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...
Wednesday, May 16, 2007
Catching Up and Preview of Coming Attractions
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5 comments:
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?
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.
Dave,
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.
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?
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:
x²−100x+1=0
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.
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