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140Bquiz6csols

140Bquiz6csols - 1 MATH 140B QUIZ#6 Solutions Due Mon Oct 8...

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Unformatted text preview: 1 MATH 140B QUIZ#6 Solutions Due: Mon Oct 8, 2007 1. i) True . ( f Β· g ) β€² (0) = f (0) g β€² (0) + f β€² (0) g (0) = (3)(0) + (2)( βˆ’ 3) = βˆ’ 6 . ii) False . ( f/g ) β€² (4) = g (4) f β€² (4) βˆ’ f (4) g β€² (4) g (4) 2 = (2)(5) βˆ’ (1)( βˆ’ 2) 2 2 = 12 4 = 3 . iii) False . ( f β—¦ g ) β€² ( βˆ’ 1) = f β€² ( g ( βˆ’ 1)) Β· g β€² ( βˆ’ 1) = f β€² (4) Β· ( βˆ’ 3) = 5 Β· ( βˆ’ 3) = βˆ’ 15 . Therefore, A) is the answer. 2. i) True . By definition of local maximum and local minimum. ii) True . At all points x in the interval (2 , 5) , f β€² ( x ) = 0 (horizontal tangent), so these points are all critical points. That certainly makes infinitely many. iii) True . Local minima occur at all points x in the interval (2 , 5) . The statement makes no claim about the endpoints x = 2 and x = 5 , so we don’t need to address them. Therefore, E) is the answer. 3. i) True . Using the product rule, f β€² ( x ) = x 2 Β· 1 2 (1 βˆ’ x 2 ) βˆ’ 1 / 2 ( βˆ’ 2 x ) + 2 x (1 βˆ’ x 2 ) 1 / 2 = x (1 βˆ’ x 2 )...
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