BARUCH COLLEGE MATH 003 SPRING 006 MANUAL FOR THE UNIFORM FINAL EXAMINATION The final examination for Math 003 will consist of two parts. Part I: Part II: This part will consist of 5 questions similar to the questions that appear in Problem Section A and Section M (Matrices). No calculator will be allowed on this part. This part will consist of 10 questions. The graphing calculator is allowed on this part. At least 6 of the questions will be similar to questions that appear in Problem Sections C and CM (Calculator allowed). The remaining 4 questions will be similar to questions that appear in all Sections: A, M, C and CM. (Note that all 10 questions might come from Section C.) There may be a few new problem types on the exam that are not similar to the problems in this review. If such problems appear, they will be similar to problems that you have seen during the semester. GRADING: Each question will be worth 3 points. Anyone who gets 34 or 35 questions correct will be assigned a grade of 100. No points are subtracted for wrong answers. CONTENTS OF THIS MANUAL: Page shows the sample questions that correspond to each section of the current text. (When a section has been covered in class, the list indicates the problems that can be used in studying for the exam that includes that section during the semester.) Problem Section A. (Problems for which a calculator may not be used.) Problem Section M. (Problems on matrices for which a calculator may not be used.) Problem Section C. (Problems for which a calculator should be used.) Problem Section CM. (Problems on matrices for which a calculator can be used.) Answers to the problems. 1
Math 003 Textbook Sections Corresponding to Sample Uniform Final Exam Questions Textbook: Precalculus and Elements of Calculus, Warren B. Gordon, Walter O. Wang, Baruch College, CUNY Section Problems 1.1 9, 6, 35, 76, 11, 113, 119, 153 1. 16,, 31, 33, 61 (also from 0.5), 64, 69, 73, 86, 139, 15 1.3 71, 14, 150, 164, C8 1.4 0, 7, 8, 101, 10, 103, 143, 146, C5, C15 1.5 9, 67, 144, 145, 154, 155, C9 1.6 90, 104, 105, 108, 147, 165, C1, C16, C17, C3, C5 1.7 30, 3, 34, 59, 63, 70, 74, 75, 100, 106, 107, 109, 110, 111, 130, 13, 133, 136, 149, 168, C, C10, C8 1.8 6, 156, 157, C6, C7, C3.1 M1, M14, M18. M3, M4, M5, M6, M9, M10, M15, M16, M1, M5, CM, CM4.3 M7, M8, M13, M17, M0, M, CM3, CM5.4 M1, M, M11, M19, M3, M4, CM1, CM6 3.1 8, 36, 37, 60, 139, 158, C6, C18, C4 3. 10, 1, 46, 47, 88, 89, 93, 95, 10, 11, 151, C11 3.3 1,, 3, 4, 17, 38, 39, 40, 4, 45, 51, 7, 78, 84, 85, 114, 115, 116, 117, 14, 15, 137, 138, 141, 159, 160, 166, C13, C0, C1, C35 3.4 5, 1, 3, 41, 43, 44, 68, 79, 80, 81, 8, 83, 99, 118, 134, 135, 140, 141, 163, C1, C7 3.5 6, 7, 13, 48, 50, 5, 58, 87, 9, 18, 19, 131, 16, C34 3.6 14, 53, 54, 94, 1, 161, C18, C19 3.7 11, 5, 49, 65, 66, 91, 13, 17, 148, C3, C14, C9, C30, C33 3.8 15, 18, 55, 57, 96, 16 3.10 19, 4, 56, 77, 86, 97, 98 3.11 C31
SECTION A 1. Find a) 3 b) 4 c) 0 d) 1 e) (does not exist). Given find if it exists. a) 0 b) 1 c) -1 d) it does not exist e) x 3. At x = 1 the function is not continuous because a) f(1) does not exist b) does not exist c) and f(1) exist but d) f(1) = 1 e) 4. In order to make the function continuous at x = 1 we must define a) f(1) = 0 b) f(1) = -1 c) f(1) = d) f(1) = 1 e) f(1) = x 5. Find the horizontal asymptote for the graph of a) y = 0 b) y = 1 c) y = 3 d) y = e) y = -1 6. Find the derivative of a) b) c) d) e) 7. Find the derivative of a) 3x - 6x - 9 b) x + 4x - 11 c) x - 6x - 9 d) x - 5 e) 3x - 14x + 9 3
8. If f(x) = -x + x, which of the following will calculate the derivative of f(x)? a) b) c) d) e) 9. Which of the following is the equation of the line that has x-intercept (3,0) and y-intercept (0,-4)? a) 4x + 3y = 1 b) 3x - 4y = 0 c) 3x - y = 13 d) 3x - 4y = 5 e) 4x - 3y = 1 10. Find the equation of the line tangent to the graph of y = x - x +3 at the point (1, 3). a) y = x - b) y = 4x - 6x + 5 c) y = x + 1 d) x + y = 5 e) y = x + 11. The position equation for a moving body is for t 0. Find the time when the velocity is 0. a) 1 b) 3/ c) 0 d) 5/6 e) /3 4 1. Find all of the points at which the graph of f (x) = x - 4x + 5 has horizontal tangent lines. a) (,0) only b) (1,0) only c) (1,) and (-1,10) d) (1,0) and (,13) e) (1,) only 13. The derivative of is a) b) c) d) e) (x - 6)(x + x) 14. The derivative of is 3 a) b) -9x (1-3x ) c) d) e) -9x 4
