Teaching Differential Calculus as the Limit of Discrete Calculus

[jambaugh]

I'm teaching Calc I. this semester and we're now covering the derivatives of power function and exponential functions as well as the basic rules, e.g. linearity and product rule. Some years back I ran across an exposition of umbral calculus in the appendix of a reference. I cannot help but wonder if it would be useful for my students to be shown the discrete versions of the basic differential formulas and then to take one big whopping limit of everything to derive the corresponding derivative formulas:

Specifically...

We define the finite difference and the difference quotient of a function for some fixed non-zero parameter $h$ as...
$$\Delta f(x) = f(x+h)-f(x)$$ For the identity function in particular: $\Delta x = h$
Then the difference quotient is:$$\frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{h}$$
We then define the "preternatural" (any ideas on a better name?) base $\tilde{e} = (1+h)^{1/h}$
and the $n^{th}$-degree polynomials (generalized powers):$$p_n(x,h) = x(x-h)(x-2h)\cdots (x-nh+h)$$
and for $n<0$ the rational functions: $$p_n(x) = \frac{1}{(x+h)(x+2h)\cdots (x-nh)}$$

The idea is to show the usual derivative formulas, but for the difference quotients:
$$\frac{\Delta \tilde{e}^x}{\Delta x} = \tilde{e}^x \quad \text{ and } \quad \frac{\Delta p_n(x)}{\Delta x} = n p_{n-1}(x)$$ we also have a product rule:
$$\frac{\Delta [u\cdot v]}{\Delta x} = \frac{\Delta u}{\Delta x}\cdot v + u \frac{\Delta v}{\Delta x} + h\frac{\Delta u}{\Delta x}\frac{\Delta v}{\Delta x}$$

Then "in the limit as $h\to 0$:
$$\frac{\Delta}{\Delta x}\to \frac{d}{dx}, \quad p_n(x,h)\to x^n,\quad \tilde{e}^x \to e^x$$
And the power rule, product rule, and derivative of the exponential function all manifest.

I'd be interested in hearing from anyone if they think this would be conceptually useful approach, or a horrible idea, or if there were any suggestions. At the very least I thought it might be a good undergraduate special topics course.

There are, further extensions as we can express the perturbed versions of any other analytic functions using the generalized powers in their usual power series expansion, e.g.$$\widetilde{\sin}(x) = p_1(x)-\frac{p_3(x)}{3!} + \frac{p_5(x)}{5!} + \cdots; \quad \frac{\Delta \widetilde{\sin}(x)}\Delta{x} = \widetilde{\cos}(x)$$
Too much? Comments encouraged!

 

[scottdave]

How is "just learning derivatives" taught now?

Some of this might be too much. I'm trying to remember when I first learned Calculus. I have not seen this Generalized form for polynomials p_n(x,h) before. I'm not sure that I could take that for granted - especially as a student new to Calculus.

The power series came later when I learned them - using derivatives to derive them.

Just some thoughts.

 

[mpresic3]

I hated delta-epsilon proofs in early calculus for definitions of derivatives etc. As difficult as these concepts and proofs are, I think they are far more approachable than what you are proposing. For example, without the elemetary calculus, how do you motivate e = (1+h) to the (1+h) power? This is better left to a special projects course.
I see you as a interested and astute teacher, on how best to present material to students in novel ways. You are to be congratulated. However, in this case, I think this is best left to a higher math course.

 

[kith]

I don't think this is a good approach because dragging the $h$ along longer makes the formulas more complicated.

If I wanted to introduce a discrete version of Calculus first, I would try what Donald Knuth calls "finite calculus" in his book "Concrete Mathematics" where he uses the simple difference operator $\Delta f(x) = f(x+1) - f(x)$. He gets many interesting formulas but has a different goal. I'm not sure if this can really be made into an illuminating precursor to Calculus or if knowing Calculus is necessary to appreciate it.

In any case, Mathologer also has an interesting video on it which contains delightful analogies like $2^n$ corresponding to $e^x$:

 

[jambaugh]

How is "just learning derivatives" taught now?

Some of this might be too much. I'm trying to remember when I first learned Calculus. I have not seen this Generalized form for polynomials p_n(x,h) before. I'm not sure that I could take that for granted - especially as a student new to Calculus.

The power series came later when I learned them - using derivatives to derive them.

Just some thoughts.

"just learning derivatives" in my course is, first the geometric definition as slope of tangent line on the graph of the function, then we do the limit of the difference quotient as the "formal definition". Then we start running through various properties an functions, e.g. linearity, product rule, powers exponentials....

As to "this might be too much", yes I think so for a conventional Calc. I course. But I think for an honors calculus series it would be within the student's grasp if the three course sequence is constructed appropriately. In my ambitious moments I fiddle with the idea of writing a textbook.

As for the "generalized derivative", see umbral calculus, falling (and rising) factorials, and Pochammer symbol. In most expositions the $h = \Delta x$ value is fixed at 1.