Test 2 (PreTest) - Limits - AP Calculus AB PreTest #2 -...

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Unformatted text preview: AP Calculus AB PreTest #2 - Limits Objectives: You will be able to determine average the rate of change of a function You will be able to determine limits with an algebraic approach and limit laws You will be able to determine a limit with an analytical method You will be able to determine a limit with a graphical method You will be able to recognize when a limit does not exist You will be able to evaluate limits with an algebraic approach by simplifying. You will be able to determine limits using the squeeze theorem You will be able to determine the limits of trigonometric functions You will be able to evaluate one-sided limits You will be able to describe the continuity of a function. You will be able to test for the existence of a limit You will be able to test for continuity at a point You will be able to apply the intermediate value theorem to predict the behavior of a function. You will be able to determine infinite limits of a function. Name___________________________________ Date: _______________________ 3) Find lim f(x) and lim f(x). x2 x2 + for the given x 0 and 8 6 4 2 -4 -3 -2 -1 -2 -4 -6 -8 1 2 3 4 x y Evaluate lim h0 function f. f(x 0 + h) - f(x 0 ) h x 1) f(x) = + 10 for x0 = 7 5 Determine the limit graphically, if it exists. 2) Find lim f(x) and lim f(x). x-1 x-1 + y 4 2 -4 -2 -2 -4 -6 -8 -10 2 4 6 x -6 4) lim f(x) x0 1 5) lim x0 x3 - 6x + 8 x - 2 11) lim x0 x3 + 12x2 - 5x 5x 6) lim x2 + 8x + 16 x3 12) lim x -6 x2 + 16x + 60 x + 6 7) lim x18 10 13) x2 - 16 lim 2 - 6x + 8 x 4 x 14) 8) Let lim f(x) = 144. Find lim x -6 x -6 f(x). lim h 0 (x + h)3 - x3 h Find the limit. 9) Let lim f(x) = -1 and lim g(x) = 4. Find x 9 x 9 lim [f(x) + g(x)]2 . x 9 15) lim h0 + h 2 + 13h + 5 - 5 h 16) 10) Let lim f(x) = 8 and lim g(x) = -4. Find x -5 x -5 lim x -5 8f(x) - 2g(x) . 1 + g(x) lim x 11 11 - x 11 - x 17) lim x0 1 1 - x + 3 3 x 2 18) lim x0 7 sin x 9x 24) cos 19) lim 0 + 3 Find the intervals on which the function is continuous. x + 3 25) y = 2 - 14x + 48 x 20) lim 6x2 (cot3x)(csc2x) x0 sin 3() 3 26) y = sin 5x 21) lim x0 x sec x 27) y = 3 (x + 4)2 + 8 22) lim t0 tan2 5t 6t State whether the function is continuous at the indicated point. 28) State whether r(t) is continuous at the point t = 3. t2 - 9 if t 3 r(t) = t - 3 6 if t = 3 Find all points where the function is discontinuous. 23) 29) State whether f(t) is continuous at the point t = 9. 2 f(t) = t - 81 if t 9 (t - 9)2 if t > 9 3 Determine the points at which the function is discontinuous. x + 2 30) g(x) = 2 - 8x + 15 x Find the limit. 35) 1 lim x + 2 x(-2)- x2 - 9 for x < -1 31) h(x) = 0 for -1 x 1 x2 + 9 for x > 1 36) lim x2 + x2 + 2x - 8 x2 - 4 -x2 + 1, 32) f(x) = 2x, -2x + 4 2, 37) x 0 0 < x 1 1 < x < 3 x 3 t2 lim 2 t8 + 64 - t Provide an appropriate response. f(x) - 3 38) If lim = 4, find lim f(x). x1 x - 3 x1 Find a value for a so that the function f(x) is continuous. 2 33) f(x) = x + x + a, x < -5 x3 , x -5 f(x) - 3 39) If lim = 2, find lim f(x). x1 x - 1 x1 Provide an appropriate response. 34) A function y = f(x) is continuous on [1, 3]. It is known to be positive at x = 1 and negative at x = 3. What, if anything, does this indicate about the equation f(x) = 0? Illustrate with a sketch. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 40) Find lim x 0 1 1 - x + 7 7 x . Find the limit. 41) 1 lim x + 3 x-3 - 4 42) x3 lim x - 5 x5 + 46) lim f(x) x1 5 4 3 f(x) 4t2 43) lim sin t t-5 -4 -3 -2 -1 2 1 1 -1 -2 -3 2 3 4 5 x MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate lim h0 function f. 44) f(x) = 3x2 - 2 for x0 = -1 A) -6 B) Does not exist C) -8 D) 3 f(x 0 + h) - f(x 0 ) h for the given x 0 and -4 -5 A) Does not exist 1 B) 2 C) 2 D) -1 47) Let lim f(x) = -9 and lim g(x) = -2. Find x 5 x 5 Determine the limit graphically, if it exists. 45) lim f(x) x -1/2 f(x) . lim x 5 g(x) A) -7 2 B) 9 C) 5 9 D) 2 A) Does not exist B) -2 C) -1 D) 0 5 48) Let lim f(x) = -6 and lim g(x) = 8. Find x -1 x -1 Find the limit. 