brief tutorial
OK, suppose you've gotten some increasing function. Increasing means that the bigger x is (the number you plug in), the bigger y is (the number you get out).
It might be a power function: y=x, y=x^2, y=x^3, y=x^4, and so forth. These form a hierarchy. The higher the exponent, the faster the function grows. Now sometimes we talk more generally about linear growth, quadratic growth, cubic growth, quartic growth, etc. That means a function that looks somewhat like x, x^2, x^3, x^4 respectively. A quadratic function might not be x^2, it might be 10000*x^2. But it amounts to (almost) the same thing. If you multiply by a constant, you're not moving up the hierarchy, you're staying at the same level. If you compare, say, x^3 to 10000*x^2, the factor of 10000 makes the latter much bigger for small values of x. But once x gets big enough, x^3 is going to be much bigger. That's why we consider it a higher level of the hierarchy.
You might also tack a smaller term on. It's basically the biggest term that tells you where in the hierarchy you are. y=x^2+500x+20 is quadratic, because the x^2 term is the biggest -- it basically controls the approximate size of the function when x gets big.
If you're looking for a moderately simple function to approximate some data, you can probably get away with making it a single-term function. If you have a quadratic function, just assume it's y=b*x^2, no smaller terms. If cubic, y=b*x^3, no smaller terms. The smaller terms, if they exist, probably don't have that much effect, at least for large x. (For small x, they may make a much bigger difference).
So if you think your function is anywhere in this hierarchy, think of it as y=b*x^n. The task then is to find b and n that most closely approximate your data.
Now, your function might be somewhere above this whole hierarchy -- growing faster than x^2, faster than x^3, faster than x^10000. An example is an exponential function: y=2^x, y=3^x, y=10^x, etc. A lot of students confuse exponential functions (the base is constant, the variable is the exponent) with power functions (the exponent is constant, the variable is the base). There's a huge difference. Any exponential function eventually grows much faster than any power function.
The flip side (the technical term is "inverse") of an exponential function is a logarithm function. If y=10^x, then x=log
. The inverse of a power function is another power function; e.g., if y=x^2 then x=y^0.5. Note that the faster a function grows, the slower its inverse grows. So logarithm functions grow slower than x, slower than x^0.5 (square root of x), slower even than x^0.000001.
So you have some data and want to find a function to fit it. First thing is, decide where in the hierarchy the function belongs. One thing you might do is to look at differences between successive terms. If you have any power function, its differences will give you a power function of one lower degree. If the differences are constant, you have a linear function. For example, the sequence 1,3,5,7,9,... gives you differences 2,2,2,2,2,..., so it's linear. If you have a quadratic function like 1,4,9,16,25,..., the differences are linear (in this case 1,3,5,7,...). So the differences of the differences are constant. With a cubic sequence, the differences of the differences of the differences are constant. And so forth.
On the other hand, if you have an exponential sequence, its differences will give you... another exponential sequence. For example, the sequence 1,2,4,8,16,... has differences 1,2,4,8,... So no matter how often you take differences, you don't stop seeing the growth rate.
The way to recognize an exponential sequence is by looking at the ratio of successive terms. In the sequence 1,2,4,8,... if you divide two consecutive terms you get a ratio of 2. So the sequence is something like y=2^x. If the ratios between consecutive terms is constant, you're looking at an exponential function. On the other hand, try dividing consecutive terms of a power sequence, and you'll find the ratios get smaller and smaller, until they're almost 1. In other words, power functions don't grow nearly as fast as exponential functions.
! Gods, I love Math! (What a geeky statement that was.)
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