Typical Errors(2) - Typical Errors (2) 1. y = y' = 1 tan 2...

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Unformatted text preview: Typical Errors (2) 1. y = y' = 1 tan 2 θ + tan θ , ? dy 2 1 ⋅ 2 tan θ + sec 2 θ = tan θ + sec 2 θ , dy = (tan θ + sec 2 θ ) dθ 2 2. y = 3 − cos 2 x , y' = sin 2 x . 2 3 − cos 2 x 3. y = cos( x 2 − y 2 ), ? y '. y ' = − sin( x 2 − y 2 ), 或 y ' = − sin( x 2 − y 2 )(2 x − 2 y ). 1 1 ( x − sin x )' 1 − cos x (sin x)' − ) = lim = lim = lim 4或 lim( x →0 sin x x →0 ( x sin x )' x →0 sin x + x cos x x →0 ( 2 cos x − x sin x )' x cos x = −∞ . − 3 sin x − x cos x 1 1 x − sin x (1 − cos x ) x sin x − ( x − sin x )( x sin x)' − ) = lim = lim = ...... 5. lim( x →0 sin x x →0 x sin x x →0 x ( x sin x ) 2 = lim x →0 6.或 f ( x) = 6x 或或或 x +1 2 6(1 − x 2 ) = 0, x = ±1. 或. f ' ( x ) = 2 ( x + 1) 2 f (1) = 3, f (−1) = −3, f (1) > f (−1),∴ f ( x)? x = 1? ? ? ? ? 3; ? x = −1? ? ? ? ? − 3. x + sin x 1 + cos x − sin x = lim = lim = −1. x →∞ 1 − cos x x →∞ sin x x − sin x ) 8.或 f ( x)? (−∞ ,+ ∞ 或或或或或或 f (0) = f ' (0) = 0, 7. lim x →∞ f ( x) ,x ≠ 0 g ( x) = x 或 g ' (0). 0, x = 0. f ( h) −0 f ( h) f ' ( h) f ' ' ( h) 1 或. h g ' (0) = lim = lim 2 = lim = lim = f ' ' (0). h →0 h →0 h h →0 2h h →0 h 2 2 9.或 f ( x) 或 x = a 或或或或或 lim x →0 或. f (a + x) − f (a − x) . x 0 f ' (a + x)(a + x)'− f ' (a − x)(a − x)' ? , 或或或或或或或,或或= lim x →0 0 1 1 = lim x →0 f ' (a + x) + f ' (a − x ) = 2 f ' ( a). 1 2 ...
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