1.6.1 A Cautionary Tale

You need to know the answer to the question in the section title when your calculus or chemistry instructor asks you to report a numerical answer to, say, four significant digits. This question is not specifically about functions, the central theme of this chapter, but it is related to our numeric representations of functions — and to how we make sense out of those representations.

Here's a true story about a student, a calculator, and a calculus exam. We will call the student “Anna” — not her real name. To protect the innocent, both the calculator and the exam shall remain nameless. While the story comes from the “calculator era” — when every calculus student had a graphing/numeric calculator in hand — its cautions apply equally well to the study of calculus with a computer algebra system, as we will see. You can follow Anna's calculations by clicking on the calculator icon at the right to pop up a numeric calculator.

A question on the exam asked Anna to estimate the rate of change, on a very short interval starting at 0.5, of a function `ftext[(]x text[)]` known only by a calculator button that had never been discussed in class: . If you are not sure yet what the question means, think of it as asking for the slope of a line through

( 0.5 , f ( 0.5 ) ) ,

which will be calculated as rise over run.

Being a good student, Anna knew exactly what to do. First, she chose her very short interval to be [0.5, 0.501]. Given this choice, the rate of change (think slope) of the function over the interval can be calculated as the change in function values (think rise) divided by the length of the interval (think run):

f ( 0.501 ) - f ( 0.5 ) 0.501 - 0.5 .

Anna's calculator told her that f (0.5) was 0.523598775598, which she wrote down as 0.524 — and that f (0.501) was 0.524753861551, which she recorded as 0.525. [Note: Anna was using a 12-digit calculator. The one we provide has 16 digits, but the rounded answers are the same.]

Next, she calculated the rate of change:

f ( 0.501 ) - f ( 0.5 ) 0.501 - 0.5 = 0.001 0.001 = 1.

Actually, Anna wrote the final step as “$\approx 1$ ” — she knew that this was only an estimate, and that was all the problem asked for. However, by her calculator procedure, she forced the answer to have less accuracy than was available. If the problem had specified an accuracy, her answer might have been wrong.

What was wrong with Anna's procedure? There was no need for her to write down, round off, or re-enter either of the function values. The calculator's 12-digit value for f ( 0.501 ) - f ( 0.5 ) is 0.001155085952, so the end of the calculation might have been

0.001155 0.001 = 1.155 ,

an answer that differs from Anna's by about 15%. There was no way Anna could have known exactly what she was trying to estimate — that's why we sometimes have to estimate! But she should have known — more important, you should know — that sloppy use of meaningful digits can lead to wrong answers.

In this section we will explore some examples and exercises on significance and then describe a test by which you can decide whether you are responding appropriately to assignments that call for numerical answers — which means most assignments.

There are at least these three sources of concern about significance of digits:

• The accuracy of available data
• The finite precision of your calculating device
• Loss of significance due to the way we manipulate the numbers

(Anna's calculation is in the third category.) On the following pages, we comment briefly on each of these and provide examples of each.

Note 1 – Source

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