! and further on there'll be x¹⁰⁰⁰/1000! and so on.......

We can use this series to calculate the value of e^x for ANY value of x. You might think that as it's an infinite series that may take some time, but in fact the size of terms decreases very quickly as you go on, so just using the first few terms gives a reasonable approximation.

The reason the size of the terms gets smaller, even though they contain larger and larger powers of x, is because the factorials on the bottom get very large very quickly, so overall the terms are reduced.

If we choose the value of x to be 1, then the expression e^x is just equal to e. This means that if we replace x by 1 everywhere in that series for e^x it will tell us the approximate value of e itself.

So try putting x=1 in the series, just use the first four terms. We can extend the usefulness of the exponential series by allowing other arguments for the exponential function, e.g. we can write down the series for e^3x simply by replacing x by 3x everywhere:

e^3x = 1+3x/1!+(3x)²/2!+(3x)³/3!+(3x)⁴/4!+....,

which simplifies to:

e^3x = 1+3x+9x²/2!+27x³/3!+81x⁴/4!+.....

To write e^x+2 as a series, it is best to rewrite e^x+2 as e^x. e² so:

e^x+2 = e²(1+x/1!+x²/2!+x³/3!+x⁴/4!+.....)

This is true for any constant c in e^x+c, try writing down the first four terms of the series for e^x+4.

Know the Facts

 

              
 

  

Derivatives and differential equations

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,

        

That is, e^x is its own derivative and hence is a simple example of a pfaffian function. Functions of the form Ke^x for constant K are the only functions with that property. (This follows from the Picard-Lindelof theorem with y(t) = e^t, y(0)=K and f(t,y(t)) = y(t).) Other ways of saying the same thing include:

• The slope of the graph at any point is the height of the function at that point.
• The rate of increase of the function at x is equal to the value of the function at x.
• The function solves the differential equation y ′ = y.
• exp is a fixed point of derivative as a functional.

In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and Laplace's equation as well as the equations for simple harmonic motion.

For exponential functions with other bases:
     

        

A proof being,

y = a^x

lny = lna^x

lny = xlna 

Thus, any exponential function is a constant multiple of its own derivative.

If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time.

Furthermore for any differentiable function f(x), we find, by the chain rule: