Walter Spunde Department of Mathematics and Computing U.S.Q.
 Research Publications Teaching  Professional Personal

Research Interests:


20th Derivatives of 
(x csc x) / ln arctan exp x
  • Multi-point automatic differentiation techniques applied to the solution of differential equations.
Ask a computer algebra system to compute the twentieth derivative of a mildly complicated quotient of elementary functions, such as that illustrated at left, and you may have to wait a little while for an answer. The tenth derivative of the function illustrated took a little over a minute with each additional derivative roughly doubling the time required. The formulae, of course,  take minutes to scroll down the screen. 
But, ask for the first twenty Taylor coefficients for this same function at any (regular) point and the numbers are returned in less than half a second. So how are those coefficients calculated without the formulae for the derivatives? 

The graph to the left is not obtained from multiplying the 21st Taylor coefficient from a CAS by 20! at each of the 60 points used to generate the graph--although the numbers do agree. It is done with an automatic differentiation technique implemented in APL (see Publications, Articles 7, 8, 9). Using automatic differentiation techniques to generate high degree Taylor approximations provides an interesting approach to the solution of differential equations.
Leibniz used the operators d and S to apply to sequences and to return sequences, although he didn't have an adequate notation for them. For example, he wrote 

Supposing that 1+1+1+1+1+etc = x or that x represents the natural numbers,  for which   d x = 1,  then  1+2+3+4+5+etc = S x 
[ See: Florian Cajori, A History of Mathematical Notations, Vol.II, p.264, Open Court Publishing, Chicago, (1929). (The quotation is slightly modified to correct a small error.)] 

We might write:   if       x = 0 1 2 3 4 5 ... 
then        d x = 1 1 1 1... 
and       S x = 0 1 3 6 10 15 ... 
for which       d S x = x
and      S d x = x.

If we think of a curve as made up of a sequence of points infinitely close together, as Leibniz did (but which, of course, it isn't, since the points on a curve are uncountable) then we can compute the 'integral' S y*d x  and deduce that, since 
     d (S y*d x) = y*d x, 
we have, 
     d(S y*d x)/d x = y 
and also that 
     S(d y/ d x)*d x = S d y = y - y0 . 
  • The effect of notation on concept development in calculus; 

    • and the effect of computing power on learning and teaching styles in mathematics.
Notation can clarify or obscure. Conventional notation of calculus is particularly rich in inconsistencies and context sensitive interpretations. Why for example is nl the product of n and l, but  ln  is not the product of l and n; why does cos(3+x^2) require parentheses but cos 3x^2 not; what do we mean by d^2/dx^2 -- shouldn't it be d^2/(dx)^2 -- or should it be d(d/dx)/dx -- is the operator the d , in Leibniz's sense, or the d/dx in Newton's sense? What sense can one make of it if one doesn't know anything about calculus? 

Making sense of something depends on the experiences one has on which to base one's judgement. In classical mathematical instruction, experience is often restricted to listening to verbal descriptions, reading descriptions, observing procedures and being told that the procedure one has followed is correct. Computing power allows one to interpret and evaluate numerical results and provides a whole new set of concrete experiences on which to base expectations. 

There are many sources of misconceptions in the learning of calculus. Computing power not only opens new avenues for the delivery of material but new experiences in numerical experimentation associated with mathematical concepts.  (See Articles 1-6, 10-18 under Publications.)

As one calculates points (x,y) for the graph of a function and joins them together with little straight lines, add on also to another list, Ay, the average y value over the interval times the width, dx, of the interval between successive x values.  Graph Ay, the integral of y with respect to x, concurrently. Alternatively the solid graph (y vs x) can be  interpreted as the derivative of the dotted graph (Ay vs x)  giving the slope of the dotted graph. 

The Sample Calculus identifies the key tools for making basic computations such as the above easy and intuitive.
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http://www.usq.edu.au/faculty/sci/staff/spunde/research.htm July 1997