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PRECALCULUS FINAL EXAM REVIEW Evaluate the function at the indicated value of x. Round your result to three decimal places. 1. f (x) = −14(5 x ); 2. f (x) = e x ; x = −0.8 x = 0.278 Use the graph of f to describe the transformation that yields the graph of g. 3. f (x) = 4 x , g(x) = 4 x − 3 4. 1 1 f (x) = , g(x) = − 2 2 x x+2 Use the One-to-One Property to solve the equation for x. 5. 3x + 2 = 27 6. e5 x − 7 = e15 Write the exponential equation in logarithmic form. 0.8 8. e = 2.2255... 7. 4 3 = 64 Evaluate the function at the indicated value of x without using a calculator. 9. f (x) = log x; 10. log 4 x; x = x = 1000 1 4 Use the One-to-One Property to solve the equation for x. 11. log 4 (x + 7) = log 4 14 12. The antler spread a (in inches) and shoulder height h (in inches) of an adult male American elk are related by the model h = 116 log(a + 40) − 176 . Approximate the shoulder height of a male American elk with an antler spread of 55 inches. Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 13. log 3 6 x 14. ln x 2 y 2 z 3 Condense the expression to the logarithm of a single quantity. 1 log 8 (x + 4) + 7 log 8 y 15. 3 16. 5ln(x − 2) − ln(x + 2) − 3ln x Solve for x. When necessary, approximate your result to three decimal places. 17. 8 x = 512 19. e4 x = ex 2 +3 18. log 4 x = 2 20. 2 x + 13 = 35 4π 3 a. Sketch the angle in standard position. 21. Consider the angle − b. Determine the quadrant in which the angle lies. c. Determine one positive and one negative coterminal angles. 22. Consider the angle 280°. a. Sketch the angle in standard position. b. Determine the quadrant in which the angle lies. c. Determine one positive and one negative coterminal angles. Convert the angle measure from degrees to radians. If necessary, round your answer to three decimal places. 23. 40˚ Convert the angle measure from radians to degrees. If necessary, round your answer to three decimal places. 24. 7π 6 25. − 2π 3 Evaluate the sine, cosine, and tangent of the angle without using a calculator. 26. π 4 28. −210o 27. − 3π 4 29. 330o 30. Find the length of the arc on a circle with a radius of 10 inches intercepted by a central angle of 145°. Find the point (x, y) on the unit circle that corresponds to the real number t. 31. t = 5π 4 32. t = − 2π 3 Evaluate (if possible) the six trigonometric functions of the real number. 33. t = 5π 6 34. t = − 5π 3 Evaluate the trigonometric function using its period as an aid. 35. sin 7π 4 19π 36. cos − 6 Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 37. tan130o 38. csc 3π 5 Find the exact values of the six trigonometric functions of the angle θ shown in the figure. 39. 40. 6 2 8 θ θ 15 Use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions. 13 13 , secθ = 2 3 a. sinθ b. cosθ c. tanθ d. sec(90°−θ) 41. cscθ = 42. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33º. How tall is the tree? 43. An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the airport? State the quadrant in which θ lies. 44. tan θ < 0 and cosθ < 0 45. cosθ > 0 and csc θ < 0 Find the values of the six trigonometric functions of θ. 3 46. cosθ = − , 5 1 47. sin θ = , 5 θ lies in Quadrant II cosθ > 0 48. You accidentally threw your football on top of a building. From a distance 10 feet away, the angle of elevation to the roof is 56°. How tall is the building? Match the function with its graph. State the amplitude (if it applies) and the period of the function. A y= 1 csc x 2 D y = tan(x + π 2 ) B y = −2sin x − 2 C y = cot 2x E y = 2sin(x − π ) F y = csc(3x) 49. 50. 51. 52. 53. 54. 55. Sound waves can be modeled by sine functions of the form y = a sin bx , where x is measured in seconds. a. Write an equation of a sound wave whose amplitude is 2 and whose period is 1 second. 264 b. What is the frequency of the sound wave described in part (a)? True or False? Determine whether the statement is true or false. Justify your answer. 56. The graph of y = csc x can be obtained on a calculator by graphing the reciprocal of y = sin x . 57. The function given by y = given by y = cos x . 1 cos 2 x has an amplitude that is twice that of the function 2 58. The graph of the function given by f ( x) = sin( x + 2π ) translates the graph of f ( x ) = sin x exactly one period to the right so that the two graphs look identical. Evaluate the expression without a calculator. 