15. Find dy/dx by using implicit differentiation if xy + y = 5x. a) b) c) d) 10x - 1 e) 10x - y 16. Given f(x) = 3x - and g(x) = 5x + 1, find the composite function f(g(x)) 3 3 a) -10x + 3x - b) 15x - 10x + 3x - c) 15x + 1 d) 45x - 60x + 1 e) 15x - 17. has a hole, that is a missing point where the limit exists. What are the coordinates of the hole? a) (5, -4/3) b) (5, -19) c) (9/, 5) d) (-5, -11/19) e) (-4/3, 9/) 3 18. What is the slope of the line tangent to xy = 8 at the point (1, )? a) -/3 b) 1 c) 1 d) 8 e) -8 19. At what rate is the area of a circle increasing if the radius is increasing at the rate of 10 feet per minute when the radius is 6 feet long? (A = r ) a) ft /min b) 60 ft /min c) 10 ft /min d) 1 ft /min e) 10 ft /min 0. A factory has a profit function for a production level of x units. For what production level will the factory break even? a) 10 b) 0 c) 30 d) 6000 e) 0 1. The graph of has the horizontal asymptote: a) y = 1/ b) y = 1/4 c) y = 0 d) y = 4 e) It does not have a horizontal asymptote. If and then = a) -1 b) -9 c) -1/ d) /3 e) 3 3. Evaluate the limit: a) 0 b) 4 c) 4 d) e) 5
4. The area of a square is increasing at the rate of 7 square inches per minute. At what rate is the length of one of the sides increasing at the moment when the side has length 4? a) 7/4 b) 8 c) ½ d) 7/16 e) 7/8 5. Let the profit function for a particular item be P(x) = -10x + 160x - 100. Use the marginal profit function to approximate the increase in profit when production is increased from 5 to 6. a) 30 b) 40 c) 55 d) 60 e) 70 6. The equation of the straight line through the point (-6, 4) with x-intercept 3 is: a) b) c) d) e) 7. Describe the solution(s) to the equation : a) One rational solution b) Two rational solutions c) Two irrational solutions d) Two complex solutions e) Cannot be determined 8. The axis of symmetry of is: a) -4 b) y = 5 c) y = d) x = -4 e) x = 9. Find the center and radius of the circle given by (x + 3) + (y ) = 6 a) Center (3, -); radius 36 b) Center (-3, ); radius c) Center (3, -); radius d) Center (-3, ); radius 36 e) Center (9, 4); radius 6 30. Find a possible formula for the rational function with the following properties: Zeros at x = -3 and x = Vertical asymptotes at x = -5 and x = -7 Horizontal asymptote at y = - a) b) c) d) e) 6
31. For the function, which of the following are possibilities for p(t) and q(t)? A) and B) and C) and D) and a) A and B only b) A and C only c) B, C and D only d) C only e) A, B, C, D 3. The zeros of are a) x = 5/3 only b) x = 5/3 and x = c) x = only d) x = -5/3 only e) x = -5/3 and x = 3 33. Find f(, -3) for f (x, y) = 5x - 4xy + y a) 17 b) (-3, ) c) -31 d) (3, -) e) 3 34. The graph to the right could be the graph of the function: a) b) c) d) e) 35. The distance from the origin to the midpoint of the line segment joining P(5, 5) and Q(1, 3) is a) 5 b) c) 0 d) 5 e) 36. Given f (x) = 7x, find. a) 14x + 7 x b) 14x + ( x) c) 7 x d) 1 e) 14x + 7( x) 7
37. Find. 9a) 0 b) 5x + c) 5x d) e) undefined 38. What is? a) b) 0 c) d) 11 e) 3 39. What is? a) 5 b) 0 c) d) 5/4 e) -5/4 40. The intervals on which is continuous are a) (-, ) (all real x) b) (-4, 4) (-4 < x < 4) c) (-, -4) (4, 8) (x < -4 or 4 < x < 8) d) (-, ) (- < x < ) e) (-, -) (-, ) (, ) (all real x -, ) 41. Find. a) 1 b) 1/4 c) -3/8 d) e) - 4. Find the intervals on which f (x) = x - x + 1 is continuous. a) (0, ) (0 < x < ) b) (-, ) (all real x) c) (0, 1) (0 < x < 1) d) (0, ) (x > 0) e) (-4, 4) (4, ) (-4 < x < 4 or x > 4) 43. has as vertical asymptotes a) x = -1 and x = b) x = 1 only c) x = 0 only d) x = -1 only e) x = 1 and x = - 44. has as a horizontal asymptote a) y = 1/3 b) y = ½ c) y = 3 d) y = 0 e) y = 3/ 8
45. Find. a) 1 b) c) - d) /3 e) 5/6 46. Find the derivative of a) b) c) d) e) 47. The slope of the line tangent to at (4, 1) is a) 16 b) 3 c) 1/4 d) -1/4 e) 1 48. Find f (-1) for a) 5 b) -5 c) 3 d) -3 e) -1 49. What is the average rate of change of f (x) = x - 4x + 1 on the interval [-1, 1]? a) -4 b) 4 c) -6 d) - e) 50. At which of the following points is the slope of the tangent to equal to ½? a) (-1, -) b) (1, ½) c) (1, 0) d) (-3, ½) e) (3, ½) 51. The function is differentiable for all values of x except a) x = 0 b) 0 < x < 1 c) -1 < x < 0 d) x < 0 e) x < -1 5. If g(x) = (x + 1)(x - 3x + 1), then the slope of the line tangent to the graph of g(x) at the point where x = - is a) 0 b) c) -3 d) -11 e) 18 7 53. The derivative of f (x) = (x - 5x ) is 6 6 6 6 6 a) 7(x - 5x ) b) 7(1-10x) c) 7(1-10x)(x - 5x ) d) 70x(x - 5x ) e) (1-10x) 9
