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Test Bank for Precalculus, 6th Edition

Test Bank for Precalculus, 6th Edition helps you familiarize yourself with exam formats and key concepts.

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    Ch. 11 Introduction to Calculus 11.1 Finding Limits Using Tables and Graphs 1 Understand Limit Notation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the table which contains the best values of x for finding the requested limit of the given function. 1) lim x → 2 (x2 + 8x - 2) A) x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) B) x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) C) x - 1.9 - 1.99 - 1.999 2.001 2.01 2.1 f(x) D) x - 0.9 - 0.99 - 0.999 1.001 1.01 1.1 f(x) 2) lim x → 1 x4 - 1 x - 1 A) x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) B) x 1.9 1.99 1.999 1.001 1.01 1.1 f(x) C) x - 0.9 - 0.99 - 0.999 - 1.001 - 1.01 - 1.1 f(x) D) x 0.1 0.19 0.119 1.99 1.09 1.9 f(x) Page 1

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    3) lim x → 0 sin 2x x A) x - 0.03 - 0.02 - 0.01 0.01 0.02 0.03 f(x) B) x - 0.01 - 0.02 - 0.03 0.03 0.02 0.01 f(x) C) x - 0.3 - 0.2 - 0.1 0.1 0.2 0.3 f(x) D) x 0.03 0.02 0.01 0.001 0.002 0.003 f(x) Translate the given limit notation into a sentence. 4) lim x → 4 ( x - 2) = 0 A) The limit of x - 2 as x approaches 4 equals the number 0. B) The limit of x - 2 as x approaches 0 equals the number 4. C) The limit of x - 4 as x approaches 2 equals the number 0. D) The limit of x - 4 as x approaches 0 equals the number 2. 5) lim x → 0 sin 2x x = 2 A) The limit of sin 2x x as x approaches 0 equals the number 2. B) The limit of sin 2x x as x approaches 2 equals the number 0. C) The limit of sin 2x x as x approaches 0 from the left equals the number 2. D) The limit of sin 2x x as x approaches 0 from the right equals the number 2. 2 Find Limits Using Tables MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Complete the table for the function and find the indicated limit. 1) lim x → 2 (x2 + 8x - 2) x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) A) 16.810; 17.880; 17.988; 18.012; 18.120; 19.210 limit = 18.0 B) 5.043; 5.364; 5.396; 5.404; 5.436; 5.763 limit = 5.40 C) 16.692; 17.592; 17.689; 17.710; 17.808; 18.789 limit = 17.70 D) 6.810; 7.880; 7.988; 8.012; 8.120; 9.210 limit = 8.0 Page 2

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    2) lim x → 1 x4 - 1 x - 1 x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) A) 3.439; 3.940; 3.994; 4.006; 4.060; 4.641 limit = 4.0 B) 1.032; 1.182; 1.198; 1.201; 1.218; 1.392 limit = 1.210 C) 4.595; 5.046; 5.095; 5.105; 5.154; 5.677 limit = 5.10 D) 7.439; 7.940; 7.994; 8.006; 8.060; 8.641 limit = 8.0 3) lim x → 0 x3 - 6x + 8 x - 2 x - 0.1 - 0.01 - 0.001 0.001 0.01 0.1 f(x) A) - 4.09476; - 4.00995; - 4.00100; - 3.99900; - 3.98995; - 3.89526 limit = - 4.0 B) - 1.22843; - 1.20298; - 1.20030; - 1.19970; - 1.19699; - 1.16858 limit = - 1.20 C) - 2.18529; - 2.10895; - 2.10090; - 2.09910; - 2.09096; - 2.00574 limit = - 2.10 D) 4.09476; 4.00995; 4.00100; 3.99900; 3.98995; 3.89526 limit = 4.0 4) lim x → 0 (x2 - 5) x - 0.1 - 0.01 - 0.001 0.001 0.01 0.1 f(x) A) - 4.9900; - 4.9999; - 5.0000; - 5.0000; - 4.9999; - 4.9900 limit = - 5.0 B) - 1.4970; - 1.4999; - 1.5000; - 1.5000; - 1.4999; - 1.4970 limit = - 1.50 C) - 2.9910; - 2.9999; - 3.0000; - 3.0000; - 2.9999; - 2.9910 limit = - 3.0 D) 1.4970; 1.4999; 1.5000; 1.5000; 1.4999; 1.4970 limit = 1.50 5) lim x → 0 sin 2x x x - 0.03 - 0.02 - 0.01 0.01 0.02 0.03 f(x) A) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 2 B) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 0 C) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = 1 D) 1.9988, 1.9995, 1.9999, 1.9999, 1.9995, 1.9988 limit = - 2 Page 3

