Question

How do you use linear approximation to the square root function to estimate square roots `sqrt 3.60`?

Answer 1

`1.9-0.01/(2 xx 1.9)`
#1.8973684 ...#
compared to exact value of
#1.8973665... #

Explanation:

Noting that `2.0^2=4.0` exactly and `1.9^2=3.61` exactly, do a Taylor Series or Binomial Series to get a linear approximation to `sqrt(x)`around `3.61`:

Using the Taylor series we get
`f(a+h) = f(a)+hf'(a)...`
Setting `f(x)=sqrt(x)`, `f prime(x)=(1/2) xx 1/sqrt(x)`, `a=3.60`, `h=-0.01` and truncating after the term in `h`we get.
`sqrt(3.61-0.01)approx sqrt(3.61) - 0.01 xx (1/2) xx 1/(sqrt(3.61))`
`sqrt(3.60) approx 1.9-0.01/(2 xx 1.9)`
which is about `.0001%` too high.

If you prefer the Binomial Series:
`(3.61-0.01)^(1/2)`
`=1.9 xx ( 1-0.01/3.61)^(1/2)`
`=1.9 xx (1-(1/2)xx(0.01/3.61)...)`
`approx1.9-(1/2)xx(0.01/1.9)`
as before.