Chap 2 Prob-Answers for Exam 1

Thomas' Calculus, Media Upgrade (11th Edition)

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Math2253 Review Problems for Exam 1 (Covers chapter 2) 1. If 2 2 ) ( 2 + - = x x x f , find (a) ) 3 ( f (5) (b) ) 3 ( h f + ( 5 4 2 + + h h ) (c) h f h f ) 3 ( ) 3 ( - + ( 4 + h ) 2. Complete the table and use the result to find the limit. a) 2 2 3 lim 2 1 - + - x x x x (0) x 0.99 0.999 0.9999 1.0001 1.001 1.01 ) ( x f b) x x x 1 1 lim 0 - + (1/2) x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 ) ( x f 3. Find the average rate of change of the function over the given interval or intervals. 1 ) ( 3 + = x x f ; a) ] 3 , 2 [ b) ] 2 , 2 [ h + 4. Find the limit. a) 4 2 lim 2 2 - + - x x x (-1/4) b) ) 8 5 3 ( lim 2 2 + - x x x (10) c) 7 lim 2 3 + x x (4) d) x x x 3 9 lim 0 - + (1/6) e) 1 1 lim 3 1 - - x x x (1/3) f) 12 tan lim 3 x x π (1) 1
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g) x x x 4 5 sin lim 0 (5/4) h) x x x tan lim 0 (1) i) x x x sin cos 1 lim 0 - (0) j) + - = 2 , 3 2 , 1 2 ) ( 2 x x x x x f ) ( lim 2 x f x - (3) ) ( lim 2 x f x - (7) k) x x x 2 sin lim (0) l) 1 5 3 2 lim + - x x x (2/5) m) 2 1 lim 2 - + x x ( ) 5. Discuss the continuity of each of the following functions. a) 1 2 3 ) ( 2 - + - = x x x x f (continuous everywhere but x=1) b) + - = 2 , 3 2 , 1 2 ) ( 2 x x x x x f (continuous everywhere but x=2) 6. Determine the value of c such that the function is continuous on the entire real line. + + = 4 , 6 4 , 3 ) ( x cx x x x f (c = ¼) 7. Find the vertical asymptote (if any) 1 ) ( 2 2 - = x x x f (x = -1 and x = 1) 8. The function f is defined as follows 0 , 2 tan ) ( = x x x x f a) Find ) ( lim 0 x f x
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