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FinalExamRev

# FinalExamRev - MATH 1431 Problem 1 FINAL EXAM REVIEW a In...

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MATH 1431 FINAL EXAM REVIEW Problem 1. a. In each of the following, determine whether the given limit exists. If the limit exists, give its value: ( i ) lim x 0 x 2 + 3 + cos x ( ii ) lim x 0 sin 2 x cos x x ( iii ) lim x 2 + x 2 - 9 x 2 - 3 x + 2 ( iv ) lim x 1 f ( x ) where f ( x ) = 2 x + 1 , x 1 x 2 - 2 x, x > 1 . b. Let f ( x ) = x 2 - 4 x 3 - 5 x 2 + 6 x . ( i ) Determine the intervals on which f is continuous. ( ii ) Determine the vertical asymptotes of the graph of f , if any. ( iii ) Determine the horizontal asymptotes of the graph of f , if any. Problem 2. a. Let f ( x ) = 3 x 2 - 1 , x < 1 A, x = 1 x 3 + 2 Bx, x > 1 . Find A and B so that f will be continuous at x = 1. b. Let g ( x ) = x + 2 , x < - 1 x 2 , - 1 x 1 1 / ( x - 1) , x > 1 . ( i ) Sketch the graph of g . ( ii ) What is lim x →- 1 g ( x ) ? What is lim x 1 g ( x )? ( iii ) At what numbers x , if any, does g fail to be continuous? ( iv ) At what numbers x , if any, does g fail to be differentiable? Problem 3. a. Use the definition of the derivative to calculate f (3) where f ( x ) = 1 + 3 x - 2 x 2 . b. Calculate the derivative of each of the following functions: ( i ) f ( x ) = 3 x 4 - 3 x 2 + x + 2 ( ii ) g ( x ) = x 2 + cos x sin 2 x ( iii ) y = x 3 tan( x 2 + 1) ( iv ) h ( x ) = ( x 3 / 2 + sec 3 x ) 4 1

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Problem 4. a. Determine dy dx if y is defined implicitly by the equation: x 3 - 3 x 2 y 2 + x 2 sin y = y 3 .
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FinalExamRev - MATH 1431 Problem 1 FINAL EXAM REVIEW a In...

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