|
There is a difficulty with finding anti-derivatives akin
to the problem of defining the inverse to a function
that takes on some of its values at more than one argument
each. Knowing that the original function attains one of those
values does not determine the inverse function; additional
information in the form of an additional condition is necessary
to distinguish among the more than one possible original arguments
that could be called the inverse function under such circumstances.
In going from the derivative to the function we must contend
with the fact that a constant has zero derivative: thus we
can add any constant to any possible anti-derivative and get
another equally valid one. Which is really to say that to
determine an anti-derivative completely you must add additional
information. In particular the value of the function at any
one argument will be enough to determine it from its derivative
over its domain.
Thus we must recognize the fact that without an additional
condition on f, there is no one single anti-derivative f for
given g; we can find an anti-derivative, or describe all anti-derivatives
(in which case you should add +c to any one anti-derivative)
but it is not quite right to use the terminology "the
anti-derivative of g" until an additional condition has
been specified. (I recall that when I studied calculus in
ancient times we had to play a game reminiscent of "Simon
Says" with +c; if the question at hand was phrased one
way the answer had to have a +c; otherwise not. As I recall
from even longer ago, it is very easy to distract a person
to lose at Simon Says and it is equally easy to forget to
write +c when it is called for.)
|
