Finding the square root of an integer using Calculus

Have you ever been in a situation where you needed to find the square root of an integer (especially imperfect squares), but had no calculator close by? I have. I however remember learning a Calculus trick (in senior high) that has always come to my aid in such situations. Grab a piece of paper and a pen, and come along as I share this trick. The trick requires that you identify some things:

• The integer (Obviously!)
• A perfect square that can be got from the integer
• The remainder in subtracting the perfect square from the integer

The steps involved in deriving the formula for finding the square root of an integer is shown below:

 

As an example, let's find the square root of the integer 7:

PS: You will notice (after a number of practices) that the values got using this method differ slightly from the ones got from the calculator. Do not be alarmed, the values are still correct nonetheless.

I do hope this is helpful and would come in handy when we need it.
Please let me know what your thoughts on any other trick you know. Waiting for your comments. ;-)