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    6) lim x → 0 x 2 sin x x - 0.03 - 0.02 - 0.01 0.01 0.02 0.03 f(x) A) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = 0 B) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = 1 C) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = 0.1 D) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = - 1 7) lim x → - 7 x 2 - 49 x + 7 x - 7.1 - 7.01 - 7.001 - 6.999 - 6.99 - 6.9 f(x) A) - 14.1, - 14.01, - 14.001, - 13.999, - 13.99, - 13.9 limit = - 14 B) 7.1, 7.01, 7.001, 6.999, 6.99, 6.9 limit = 7 C) - 14.1, - 14.01, - 14.001, - 13.999, - 13.99, - 13.9 limit = - 14 D) - 7.1, - 7.01, - 7.001, - 6.999, - 6.99, - 6.9 limit = - 7 8) lim x → 0 1 - cos x sin x x - 0.03 - 0.02 - 0.01 0.01 0.02 0.03 f(x) A) - 0.0150, - 0.0100, - 0.0050, 0.0050, 0.0100, 0.0150 limit = 0 B) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = 1 C) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = 0.1 D) - 0.0300, - 0.0200, - 0.0100, 0.0100, 0.0200, 0.0300 limit = - 1 9) lim x → 0 f(x), where f(x) = x + 5 if x < 0 3x + 5 if x ≥ 0 x - 0.1 - 0.01 - 0.001 0.001 0.01 0.1 f(x) A) 4.9, 4.99, 4.999, 5.003, 5.03, 5.3 limit = 5 B) 2.9, 2.99, 2.999, 3.003, 3.03, 3.3 limit = 3 C) 0.9, 0.99, 0.999, 1.003, 1.03, 1.3 limit = 1 D) 1.9, 1.99, 1.999, 2.003, 2.03, 2.3 limit = 2 Page 4

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    3 Find Limits Using Graphs MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. 1) a. lim x → 0 f(x) b. f(0) x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 y 4 3 2 1 -1 -2 -3 -4 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 y 4 3 2 1 -1 -2 -3 -4 A) a. lim x → 0 f(x) = 0 b. f(0) = 0 B) a. lim x → 0 f(x) = 0 b. f(0) = 2 C) a. lim x → 0 f(x) = - 2 b. f(0) = - 2 D) a. lim x → 0 f(x) does not exist b. f(0) does not exist 2) a. lim x → 0 f(x) b. f(0) x -10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 y 1 -1 x -10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 y 1 -1 A) a. lim x → 0 f(x) = 1 b. f(0) = 1 B) a. lim x → 0 f(x) = 0 b. f(0) = 0 C) a. lim x → 0 f(x) = - 1 b. f(0) = - 1 D) a. lim x → 0 f(x) does not exist b. f(0) does not exist Page 5

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    3) a. lim x → 0 f(x) b. f(0) A) a. lim x → 0 f(x) = 1 b. f(0) does not exist B) a. lim x → 0 f(x) = - 1 b. f(0) does not exist C) a. lim x → 0 f(x) = 0 b. f(0) = 0 D) a. lim x → 0 f(x) does not exist b. f(0) does not exist 4) a. lim x → 0 f(x) b. f(0) x -2 -1 1 2 3 4 5 y 10 8 6 4 2 -2 x -2 -1 1 2 3 4 5 y 10 8 6 4 2 -2 A) a. lim x → 0 f(x) = 0 b. f(0) = 6 B) a. lim x → 0 f(x) = 6 b. f(0) = 6 C) a. lim x → 0 f(x) = 6 b. f(0) = 0 D) a. lim x → 0 f(x) does not exist b. f(0) = 6 Page 6

