MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

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1 Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the intervals on which the function is continuous. 2 1) y = ( + 5) A) (-, ) B) (-, 35), (35, ) C)(-, -0),(-0, ) D) (-, -5),(-5, ) + 5 2) y = A) (-, 3), (3, 8), (8, ) B) (-, -8),(-8, 3), (3, ) C)(-, 3), (3, ) D) (-, -3),(-3, 8), (8, ) 1) 2) 3) y = A) 5, B) - 8 5, C) - 8 5, D) -, ) ) y = A) - 1, B) 1, C) 1, D) -, 1 ) 5) y = e1/ A) (-, 0), (0, ) B) (-, ) C)(-, 1), (1, ) D) (-,-1),(-1, ) 5) 6) y = ln(9-9) A) -, 1 B) 1, C) 1, D) - 1, 6) 7) y = cot A) (-2π, 2π) B) (-, ) 7) C) = π 2 + nπ, n is any integer D) (-, ) ecept where = nπ, n is any integer Find the points of discontinuity. Identify each type of discontinuity. 5 8) y = ( + 5) A) = -5, infinite discontinuity B) = 35 C)None D) = -5, jump discontinuity + 1 9) y = A) = 7, infinite discontinuity B) = 5, = 7, both infinite discontinuities C) = 5, infinite discontinuity D) = -5, = -7, both infinite discontinuities 8) 9) 1

2 10) y = + 1 A) > - 1, all points not in the domain B) = - 1, jump discontinuity 10) C) = - 1, infinite discontinuity D) < - 1, all points not in the domain 11) y = 5-9 A) < - 9 5, all points not in the domain B) > 9, all points not in the domain 5 11) C) < 9, all points not in the domain 5 D) > - 9, all points not in the domain 5 12) y = e1/ A) = 1, infinite discontinuity B) = -1, infinite discontinuity C) None D) = 0, infinite discontinuity 12) 13) y = ln( - 9) A) > - 9, all points not in the domain B) < 9, all points not in the domain 13) C) > 9, all points not in the domain D) < - 9, all points not in the domain 1) y = cot A) = nπ, n is any integer, jump discontinuity 1) B) = π 2 + nπ, n is any integer C) = nπ, n is any integer, infinite discontinuity D) None Find all points where the function is discontinuous. 0, < 0 15) f() = 2-3, 0 3 3, > 3 A) = 0 B) = 0 and = 3 C) = 3 D) Nowhere 15) 16) f() = , -3 8, = -3 16) A) Continuous for all B) = 8 C) = 3 D) = -3 2

3 17) f() = (2-9), , = -3 17) A) = -7 B) Continuous for all C) = 3 D) = -3 18) 3, = -1, 18) f() = , -1; A) = 3 B) = -1 C)Continuous for all D) = ) 19) A) = B) =, = 2 C) = 2 D) None 20) 20) A) = -2 B) = 1 C)None D) = -2, = 1 Give a formula for the etended function that is continuous at the given point. 21) f() = 2-25 ( + 5), = - 5 A) y = 2 B) y = + 5 C)y = - 5 D) y = ) 22) f() = , = 10 22) A) y = B) y = C)y = + 10 D) y =

4 23) f() = sin 2, = 0 23) A) y = sin 2, = 0 B) y = sin 2, 0 2, 0 2, = 0 C)y = sin 2 D) y = 2 Provide an appropriate response. 2) Decide whether the function f() = is continuous for all, and provide a short statement supporting your conclusion. A) Yes, polynomial functions are continuous; there are no breaks in the graph of a polynomial function. B) Yes, polynomial functions are defined for all. C)No, there is a break in the graph of this function at = 0. D) No, this polynomial is not defined for all. 25) Given f() = + 2 and g() = - 5, where is the function f()/g() continuous? A) The function f()/g() is continuous for all. B) The function f()/g() is continuous for all ecept = -2 and = 5. C)The function f()/g() is continuous for all ecept = -2. D) The function f()/g() is continuous for all ecept = 5. 2) 25) 26) Given f() = 3 6 and g() = - 3, where is the function f()/g() continuous? A) The function f()/g() is continuous for all. B) The function f()/g() is continuous for all ecept = 3. C)The function f()/g() is continuous for all ecept < 0 and = -3. D) The function f()/g() is continuous for all ecept = ) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) If functions f and g are continuous for 0 6, could f g at a point of [0,6]? Provide an eample. possibly be discontinuous 27) 28) A function y = f() is continuous on [-3, -1]. It is known to be positive at = -3 and negative at = -1. What, if anything, does this indicate about the equation f() = 0? Illustrate with a sketch. 28)

