Intro to Analysis in-class_Part_33

Intro to Analysis in-class_Part_33 - Chapter 5...

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Chapter 5 Differentiation 5.1 Concepts of the derivative 1 5.1.1 Definitions Definition 5.1.1. f is differentiable at x 0 iff ε > 0 , δ > 0 | x - x 0 | < δ = fl fl fl fl f ( x ) - f ( x 0 ) x - x 0 - L fl fl fl fl < ε, in which case f 0 ( x 0 ) = L . Since x 6 = x 0 , multiply the inequality to obtain Definition 5.1.2. f is differentiable at x 0 iff m > 0 , n > 0 | x - x 0 | < 1 n = | f ( x ) - ( f ( x 0 ) + f 0 ( x 0 )( x - x 0 )) | < | x - x 0 | m . So f g for g ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ). Definition 5.1.3. If f ( x ) g ( x ) x x 0 -----→ ∞ , then f “blows up” faster than g . If f ( x ) g ( x ) x x 0 -----→ 0, then g “blows up” faster than f ; write f ( x ) = o ( g ( x )). Write f ( x ) = O ( g ( x )) iff f ( x ) g ( x ) b < as x x 0 . 1 May 2, 2007
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72 Math 413 Differentiation Then “ f is differentiable” means f ( x ) - g ( x ) = o ( | x - x 0 | ), where g is the affine approximation to f : g ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ). 5.1.2 Continuity and differentiability Theorem 5.1.4. f is differentiable at
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