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    5) a. lim x → 0 f(x) b. f(0) A) a. lim x → 0 f(x) = 0 b. f(0) does not exist B) a. lim x → 0 f(x) = 1 b. f(0) = 1 C) a. lim x → 0 f(x) = - 1 b. f(0) does not exist D) a. lim x → 0 f(x) does not exist b. f(0) does not exist 6) a. lim x → 1 f(x) b. f(1) x -3 -2 -1 1 2 y 2 1 -1 -2 -3 -4 x -3 -2 -1 1 2 y 2 1 -1 -2 -3 -4 A) a. lim x → 1 f(x) = - 3 b. f(1) = - 3 B) a. lim x → 1 f(x) = 3 b. f(1) = 3 C) a. lim x → 1 f(x) = 1 b. f(1) = 1 D) a. lim x → 1 f(x) does not exist b. f(1) = - 3 Page 7

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    7) a. lim x → 1 f(x) b. f(1) x 1 2 3 4 5 y 4 3 2 1 -1 x 1 2 3 4 5 y 4 3 2 1 -1 A) a. lim x → 1 f(x) = 3 b. f(1) = 3 B) a. lim x → 1 f(x) = 1 b. f(1) = 3 C) a. lim x → 1 f(x) = 0 b. f(1) = 0 D) a. lim x → 1 f(x) does not exist b. f(1) does not exist 8) a. lim x → 2 f(x) b. f(2) x 1 2 3 4 5 y 5 4 3 2 1 x 1 2 3 4 5 y 5 4 3 2 1 A) a. lim x → 2 f(x) = 2 b. f(2) = 1 B) a. lim x → 2 f(x) = 1 b. f(2) = 1 C) a. lim x → 2 f(x) = 1 b. f(2) = 2 D) a. lim x → 2 f(x) does not exist b. f(2) = 1 Page 8

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    9) a. lim x → 1 f(x) b. f(1) x -3 -2 -1 1 2 y 2 1 -1 -2 -3 -4 x -3 -2 -1 1 2 y 2 1 -1 -2 -3 -4 A) a. lim x → 1 f(x) = - 3 b. f(1) = 1 B) a. lim x → 1 f(x) = - 3 b. (1) = - 3 C) a. lim x → 1 f(x) = 1 b. f(1) = - 3 D) a. lim x → 1 f(x) = - 3 b. f(1) does not exist Use the graph and the viewing rectangle shown below the graph to find the indicated limit. 10) lim x → - 2 (x 2 - 2) [ - 6, 6, 1] by [ - 6, 6, 1] A) lim x → - 2 (x 2 - 2) = 2 B) lim x → - 2 (x 2 - 2) = - 6 C) lim x → - 2 (x 2 - 2) = 6 D) lim x → - 2 (x 2 - 2) = - 2 Page 9

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    11) lim x → π sin x - π , π , π 2 by [ - 2, 2, 1] A) lim x → π sin x = 0 B) lim x → π sin x = 1 C) lim x → π sin x = - 1 D) lim x → π sin x = 2 Graph the function. Then use your graph to find the indicated limit. 12) f(x) = 3x + 1, lim x → 6 f(x) x y x y A) 19 B) 6 C) 18 D) 7 Page 10

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    13) f(x) = 2x - 3, lim x → - 1 f(x) x y x y A) - 5 B) - 1 C) - 2 D) - 4 14) f(x) = 3 - x 2 , lim x → 2 f(x) x y x y A) - 1 B) 3 C) 1 D) 4 15) f(x) = 5x , lim x → - 3 f(x) x y x y A) 15 B) 3 C) - 15 D) - 3 Page 11

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    16) f(x) = sin x - 2, lim x → π /2 f(x) x y x y A) - 1 B) - 2 C) 3 D) 1 17) f(x) = x 2 - 36 x - 6 , lim x → 6 f(x) x y x y A) 12 B) does not exist C) 6 D) 1 18) f(x) = x 2 - 4 x + 2 , lim x → - 2 f(x) x y x y A) - 4 B) does not exist C) - 2 D) 1 Page 12