54. The derivative of is a) b) c) d) e) 3 55. Use implicit differentiation to find dy/dx when x = 1 and y = - for x + xy + y = 1. a) -1/ b) -1 c) 0 d) 1/6 e) undefined 56. The radius r of a circle is increasing at the rate of 3 inches per minute. What is the rate of change in the area when r = 0 inches? a) 6 inch/min b) 400 in /min c) 40 in /min d) 60 in /min e) 10 in /min 57. If xy = 3, find dy/dt when x = 6 and dx/dt = 4. a) 1/3 *b) -1/3 c) 0 d) ½ e) undefined 4 58. The derivative of f (x) = x(4x - 7) is 4 3 3 3 3 a) (4x - 7) b) 4x(4x - 7) c) 16x(4x - 7) d) (0x - 7)(4x - 7) e) 17x(4x - 7) 59. Suppose u(x) is a polynomial of degree 6. What is the minimum number of zeros u(x) could have? a) 0 b) 1 c) d) 4 e) 6 60. Which is the best choice for describing? a), b) is approximately equal to as x gets very small c) is the best approximation of the tangent line to f(x) d) gives the slope of the secant line connecting a fixed point and a variable point e) At the fixed point, measures the rate of change of with respect to x. 61. The domain of is a) b) c) (-7, 7) d) [-7, 7] e) 10
6. The following tables give values of functions f, g, u and v. Determine which of these functions could be linear. x f(x) x g(x) x u(x) x v(x) 50 0.10 1 5 0 50-3 5 55 0.11 4 100 100-1 1 60 0.1 3 5 00 150 0-1 65 0.13 4 4 300 00 3-7 70 0.14 5 5 a) u and g only b) f and v only c) g only d) f only e) f, u, and v only 63. Given the zeros of f are: a) x = -5/, 3 b) x = -3, 5/ c) x = -5/, 0, 3 d) x = -3, 0, 5/ e) x = -3, -5/, 0 64. Let. Then a) b) c) d) e) 65 and 66. The height in feet above the ground of a ball thrown upwards from the top of a building is given by s = -16t + 160t + 00, where t is the time in seconds 65. What is the average velocity of the ball between and 4 seconds? a) 96 b) 64 c) 3 d) 18 e) 515 66. What is the instantaneous velocity of the ball at t = seconds? a) 96 b) 64 c) 3 d) 18 e) 515 67. The y-axis is tangent to a circle centered at (5, ). Find the equation of the circle a) x - 10x + y - 4y + 9 = 5 b) x - 10x + y - 4y + 9 = 5 c) x - 4x + y - 10y + 9 = 4 d) (x + 5) + (y + ) = 5 e) (x + 5) + (y + ) = 5 11
68. Evaluate ` a) 0 b) c) d) 0 e) DNE 69. Suppose that f (x, y) = 30 - (y + 4x). Find f (3, -). a) 9 b) 38 c) 16 d) 13 e) 14 70. Give a possible formula for the polynomial graphed on the right: a) b) c) d) e) 71 and 7. For problems 71 and 7, use the function 71. The coordinates of the y-intercept of the graph of f(x) are: a) (0 ) b) (-, 0) c) (1/3, 0) d) (0, -1) e) the graph has no y-intercept 7. In order to make f(x) continuous on its entire domain, one must: a) redefine f(-) = 0 b) redefine f(0) = -1 c) redefine f(0) = d) redefine f(1/3) = 0 e) f(x) cannot be made continuous by redefining one point 73. Find two functions f and g such that where. a) and b) and c) and d) and e) and 1
74. Find a possible formula for the rational function with the following properties: Zeros at x = -4 and x = 3 Vertical asymptote at x = - Horizontal asymptote at y = 4 Hole (removable discontinuity) at x = -1 a) b) c) d) e) 75. Suppose m(x) is a polynomial of degree 5. What is the maximum number of zeros m(x) could have? a) 0 b) 1 c) 4 d) 5 e) 6 76. If the point (-4, 7) lies on the graph of the equation, then the value of b must be a) 1/ b) -3 c) 5 d) 3/ e) 9 3 77. For x + y = 9, find at the point (, -1) if = - a) -3 b) 1/6 c) 13/4 d) 8 e) -1 78. The function is discontinuous when a) x = 0 b) x > 0 c) x = d) x > e) never 79. The function has vertical asymptotes a) t = 1 only b) t = -1 only c) t = 3 only d) t = 1 and t = 3 only e) t = -1, t = 1 and t = 3 80. Find ALL of the vertical asymptotes for a) x = ± 4 b) x = ± 3 c) y = 1 d) y = 9/16 e) x = ±3, ±4 13
81. Find the horizontal asymptote of a) y = 5 b) y = 7 c) y = 0 d) y = 3/8 e) y = 5/7 8. Find the horizontal asymptote of a) y = 0 b) y = 7/5 c) y = -7/3 d) y = 7 e) there is no horizontal asymptote 83. Find a) 1/8 b) 3/ c) d) -7/ e) 0 84. Find a) b) 1 c) -1 d) 0 e) 85. Find a) 1 b) -6 c) d) x e) -3 86. Given Determine a) b) c) d) e) 87. Find the derivative of. a) b) c) d) e) /3 88. Find the slope of the line tangent to the graph of f (x) = x - x + 9 at the point (8, 1). a) 11 b) -11/6 c) 1 d) -4/3 e) -8 89. Determine the x coordinate of the point(s), if any, at which the function has a horizontal tangent. a) x = 0 b) x = -3, 5 c) x = 3, -5 d) No such points e) x = -15, 1 14
