How
do we know that if y = sin(x), then dy/dx = cos(x), when x is measured
in radians?
There
are two ways to approach it.
In "pure" mathematics, the only acceptable method is to provide a proof, namely, a
logical derivation of the formula at hand from already accepted theorems of a
given domain of mathematics.
But in applied mathematics, we often use a different justification. First, a proof is often not sufficient because of a possible mismatch between the mathematical model and the physical event that is described. Second, experimental evidence, which consists of checking the equality for some specific values, can disprove a false formula or provide a plausible and believable hypothesis that the formula is true, even if we do not know how to prove it.
When we work with polynomials, trigonometric, and other "elementary" functions, it is usually the case that when the equality is false, then its left and right sides have different values for most values of the variables involved.
For the problem at hand we will
proceed as follows. We will compute dy/dx - cos(x) for twenty numbers
randomly chosen between 0 and 2*π. If the computation were "exact",
we would get twenty zeros. But because the computation is only approximate, we can be satisfied when all the values
are "close" to zero.
Set
MODE to Radian.
Define:
\Y1=nDeriv(sin(X),X,X)
Enter
2π
*rand(20)→L1 ENTER (these numbers will vary
between 0 and 2π)
Y1 (L1)-cos(L1) ENTER
(Use
left and right arrows to view the list.)
The
differences are on the order of 10-7, which is close to the error in computing
derivatives. So we have good reason to
believe that the formula is true, even if we do not prove it.
Using
a random number generator to create our sample is a standard statistical
procedure that should be used in problems of this type.