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    19) f(x) = 10e x , lim x → 0 f(x) x y x y A) 10 B) - 10 C) 1 D) 0 20) f(x) = x + 1 if x < 0 2x + 1 if x ≥ 0 , lim x → 0 f(x) x y x y A) 1 B) does not exist C) 3 D) 0 21) f(x) = x + 7 x < - 7 7 - x x ≥ - 7 , lim x → - 7 f(x) x y x y A) does not exist B) 0 C) 14 D) - 7 Page 13

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    4 Find One - Sided Limits and Use Them to Determine If a Limit Exists MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist. 1) a. lim x → 3 f(x) b. f(3) x 1 2 3 4 5 y 5 4 3 2 1 x 1 2 3 4 5 y 5 4 3 2 1 A) a. lim x → 3 f(x) does not exist b. f(3) = 5 B) a. lim x → 3 f(x) = 5 b. f(3) = 5 C) a. lim x → 3 f(x) = 3 b. f(3) = 5 D) a. lim x → 3 f(x) = 4 b. f(3) does not exist 2) a. lim x → 1 f(x) b. f(1) x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 A) a. lim x → 1 f(x) does not exist b. f(1) = 2 B) a. lim x → 1 f(x) = 1 b. f(1) = 2 C) a. lim x → 1 f(x) = 2 b. f(1) = 2 D) a. lim x → 1 f(x) = 1 b. f(1) = 0 Page 14

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    3) a. lim x → 1 f(x) b. f(1) A) a. lim x → 1 f(x) = 1 b. f(1) does not exist B) a. lim x → 1 f(x) = 0 b. f(1) = 1 C) a. lim x → 1 f(x) = - 1 b. f(1) = 1 D) a. lim x → 1 f(x) does not exist b. f(1) does not exist 4) a. lim x → - 1 f(x) b. f( - 1) A) a. lim x → - 1 f(x) does not exist b. f( - 1) = - 1 B) a. lim x → - 1 f(x) = 0 b. f( - 1) = - 1 C) a. lim x → - 1 f(x) = - 1 b. f( - 1) does not exist D) a. lim x → - 1 f(x) = - 2 b. f( - 1) = - 1 Page 15

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    5) a. lim x → - 1/2 f(x) b. f( - 1/2) A) a. lim x → - 1/2 f(x) = - 1 b. f( - 1/2) = - 1 B) a. lim x → - 1/2 f(x) = - 2 b. f( - 1/2) = - 1 C) a. lim x → - 1/2 f(x) = 0 b. f( - 1/2) does not exist D) a. lim x → - 1/2 f(x) does not exist b. f( - 1/2) = - 1 6) a. lim x → 3 - f(x) b. f(3) x 1 2 3 4 5 y 5 4 3 2 1 x 1 2 3 4 5 y 5 4 3 2 1 A) a. lim x → 3 - f(x) = 3 b. f(3) = 5 B) a. lim x → 3 - f(x) = 3 b. f(3) = 3 C) a. lim x → 3 - f(x) = 3 b. f(3) does not exist D) a. lim x → 3 - f(x) does not exist b. f(3) = 5 Page 16

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    7) a. lim x → 3 + f(x) b. f(3) x 1 2 3 4 5 y 5 4 3 2 1 x 1 2 3 4 5 y 5 4 3 2 1 A) a. lim x → 3 + f(x) = 5 b. f(3) = 5 B) a. lim x → 3 + f(x) = 3 b. f(3) = 3 C) a. lim x → 3 + f(x) = 3 b. f(3) does not exist D) a. lim x → 3 + f(x) does not exist b. f(3) = 5 8) a. lim x → 1 + f(x) b. f(1) x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 A) a. lim x → 1 + f(x) = 2 b. f(1) = 2 B) a. lim x → 1 + f(x) = 2 b. f(1) = 1 C) a. lim x → 1 + f(x) = 1 b. f(1) = 0 D) a. lim x → 1 + f(x) does not exist b. f(1) = 2 Page 17

