Wednesday, May 16, 2007

Catching Up and Preview of Coming Attractions

Gee, so why don't I have answers to the parabola-circle problem posted yet? It's coming later tonight or tomorrow -- hang in there...

Also, I'm working on a major exploration of exponential functions and geometric series via an analysis of Mortgage Calculations. I set a very high standard for myself so it's still under construction. I think it will be worth waiting for since it involves derivation of formulas, analysis in special cases, and graphing calculator investigation (not merely relying on a built-in application), etc...


Eric Jablow said...

Then here's a suggestion for a lesson plan:

Find eht least positive root of f(x) = x³−x−1 by:

1. The bisection method on [0,1],

2. The method of false position on [0,1],

3. Newton's method on [0,1].

How fast do these three methods converge to the root? When can Newton's method fail?

Eric Jablow said...

I am an idiot today. That's [0, 2] in all three cases.

You might also mention the old approximation algorithm for finding √x:

Let r_0 be a guess at √x. Then, refine the guess as follows:

r_{n+1} = (r_n + (x/r_n))/2.

Show that this is what Newton's method would give for finding the roots of g(t) = t² - x. While you're at it, point out that the x and y of the x-axis and y-axis are just symbols.

jonathan said...


I am cruising through lighter stuff with conic sections while the AP exams are finishing up. Late next week we will look at some systems, and I have this set aside for the "extra" day (how many units really could use one "extra" day to challenge, tie together, spur imagination...)

I love being slightly off-topic, challenging the kids to reach into other parts of their mathematical thinking. So far, that's what's grabbed me most about your explorations.

I will let you know if anything interesting pops up after we try this one.

Dave Marain said...

Great stuff as always, Eric! Sorry, I didn't respond sooner...
You obviously love recursive and iterative processes and so do I (that's the programmer in me). When teaching Newton's method in Calculus in the past, I always showed the connection to the square root technique you alluded to. Now does anyone remember ever learning Horner's Approximation Method back in the Dark Ages?

Eric Jablow said...

I just picked up an old favorite yesterday: Real Computing Made Real, by Forman S. Acton. He points out that the efficient way to find the smallest root of x²−100x+1=0 definitely isn't the standard quadratic formula solution, (100−√9996)/2, isn't the alternative solution, 2/(100+√9996), but treats the equation as a perturbed linear equation:


100x = 1 + x²

x = 1/100 + x²/100

x → 1/100 + x²/100.

Guess x = 0, and recurse. After 3 steps, you're close enough for practical purposes.

I like these techniques because they're practical, they provide insight, and they aren't cookbook. You should look at Acton's other book, Numerical Methods That [Usually] Work too.