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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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). ...
View Full Document

This note was uploaded on 08/12/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

Ask a homework question - tutors are online