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    9) a. lim x → - 1 - f(x) b. f( - 1) A) a. lim x → - 1 - f(x) = - 1 b. f( - 1) does not exist B) a. lim x → - 1 - f(x) = 0 b. f( - 1) = 1 C) a. lim x → - 1 - f(x) = - 1 b. f( - 1) = 1 D) a. lim x → - 1 - f(x) does not exist b. f( - 1) does not exist 10) a. lim x → - 2 + f(x) b. f( - 2) A) a. lim x → - 2 + f(x) = - 2 b. f( - 2) = - 2 B) a. lim x → - 2 + f(x) = - 1 b. f( - 2) does not exist C) a. lim x → - 2 + f(x) = - 2 b. f( - 2) = - 1 D) a. lim x → - 2 + f(x) does not exist b. f( - 2) = - 1 Page 18

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    11) a. lim x → 2.5 - f(x) b. f(2.5) A) a. lim x → 2.5 - f(x) = 1 b. f(2.5) = 1 B) a. lim x → 2.5 - f(x) = 2 b. f(2.5) = 1 C) a. lim x → 2.5 - f(x) = 0 b. f(2.5) = 2 D) a. lim x → 2.5 - f(x) does not exist b. f(2.5) = 2 12) a. lim x → 1 - f(x) b. f(1) x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 x -2 -1 1 2 3 4 5 y 5 4 3 2 1 -1 -2 A) a. lim x → 1 - f(x) = 1 b. f(1) = 2 B) a. lim x → 1 - f(x) = 2 b. f(1) = 2 C) a. lim x → 1 - f(x) = 1 b. f(1) = 1 D) a. lim x → 1 - f(x) does not exist b. f(1) = 2 Page 19

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    5 Solve Apps: Use Limits MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) You are riding your motorcycle along a freeway traveling x miles per hour. The function f(x) = 0.0014x 2 + x + 9 describes the recommended safe distance, f(x), in feet, between your motorcycle and the other motorcycles on the freeway. Use the values in the table below to solve this exercise. x 49.9 49.99 49.999 → ← 50.001 50.01 50.1 f(x) 93.76 93.976 93.998 → ← 94.002 94.024 94.24 Find lim x → 50 f(x). Describe what this means in terms of your motorcycle's speed and the recommended safe distance. A) 94; This means that the recommended safe distance between motorcycles traveling at 50 miles per hour is 94 feet. B) 50; This means that the recommended safe distance between motorcycles traveling at 50 miles per hour is 50 feet. C) 93.999; This means that the recommended safe distance between motorcycles traveling at 50 miles per hour is 93.999 feet. D) lim x → 50 f(x) does not exist. Page 20

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    2) You rent a moving van from a company that charges $10 per hour plus $0.50 per mile. The van is driven 40 miles in the first hour. The figure below shows the graph of the cost, f(x), in dollars, as a function of the miles, x, that you drive the van. (i) Find lim x → 20 f(x). Interpret the limit, referring to miles driven and cost. (ii) For the first hour hour only, what is the rental cost approaching as the mileage gets closer to 40? (iii) What is the cost to rent the van at the start of the second hour? A) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20. (ii) $30 (iii) $40 B) (i) 30; the cost to rent the van for one hour and drive it 20 miles is $30. (ii) $40 (iii) $40 C) (i) 40; the cost to rent the van for one hour and drive it 20 miles is $40. (ii) $30 (iii) $40 D) (i) 20; the cost to rent the van for one hour and drive it 20 miles is $20. (ii) $30 (iii) $30 3) You are building a screened - in porch attached to your house. Because the house will be used for one side of the enclosure, only three sides will need to be enclosed. You have 24 yards of material to enclose the three walls. The function f(x) = x(24 - 2x) describes the area of the screened - in porch that you can enclose, f(x), in square yards, if the width of the screened - in porch is x yards X Y1 5.7 5.8 5.9 6 6.1 6.2 6.3 71.82 71.92 71.98 71.98 71.92 71.82 Y1 = X * (24 - 2X) (i) Use the table shown to find lim x → 6 f(x). (ii) Use the graph of f(x) = x(24 - 2x) shown to find lim x → 6 f(x). Do you get the same limit as you did in part (i)? What information about the limit is shown by the graph that might not be obvious from the table? Page 21

