Exponential Functions
There are many different exponential functions and they range from the fairly simple to the rather complex. We will take them in turn.
If the interest on a principal amount is compounded once at the end of an interval the amount is A = P(1 + r) where r is the rate of return written as a decimal. A one time 8% interest payment on $1000 would produce A = $1000(1 + 0.8)=$1080. If this $1080 remained at the 8% and the interest was compounded again at the end of the next interval the amount would be A = [P(1 + r)](1 + r) = $1080(1 + 0.8) = $116.64. The expression in brackets represents the amount after one compounding and the entire expression represents the amount after two compoundings. The following problems illustrate compound interest problems.
EXPF04 Instantaneous compounding
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In many real-life problems growth is limited. Exponential models often are used to describe this limited growth. The simplest model for limited growth involving exponentials is one in the form
This growth model comes from a differential equation that is beyond the typical calculus course. Therefore, we will discuss limited growth exponentials starting with equations of this form. This equation is shown in the accompanying graphic. |
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Note that the N starts at zero at time zero.
The slope of the curve is the derivative or
This type of curve is sometimes called the learning curve because it describes someone learning a skill and eventually reaching a limit in productivity with that skill. the following problems illustrate the application of this type of curve to practical problems.
EXPF10 Assembling sewing machines
There is another type of exponential function used to describe limited growth that has the form
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is the present rate or number, whatever R represents.
As t goes to infinity, e-kt = 1/ekt goes to 0 and R approaches A. A is the maximum rate or number. This curve has the general shape shown.
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Many industries follow this type of a growth curve. When a new product is introduced demand increases, but as more and more people acquire the product sales drop to a level determined by the number of new people entering the marketplace and replacement of old or outdated product. The automobile industry is an excellent example of this type of growth.
There are several variations of this curve depending on the application. This, however, is a basic growth curve.
The function shown above is often called the Logistic Function. It is used in business to describe the growth of an industry or individual business. It is also applicable to populations and epidemics.
Often the return on an investment can be described with an exponential function. This usually works for a finite time early in the life of the investment. The return from an investment in an oil or gas well declines over time and the decline can be modeled with an exponential. The return over a period of time can be determined by the integral of the exponential model over the time period involved.
EXPF40 Return on WiFi investment
There is another relationship law, especially popular in manufacturing, that while it is not strictly an exponential function is very useful in understanding the interaction of components of the manufacturing process. The relationships are generally known as the Cobb-Douglas model and they involve the result of a manufacturing process being dependent on two variables each raised to a fractional power.
EXPF50 Cobb-Douglas employee-budget
TEXPF51 Cobb-Douglas robot-employee
Inflation and deflation, generally described as an increase or decrease in the amount of paper money required to by a typical "basket of necessities," can be modeled with rates. In the examples presented below the inflation rate is taken as a constant. One thing to note in the problems is that even a constant rate of either inflation or deflation has a large long term effect. The two problems below will give you a feel for how to apply growth and decay to inflation and deflation problems.
