AP Calculus AB Exam Review Limits and Continuity MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the watermelon's average speed during the first 6 sec of fall. A) 97 ft/sec B) 48 ft/sec C) 96 ft/sec 1) D) 192 ft/sec Determine the limit by substitution. 2) lim x→4 x2 + 14x + 49 2) A) 121 B) Does not exist C) ± 11 D) 11 Determine the limit algebraically, if it exists. x3 + 12x2 - 5x 3) lim 5x x→0 A) 0 3) B) Does not exist C) -1 D) 5 Determine the limit graphically, if it exists. 4) lim f(x) x→1 - 4) 5 f(x) 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 A) Does not exist B) 1 2 C) -1 D) 2 Find the indicated limit. 5) lim int x x→-3A) -3 5) B) -4 C) 4 1 D) 0 Match the function with the correct table values. x2 - 2x - 8 6) f(x) = x2 - 9x + 20 6) x 3.9 3.99 3.999 4.001 4.01 4.1 f(x) A) -5.2636; -5.8307; -5.8930; -5.9070; -5.9707; -6.6778 B) -5.4636; -6.0307; -6.0930; -6.1070; -6.1707; -6.8778 C) 0.2308; 0.2231; 0.2223; 0.2221; 0.2214; 0.2135 D) -5.3636; -5.9307; -5.9930; -6.0070; -6.0707; -6.7778 Find the limit. 7) Let lim f(x) = 49. Find lim x → -1 x → -1 A) -1 f(x). 7) B) 7 C) 2.6458 Evaluate or determine that the limit does not exist for each of the limits (a) D) 49 lim f(x), (b) lim f(x), and (c) lim f(x) x→dx→d+ x→d for the given function f and number d. 8) 8) f(x) = x2 - 4, 3, for x < 0, for x ≥ 0 d = -4 A) (a) -4 (b) 3 (c) Does not exist B) (a) 12 (b) 12 (c) 12 C) (a) -4 (b) 3 (c) 3 D) (a) 3 (b) -4 (c) Does not exist Provide an appropriate response. 9) It can be shown that the inequalities 1 -x ≤ x cos( ) ≤ x x 9) 1 hold for all values of x ≥ 0. Find lim x cos( ) if it exists x x→0 A) Does not exist B) 0.0007 C) 1 D) 0 Find the limit, if it exists. cos 3x 10) lim x x→-∞ A) 0 10) B) -∞ C) 3 2 D) 1 Find the indicated limit. sin x 11) lim x→-∞ x A) 1 11) B) ∞ C) 0 D) Does not exist Find the limit. 12) lim (1 + csc x ) x→0 + A) -∞ 12) B) 1 C) 0 D) ∞ Find the vertical asymptotes of the graph of f(x). 13) f(x) = sec x π A) x = + nπ, n is any integer 2 13) B) x = 0 C) x = nπ, n is any integer D) x = π 3 Match the function with the graph of its end behavior model. 2x3 - 5x2 + 1 14) y = x+9 A) 14) B) y -40 y 400 400 200 200 -20 20 40 x -40 -20 -200 -200 -400 -400 C) 20 40 x 20 40 x D) y y -40 400 400 200 200 -20 20 40 x -40 -20 -200 -200 -400 -400 Find a power function end behavior model. 15) y = 4x2 - 2x + 7 A) y = 4x2 15) B) y = 4x2 + 7 C) y = 4x2 - 2x D) y = 4x - 2 Find a simple basic function as a right-end behavior model and a simple basic function as a left-end behavior model. 16) y = 4x + ln x 16) A) y = 4x; y = -4x B) y = 4x; y = 4x C) y = 4x; y = ln x D) y = x; y = -ln x Find the limit, if it exists. 17) lim x4ex x→-∞ A) 0 17) B) 4 C) 1 4 D) ∞ Find the limit of f(x) as (a) x→-∞, (b) x→∞, (c) x→0-, and (d) x→0+. x-4 , x≤0 x-2 18) f(x) = 1 , x2 18) x>0 A) (a) -∞ (b) ∞ (c) 4 (d) 2 B) (a) 1 (b) 0 (c) 2 (d) ∞ C) (a) 2 (b) Does not exist (c) 1 (d) 0 D) (a) ∞ (b) 2 (c) 0 (d) 1 Find the intervals on which the function is continuous. 4 19) y = 9x - 8 8 8 A) - , ∞ B) -∞, 9 9 19) 8 C) , ∞ 9 8 D) , ∞ 9 Find the points of discontinuity. Identify each type of discontinuity. x+4 20) y = 2 x - 14x + 48 20) A) x = 6, x = 8, both infinite discontinuities B) x = 6, infinite discontinuity C) x = -6, x = -8, both infinite discontinuities D) x = 8, infinite discontinuity Find all points where the function is discontinuous. 21) 21) A) x = 4 B) None C) x = 2 D) x = 4, x = 2 Give a formula for the extended function that is continuous at the given point. sin 5x 22) f(x) = ,x=0 x A) y = sin 5x , x≠0 x 5, B) y = x=0 sin 5x , x=0 x 5, C) y = 5x D) y = sin 5x 5 22) x≠0 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 23) Verify that the function f(x) = -4x2 + x is continuous. Indicate which theorems are needed x 23) and which functions are assumed to be continuous for all x in the domain. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a value for a so that the function f(x) is continuous. 2 24) f(x) = x - 5, x < 4 5ax, x≥4 A) a = 9 24) B) a = 11 C) a = 11 20 D) a = 4 5 Solve the problem. 25) Consider the learning curve defined in the graph. Depicted is the accuracy, p, expressed as a percentage, in performing a series of short tasks versus the accumulated amount of time spent practicing the tasks, t. Is p(t) continuous at t = 25? at t = 40? at t = 45? A) Yes; no; no B) No; no; no C) Yes; no; yes 6 D) Yes; yes; yes 25) Answer Key Testname: LIMITS AND CONTINUITY 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) C D C D B D B B D A C D A D A B A B C A A A Assume y = x and the square root function are continuous. Use the sum, constant multiple, product, and quotient theorems of continuity. 24) C 25) C 7
© Copyright 2024 Paperzz