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    [ - 5, 15, 5] by [0, 80, 10] A) (i) 72 (ii) 72; Yes, the same limit is obtained. The graph shows that the limit is the maximum area. B) (i) 80 (ii) 80; Yes, the same limit is obtained. The graph shows that the limit is the maximum area. C) (i) 72 (ii) 80; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of material that can be used. D) (i) 80 (ii) 72; No, the same limit is not obtained. The graph shows that the limit is the maximum amount of material that can be used. 11.2 Finding Limits Using Properties of Limits 1 Find Limits of Constant Functions and the Identity Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. 1) lim x → 10 - 6 A) - 6 B) 10 C) 6 D) 0 2) lim x → - 8 x A) - 8 B) 8 C) 0 D) 1 2 Find Limits Using Properties of Limits MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. 1) lim x → - 7 15x 3 A) - 5145 B) - 7 C) - 343 D) 15 2) lim x → 0 (x2 - 5) A) - 5 B) 5 C) 0 D) does not exist Page 22

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    3) lim x → 2 (x2 + 8x - 2) A) 18 B) - 18 C) 0 D) does not exist 4) lim x → 2 (x 3 + 5x 2 - 7x + 1) A) 15 B) 29 C) 0 D) does not exist 5) lim x → 0 ( x - 2) A) - 2 B) 0 C) 2 D) does not exist 6) lim x → 1 (3x 2 - 19) 2 A) 256 B) - 352 C) 144 D) 100 7) lim x → 4 (x - 4)( x - 2) A) 0 B) 2 C) 4 D) does not exist 8) lim x → 1 (x 2 - 2) 3 A) - 1 B) 1 C) 3 D) - 3 9) lim x → 3 (3x 2 - 2x + 2) 2 A) 529 B) - 529 C) 81 D) does not exist 10) lim x → 1 3x - 2 A) 1 B) 2 C) - 1 D) does not exist 11) lim x → 1 2x - 7 4x + 5 A) - 5 9 B) - 1 2 C) - 7 5 D) does not exist 12) lim x → 1 2x - 7 4x + 5 A) - 5 9 B) - 1 2 C) - 7 5 D) does not exist 13) lim x → 5 x 2 - 2x - 15 x + 3 A) 0 B) - 8 C) 5 D) does not exist 14) lim x → 0 x3 - 6x + 8 x - 2 A) - 4 B) 0 C) 4 D) does not exist Page 23

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    15) lim x → 1 3x 2 + 7x - 2 3x 2 - 4x + 2 A) 8 B) - 7 4 C) 0 D) does not exist 16) lim x → 2 [(x - 3) 3 (5x + 1) 2 ] A) - 121 B) 121 C) - 288 D) does not exist 17) lim x → 7 x - 7 x - 49 A) 1 6 - 7 42 B) 0 C) 7 42 - 1 6 D) does not exist 3 Find One - Sided Limits Using Properties of Limits MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does not exist. 1) f(x) = 4x - 3 if x < 1 - 3x + 4 if x > 1 a. lim x → 1 - f(x) b. lim x → 1 + f(x) c. lim x → 1 f(x) A) a. 1 b. 1 c. 1 B) a. 1 b. 1 c. does not exist C) a. - 3 b. 4 c. does not exist D) a. 4 b. - 3 c. does not exist 2) f(x) = x 2 - 1 if x < 0 - 4 if x ≥ 0 a. lim x → - 5 - f(x) b. lim x → - 5 + f(x) c. lim x → - 5 f(x) A) a. 24 b. 24 c. 24 B) a. - 1 b. - 4 c. does not exist C) a. - 4 b. - 1 c. does not exist D) a. - 1 b. - 4 c. - 4 3) f(x) = 1 x - 5 if x > 5 x 2 - 2x if x ≤ 5 a. lim x → 5 - f(x) b. lim x → 5 + f(x) c. lim x → 5 f(x) A) a. 15 b. does not exist c. does not exist B) a. does not exist b. 15 c. does not exist C) a. 15 b. does not exist c. 15 D) a. does not exist b. 15 c. 15 Page 24

