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There are mathematical functions besides linear and polynomial functions that are very useful. For example, it may very well be the case that exponential functions arise even more frequently in nature than linear ones. The exponential function is by far the most important function in all of mathematics (this becomes especially true when one considers generalizations of exponential functions from real to complex numbers and beyond to more exotic situations). For the moment, we will stick to real-valued functions.
Pure exponential functions are often written in the form y = abx where a and b are given constants, and b must be positive. Examples would be 3(6x) or 5(2/3)x. The point is that the variable x appears in the exponent.
Before proceeding, do exercise
1-20 to find out more
about the properties of exponential functions.
In exercise 1-20, you should have discovered that exponential curves all have basically the same shape. For values of b bigger than one and positive values of a the value of y rises slowly at first (for negative x), but then begins to rise more and more rapidly, until finally it is shooting almost straight up. For values of b less than one, the value of y decreases - quickly at first and then more and more gradually, approaching but never quite reaching zero.
Recall that the special thing about linear functions was that the slope of their graphs is well-defined, i.e. the quantity (y2 - y1)/(x2 - x1) does not depend on which two points (x1, y1) and (x2, y2) on the line are chosen. The graphs of exponential curves satisfy an analogous property, which can be verified using the rules of exponents: If (x1, y1) and (x2, y2) are points on the graph of y = abx, then the quantity
is independent of which two points on the graph are chosen.
Do exercise
1-21 before proceeding.
When b > 0, so the exponential function is an increasing one, the equation y = abx is called an equation of exponential growth. When b < 0, the phenomenon being modelled is one of exponential decay.
In an exponential growth situation, the rate at which the quantity y is increasing is proportional to the value of y itself - in other words, the more y you have, the faster you get more. This is the case, for example, for populations of animals or bacteria. The larger the population, the more organisms there are to produce still more, so the population increases faster, etc. Another situation which produces exponential growth is an exothermic (i.e. heat-producing) chemical reaction: the more of the reaction that is happening, the more heat is produced, which in turn causes the reaction to proceed faster, which in turn produces still more heat, etc. etc. etc... This process is sometimes described as a chain reaction.
An example of an exponentially growing population appears to be the population of Mexico during the 1980s. The estimated population of Mexico for the early 1980s is given by the following table:
| Year | Population of Mexico |
|---|---|
| 1980 | 67,380,000 |
| 1981 | 69,130,000 |
| 1982 | 70,930,000 |
| 1983 | 72,770,000 |
| 1984 | 74,660,000 |
| 1985 | 76,600,000 |
| 1986 | 78,590,000 |
Use the data from this table to do exercise
1-22.
Lecture Question #4: What Determines The Rate of Exponential Population Growth?
For an example of exponential decay, consider the process by which kerosene is purified to make jet fuel - pollutants are removed by passing the kerosene through a special clay filter. Suppose the clay is in a pipe and that each foot of the pipe removes 20% of the pollutants that enter it. Therefore, each foot leaves 80% of the pollutants. If P0 is the initial amount of pollutant, and P = f(n) is the quantity left after n feet of pipe, then f(1) = (0.8)P0, f(2) = (0.8)(0.8)P0, etc., and one reasons that f(n) = P0(0.8)n.
Use the data given above to do exercise
1-23.
You have seen how the value of b in the exponential function abx affects the steepness of the exponential's graph. Another standard measure of the relative speed with which an exponential function grows (or decays) is to give the length of the interval on the x axis that it takes for the value of y to exactly double (or divide in half). An exceptional characteristic of exponential functions and graphs is that the length of this interval does not depend on its location. To verify this, let y = abx and assume that a > 0 and b > 1 so we are talking about exponential growth. We search for two values of x, call them x1 and x2 such that y(x2) = 2y(x1). To find them, do some algebra:
Notice that we end up solving for the length of the interval between x1 and x2 - it is called the doubling time of the quantity y. To solve this last equation, we should take the logarithm of both sides to get x2 - x1 = logb(2). Even though Maple is capable of computing logarithms with respect to any base, it is convenient to express logarithms in terms of a standard base, usually 10 or e (i.e. natural logarithms). If we do this, we get
where log stands for either of the standard logarithms.
Do exercise
1-24.
Another situation in which exponential decay is the appropriate model is radioactive decay. Scientists usually describe the rate at which a radioactive substance decays by giving its half-life, i.e. the length of time interval required for the value of y in y = abx with b < 1 to be reduced by half.
Do exercises
1-25,
1-26, and
1-27
to compare and contrast the behavior of
polynomials and exponentials.
