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MAC 2311 Test Two Review Answers, Fall 2010 1. a) ¡ 2 sin(2 x ) b) ¡ 2 x 3 2 2. f 0 ( x ) = 3 (3 ¡ 2 x ) 2 3. (a) f ( x ) is continuous at x = 0 since lim x ! 0 f ( x ) = 2 = f (0) (b) f 0 (0) does not exist since lim x ! 0 f ( x ) ¡ f (0) x ¡ 0 = f ( x ) ¡ 2 x does not exist. (c) f 0 ( x ) = ( 2 x x < 0 cos x x > 0 (d) 4. a) false (see problem #3) b) true c) false ( df dx is undeflned) 5. a) ¡ 14 b) 51 12 6. y = 2 x 2 ¡ x 7. a) p x ¡ 1 x 2 b) 2( x + 4)(6 x ¡ 2) 2 (15 x + 34) c) 2 x ln 4 ¡ 2 tan x ln 4 d) 1 + 1 6 + 3 x ¡ 6 3 x + 1 e x ¡ 3 3 p 6 + 3 x (3 x + 1) 2 1

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8. dy dx = ¡ y x + 2 y ; d 2 y dx 2 = 2 xy + 2 y 2 ( x + 2 y ) 3 9. a = 15 10. a) f 0 ( x ) = 6 + 6 x ( x 2 + 3) 3 2 ; horizontal tangent line is y = ¡ 4 at x = ¡ 1 There are no vertical tangent lines. b) f 0 ( x ) = 4 + 2 x 1 3 3 x 2 3 ; horizontal tangent line is y = ¡ 4 occurring at x = ¡ 8; vertical tangent line: x = 0 11. tangent line: y = 4 …x + 2 ¡ , normal line: y = ¡ 1 4 x + 1 16 + 2 12. a) 2 xy ¡ x 2 y 2 ¡ x 2 b) ¡ 2 x sin( x 2 )[sin(cos( x 2 )) + cos(cos( x 2 ))(cos( x 2 ))] c) e tan x x x sec 2 x ¡ tan x x 2 d. (ln 3)(3 x 3 +2 x )(3 x 2 + 2) e.
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