Answer to Question #1751 in Calculus for Imee Carla

Question

How can I use the concept in differentials to approximate a certain number which is not a perfect square?

Answer 1

You can expand the function of square root in a series :
We can represent the number as a sum of nearest perfect square root R and the difference of the number and this sq.root (N-R), it can be both positive and negative.
f(x) = √N = √(R + (N-R)) = √R √(1 + (N-R)/R).

Denote (N-R)/R as α , |α|<1. The function √(1 +α) can be represented as a sum:
√(1 +α)= 1 + 1/2 α - 1/8 α² + 1/16 α³ - 5/128 α⁴ + ... This expression was obtained by expanding the function √(1 +α) in a Teylor series in neiborhood of 1. √(1 +α) = sum from n=0 to infinity ( f⁽ⁿ⁾(1)/n! αⁿ ), where f⁽ⁿ⁾ (1) the n-th derivative of the function f(a)=√x at the point x = 1.
Thus the final fomula would be as
√N = √R (1 + 1/2 α - 1/8 α² + 1/16 α³ - 5/128 α⁴ +...), where α = (N-R)/R