MAT371defs4

MAT371defs4 - Chapter 4. 1. Three equivalent definitions of...

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Unformatted text preview: Chapter 4. 1. Three equivalent definitions of differentiable and derivative: Let f : D R with x0 an accumulation of D, and x0 D. The following are three are equivalent definitions of differentiable: (a) The limit L := lim exists. (b) The limit L := lim exists. (c) There exists a number L and a function such that limt0 (t) = 0 and f (x0 + t) = f (x0 ) + Lt + t(t), t {t | t + x0 D} t0 xx0 f (x) - f (x0 ) x - x0 f (x0 + t) - f (x0 ) t The limit L, if it exists, is called the derivative of f at x0 and is denoted by f (x0 ). If a function has a derivative at each x S then we say the function is differentiable on S. If a function is differentiable on its domain, we simply say that the function is differentiable. 2. Let f : D R, x0 D such that there exist a neighborhood Q of x0 where x Q D we have f (x) ( resp. )f (x0 ), then we say that f has a relative minimum (resp. relative maximum) at x0 . Note, a relative maximum or a relative minimum can be referred to as a relative extremum. 3. Chain Rule. 4. Rolle's Theorem. 5. Mean Value Theorem (for derivatives). 6. Cauchy Mean Value Theorem. 7. L'H^pital's Rule. o 8. Inverse Function Theorem: Let D be an interval and f : D R be a differentiable function such that x D we have f (x) = 0. Then f is a one-to-one function from D onto an interval J. Its inverse function f -1 : J D exists and is differentiable and for all y J we have f -1 (y) = 1 , f (x) where y = f (x). ...
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This note was uploaded on 08/12/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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