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02_Final1-Sample

# 02_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 2 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. Evaluate the following limits. (a) (5 points) lim x ! 1 x ° 1 p 3 x +1 ° 2 (b) (5 points) lim x ! 3 x 2 ° 6 x +9 x 2 ° 9 (c) (5 points) lim x ! 0 sin(4 x ) 3 x cos(5 x ) (d) (5 points) lim x !1 2+3 x 2 5 x 2 ° 4 x 2. Find the derivatives of the following functions. (a) (6 points) f ( x ) = x 2 ° 1 x 2 +1 (b) (7 points) g ( x ) = 3 sin(4 x ) + 4 cos(3 x ) (c) (7 points) h ( x ) = tan( p 1 + x 3 ) 3. Given the function f ( x ) = 5 x ° 2 + sin( x ) ,show that the equation f ( x ) = 0 has exactly one real root with the following two steps.

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02_Final1-Sample - University of California Riverside...

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