90. Determine the average cost of producing 50 items if the total cost is given by the function where x is the number of units produced. a) 95,000 b) 9,500 c) 1,900 d) 190 e) Not enough information is given to determine the average cost. 3 91. Find the average rate of change of the function f (x) = -x + 7x + 1 on the closed interval [1, 4]. a) -14 b) 7 c) -6 d) -4 e) 4 9. Find the slope of the line tangent to the graph of the function f (x) = x (x + 5) at x = 1. a) 4 b) 16 c) 14 d) 10 e) 0 3 93. What is the EQUATION of the line tangent to y = 4x at (, 3)? a) y = 1x - 16 b) y = 1x c) y = 48 d) y = 1x + 8 e) y = 48x - 64 94. Find the derivative of. a) b) c) d) e) 95. Given. At what point will the curve have zero slope? a) (0,5) b) (-7,) c) (, 0) d) (, -7) e) (0,) 96. If 5x - 3xy + y =, find dy/dx by using implicit differentiation and evaluate dy/dx at the point (0, ). a) - b) -5/4 c) 0 d) 6 e) Does not exist. 97. The side of a square is increasing at the rate of feet per minute. Find the rate at which the area is increasing when the side is 7 feet long. a) 8 ft /min b) 14 ft /min c) 49 ft /min d) 8 ft /min e) 49 ft /min 3 98. If y = 3x - 6x + 4, find dx/dt when x = and dy/dt = 60. a) ½ b) c) 4 d) 30 e) 1800 99. The graph of the function has as its vertical asymptote(s) a) x = -1 only b) x = 1 only c) x = 1 and x = -1 d) No vertical asymptotes e) y = 1 only 15
100. Which of the following represents the graph of an odd function? a) b) c) d) e) 101. A cosmetics company has analyzed that their profit depends on the percent of their sales that they spend on advertising, according to the formula Profit where x is the percent of their sales spent on advertising. What percent of sales spent on advertising will produce the largest Profit? a) 45% b) 5% c) 15% d) 0% e) 33 1/3% 10. The graph of f (x) = -x - x + 3 resembles a) b) c) d) e) 103. The function whose equation is has a graph which is a parabola whose vertex is: a) (, 5) b) (-, 5) c) (, 9) d) (-, -9) e) (, 1)) 104. For a production level of x units of a commodity, the cost function in dollars is C = 00x + 4100. The demand equation is p = 300-0.05x. What price p will maximize the profit? a) $100 b) $50 c) $900 d) $1500 e) $6000 105. For a production level of x units of a commodity, the demand equation is p = 300-0.05x. What quantity x will maximize revenue? a) 3000 b) 6000 c) 0 d) 4000 e) 500 106. Which of the following define functions which are even: A:, B: C: a) A only b) B only c) A and B d) A and B and C e) A and C only 16
107. Which of the following define functions which are odd: A:, B: C: a) A only b) B only c) A and B d) A and B and C e) A and C only 108. For the demand function define by for and the supply function for, the market equilibrium price is: a) 76 b) 70 c) 7 d) 8 e) 4 109. The zeros of are: a) 1 and -1 b) -1 only c) 1 only d) 5 and 10 e) there are none 110. The horizontal asymptote of is: a) x = b) y = c) x = -1 d) y = -1 e) there is none 111. For the composite function defined by, we could decompose the function into. A possible decomposition for h(x) would be: a) and b) and c) and d) and e) and 11. The equation of the line parallel to passing through the point (-5, 4) is: a) b) c) d) e) 113. The equation of the line perpendicular to passing through the point (-5, 4) is: a) b) c) d) e) 114. Find a) 0 b) 1 c) 3 d) e) 3/ 17
115. Find. a) 4 b) 3.5 c) d) 1 e) 0 116. Describe the intervals on which is continuous. a) All real x b) All real x 3 c) All real x d) All real x 1 e) All x in the interval (0,4) 117. Determine the values of x for which is continuous. a) All real x b) All real x 4 c) All real x 1 d) All real x 1, 4 e) All x in the interval (0,5) 118. Find a) 0 b) Does not exist (undefined) c) 4 d) -10 e) - 119. When the price is $40.00, a company can sell 300 chairs but for every 1.00 increase in price, the demand drops by 3 chairs. Find the equation for the demand, x as a function of price, p. a) b) c) d) e) 10. The equation of the line tangent to the curve y = 3x - x + 7 at the point (1, 8) is a) y = 4x b) y = 6x c) y = 4x + 4 d) y = 0 e) y = -4x + 8 11. If then g (x) is a) b) c) d) e) x + x 1. Find the slope of the line tangent to at (, 4). a) 3 b) 1/8 c) 6 d) e) 3/8 3 13. Find the values of x for which the rate of change with respect to x of y = 10x - 10x + 450x -130 is 0. a) x = ± 300 b) x = 3, 5 c) x = 100 d) x = ± 3 e) x = 100, 500 18
