Noticethat this equationis mucheasierto solvethan Settingf(x) = 0 and solvingforx,
we obtain,
If you use a calculator, you will getx= 2.236...As you can see, this is a fairlygoodapproximation.To be
sure,calculatethepercentdifference[%diff] betweenthe actualvalueand the approximatevalue,
whereAis the acceptedvalueandXis the calculatedvalue.
whichis less than 1%.
We can actuallymakeour approximationevenbetterby repeatingwhatwe havejust donenot forx= 2, but
forx 1 = 2.25 = , a numberthat is evencloserto the actualvalueof. Usingthe linearapproximation
again,
Solvingforxby settingf(x) = 0, we obtain
x = x 2 = 2.236111,
whichis evena betterapproximationthanx 1 = 9/4. We couldcontinuethis processgeneratinga betterap-
proximationto. This is the basicidea ofNewton’s Method.
Hereis a summaryof Newton’s method.
Newton’s Method1. Guessthe first approximationto a solutionof the equation
f(x) = 0. A graphwouldbe very helpfulin findingthe first approximation(see Figure
below).2. Use the first approximationto find the second,the secondto find the
third and so on by usingthe recursionrelation