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Re: A differential equations based course
Posted:
May 14, 1999 2:39 PM
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This sounds like Project CALC.
MARK BRIDGER AT NORTHEASTERN UNIVERSITY wrote:
>
> I think that Kaz has an excellent point which seems to
> have been drowned out by the currently fashionable push for
> the integral as an "accumulator" and what seems to me to be an
> often mindless desire to present numerical calculations. Is
> there some sort of puritan ethic at play here? In terms of
> computing anything, Riemann sums are losers. They have their
> theoretical and pedagogical place, but it may be overrated.
>
> It makes perfectly good sense, on many counts, to approach
> calculus by making derivatives and differential equations
> central. Here's the schema:
>
> 1. Rates of change and derivatives
> 2. Applications of derivatives
> 3. Flow fields and differential equations
> 4. Anti-derivatives
> 5. Euler's method
> 6. Intuition: the derivative of area
> 7. Euler's method and Riemann sums
> 8. The definite integral
> 9. Etc.
> 10. Infinite series as D.E. solutions
>
> If I were writing a calculus book for scientists, I might
> very well take this approach. Slope fields can be understood
> by students who don't know anything about calculus;
> differential equations are central to much of applied
> mathematics; Euler's method has as much intuitive content,
> especially with the availability of slope-field plotters, as
> the definite integral, and can be applied to all ordinary
> first-order differential equations. My students, even the
> weakest, have absolutely *no* trouble doing Euler's method,
> and even improved Euler's method, on the Excel spreadsheet,
> which also give them plots of the solution. And, of course,
> there are numbers galore.
>
> This order of topics allows students to learn things as
> sophisticated as predator-prey equations and phase-plane
> portraits very early.
>
> The last topic, infinite series, has a far more realistic
> motivation from differential equations than from
> approximations of the standard functions. Students can find
> sin(x) at the touch of a button so this is hardly a compelling
> selling point for infinite series. In any case, all the
> standard functions of calculus are solutions to simple
> differential equations, and are perhaps best *defined* in this
> way. Their power series expansions follow immediately.
> Formal Taylor series, with their usually impossible to
> compute "nth derivatives at P", are mostly of theoretical
> interest. Any convergent power series, no matter how its
> obtained, is a "Taylor" series. That it converges to a
> particular function is a result of the fact that it and the
> function satisfy the same differential equation. Its
> truncation error (remainder) is best estimated by comparison
> with a geometric series, *not* by the generally unusable
> "Lagrange" (or other) formula.
>
> This is how I'd do calculus for science majors. A lot of
> these ideas have been tried and found successful; I know some
> of them are used by Ostebee and Zorn, and the Duke calculus
> project; I'm sure they appear, in pieces, elsewhere as well.
>
> --- Mark
>
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