51) lim h0 A) B) -3 6 3 2 3 3 - 3h 2 + 3h + 3 h [f(x)]2 . lim x -1 7 + g(x) A) 12 5 2 B) - 5 C) 4 25 C) Does not exist -3 D) 2 3 D) -1 52) x + 6 49) lim x6 (x - 6)2 A) -6 B) 6 C) 0 D) Does not exist lim x + 3 x-1 + A) 2 B) 4 x + 1 x + 1 C) Does not exist D) -2 2 50) lim 3h+4 + 2 h0 A) 2 B) Does not exist C) 1/2 D) 1 53) lim x0 6x cos x sin x A) 1 1 B) 6 C) 0 D) 6 6 Find all points where the function is discontinuous. sin3x cot4x 54) lim cot5x x0 A) Does not exist B) 15/4 C) 12/5 D) 0 A) x = 4, x = 2 B) x = 4 C) x = 2 From the graph of f, indicate the intervals on which f is continuous. 55) y 12 8 4 56) D) None Find the intervals on which the function is continuous. x + 3 57) y = x2 - 5x + 6 A) (-, 2), (2, ) -6 -4 -2 -4 -8 -12 2 4 6 x B) (-, 2), (2, 3), (3, ) C) (-, -3), (-3, 2), (2, ) D) (-, -2), (-2, 3), (3, ) A) (-, -1), [-1, 1], (1, ) B) (-, -1), (-1, 1), (1, ) C) (-, -1], (-1, 1), [1, ) D) (-, 1], (1, ) 4 58) y = 6x - 2 1 A) , 3 1 B) - , 3 C) -, D) 1 3 1 , 3 7 State whether the function is continuous at the indicated point. 59) State whether f(t) is continuous at the point t = 4. f(t) = 4t - 6 if t 4 -8 if t = 4 A) Not continuous; lim f(t) and f(4) exist t4 but lim f(t) f(4) t4 B) Not continuous; f(4) does not exist C) Not continuous; lim f(t) does not exist t4 D) Continuous Find a value for a so that the function f(x) is continuous. 2 62) f(x) = x - 7, x < 4 5ax, x 4 A) a = 9 B) a = 11 9 C) a = 20 D) a = 4 5 Find the limit. 63) 1 lim x + 2 x(-2)+ A) -1/2 Determine the points at which the function is discontinuous. 5x - 1 60) h(x) = x + 4 A) -4 B) (-, -4) C) (-, -4] D) [-4, ) 64) B) 1/2 C) - D) x2 - 81 lim x + 9 x-9 + A) 1 B) -9 C) D) -18 x3 , 61) f(x) = -2x, 5, 0, A) 0, 2 B) 0 C) 2 D) None x < 0 0 x < 2 x > 2 x = 2 8 Provide an appropriate response. 65) Given lim f(x) = Ll , lim f(x) = Lr , and x0 x0 + Ll Lr, which of the following statements is true? 67) If lim f(x) = L, which of the following x0 expressions are true? I. lim f(x) does not exist. x0 II. lim f(x) does not exist. x0 + III. lim f(x) = L x0 IV. lim f(x) = L x0 + A) I and II only B) I and IV only C) II and III only D) III and IV only I. lim f(x) = Ll x0 II. lim f(x) = Lr x0 III. lim f(x) does not exist. x0 A) II B) I C) III D) None 66) Given lim f(x) = Ll, lim f(x) = Lr , and x0 x0 + Ll = Lr, which of the following statements is false? 68) If lim f(x) = 1, lim f(x) = -1, and f(x) is x1 x1 + an even function, which of the following statements are true? I. lim f(x) = -1 x-1 lim f(x) = -1 x-1 + I. lim f(x) = Ll x0 II. lim f(x) = Lr x0 III. lim f(x) does not exist. x0 A) II B) III C) I D) None II. III. lim f(x) does not exist. x-1 A) II and III only B) I and II only C) I and III only D) I, II, and III 9 69) If lim f(x) = 1 and f(x) is an odd function, x0 which of the following statements are true? I. lim f(x) = 1 x0 II. lim f(x) = -1 x0 + III. lim f(x) does not exist. x0 A) I and II only B) I and III only C) II and III only D) I, II, and III 10 Answer Key Testname: TEST 2 (PRETEST) LIMITS 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 1 5 34) The Intermediate Value Theorem implies that there is at least one solution to f(x) = 0 on the interval 1, 3 . Possible graph: 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 -2; -7 4; -3 Does not exist -4 7 10 12 9 - 24 -1 4 4 14) 3x2 13 15) 2 5 16) Does not exist 1 17) - 9 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 7 9 1 3 1 5 0 x = -2, x = 0, x = 2 x = 3 (-, 6), (6, 8), (8, ) (-, 0), (0, ) (-,) Continuous Continuous 3, 5 -1, 1 0, 3 a = -145 35) - 3 36) 2 37) - 38) -5 39) 3 1 40) - 49 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64) 65) 66) 11 - A C C D A D C D A D B A B B D A C C C D D C B Answer Key Testname: TEST 2 (PRETEST) LIMITS 67) D 68) C 69) C 12 ...
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This note was uploaded on 02/03/2012 for the course MATH 101 taught by Professor Lee during the Spring '11 term at International Institute for Higher Education.

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