59. arcsin 1 2 ( ) 60. arccos 0 2 2 61. tan −1 − 3 62. arcsin 63. tan −1 0 1 64. cos −1 − 2 Evaluate the expression. Round your answer to two decimal places. 65. arcsin 0.4 66. cos −1 0.213 Find the exact value of the expression. 4 67. cos arctan 5 5 68. csc arctan − 12 69. Your football landed at the edge of the roof of your school building. When you are 25 feet from the base of the building, the angle of elevation to your football field is 21°. How high off the ground is your football? 70. A plane is 80 miles south and 95 miles east of Cleveland Hopkins International Airport. What bearing should be taken to fly directly to the airport? Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. o o 71. B = 72 , C = 82 , b = 54 o o 72. A = 95 , B = 45 , c = 104.8 Find the area of the triangle having the indicated angle and sides. o 73. A = 27 , b = 5, c = 7 o 74. C = 123 , a = 16, b = 5 75. From a certain distance, the angle of elevation to the top of a building is 17º. At a point 50 meters closer to the building, the angle of elevation is 31º. Approximate the height of the building. Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. 76. a = 5, b = 8, c = 10 77. a = 2.5, b = 5.0, c = 4.5 78. The lengths of the diagonals of a parallelogram are 10 feet and 16 feet. Find the lengths of the sides of the parallelogram if the diagonals intersect at an angle of 28º. Use Heron’s Area Formula to find the area of the triangle. 79. a = 4, b = 5, c = 7 80. a = 12.3, b = 15.8, c = 3.7 Determine whether each ordered pair is a solution of the system of equations. 4 x − y = 1 81. 6 x + y = −6 a) (0,− 3) b) (−1,− 4) 3 c) − , − 2 2 1 d ) − , − 3 2 Solve the system by graphing. 2 x + y = 6 82. − x + y = 0 Solve each system by substitution. x = y 83. 5 x − 3 y = 10 x − 4 y + 3z = 3 85. − y + z = −1 z = −5 x2 − y = 0 2x + y = 0 84. Solve the system by elimination. x + 2 y = 4 86. x − 2 y = 1 9 x + 3 y = 1 87. 3x − 6 y = 5 3x + 2 y + z = 17 88. − x + y + z = 4 x − y + z = 2 Set up and solve the system using a method of your choice. 89. The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week. At the same time, the weekly rentals for a newly released DVD of a horror film increased each week. Models that approximate the weekly rentals R for each DVD are R = 360 − 24 x R = 24 + 18x where x represents the number of weeks each DVD was in the store, with x=1 corresponding to the first week. a) After how many weeks will the rentals for the two movies be equal? 90. The perimeter of a rectangle is 280 centimeters and the width is 20 centimeters less than the length. Find the dimensions of the rectangle. 91. At a local high school city championship basketball game, 1435 tickets were sold. A student admission ticket cost $1.50 and an adult admission ticket cost $5.00. The sum of all the total ticket receipts for the basketball game were $3552.50. How many of each type of ticket were sold? 92. Write the first five terms of the sequence an = 2 + 6 . (Assume that n begins with 1) n 93. Write an expression for the nth term of the sequence: − 1, 2, 7, 14, 23, ... 94. Find the next three terms of the series. Then find the fifth partial sum of the series. 1 + 4 + 7 + 10 + 13 + ... 95. The sixth term of an arithmetic sequence is 65, and the second term is 93. Find the a formula for the nth term. 96. Write the first five terms of the sequence an = 6(3) n −1 . (Assume that n begins with 1) Find the sum. 5 97. ∑ (2i 3 + 4) i =1 7 98. ∑ (4n + 2) n =1 ∞ 1 99. ∑ 2 i =1 3 i −1 Determine whether the sequence is arithmetic. If so, find the common difference. 100. 0, 2, 5, 9, 14, ... 101. 5, 3,1, − 1, − 3, ... 102. In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Because each round gets shorter, the farmer estimates that the same pattern will continue. Estimate the total number of bales made if the farmer takes another six trips around the field. Determine whether the sequence is geometric. If so, find the common ratio. 103. 54, − 18, 6, − 2, ... 104. 3, 5, 7, 9,11, ... Write an expression for the nth term of the geometric sequence. Then find the 20th term of the sequence. 105. a1 = 100, r = 1.05 106. a1 = 2, a3 = 12