Midterm Exam 1, Version A

Midterm Exam 1, Version A - f is differen-tiable at x = 0...

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MATH150 Midterm Exam 1, Version A 10/05/05 GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (15 marks) Compute the following limits: (a) lim x →-∞ 3 x 2 - x - 7 - 5 x 2 + x + 3 (b) lim x 5 x 2 + 5 - 30 x - 5 (c) lim x 0 1 - cos x sin x Using L’Hˆopital’s rule is not permitted! 2. (15 marks) (a) Compute the following derivatives. You may use all the differentiation rules. (i) d dx q sin x (ii) d 2 dx 2 sec x (b) Let f ( x ) = 3 x 2 . (i) Show that f is continuous at x = 0. (ii) Use the limit definition of the derivative to determine whether
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Unformatted text preview: f is differen-tiable at x = 0 or not. 3. (15 marks) Consider the curve C defined by the equation y 3-x 2 y = 1. (a) Show that the point P (0 , 1) lies on C. (b) Find the equation of the tangent line to C at P . (c) Compute d 2 y dx 2 at P . 4. (15 marks) Let f ( x ) = 2 x 4-9 x + 1. (a) Show that f has exactly one root in the interval [0 , 1]. Carefully explain all your steps! (b) Show that f has exactly two (real) roots. Carefully explain all your steps!...
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This note was uploaded on 08/23/2008 for the course MATH 150 taught by Professor Hundemer during the Fall '04 term at McGill.

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