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MTH 207 Lab Lesson 11

Exponential Functions

Previous Lesson - Secants, Tangents and Derivatives

Exponential Functions

Exponential functions are those of the form f(x) = a^x.
Exponential functions are always increasing (for a > 0) and concave up.
> plot(2^x, x=-2..2);
> plot(2^x, x=-5..5);
> plot(2^x, x=-10..10);

Plot the following on the same graph.

2^x and 3^x between -2 and 2.
2^x and 2^2x between -2 and 2.
2^x and 2^x+1 between -2 and 2.
2^x and 3*2^x between -2 and 2.
2^x and x*2^x between -2 and 2.

Let f(x) = a^x and consider f(x+h)/f(x) = a^x+h/a^x = a^h. 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:

Year 1750 1800 1850 1900 1950
Population 700 900 1200 1600 2200
We suspect that this relation is an exponential one. We can test this assertion by considering the ratio of terms.
> L := [ 700, 900, 1200, 1600, 2200 ];
> evalf(L[i]/L[i+1] $ i = 1..4);

.7777777778, .7500000000, .7500000000, .7272727273 While this is not perfect, it does tend to indicate an exponential relation. Since h is 50 in this case we might geuss that with t mearured in years
f(t) = 700 2^{.4344028242(t -
1750)/50}
(2^-.4344028242 = 0.74)
> f := t -> 700 * 2^(.4344028242 * (t - 1750)/50);
> f(1800);
> f(1850);
> f(1900);
> f(1950);
Use this approximation to estimate the population in 2000.
In fact this estimate is somewhat low.

Use the method above to check if the following data for the population of the United States, in millions, has an exponential form. If it does try to guess what this function might be and use it to estimate the population in 2000.

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

While an exponential function has the form f(x) = a^x
The exponential function is the function f(x) = e^x.
Where e = 2.718281828459...
Maple represents the exponential function by exp(x).
> plot(exp(x), x=-3..3);
> evalf(exp(1));

Plot the following graphs.

exp(-x)
exp(x) + exp(-x)
exp(x^2)
exp(-x^2)

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 form
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.

Plot the following normal distributions on the same graph as exp(-x^2).

mean 0, s.d. 2
mean 0, s.d. 3
mean 1, s.d. 2
mean -1, s.d. 3

Over the complex plane the exponential function is related to the trigonometric functions:
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

Plot exp(x), exp(-x) and (exp(x) + exp(-x))/2 on the same graph.

Plot exp(x), exp(-x) and (exp(x) - exp(-x))/2 on the same graph.

Logarithms

The logarithm function is the inverse of the exponential function. In particular the natural logarithm ln(x) is the inverse of the exponential function.
> exp(ln(x));
> plot(ln(x), x =-1..20);

Plot exp(x), ln(x) and the line y = x on the same graph.

Logarithms are useful for placing one exponential in terms of another.
If c = ln(a) then a^x = e^cx

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Maintained by: P. Danziger, Febuary 1998