



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 