14. Find a) 8 b) -8 c) 0 d) Does not exist (undefined) e) 15. The function defined by a) is continuous for real x b) is discontinuous at x = 4 only c) is discontinuous at x = 5 only d) is discontinuous at x = 4 and x = 5 e) is discontinuous for 4 < x < 5 16. The slope of the line tangent to the curve x + y = 4 at the point (-4, 3) is a) 4/3 b) -4/3 c) 3/4 d) -3/4 e) 1 17. If the demand equation is given by, then the marginal revenue is 0 when a) x = 9 b) x = c) x = 6 d) x = 3 e) x = 5 18. Find the derivative of a) b) c) d) e) 19. For what value(s) of x is the derivative of equal to zero or undefined? a) x = 0 only b) x = only c) x = - only d) x = -, only e) x = -, 0, 130. For and Find all x such that a) x = 0 b) x = 1 c) x = -1 d) x = 0 or x = 1 e) x = 0 or x = -1 or x = 1 4 131. Find the value of the derivative of y = 5x(x - 4) at x = 3. a) 1040 b) 560 c) 30 d) 160 e) 960 13. For the rational function the horizontal asymptote is: a) x = and x = - b) y = and y = - c) y = 4 d) x = 4 e) There is none 19
133. For the rational function defined by the vertical asymptotes are: a) x = and x = - b) y = and y = - c) y = 4 d) x = 4 e) There are none 134. Find a) 0 b) -1/3 c) d) e) 135. Find a) 0 b) -1/3 c) d) e) 136. If, find f(1, -6, 5). a) 4 b) c) 9 d) 5 e) 7 137. Find a) 7/16 b) 0 c) 5/8 d) e) Does not exist. 138. For what values of x is the function defined by not continuous? a) 1 b) - and 1 c) -5, - and 1 d) -1 and e) -1, 1 and 139. Evaluate for f (x) = x + 5x. a) 7x + x b) 3x + x c) 5 + x d) x + 5 + x e) 10x + x 140. Find a) 1/4 b) 4 c) 0 d) 1 e) 141. Find a) 0 b) c) 7 d) 3 e) Does not exist 0
14. ABC Inc. buys a new truck for $3,000 The truck will have a scrap value of $8,000 after 8 years. If they use the straight line method for depreciating the truck, find the value of the truck after years. a) 0,000 b),000 c) 4,000 d) 6,000 e) 8,000 143. Find the maximum profit for the profit function defined by P(x) = -x + 10x - 3. a) 10 b) 19/ c) (5 + 19)/ d) 7/4 e) 67/8 144. Find the center of a circle given by the equation a) (0, 0) b) (-1, 1) c) (1, -1) d) (, -) e) (-, ) 145. Find the radius of a circle given by the equation a) b) 4 c) d) 0 e) 3 146. The vertex of the parabola is: a) (4, 1) b) (6, 37) c) (-6, 181) d) (, -11) e) (-, 37) 147. For the functions defined by for and g(x) = for, which could be the supply function? a) f(x) b) g(x) c) either f(x) or g(x) d) neither f(x) nor g(x) e) insufficient information to answer the question 148. The demand function and cost function for x units of a product are defined by and C = 0.65x + 400. Find the marginal profit when x = 100. a) $.35 per unit b) $4.58 per unit c) $193.50 per unit d) $187.35 per unit e) $3.65 per unit 149. The function defined by moves (translates) the graph of a) 3 units up and 8 units to the right b) 3 units down and 8 units to the left c) 3 units to the right and 8 units down d) 3 units to the left and 8 units up e) none of the other answers 150. If the Revenue function is defined by R(x) = 5x and the Cost function is defined by C(x) = 0x + 500, the break even quantity will be: a) 100 b) 40 c) 50 d) 1000 e) 00 1
151. Determine the x coordinate of the point(s), if any, at which the graph of has a horizontal tangent. a) x = -1, 9 b) None c) x = 0, -1 d) x = 0, 1, 9 e) x = 11 15. Given and, find g(f()) a) 84 b) 11 c) 14 d) 17 e) 137 153. The population, P, of a town of 30,000 people grows at a constant rate of,000 people every year. Find a formula for P as a function of time t (the number of years from the year the population was 30,000). a) P = 30,000 +,000t b) P =,000 + 30,000t c) P = 30,000 -,000t d) P =,000t - 30,000 e) Not enough information is given to determine the function. 154. The x coordinate of the center of the circle given by the equation is a) 5 b) -5 c) 3 d) -3 e) 4 155. The radius of the circle given by the equation is a) -5 b) 3 c) 8 d) 16 e) 4 156. For a linear regression between the variables, x = number of practice problems done the day before the final exam and y = score on the final exam, you would expect the value of the correlation coefficient, r, to be close to: a) - b) -1 c) 0 d) 1 e) 157. For a linear regression between the variables, x = number of hours watching television the night before the final exam and y = score on the final exam, you would expect the value of the correlation coefficient, r, to be close to: a) - b) -1 c) 0 d) 1 e) 158. For the function defined by, find a) 0 b) 1 c) 1/ d) e) 4
