252 Appendix E: A Detailed Look at f
The graph is a spike very close to the origin. (Actually, to illustrate f(x) the
width of the spike has been considerably exaggerated. Shown in actual
scale over the interval of integration, the spike would be indistinguishable
from the vertical axis of the graph.) Because no sample point happened
to discover the spike, the algorithm assumed that f (x) was identically
equal to zero throughout the interval of integration. Even if you increased
the number of sample points by calculating the integral in i 9, none
of the additional sample points would discover the spike when this
particular function is integrated over this particular interval. (Better
approaches to problems such as this are mentioned at the end of the next
topic, Conditions That Prolong Calculation Time.)
You’ve seen how the f algorithm can give you an incorrect answer when
f (x) has a fluctuation somewhere that is very uncharacteristic of the
behavior of the function elsewhere. Fortunately, functions exhibiting
such aberrations are unusual enough that you are unlikely to have to
integrate one unknowingly.
A function that could lead to incorrect results can be identified in simple
terms by how rapidly it and its low-order derivatives vary across the
interval of integration. Basically, the more rapid the variation in the
function or its derivatives, and the lower the order of such rapidly varying
derivatives, the less quickly will the f algorithm terminate, and the less
reliable will the resulting approximation be.
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