2413labI6 - Laboratory I.6 Investigating the Intermediate...

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Investigating the Intermediate Value Theorem and Fixed Points Goals • The student will discover and acquire a feel for one of the major theorems in calculus. • The student will apply the theorem in practical and theoretical ways. • The student will understand why continuity is required for the theorem. Before the Lab The Intermediate Value Theorem (IVT) is an important theorem in later mathematics. (A theorem is a mathematical fact which has been proved to be true.) We’re going to have you work some examples related to the theorem, and you ought to be able to figure out what the theorem says by thinking about the examples. (Frankly, this is one of those “deep insights” which often inspires reactions like “Well, duh, of COURSE it’s true, what’s the big deal?” As you go on in math, though, you’ll find more and more “obvious” things that are hard to prove true.) Anyhow, here all we’re asking you to do is discover the IVT. We’re going to spend more time here in “Before the Lab” talking about fixed points , which turn out to have important significance in both theory and practice. Some of the really slick methods computers use to solve equations depend on fixed points. We want to define them, and help you see them. Let f ( x ) = ( x – 1) 2 . When you plug in x = (3 5)/2, you find that f( (3 5)/2) = (3 5)/2 That is, you get the same number back that you plugged in. This is what a fixed point is: a fixed point for a function f ( x ) is some number x = c such that
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2413labI6 - Laboratory I.6 Investigating the Intermediate...

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