5 29) If f() = , show that there is at least one value of c for which f() equals π. 29) 30) Verify that the function f() = is continuous. Indicate which theorems are needed and which functions are assumed to be continuous for all in the domain. 30) 31) Show that the function f() = sin (2 + 1) is continuous. 31) 32) Sketch a possible graph for the function f that has the following properties: f(-3) eists but lim 3 f() does not eist. 32) 33) Sketch a possible graph for the function f that has the following properties: f() is continuous for all ecept = -5, where f has a non removable discontinuity. 33) 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() is continuous. 3) f() = 2-2, < 3 5a, 3 A) a = 3 5 B) a = 7 15 C)a = 7 D) a = 5 3) 35) f() = a, < 5 3, 5 A) a = 155 B) a = 95 C)a = 125 D) a = 30 35) 5

6 Solve the problem. 36) The graph below shows the amount of income ta that a single person must pay on his or her income when claiming the standard deduction. Identify the income levels where discontinuities occur and eplain the meaning of the discontinuities. 36) Income Ta, 1000's of dollars Income, 1000's of dollars A) Discontinuities at = $,000 and = $60,000. Discontinuities represent boundaries between ta brackets. B) Discontinuities at = $22,000, = $,000, and = $60,000. Discontinuities represent ta cheating on the part of high-income earners. C) Discontinuities at = $,000 and = $60,000. Discontinuities represent ta shelters. D) Discontinuities at = $22,000, = $,000, and = $60,000. Discontinuities represent boundaries between ta brackets. 37) In order to boost business, a ski resort in Vermont is offering rooms for $125 per night with every fourth night free. Let C() represent the total cost of renting a room for days. Sketch a graph of 37) C() on the interval (0, 6] and determine the cost for staying 1 2 days. 6

7 A) B) C() C() C 1 2 = $625 C 1 2 = $500 C) D) C() C() C 1 2 = $500 C 1 2 = $375 38) Suppose that the cost, p, of shipping a 3-pound parcel depends on the distance shipped,, according to the function p() depicted in the graph. Is p continuous at = 50? at = 500? at = 1500? at = 3000? 38) A) Yes; yes; yes; no B) No; no; yes; no C)Yes; no; no; no D) Yes; no; yes; no 7

8 39) Suppose that the cost, p, of shipping a 3-pound parcel depends on the distance shipped,, according to the function p() depicted in the graph. Find each limit, if it eists: 39) lim p(), lim p(), lim p() A) 5; does not eist; does not eist B) 5; does not eist; 15 C)5; 5; 15 D) 5; 10; 15 0) Suppose that the cost, C, of producing units of a product can be illustrated by the given graph. Find each limit, if it eists: 0) lim p(), lim p(), lim p() A) 200; 200; 200 B) 200; does not eist; does not eist C)200; 300; 200 D) 200; 300; does not eist 8

9 1) Suppose that the cost, C, of producing units of a product can be illustrated by the given graph. Is C() continuous at = 50? = 100? = 150? 1) A) Yes; no; no B) Yes; no; yes C)Yes; yes; yes D) No; no; no 2) Suppose that the unit price, p, for units of a product can be illustrated by the given graph. Find each limit, if it eists: 2) lim p(), lim p(), lim p(), lim p() A) 8; 8; does not eist; 8 B) 10; 8; 8; 8 C)8; 8; 8; 8 D) 10; 8; does not eist; 8 9

10 3) Suppose that the unit price, p, for units of a product can be illustrated by the given graph. Is p continuous at = 50? = 100? = 150? 3) A) No; yes; no B) No; yes; yes C)Yes; no; yes D) No; no; no ) Consider the learning curve defined in the graph. Depicted is the accuracy, p, epressed 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 = 0? at t = 5? ) A) Yes; yes; yes B) No; no; no C)Yes; no; yes D) Yes; no; no 10

11 5) Consider the learning curve defined in the graph. Depicted is the accuracy, p, epressed as a percentage, in performing a series of short tasks versus the accumulated amount of time spent practicing the tasks, t. Find each limit, if it eists: 5) lim p(), lim p(), lim p() A) 0; 100; 100 B) 100; 100; 100 C)0; 100; does not eist D) 0; 0; 0 11

12 Answer Key Testname: UNTITLED2 1) A 2) A 3) B ) C 5) A 6) C 7) D 8) C 9) B 10) D 11) C 12) D 13) B 1) C 15) C 16) D 17) D 18) B 19) A 20) B 21) C 22) B 23) B 2) A 25) D 26) B 27) Yes, if f() = 1 and g() = - 3, then h() = 1 is discontinuous at = ) The Intermediate Value Theorem implies that there is at least one solution to f() = 0 on the interval -3, -1. Possible graph: ) Notice that f(0) = 5 and f(1) = 2. As f is continuous on [0,1], the Intermediate Value Theorem implies that there is a number c such that f(c) = π. 30) Assume y = and the square root function are continuous. Use the sum, constant multiple, product, and quotient theorems of continuity. 31) f is a composite of two functions g h where g() = sin and h() = Since g and h are known to be continuous, and the composition of continuous functions are continuous, then f is continuous. 12

13 Answer Key Testname: UNTITLED2 32) Answers may vary. One possible solution: 10 y ) Answers may vary. One possible solution: 10 y ) B 35) B 36) D 37) B 38) D 39) B 0) D 1) B 2) D 3) A ) C 5) C