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PRECALCULUS HONORS FINAL EXAM REVIEW 4π 3 a. Sketch the angle in standard position. 1. Consider the angle − b. Determine the quadrant in which the angle lies. c. Determine one positive and one negative coterminal angles. 2. 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. 3. −127.5° Convert the angle measure from radians to degrees. If necessary, round your answer to three decimal places. 4. 7π 6 5. − 2π 3 Evaluate the sine, cosine, and tangent of the angle without using a calculator. 6. π 7. − 4 8. −210o 3π 4 9. 330o Find the point (x, y) on the unit circle that corresponds to the real number t. 10. t = 5π 4 11. t = − 2π 3 Evaluate (if possible) the six trigonometric functions of the real number. 12. t = 5π 6 13. t = − 5π 3 Evaluate the trigonometric function using its period as an aid. 14. sin 19π 15. cos − 6 7π 4 Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. 16. tan130o 17. csc 3π 5 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 18. cscθ = b. cosθ c. tanθ d. sec(90°−θ) Find the exact values of the six trigonometric functions of the angle θ shown in the figure. 19. 20. 6 θ 2 8 15 θ 21. Find the length of the arc on a circle with a radius of 10 inches intercepted by a central angle of 145°. 22. On the top of the Sears Building in Chicago, is a TV tower. From a point 300 ft. away the bottom of the tower is at an angle of elevation equal to 42.14 degrees and the top of the tower is at an angle of elevation equal to 45.81 degrees. How tall is the tower? 23. 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? 24. 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? 25. Find cosθ if tan θ = − 4 and sin θ < 0 . 3 In 26-28 sketch the graph of the function. (Include two full periods.) 26. f (x) = 3 − 2sin π x 27. g(x) = 1 π tan x − 2 2 28. h(x) = −sec(x + π ) In 29-30, find the exact value of the expression without using a calculator. 29. tan(arctan 6.7) 3 30. tan arcsin 5 31. Write an algebraic expression equivalent to sin(arccos2x) . π 32. Use the fundamental identities to simplify: cos − x csc x . 2 33. Subtract and simplify: sin θ − 1 cos θ . − cos θ sin θ − 1 In 34-36, verify the identity. 34. cot 2 α (sec 2 α − 1) = 1 35. sin(x + y)sin(x − y) = sin 2 x − sin 2 y 36. sin 2 x cos 2 x = 1 (1 − cos 4 x ) 8 37. Write 4 cos 2θ sin 4θ as a sum or difference. In 38-39, find all solutions of the equation in the interval [0, 2π ) . 38. 2 cos2 β − cos β = 0 39. 3tan θ − cot θ = 0 40. Use the Quadratic Formula to solve the equation in the interval [0, 2π ) : sin 2 x + 2sin x + 1 = 0 . 41. Given that sin u = 12 3 , cos v = , and angles u and v are both in Quadrant I, find 13 5 tan(u − v) . 42. If tan θ = 1 , find the exact value of tan(2θ ) . 2 In 43-44, use the information to solve the triangle shown in the figure. Round your answers to two decimal places. C b A a c B 43. A=30°, a=9, b=8 44. a=4, b=8, c=9 45. Two sides of a triangle have lengths 7 inches and 12 inches. Their included angle measures 60°. Find the area of the triangle. 46. Write the first five terms of the sequence an = 2 + 6 . (Assume that n begins with 1) n 47. Write an expression for the nth term of the sequence: − 1, 2, 7, 14, 23, ... 48. Find the next three terms of the series. Then find the fifth partial sum of the series. 1 + 4 + 7 + 10 + 13 + ... 49. The sixth term of an arithmetic sequence is 65, and the second term is 93. Find the a formula for the nth term. 50. Write the first five terms of the sequence an = 6(3)n −1 . (Assume that n begins with 1) Find the sum. 51. 