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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 > 0x − 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 > 0x − 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.
 Fall '11
 Wong
 Derivative

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