111F08Exam1Sol - Math 1110 (Fall 2008) Prelim 1 (9/30/2008)...

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Math 1110 (Fall 2008) Prelim 1 (9/30/2008) 1 Question 1. (20 points, 5 points per part) Calculate the following derivatives: (a) d (sin θ cos θ ) = ± d sin θ ² cos θ + sin θ ± d cos θ ² = cos θ cos θ - sin θ sin θ = cos 2 θ (b) d dt sin 2 t + t = 1 2 sin 2 t + t d dt (sin 2 t + t ) = 2 sin t cos t +1 2 sin 2 t + t = sin 2 t +1 2 sin 2 t + t (c) d dx x 1+ x 2 = 1(1+ x 2 ) - 2 x 2 (1+ x 2 ) 2 = 1 - x 2 (1+ x 2 ) 2 (d) d dx ± f ( x ) x 2 ² ³ ³ ³ ³ ³ x =2 = ± f 0 ( x ) x 2 - 2 xf ( x ) x 4 ² ³ ³ ³ ³ ³ x =2 = ± ( - 1)2 2 - 2 * 2 * 3 2 4 ² = - 1 given that f (2) = 3 and f 0 (2) = - 1, Question 2. (15 points, 7.5 points per part) Evaluate the following limits, if they exist. If you use a theorem to help you get an answer, be sure to reference it in your solution. If a limit does not exist, indicate in what way it fails to exist. (For example, one sided limits don’t agree, or the function approaches + or - ). (a) lim x 5 x 2 - 25 | x - 5 | . lim x 5+ x 2 - 25 | x - 5 | = lim x 5+ ( x - 5)( x +5) ( x - 5) = 10 lim x 5 - x 2 - 25 | x - 5 | = lim x 5 - ( x - 5)( x +5) - ( x - 5) = - 10 The one-sided limits do not agree.
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).

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111F08Exam1Sol - Math 1110 (Fall 2008) Prelim 1 (9/30/2008)...

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