Let’s say you need to find the square root of 4.7.  You pull out your hand calculator (or the Calculator app on your phone), type in , and you have the answer.

But how did the calculator know it?

In fact, we take modern computational conveniences for granted.  The hand calculator (or the corresponding app) is a fantastic product of engineering; and in this exercise, we’re going to apply one of the techniques they use to calculate square roots: Taylor Series.

Taylor series allow us to write any differentiable function f(x) as an infinite series of the form

for values of x near x = a.  Note that  represents the n’th derivative of the function f(x) evaluated at x = a.

Clearly, the details of a Taylor series expansion depend upon the function involved; for the square root function  , it can be shown that:

Combined, these two facts tell us that the Taylor series for   converges; and if we truncate the series after n terms, then the error in our approximation will be smaller than the absolute value of term n+1 in the series.


With this background, here is your assignment:

Use this table to organize your work:


Function and derivatives Evaluate function and derivatives  term of Taylor Series term of Tayler Series evaluated at value of interest within Cumulative sum of Taylor Series terms Approximation accurate to within Error estimate




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