The Calculator Puzzler

In this Activity we will investigate how your calculator's [] (squared-root) key works. You might wonder how you would compute a square-root yourself, or at least how you would approximate it. If you answered "I would use my calculator", remember that someone had to program your calculator to approximate the square root of a number, so that doesn't really answer the question of how one might carry out such an approximation.

• Notice what happens when you repeatedly press the [] key on your calculator. Let's try this with 5 as the initial input number and see what sequence of numbers we get:
5
2.23606...
1.49534...
1.22284...
1.10582...
1.05158...
1.02546...
. . . . . .

• It looks like the fractional part is roughly halved at each successive stage, that is, each time we take the square-root, the next number we get seems to have a fractional part about half that of the preceding number. But it can't be that that is all our calculator really does!

What is going on?

Let's investigate...

• Does this process take place if we start with a number other than 5? Do you think this happens for all? most? some? choices for initial value? Experiment.

• When does this "halving" phenomenon happen in general? How big or small does the input number have to be?

• Did you investigate also what happens when your input number is a fraction?

• Follow the steps below, assuming you are dealing with input numbers of the sort which you have experimentally determined would produce the "roughly half the fractional part" phenomenon.

• Part A: Non-calculus Method

• (1)Let s be a "small", positive number. Then our observation that "the fractional part is roughly halved at each stage" can be mathematically described by:

• The square-root of 1+s is approximated by (1+s) __________

• (2) Give an informal proof/justification for this "approximate equality" by analyzing what you get by squaring both sides.

• (3) If x=1+s then another way to describe our observations mathematically is:

• The square-root of x is approximated by (x) _____________
(You should fill in the blank with an expression in terms of x)

• (4) Give an informal proof/justification of (3)

• Part B: Calculus to the Rescue!

• (1) Use the Taylor series of f(x)= (x) at c=1 to give a more formal justification of your approximation in part (A)(3). You may assume that f(x)is equal to it's Taylor series, T(x),so that you need only show that T(x)is approximated by the right-hand side of the approximation (A)(3).

• (2) Now use Taylor's Theorem with Remainder to quantify how good an approximation the approximate-equality in (A)(1) is. For example, if x is within 1/100 of 1, how good an approximation of the square-root of x is the right-hand side of (A)(3)?

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