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    4) f(x) = x 2 - 9 x - 3 if x ≠ 3 5 if x = 3 a. lim x → 3 - f(x) b. lim x → 3 + f(x) c. lim x → 3 f(x) A) a. 6 b. 6 c. 6 B) a. 6 b. 6 c. 5 C) a. 3 b. 3 c. does not exist D) a. 3 b. 3 c. 5 5) f(x) = 4x - 5 if x < 1 1 if x = 1 - 3x - 4 if x > 1 a. lim x → 1 - f(x) b. lim x → 1 + f(x) c. lim x → 1 f(x) A) a. - 1 b. - 7 c. does not exist B) a. - 7 b. - 1 c. does not exist C) a. - 1 b. - 7 c. - 8 D) a. - 7 b. - 1 c. - 8 6) f(x) = 6 - x if x < 1 4 if x = 1 x 2 + 4 if x > 1 a. lim x → 1 - f(x) b. lim x → 1 + f(x) c. lim x → 1 f(x) A) a. 5 b. 5 c. 5 B) a. 5 b. 4 c. does not exist C) a. 4 b. 6 c. does not exist D) a. 5 b. 5 c. 4 4 Find Limits of Fractional Expressions in Which the Limit of the Denominator is Zero MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied. 1) lim x → 1 x4 - 1 x - 1 A) 4 B) 2 C) 0 D) does not exist 2) lim x → - 3 x 2 - 2x - 15 x + 3 A) - 8 B) 0 C) 5 D) does not exist 3) lim x → 0 x 3 + 12x 2 - 5x 5x A) - 1 B) 0 C) 5 D) does not exist 4) lim x → 1 x3 + 5x2 + 3x - 9 x - 1 A) 16 B) - 16 C) 0 D) does not exist Page 25

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    5) lim x → 0 4 + x - 2 x A) 1 4 B) 1 2 C) 4 D) 0 6) lim x → 9 1 x - 1 9 x - 9 A) - 1 81 B) 1 9 C) 0 D) does not exist 7) lim x → 4 x - 4 x - 2 A) 4 B) 1/4 C) 0 D) does not exist 8) lim x → 4 x - 2 x - 4 A) 1/4 B) 4 C) 0 D) does not exist 9) lim x → 5 x 2 - 25 x 3 - 125 A) 2 15 B) - 2 15 C) 1 15 D) does not exist 10) lim h → 0 (x + h) 3 - x 3 h A) 3x 2 B) 0 C) 3x 2 + 3xh + h 2 D) does not exist 11.3 Limits and Continuity 1 Determine Whether a Function is Continuous at a Number MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the definition of continuity to determine whether f is continuous at a. 1) f(x) = 5x4 - 9x3 + x - 6 a = 0 A) Continuous B) Not continuous 2) f(x) = 3x4 - 9x3 + x - 5 a = 5 A) Continuous B) Not continuous 3) f(x) = 5 x + 2 a = - 2 A) Not continuous B) Continuous Page 26

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    4) f(x) = 5 x - 8 a = 0 A) Continuous B) Not continuous 5) f(x) = x + 7 x + 2 a = - 7 A) Continuous B) Not continuous 6) f(x) = x - 6 x + 3 a = 0 A) Continuous B) Not continuous 7) f(x) = x + 7 x + 3 a = - 3 A) Not continuous B) Continuous 8) f(x) = 7 x 2 + 8x a = 0 A) Not continuous B) Continuous 9) f(x) = 1 x 2 - 9x a = 9 A) Not continuous B) Continuous 10) f(x) = 1 x 2 - 3x a = - 1 A) Continuous B) Not continuous 11) f(x) = x2 - 16 x - 4 a = 0 A) Continuous B) Not continuous 12) f(x) = x2 - 64 x - 8 a = 8 A) Not continuous B) Continuous 13) f(x) = x 2 - 5, if x < 0 3, if x ≥ 0 a = - 4 A) Continuous B) Not continuous Page 27

