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Section 7.2 Thick Derivatives

In Section 7.1, we briefly discussed representing derivatives symbolically (as ratios), graphically (as slopes), and even verbally (as the ratio of small quantities). What about numerically or experimentally?  1 

Suppose you record your position every few minutes. You can make a table of your data, showing times and positions. How fast were you going?

Well, you can divide the various distances traveled by the time it took, but the results are clearly average speeds for each interval. What if you were to record your position every few seconds? Every few microseconds?

Mathematicians would argue that this question is still poorly posed: Since you only have discrete, numerical data, no limits can be taken, so you can only ever compute average speeds. On the other hand, in everyday circumstances the accuracy obtained with data taken every few microseconds is surely sufficient.

In the real world, the important question is not whether the computation yields an average speed or an instantaneous speed, but rather whether the average is “good enough”, where of course what's good enough depends on the context.

Rather than engage in a pedantic discussion of whether one can in principle compute derivatives from numerical or experimental data, we choose to embrace such computations due to their importance in applications. We therefore introduce the concept of “thick” derivatives to encompass both instantaneous rates of change and average rates of change that are “good enough”. Again, this notion depends on the context; a detailed analysis properly belongs to the fields of numerical analysis and data analysis. Nonetheless, we will not quibble about referring to the results of measurements over suitably small intervals as “derivatives”.

For further discussion, see [2].

We would argue that these are not quite the same thing, since experimental computations are limited by measurement error, whereas numerical computations are limited by roundoff error, a fundamentally different concept.