|Up to Main Lab Page||Next Lesson - Finding Roots||Previous Lesson - Secants, Tangents and Derivatives|
Let f(x) = ax and consider f(x+h)/f(x) = ax+h/ax = ah. Thus the ratio of a exponential functions at equal distance are equal. That is, for fixed h f(x+h)/f(x) is the same regardless of h. Often data is presented in the form of points, this gives us a useful way of checking whether a set of data is related by an exponential function.
Consider the following table which gives the worlds population in millions at 50 year intervals from 1750 to 1950:
|Population||700||900||1200||1600||2200|Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
Population 76 92 106 123 131 150 179 203 227 250
The exponential function has the property that it is its own derivative, i.e.
D(exp)(x) = exp(x).
The exponetial function is the underlying function in many applications.
This is particularly useful in the context of differential equations of the
dy/dx = ky.
y = A*exp(kx) satisfies this equation for any constant value of A. (See section 6.5 of Stewart.)
Consider the graph of exp(-x^2):
> plot(exp(-x^2), x=-3..3);
This is the graph of the normal distribution in probability, with mean 0
and standard deviation 1.
In general exp(-((x - m)/s)^2) is the graph of the normal distribution with mean m and standard deviation s.
Over the complex plane the exponential function is related to the
exp(I x) = cos(x) + I sin(x)
Substituting x and -x and solving gives
cos(x) = (exp(I x) + exp(-I x))/(2 I)
sin(x) = (exp(I x) - exp(-I x))/2.
Over the reals a similar expression relates the exponential function to the
hyperbolic sine and hyperbolic cosine functions, which are defined by
cosh(x) = (exp(x) + exp(-x))/2
sinh(x) = (exp(x) - exp(-x))/2
Logarithms are useful for placing one exponential in terms of another.
If c = ln(a) then ax = ecx
|Up to Main Lab Page||Next Lesson - Finding Roots||Previous Lesson - Secants, Tangents and Derivatives||Top of this Lesson|