MadSci Network: Engineering
Query:

Re: What use do the higher order derivatives serve in Calculus (4th, 5th,etc.)?

Date: Mon Oct 11 16:25:45 1999
Posted By: Lew Gramer, MIT S.B. Math (Theoretical)
Area of science: Engineering
ID: 939428248.Eg
Message:

Actually there are many uses, Bob! The particular
application your question refers to is the mathematics of
motion - most simply expressed as a one-dimensional (x-y)
function of time. Acceleration, as you note, is just the
derivative of the function for Velocity vs. Time. However
in actual fact, velocity itself is really the derivative
of the function for Position vs. Time. Thus, acceleration
is in fact a second-order derivative, and rate of change
of acceleration is third-order! Of course, this value can
also change with time - it is in fact yet another function
of the same 't' variable - and thus *its derivative* is
yet another Time function: i.e., the rate of change of the
rate of change of acceleration, which is actually the rate
of change of the rate of change of position! :)

"Higher order" derivatives like the above have a very wide
variety of applications in the physical sciences: in fact,
there's an important class of functions in physical science
called the "analytic functions", which can be proven to be
INFINITELY differentiable! (In other words, you cannot pick
a finite number N such that the Nth derivative of any one of
these analytic functions is NOT well-defined & continuous.)
Probably the most common application of such functions is in
"series approximations" (Tayler, MacLaurin, Laurent), which
are often taught in calculus and basic analysis courses.


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