HP 15c - Manual (Page 42)

42 Section 2: Working with f
42
Functions Related to Physical Situations
Functions like cos(4

- sin

) are pure mathematical functions. In this context, this means that
the functions do not contain any empirical constants, and neither the variables nor the limits
of integration represent actual physical quantities. For such functions, you can specify as
many digits as you want in the display format (up to nine) to achieve the desired degree of
accuracy in the integral.
†
All you need to consider is the trade-off between the accuracy and
calculation time.
There are additional considerations, however, when you're integrating functions relating to an
actual physical situation. Basically, with such functions you should ask yourself whether the
accuracy you would like in the integral is justified by the accuracy in the function. For
example, if the function contains empirical constants that are specified to only, say, three
significant digits, it might not make sense to specify more than three digits in the display
format.
Another important consideration—and one which is more subtle and therefore more easily
overlooked—is that nearly every function relating to a physical situation is inherently
inaccurate to a certain degree, because it is only a mathematical model of an actual process
or event. A mathematical model is itself an approximation that ignores the effects of known
or unknown factors which are insignificant to the degree that the results are still useful.
An example of a mathematical model is the normal distribution function,

 
 
t
dx
e
x
 
 
2
2
2 /
2
) (
which has been found to be useful in deriving information concerning physical measurements
on living organisms, product dimensions, average temperatures, etc. Such mathematical
descriptions typically are either derived from theoretical considerations or inferred from
experimental data. To be practically useful, they are constructed with certain assumptions,
such as ignoring the effects of relatively insignificant factors. For example, the accuracy of
results obtained using the normal distribution function as a model of the distribution of
certain quantities depends on the size of the population being studied. And the accuracy of
results obtained from the equation s = s
0
− ½gt
2
, which gives the height of a falling body,
ignores the variation with altitude of g, the acceleration of gravity.
Thus, mathematical descriptions of the physical world can provide results of only limited
accuracy. If you calculated an integral with an apparent accuracy beyond that with which the
model describes the actual behavior of the process or event, you would not be justified in
using the calculated value to the full apparent accuracy.
Round-Off Error in Internal Calculations
With any computational device—including the HP-15C—calculated results must be
“rounded off” to a finite number of digits (10 digits in the HP-15C). Because of this round-
off error, calculated results—especially results of evaluating a function that contains several
†
Provided that f(x) is still calculated accurately, despite round-off error, to the number of digits shown in the display.