Lesson 14.pdf - 14 Derivatives Derivative of a function function Definition Let I ⊂ R be an interval and let f I → R be a 1 The function f is

Lesson 14.pdf - 14 Derivatives Derivative of a function...

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14. Derivatives July 26, 2019 Derivative of a function Definition Let I R be an interval, and let f : I R be a function. 1. The function f is differentiable at c I if the limit lim x c f ( x ) - f ( c ) x - c (1) exists. 2. The limit is called the derivative of f at c and is denoted by f 0 ( c ). 3. If f is differentiable at each point x in a set S I , then f is differentiable on S , and the function f 0 : S R is called the derivative of f . [Examples and remark] 1. Another version of the limit (1): lim h 0 f ( c + h ) - f ( c ) h . 2. Let f ( x ) = k be constant. For any c R , f ( c + h ) - f ( c ) h = k - k h = 0 h = 0 . Then f 0 ( c ) = lim h 0 f ( c + h ) - f ( c ) h = 0. 116
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3. Let f ( x ) = x 3 + 2 x and c = 2. f ( c + h ) - f ( c ) h = (2 + h ) 3 + 2(2 + h ) - 2 3 - 2 · 2 h = 3 · 2 2 h + 3 · 2 h 2 + h 3 + 4 + 2 h - 2 · 2 h = 14 + 6 h + h 2 -→ 14 as h 0. Then f 0 (2) = 14. 4. Let f ( x ) = sin x , c = 0. f ( c + h ) - f ( c ) h = sin h - sin 0 h = sin h h 1 as h 0. Then f 0 (0) = 1. 5. Let f ( x ) = cos x , any c . f ( c + h ) - f ( c ) h = cos ( c + h ) - cos c h = - 2 sin 2 c + h 2 sin h 2 h → - 2 sin c · 1 2 = - sin c as h 0. Then f 0 ( c ) = - sin c . Theorem 0.1 f is differentiable at c if and only if the sequence f ( x n ) - f ( c ) x n - c converges whenever ( x n ) is a sequence in I which converges to c , with x n 6 = c for all n . Furthermore, if f is differentiable, the limit is f 0 ( c ) . [Example and remark] 1. Geometrically, for a function y = f ( x ), its derivative f 0 ( c ) equals to the slop of the tangent line of its graph at the point ( c, f ( c )). Therefore, for a given function, from its graph picture, if you find that there is no unique tangent line at some point c , it indicates that f 0 ( c ) may not exist. Then Theorem 0.1 may be used to show that the derivative f 0 ( c ) does not exist. (see the example below) 117
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2. Let f ( x ) = | x | defined on R . - 6 @ @ @ @ @ You find that there are two tangent line of the graph of this function at x = 0, which indicates that f 0 (0) may not exist. We now use Theorem 0.1 to prove it: Let x n = 1 n and e x n = - 1 n . We have x n 0 and e x n 0, and f ( x n ) - f (0) x n - 0 = 1 n - 0 1 n - 0 = 1 , f ( e x n ) - f (0) e x n - 0 = 1 n - 0 - 1 n - 0 = - 1 . By Theorem 0.1, if f 0 (0) exists, these two limits should be the same, but they are not. Therefore, f 0 (0) doest not exists. Differentiation and continuity For differentiation and continuity, they have the rela- tionship = f 0 ( c ) exists f is continuous at c 6⇐ = For example, f ( x ) = | x | is continuous at x = 0, but f 0 (0) does not exist (see the example above). Conversely, we have Theorem 0.2 If f ( x ) is differentiable at x = c , then f ( x ) is continuous at x = c .
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