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Approximation and Series

Functions f and g are said to have a good order of contact at a point, say x=0, if f and g as well as several of their derivatives are equal at that point. This is the starting point for motivating approximations like splines and Taylor polynomials. The concept is quite intuitive, and the resulting understanding is much deeper than that coming from pure manipulation of formulas.

Here's the list of lessons.

	Splines		Taylor's Formula
	Expansions in Powers of x		Barriers to Convergence
	Using Expansions		Power Series

Here are some pages with snapshots of problems from the course. Be careful if you look at these. They are chock full of graphics, and may take very long to download.

• A an introduction to spline knots from the Basics section.
• A visual look at series expansions of sine and cosine, and parametric plots of circles from the Give It A Try homework.
• Taylor's formula works both ways.

If you want, you may download pre-evaluated samplers from this course.

If you have a Macintosh computer: Click and hold and save the file to your disk.

If you have a Windows machine: Right mouse click and save.

Here's a sampler from each lesson.

Lesson 1 Sampler Lesson 2 Sampler Lesson 3 Sampler

Lesson 4 Sampler Lesson 5 Sampler Lesson 6 Sampler

If you are new to the courseware's structure, or to the structure of Mathematica notebooks, take a quick look at what to do once you download a sample lesson.

To view the samples, if you don't have a copy of Mathematica, you'll need to download a copy of MathReader from Wolfram Research, Inc