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    14) f(x) = 4x + 3, if x < 1 1, if x = 1 4x - 7, if x > 1 a = 1 A) Not continuous B) Continuous 15) f(x) = 1 x + 4 , if x > - 4 x 2 - 4x, if x ≤ - 4 a = - 4 A) Not continuous B) Continuous 2 Determine For What Numbers a Function is Discontinuous MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine for what numbers, if any, the given function is discontinuous. 1) f(x) = 2x + 5 A) None B) 5 2 C) - 5 2 D) - 5 2) f(x) = - 3x 2 + 5x A) None B) - 5 6 C) 5 6 D) - 3 3) f(x) = - 3 cos x A) None B) 0 C) 1 D) - 3 4) f(x) = 4 tan x A) n π 2 and - n π 2 B) 0 C) n π 4 and - n π 4 D) None 5) f(x) = 2x - 1 x 2 - 25 A) - 5 and 5 B) 5 C) - 5 and 5 and - 1 2 D) None 6) f(x) = x - 4 if x ≤ 4 x 2 - 8 if x > 4 A) 4 B) None C) - 4, 4 D) 0 7) f(x) = x - 3 if x ≤ 3 x 2 if x > 3 A) 3 B) [the set of numbers 1, 2, 3, . . . 8] C) [the set of numbers 1, 2, 3, . . . 9] D) None 8) f(x) = x 2 - 9 x - 3 if x ≠ 3 6 if x = 3 A) None B) 3 C) 9 D) 3, 9 Page 28

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    9) f(x) = 8x if x < 7 55 if x = 7 x 2 + 7 if x > 7 A) 7 B) 0, 7 C) 8, 7 D) None 3 Solve Apps: Limits and Continuity SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) The figure shows the cost of renting a car, f(x), as a function of distance driven, x, in miles, for distances not exceeding 70 miles. Cost (dollars) x y x y 49 47 45 43 41 39 37 20 40 60 Distance (miles) (i) Find lim x → 60 - f(x). (ii) Find lim x → 60 + f(x). (iii) What can you conclude about lim x → 60 f(x)? How is this shown by the graph? (iv) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities? Page 29

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    2) The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars. T(x) = 0.11x if 0 < x ≤ 7593 759.30 + 0.18(x - 7593) if 7593 < x ≤ 31,687 5096.22 + 0.26(x - 31,687) if 31,687 < x ≤ 72,784 15,781.44 + 0.29(x - 72,784) if 72,784 < x ≤ 149,897 38,144.21 + 0.32(x - 149,897) if 149,897 < x ≤ 325,127 94,217.81 + 0.36(x - 325,127) if x > 325,127 (i) Determine whether T is continuous at 7593. (ii) Determine whether T is continuous at 31,687. (iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income. 11.4 Introduction to Derivatives 1 Find Slopes and Equations of Tangent Lines MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the tangent line to the graph of f at the given point. 1) f(x) = - 2x + 20 at (8, 4) A) - 2 B) 20 C) - 20 D) - 1 2 2) f(x) = x2 + 5x at (4, 36) A) 13 B) 9 C) 3 D) 21 3) f(x) = 5x2 + x at ( - 4, 76) A) - 39 B) - 41 C) 6 D) - 14 4) f(x) = - 4x2 + 7x at (5, - 65) A) - 33 B) 33 C) - 13 D) 3 5) f(x) = 2x2 + x - 3 at (4, 33) A) 17 B) 15 C) 5 D) 19 6) f(x) = x2 + 11x - 15 at (1, - 3) A) 13 B) - 9 C) 26 D) 11 7) f(x) = x at (64, 8) A) 1 16 B) 4 C) 16 D) 1 4 8) f(x) = 3 x at (1, 3) A) - 3 B) 3 C) 6 D) 3 2 Find the slope - intercept equation of the tangent line to the graph of f at the given point. 9) f(x) = x2 + 5x at (4, 36) A) y = 13x - 16 B) y = 13x - 72 C) y = 2x - 5 D) y = 2x + 5 Page 30

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