159. The function defined by a) is continuous at x = b) has a removable discontinuity at x = c) has a non-removable discontinuity at x = d) is differentiable at x = e) has an asymptote at x = 160. The function defined by a) is continuous at x = b) has a removable discontinuity at x = c) has a non-removable discontinuity at x = d) is differentiable at x = e) has an asymptote at x = 161. Find the slope of the line tangent to at x = 3. a) b) c) 45 d) e) 16. Find the value of the derivative of at x = 5. a) 10 b) 4 c) 16 d) 40 e) 136 163. x = is a vertical asymptote of y = f(x), whose graph appears on the right. Find a) 0.4 b) c) 0 d) d) - 164. The EZ car rental company charges a fixed amount per day plus an amount per mile for renting a car. For two different one-day trips Ed rented a car from EZ. He paid $70.00 for a 100 mile trip and $10.00 for a 350 mile trip. Determine the function for a one-day trip that EZ car rental uses if x represents the number of miles driven. a) f(x) = 0.x + 50 b) f(x) = 0.0x + 50 c) f(x) = 0.0x + 70 d) f(x) = 0.0x + 70 e) f(x) = 0.x + 190 3
165. Select the correct mathematical formulation of the following problem. x A rectangular area must be enclosed as shown. The sides labeled x cost $0 per foot. The sides labeled y cost $10 per foot. If at most $300 can be spent, what should x and y be to produce the largest area? a) Maximize xy if 40x + 0y = 300 y y b) Maximize xy if 0x + 10y = 300 c) Maximize xy if x + y = 300 d) Maximize x + y if xy = 300 e) Maximize x + y if 00xy = 300 x 166. Given find the values of the limits shown. Choice (a) -1 4 undefined (b) (c) 3 0 undefined (d) 7 4 undefined (e) 7 and 4 7 and 4 7 and 4 167. A woman 5 feet tall is walking away from a 1 ft. tall flagpole at a rate of ft/sec. (See diagram) How quickly is the length of her shadow changing? a) decreasing at a rate of 6/5 ft/sec b) increasing at a rate of 6/5 ft/sec c) increasing at a rate of ft/sec d) decreasing at a rate of 10/7 ft/sec e) increasing at a rate of 10/7 ft/sec 168. Select the correct mathematical formulation of the following problem. A farmer wishes to construct 4 adjacent fields alongside a river as shown. Each field is x feet wide and y feet long. No fence is required along the river, so each field is fenced along 3 sides. The total area enclosed by all 4 fields combined is to be 800 square feet. What is the least amount of fence required? (a) Minimize xy if 4x + 5y = 800 (b) Minimize xy if 4x + 5y = 00 (c) Minimize 4x + 5y if xy = 800 (d) Minimize 4x + 5y if xy = 00 (e) Minimize 4(x + y) if xy = 800 4
SECTION M (MATRICES-NO CALCULATOR ALLOWED) M1. If a linear system of equations does not have a single unique solution, then we may conclude that a) it must have no solution. b) it has exactly two solutions. c) it must have an infinite number of solutions. d) it either has no solutions or it has an infinite number of solutions. e) it either has no solutions or it has exactly two solutions. M. The augmented matrix shown represents the solution of a system of equations in the variables (x, y, z, w). What is the solution? a) b) c) d) e) No solution M3. When writing the system of equations shown in the form AX = B, 3x - y = 8 what should A be? -5x + 7y + z = 4 3y + z = 7 a) b) c) d) e) M4. Given A = and B =, find A B. a) b) c) d) e) M5. If the dimensions of matrix A are 4 x 3, and the dimensions of matrix B are 3 x 1, what must the dimensions of AB be? a) 3 x 3 b) 4 x 1 c) 4 x 3 d) 3 x 1 e) 1 x 1 5
-1 M6. If is represented by AX = B and A =, find the solution X a) b) c) d) e) M7. For the system of equations shown, what is the reduced row x - y + z + 7w = 3 echelon form that results from Gauss-Jordan row reduction? x - y + 3z + 9w = 3 3x - 6y + 7z + 3w =8 a) b) c) d) e) M8. If A =, find A -1 a) b) c) d) e) Does not exist M9. If A =, find A. a) b) c) d) e) -1-1 -1 M10. If A and B are matrices with inverses A and B, respectively, then (AB) equals -1-1 -1-1 -1 a) A B b) (BA) c) B A d) BA e) AB 6
M11. The system of linear equations on the right x + y - 3z = -1 has as its solution y - z = 0 -x + y = 1 a) only (1, 1, 1) b) only (-1, 0, 0) c) only (3,, ) d) no solution e) infinitely many solutions M1. If A =, B = and C =, find AB - 5C. a) b) c) does not exist because AB does not exist. d) does not exist because AB and 5C have different dimensions. e) does not exist because all three matrices must have the same dimensions for AB - 5C to exist. -1 M13. If A =, find the inverse of A, A. a) b) c) d) e) Does not exist. M14. Perform the following matrix operation:. a) b) c) d) e) 7