5 ∑ (2i 3 + 4) i =1 52. 7 ∑ ( 4 n + 2) n =1 i −1 1 53. ∑ 2 i =1 3 ∞ Determine whether the sequence is arithmetic. If so, find the common difference. 54. 0, 2, 5, 9, 14 , ... 55. 5, 3, 1, − 1, − 3, ... 56. 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. 57. 54, − 18, 6, − 2, ... 58. 3, 5, 7, 9, 11, ... Write an expression for the nth term of the geometric sequence. Then find the 8th term of the sequence. 59. a1 = 100, r = 1.05 60. a1 = 2, a3 = 12 Classify the graph of the equation as a circle, parabola, ellipse, or a hyperbola. 61. 5x 2 − 2y 2 +10x − 4y +17 = 0 62. 4x 2 + 4y 2 − 4x + 8y −11 = 0 Find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. 63. Vertex (0, 0); Focus (4, 0) 64. Vertex (2, 2); Directrix y=0 65. A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters. How wide is the archway at ground level? Find the standard form of the equation of the ellipse with the given characteristics. Then graph the ellipse. 66. Vertices (−3, 0), (7, 0) ; foci (0, 0), (4, 0) 67. Vertices (−4, −1), (−4, 11) ; endpoints of the minor axis (−6, 5), (−2, 5) Find the center, vertices, foci, and eccentricity of the ellipse. 68. (x + 2)2 (y −1)2 + =1 81 100 69. 16x 2 + 9y 2 − 32x + 72y +16 = 0 Find the standard form of the equation of the hyperbola with the given characteristics. 70. Vertices (0, ±1) ; foci (0, ± 3) 71. Foci (0, 0), (8, 0); asymptotes y = ±2(x − 4) Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. (x − 3)2 (y + 5)2 − =1 72. 16 4 73. 9x 2 −16y 2 −18x − 32y −151 = 0 74. Use the graph to find the limit (if it exists). If the limit does not exist, explain why. 76. 77. Use the given information to evaluate each limit. lim f (x) = 4, x→c lim g(x) = 5 x→c 3 78. lim [ f (x)] x→c 79. lim [ f (x)g(x)] x→c Find the limit by direct substitution. 1 lim x + 3 x→4 2 80. 82. lim sin 3x x→π x 2 −1 81. lim 3 x→2 x + 2 83. lim(5x − 3)(3x + 5) x→2 Find the limit (if it exists). Use a graphing utility to verify your results graphically. t+2 2 84. x→−2 t − 4 x +1 85. x→−1 x − 5x − 6 x 3 − 64 lim 2 86. x→4 x −16 1 −1 lim x + 2 87. x→−1 x +1 lim 88. lim x→0 x+9 −3 x lim 89. lim x→5 2 3− x+2 1− x Graph the function. Determine the limit (if it exists) by evaluating the corresponding onesided limits. 90. lim x→3 x−3 x−3 5 − x, x ≤ 2 lim f (x) where f (x) = 2 x→2 x − 3, x > 2 92. Find lim h→0 93. f (x + h) − f (x) . h f (x) = 4x + 3 1 91. x→−3 x + 9 lim 2 Approximate the slope of the tangent line to the graph at the point (x, y). 94. Find a formula for the slope of the graph of f at the point (x, f(x)). )). Then use it to find the slope at the two given points. f (x) = x 2 − 4x 95. a) (0, 0) b) (5, 5) f (x) = x 96. a) (1, 1) b) (4, 2) Find the derivative of the function. 97. f (x) = 5 98. f (x) = 3x −1 99. f (x) = 2x 2 + 2 100. g(t) = 6 5−t Find the slope of the graph of f at the given point. Use the result to find an equation of the tangent line to the graph at the point. 101. f (x) = 2x 2 −1, (0, −1) Find the limit (if it exists). If the limit does not exist, explain why. 6x + 5 x→∞ 2x 102. lim 103. lim 104. lim 105. x lim + 5 2 x→∞ (x −1) 3x 2 −1 x→∞ x + 2 x→∞ 7x (x + 3)2 Use the summation formulas to rewrite the sum as a rational function S(n). Then find the lim S(n) n→∞ n 106. i 1 ∑ 2 − n n i=1 Approximate the area of the region using the indicated number of rectangles of equal width. 107. f ( x) = 2 x 2 [0,2] n=4 Use the limit process to find the area of the region between the graph of the function and the xaxis over the specified interval. 108. f ( x) = x + 2 [ − 2, 2 ]