An Introduction to Exponential GrowthDate: 10/07/2006 at 14:23:22 From: Sara Subject: exponential growth? Hello, What exactly is exponential growth? At this moment I am using my book's study aid and all I see is equations and a lot of exponents. Please tell me what they are and what they mean. Thank you! Date: 10/09/2006 at 15:40:27 From: Doctor Ian Subject: Re: exponential growth? Hi Sara, It's helpful to think about other kinds of growth. For example, something might grow linearly, e.g., y = 2x In this case, for a given change in x, we get the same change in y, no matter what x is. In the case above, changing x from 5 to 6 gives us a difference of 2*6 - 2*5 = 2*(6 - 5) = 2*1 = 2 A change of 1005 to 1006 gives the same difference: 2*1006 - 2*1005 = 2*(1006 - 1005) = 2*1 = 2 If we plot all the (x,y) pairs that satisfy this kind of growth, they all lie on a line, which is why we say the growth is "linear". Does that make sense so far? Next, consider something like y = x^2 Here, we would say that the growth is "polynomial", because we have a variable and constant exponents greater than 1. In this case, the change in y corresponding to a given change in x depends on the value of x. Go ahead and try evaluating that the way I did above. In the case above, plotting these points would give you a parabola, which increases more steeply the farther x gets from the origin. Now, suppose that instead of having a variable with a constant exponent, we have a constant with a variable exponent? That might look like this: y = 2^x When the exponent is the thing that's changing, we say that the growth is "exponential". Near the origin, this looks a lot like the parabola, but it grows _much_ more quickly than any polynomial. To see how much more quickly, try plotting all three functions, y = 2x y = x^2 y = 2^x using a graphing calculator. Or, you can make a table, with different values for x: x 2x x^2 2^x ----- ----- ----- --------- 1 2 1 2 2 4 4 4 3 6 9 8 4 8 16 16 5 10 25 32 10 20 100 1,024 20 40 400 1,048,576 When the values of x are small, there isn't a lot of difference; but as the value of x gets large, exponential growth overwhelms linear and polynomial growth. This has two consequences that are worth knowing about. One is that when we've identified true exponential growth, it's usually something we should be interested in--whether it's a good thing (e.g., a growth in profits) or a bad thing (e.g., a growth in the number of cases of a disease). The other is that because "exponential growth" sounds so dramatic, many people say that something is growing exponentially when they just mean it's growing quickly. It's a kind of exaggeration, so when you hear someone talking about exponential growth, you need to seriously consider which of the following is true: (1) the person is right, (2) the person is wrong, but doesn't realize it, or (3) the person is wrong and realizes it, but hopes that you don't. Does this make sense? Let me know if you need more help. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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