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

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:

· The Taylor series for  will be an alternating series; that is, the signs of consecutive terms will alternate from positive to negative and back again, with the pattern repeating indefinitely.

· If we assume  , then the absolute value of each term is smaller than the one before.

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:

· Determine the number of terms in the corresponding Taylor series expansion required to approximate the value of to within  , and state the resulting approximate value of .

· Use the absolute value of the first term you omitted to estimate the error in your approximation.

·

Use this table to organize your work: