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Exponential Growth and Decay
A function whose rate of change is proportional to its value exhibits exponential growth if the constant of proportionality is positive and exponentional decay if the constant of proportionality is negative. For exponential growth, the function is given by kbx with b >1, and functions governed by exponential decay are of the same form with b < 1. Populations might exhibit exponential growth in the absence of constraints, while quantities of a radioactive isotope exhibit exponential decay.GRAPH WITH a<1
An example of an exponential function is f(x)= 2 x , x as an element of all real numbers . How does the graph of f(x)= 2 x compare to that of f(x)= x 2 ?
The graph of f(x)= x 2 represents a parabola with its vertex at the origin, and is symmentrical about the x-axis. To determine the properties of exponential functions, we set up a table of values and used those values to sketch the graph of f(x)= 2 2 .
Properties:
1. Functions increase for all values of x.
(i.e. as x increase so too do the values of y)
2. Function is always positive (i.e. it lies above the x-axis)
3. As x --> infinity than y --> infinity
x --> negative infinity than y --> 0
That is, the x-axis is an asymptote
4. For values of
x > 0 then y > 1
x = 0 then y = 1
x < 0 then 0 < y < 1
The shape of the graph of y = b x depends on whether b < 1, b = 1, or b > 1 as shown on the right. The red graph is the graph of (b > 1), the blue graph is the graph of 1 x , and the green graph is the graph of (1/e) x (b < 1).
Sample Problem
1. A radioactive element decays in such a way that the amount present each year is 0.95 of the amount present the previous year. At the start of 1990, there were 50 mg of the element present. Produce a table of values showing how much of the element is present at the start of each year from 1990 to 1998.
Initially there is 50 mg of the elemtn present. At the start of the following year there will be 95% of that left over, i.e., there will be .95 * 50 = 47.5 mg left over. At the start of the year after there will be 95% of 47.5 mg left over. i.e., there will be 0.95 * 47.5 = 45.125 mg, and so on.
2.What happens when 0 Consider the case where a= 1/2. The general shape of the graph remains the same, except that it has been reflected about the y-axis.
