Intro to Analysis in-class_Part_33

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

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Unformatted text preview: Chapter 5 Differentiation 5.1 Concepts of the derivative 1 5.1.1 Definitions Definition 5.1.1. f is differentiable at x0 iff ∀ε > 0, ∃δ > 0|x − x0 | < δ =⇒ f (x) − f (x0 ) − L < ε, x − x0 in which case f (x0 ) = L. Since x = x0 , multiply the inequality to obtain Definition 5.1.2. f is differentiable at x0 iff ∀m > 0, ∃n > 0|x − x0 | < 1 n =⇒ |f (x) − (f (x0 ) + f (x0 )(x − x0 ))| < |x − x0 | . m So f ≈ g for g (x) = f (x0 ) + f (x0 )(x − x0 ). Definition 5.1.3. If f (x) g (x) x→x − − − ∞, then f “blows up” faster than g . If − − 0→ then g “blows up” faster than f ; write f (x) = o(g (x)). Write f (x) = O(g (x)) iff 1 May 2, 2007 f (x) g (x) ≤ b < ∞ as x → x0 . f (x) g (x) x →x − − − 0, − − 0→ 72 Math 413 Differentiation Then “f is differentiable” means f (x) − g (x) = o(|x − x0 |), where g is the affine approximation to f : g (x) = f (x0 ) + f (x0 )(x − x0 ). 5.1.2 Continuity and differentiability Theorem 5.1.4. f is differentiable at x0 implies f is continuous at x0 . Proof. f (t) − f (x) = f (t) − f (x) · (t − x) → f (x) · 0 = 0. t−x In fact, f must be Lipschitz. Definition 5.1.5. f is differentiable on an open set U iff it is differentiable at every point of U . f is C 1 on U iff f is continuous on U and C k iff f (k) is continuous on U . NOTE: f ∈ C 0 means that f is continuous. Example 5.1.1. A function which is C 2 but not C 3 on R: f (x) = x2 + x + 1, x ≤ 0 2 x e , x ≥ 0. ...
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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

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