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Maybe we want to find the square root of some number |
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| and our calculator isn't working. | ||
| Wait, Stop, Cut, Hold the presses. | ||
| Nobody is going to do this for real. | ||
| Not now anyway. Maybe once upon a time when there were dinosaurs, | ||
| but not when computers are given away in happy meals. | ||
| This page is just a cute way to get you more comfortable | ||
| with moving these symbols around. | ||
| The deal is, down the line a way we need to pile some more stuff onto these ideas. | ||
| When that happens, you need to be really comfortable with these equations. | ||
| This is just a made up way to get you more practice with this stuff. | ||
| OK, so there's a way to use this stuff to approximate roots. | ||
| If you need a number like the principal square root of 49 | ||
| (that is, the positive one), | ||
| it's no problem because it's probably one you know or can work out in you head: | ||
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| But what if you wanted to know the square root of 48? | ||
| We know it will be a bit less than 7, but what? 6.9? 6.3?? | ||
| Here's a way to get close: | ||
| We don't have 49 but have 48. | ||
| The change from something easy to deal with was -1. | ||
| The square root function is written: | ||
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| The derivative, or "Rate of Change" of this function is: | ||
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| Which we can also write as: | ||
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| Now we can split the dY/dX thing (multiply both sides by dX), | ||
| to get the rate of change of the function (dY) all by itself. | ||
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| dY is the amount that we have "changed" the function | ||
| from when it was the square root of 49 | ||
| to now when it is the square root of 48. | ||
| dX is the amount that we changed X. | ||
| We changed X from 49 to 48, so the change in X is -1. | ||
| That means dX = -1 and X = 49 | ||
| The square root of 48 is roughly equal to the square root of 49 | ||
| and the change (dY) | ||
| So: | ||
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| To 4 decimal places the answer is 6.9282. | ||
| A difference of only 0.0004! | ||
| Not too shabby, eh? | ||
| If we had wanted to approximate the square root of 47 (49 - 2), | ||
| we would have used: | ||
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| To 4 decimal places, the actual answer is 6.8557. | ||
| This is a difference of 0.0014. | ||
| This is still pretty good, but not as good as the answer for the square root of 48. | ||
| The further we get from 49 the bigger dX gets, | ||
| and the worse the approximation gets. | ||
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copyright 2005 Bruce Kirkpatrick |
