Calculus Using Approximation to Find Roots
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Dances With Approximation
Using Approximation to Find Roots

 

 Maybe we want to find the square root of some number

 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:

 

 

 
 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:

 

 
 The derivative, or "Rate of Change" of this function is:
 

 

 
 Which we can also write as:

 

 

 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.
 

 

 
 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:

 

 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:

 

 
 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.
 

   copyright 2005 Bruce Kirkpatrick

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