M15. Perform the following matrix operation: a) b) c) d) e) Undefined M16. If A =, find A. a) b) c) d) e) Undefined -1 M17. If A =, find the inverse, A. a) b) c) d) e) Inverse does not exist M18. Perform the matrix operations: 3 a) b) c) d) e) M19. Solve the following system of equations: x - y = 8 x + y = 7 x - 3y = 15 a) (10, ) b) (8, -3, -3) c) There are infinitely many solutions d) No solution exists e) (5, -3) M0. Find the reduced row echelon form that results 4x + y = 6 from using Gauss-Jordan row reduction. 6x + 3y = 9 a) b) c) d) e) 8
M1. Find the following product: a) b) c) d) e) M. The reduced row echelon form that results from x - 3y + z = -3 Gauss-Jordan row reduction of the system of x - y - z = -9 linear equations shown appears below. 3x - 4y + 3z = 3 Find all solutions to the system of equations. a) (1, 15, 0) b) (5z + 1, 3z + 15, z) c) (1, 3, 4) d) (1-5z, 15-3z, z) e) (-4, 0, 5) M3. Solve the system of equations. x + z = 3 x + y - z = 10 y - 3z = 8 a) (3, 7, 1) b) (3 - z, 7 + 3z, 0) c) (3 - z, 7 + 3z, z) d) (3 - z, 7 + 3z, 1) e) No solution exists M4. Find all solutions to the system of equations. x + w = 5 y - z = -1 x + y + w = 8 a) (5 - w, - + w, -1 + w, w) b) (5, -, -1, 0) c) (1, -1, -1, 0) d) (4, -1, 0, 1) e) No solution exists M5. What is the (, 3)-entry in the product? a) 0 b) - c) 17 d) -5 e)-3 9
SECTION C C1. Find a) b) - c) -1 d) e) 0 C. Given, write f(x) as a product of linear factors a) b) c) d) e) C3. The profit from manufacturing x items is given by P(x) = -0.5x + 46x - 10. Find the actual change in profit (not an estimate) if the production is increased from 0 to 1 items. a) $5.00 b) $5.50 c) $6.00 d) $710.00 e) $735.50 C4. Find the domain of the function a) b) c) d) e) C5. Let. To the nearest hundredth the x-intercepts of the graph are a) (0.63, -.0) only b) (0.60, 0) and (0.67, 1) c) (0.60, 0) only d) (0.63, 0) only e) (0.60, 0) and (0.67, 0) C6. Find a) 0 b) Does not exist (undefined) c) x + 3 d) e) C7. Find a) 0 b) Does not exist (undefined) c) d) - e) 1 30
C8. A business purchases a piece of equipment for $15,50. The company can depreciate the equipment s value over its useful life of 10 years, at which time it will be sold as salvage, with an estimated value of $350. Find the formula for a linear function that gives the value of equipment during its 10 years. a) V = 15,50-850t b) V = 17,750-11,050t c) V = -15,50t + 1490 d) V = -1490t + 15,50 e) V = 1490t - 15,50 C9. Which of the following tables represent both the y-values on the circle on EITHER SIDE of the point (-13, 34) and the corresponding y-values on the line tangent to the circle at (-13, 34), using? a) b) c) d) e) C10. The function has vertical asymptotes at: a) nowhere b) x = 1 only c) x = -3 only d) x = 1 and x = e) x = and x = -3 3 C11. For what values of x is the slope of the line tangent to y = x + 11x - x + 17 equal to 0? a) None b) c) 11/17 d) 0 e) C1. Let be a company s total cost, in millions of dollars, for producing x thousand cheeseburgers. Let be the revenue, in millions of dollars the company receives for selling x thousand cheeseburgers (x > 0). What is the lowest level of production, x, that allows the company to break even? a) 6000 cheeseburgers b) 3000 cheeseburgers c) 4000 cheeseburgers d) 1000 cheeseburgers e) 000 cheeseburgers 31
C13. Find a) 0 b) -1 c) -6 d) - e) undefined (does not exist) C14. The cost in dollars of producing x units of a product is given by for x 0. Determine the value of x when the marginal cost is 0. a) 0.8 b) 1.4 c) 3.5 d) 4. e) 5.3 C15. Let. There exists a value c such that f(c) = 10. The digit in the tenths place of c is: a) 1 b) c) 4 d) 7 e) 9 C16 and C17. The following are a pair of supply and demand equations: C16. Which of the following graphs BEST represents supply and demand as a function of the number of units produced? a) b) c) d) e) C17. Find the point of market equilibrium for the demand and supply equations above. a) (7.881, 1.565) b) (-16.315, 336.18) c) (-1.964, 73.857) d) (1.697, 7.881) e) (1.565, 7.881) 3
C18. Using the graphing calculator, graph and the tangent line at (0, 0) in the window xmin = -10 xmax = 10 xscl = 1 ymin = - ymax = yscl = 1 a) b) c) d) e) C19. Determine the intervals on which the function is differentiable. a) b) c) d) e) C0. Find a) Does not exist (undefined) b) 0 c) 0.06 d) 1/7 e) 1/18 C1. Find a) 5 b) -1/13 c) 4/3 d) 0 e) C. The demand function for a product is where x is the number of units of the product sold and p is the price in dollars. Find the value of x for which the marginal revenue is one dollar per unit. a) 0 b) 53.93 c) 130.5 d) 131.3 e) No solution 33
