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04_Final1-Sample - University of California Riverside...

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University of California, Riverside Department of Mathematics Final Exam Mathematics 9A - First Year of Calculus Sample 4 Instructions: This exam has a total of 200 points. You have 3 hours. You must show all your work to receive full credit. You may use any result done in class. The points attached to each problem are indicated beside the problem. You are not allowed to use books, notes, or calculators. Answers should be written as p 2 as opposed to 1 : 4142135 ::: . 1. Compute the limit if it exists, justifying all steps. If it does not exist give a valid mathematical reason why not. You may not use l°H±pital°s rule. (a) (6 points) lim t ! 9 9 ° t 3 ° p t . (b) (7 points) lim x !°1 p 5 x 2 +1 7 x +2 . (c) (7 points) lim x !1 cos x . 2. (20 points) Calculate the equation of the tangent line to the curve de²ned by x 3 + y 3 = 2 xy at the point, (1 ; 1) . 3. (20 points) Use the intermediate value theorem to show x 3 + x + 1 has a root and Rolle°s theorem to show this is the only root.
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