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101-2007&2008-2-M10-March2008

101-2007&2008-2-M10-March2008 - Kuwait University...

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Kuwait University Math 101 Date: March 25, 2008 Dept. of Math. & Comp. Sci. First Exam Duration: 90 minutes Calculators, mobile phones, pagers and all other mobile communication equipments are not allowed Answer the following questions: 1. Evaluate the following limits, if they exist: (a) lim x 1 x 3 1 x 2 3 x + 2 . (3 pts.) (b) lim x 1 · x 5 + ( x 2 2 x + 1) sin μ 1 x 1 ¶¸ . (3 pts.) 2. Find the vertical and horizontal asymptotes, if any, for the graph of f ( x ) = x 2 + x x 2 + 1 ( x + 1) 2 . (4 pts.) 3. Let f ( x ) = A + 3 | x 1 | x 2 + x 2 , if x < 1 , x 6 = 2 , B , if x = 1 , 2 x 1 , if x > 1 . (4pts.) Find the values of A and B, so that f is continuous at x = 1 . 4. (a) State The Intermediate Value Theorem. (1 pt.) (b) Use The Intermediate Value Theorem to show that there is a real solution of the equation ( x + 1) x 1 = 1 . (3 pts.) 5. Let f ( x ) = | x | ( x 3) x 3 9 x . Classify the types of discontinuity of f as removable, jump, or in fi nite. (4 pts.) 6. Use the de fi nition of the derivative to fi nd
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