C3. You want to choose one long-distance telephone company from the following options: Company A charges $0.37 per minute Company B charges $13.95 per month plus $0. per minute Company C charges a fixed rate of $50 per month Find the x-values (number of minutes you talk in one month to the nearest minute) for which Company B is the cheapest a) b) c) d) e) C4. Evaluate a) b) c) d) e) does not exist (undefined) C5. An experimental power plant has a net power output where t is the number of days after start-up. For what approximate values of t will the net power output be negative? a) t < -108.8 b) t < 0.9 c) t < 108.8 d) 0.9 < t < 87.9 e) t > 400 C6 and C7 Use the data given: x - 0 3 10 y -5-1 6 15 C6. Find the linear regression equation (round coefficients to 3 decimals): a) b) c) d) e) C7. Find the quadratic regression equation (round coefficients to 3 decimals): a) b) c) d) e) 34
C8. Find the zero of that is between - and -1 (round to 4 decimals). a) -1.4198 b) -1.400 c) -1.40 d) -1.404 e) -1.406 C9 and C30. The weekly profit that results from selling x units of a commodity weekly is given by P = 50x - 0.003x - 5000 dollars. C9. Find the actual weekly change in profit that results from increasing production from 5000 units weekly to 5015 units weekly. a) $0.00 b) $300.00 c) $99.33 d) $98.00 e) $19.96 C30. Use the marginal profit function to estimate the change in profit that results from increasing production from 5000 units weekly to 5015 units weekly. a) $0.00 b) $300.00 c) $99.33 d) $98.00 e) $19.96 C31. For the equation, use and Newton s Method to find to decimals. a) 1.18 b) 1.19 c) 1.0 d) 1.1 e) 1. C3. Find the linear regression equation for the data given (round coefficients to 3 decimals): x - 1 3 7 y 8 5 1-4 a) b) c) d) e) C33. The total profit that a company has obtained since it started business is given by where t is the number of months since the company started business. Find the average rate of change in profit with respect to time between 3 months after the company started and 7 months after the company started. a) 5997.6 b) -4,506.6 c) 998.57 d) 1499.8 e) 1499.31 C34. Find the equation of the straight line tangent to at the point (3, -9). a) None exists (false) b) c) y = -x + 57 d) y = -11x + 4 e) y = -11 35
C35. Evaluate to three decimals a).081 b).091 c).101 d).111 e).11 Matrix Exercises begin on the next page 36
SECTION CM (MATRICES-CALCULATOR ALLOWED) CM1. Solve the following system of equations: a) (0, 0, 1) b) (x + 7z, y + 5z, z) c) no solution d) (x + 7z, y + 5z, 1) e) (x - 7z, y - 5z, 1) CM. Given, and, find a) b) c) d) e) CM3. Solve the following system of equations: a) (-19/7, -9/7, 18/7) b) (1, 1, 1) c) (-19k + 7, -9k + 7, 18k + 7) d) (-19, -9, 18) e) no solution 37
CM4. and CM5. The following represents the system of equations AX = B CM4. In order to solve the system, it is necessary to evaluate: a) b) c) d) e) CM5. The solution to the system above is: a) (1, 1, 1) b) (3, 9, -15) c) (6, 5, -3) d) (18, 15, -9) e) no solution CM6. Solve the following system of equations: a) ( + 45/8, k, 4/7, -1/8) b) (k + 45/8, 4/7, k, -1/8) c) (45/8, -, 4/7, -1/8) d) (k + 45/8, k, 4/7, -1/8) e) no solution 38
ANSWERS 1. a. d 3. c 4. c 5. e 6. c 7. a 8. b 9. e 10. c 11. e 1. e 13. b 14. c 15. c 16. c 17. b 18. a 19. c 0. b 1. c. d 3. e 4. e 5. d 6. e 7. c 8. e 9. b 30. e 31. b 3. a 33. a 34. d 35. d 36. a 37. c 38. d 39. d 40. e 41. e 4. b 43. d 44. c 45. e 46. a 47. c 48. d 49. a 50. c 51. a 5. e 53. c 54. b 55. a 56. e 57. b 58.d 59. a 60. e 61. d 6. e 63. c 64. b 65. b 66. a 67. b 68. b 69. e 70. a 71. d 7. e 73. e 74. b 75. d 76. e 77. e 78. c 79. d 80. a 81. e 8. b 83. e 84. a 85. b 86. d 87. b 88. d 89. b 90. c 91. a 9. b 93. e 94. d 95. d 96. d 97. a 98. b 99. d 100. b 101. c 10. e 103. e 104. b 105. a 106. d 107. b 108. a 109. e 110. b 111. c 11. a 113. d 114. e 115. c 116. d 117. d 118. e 119. b 10. c 11. b 1. e 13. b 14. c 15. a 16. a 17. c 18. b 19. e 130. d 131. a 13. c 133. a 134. d 135. a 136. a 137. c 138. d 139. d 140. a 141. d 14. d 143. b 144. b 145. c 146. d 147. a 148. a 149. c 150. a 151. d 15. d 153. a 154. b 155. e 156. d 157. b 158. d 159. c 160. b 161. c 16. e 163. d 164. a 165. a 166. d 167. e 168. d M1. d M. d M3. a M4. d M5. b M6. d M7. e M8. d M9. b M10. c M11. e M1. a M13. d M14. a M15. a M16. b M17. c M18. d M19. d M0. c M1. c M. d M3. e M4. a M5. d C1. b C. c C3. b C4. e C5. e C6. e C7. c C8. d C9. e C10. d C11. b C1. b C13. b C14. b C15. c C16. b C17. e C18. b C19. d C0. e C1. b C. c C3. c C4. b C5. d C6. c C7. b C8. e C9. c C30. b C31. c C3. a C33. e C34. d C35. a CM1. c CM. b CM3. a CM4. d CM5. c CM6. d 39