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:
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