College Math Teaching

April 17, 2014

Rant: misconceptions we give to our students

Filed under: editorial, pedagogy — Tags: , , , — collegemathteaching @ 1:53 pm

I gave a take home test to my numerical analysis students. This was one of the problems (a “warm up”):

mth510gamma

I wanted them to choose a technique (say, a Lagrange polynomial or a cubic spline) to approximate the function at those values.

Most did fine. However one student’s answer bothered me: the student used a proper method but then rounded…to an integer each time.

Yes, in this special case, by pure chance..this turned out to be correct as this data comes from the gamma function.

But in general, that is terrible intuition. There is no reason a function should take integers to integers, even if that always happens with polynomials with integer coefficients.

Unfortunately, too many elementary calculus textbooks and too many math educators reinforce such bad intuition.

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March 25, 2014

The error term and approximation of derivatives

I’ll go ahead and work with the common 3 point derivative formulas:

This is the three-point endpoint formula: (assuming that f has 3 continuous derivatives on the appropriate interval)

f'(x_0) = \frac{1}{2h}(-3f(x_0) + 4f(x_0+h) -f(x_0 + 2h)) + \frac{h^2}{3} f^{3}(\omega) where \omega is some point in the interval.

The three point midpoint formula is:

f'(x_0) = \frac{1}{2h}(f(x_0 + h) -f(x_0 -h)) -\frac{h^2}{6}f^{3}(\omega) .

The derivation of these formulas: can be obtained from either using the Taylor series centered at x_0 or using the Lagrange polynomial through the given points and differentiating.

That isn’t the point of this note though.

The point: how can one demonstrate, by an example, the role the error term plays.

I suggest trying the following: let x vary from, say, 0 to 3 and let h = .25 . Now use the three point derivative estimates on the following functions:

1. f(x) = e^x .

2. g(x) = e^x + 10sin(\frac{\pi x}{.25}) .

Note one: the three point estimates for the derivatives will be exactly the same for both f(x) and g(x) . It is easy to see why.

Note two: the “errors” will be very, very different. It is easy to see why: look at the third derivative term: for f(x) it is e^x -10(\frac{\pi}{.25})^2sin(\frac{\pi x}{.25})

The graphs shows the story.

expsinfunction

Clearly, the 3 point derivative estimates cannot distinguish these two functions for these “sample values” of x , but one can see how in the case of g , the degree that g wanders away from f is directly related to the higher order derivative of g .

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