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Review Sheets for Final Exam Chapter 6 1. Divide using synthetic division: 2x 4 6x 3 24 x 28 x 4 3. One of the zeros of the function f (x) x 3 5x 2 9x 45 is x = –5, find the other zeros of the function. 5. Find all rational zeros of the polynomial: y = 3x 3 3x 2 75x 75 7. Find all the zeros of the function: f (x) 6x 4 5x 3 12x 2 5x 6 9. Write a polynomial function that has the zeros 2, –3, and 1 and has a leading coefficient of 1. Then graph the function to show that 2, –3, and 1 are solutions. 10. Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. (–3, 0), (0, 0), (5, 0) 12. Write the polynomial as a product of linear factors. x 3 3x 2 4 x 12 13. Solve for x: x 4 12x 2 11 = 0 14. Find the four real zeros (including multiplicity) of the polynomial f (x) x 4 5x 3 6x 2 32x 32 . (Hint: The real zeros are integers that lie between –4 and 4, inclusive.) Chapter 7 17. Evaluate. 16 5/4 18. Evaluate. (27) 2/3 19. Evaluate. 16 5/4 21. Rewrite 181/6 using radical notation. Simplify: 251/6 22. 2/3 25 23. 3 40 4 3 5 24. 54 5 54 5 10 49 5/6 25. 491/3 Simplify: 26. x1 3 x1 4 5 27. 9 5 81 5 3 28. (73/4 )2/9 30. Write the expression in simplest form. 16xy 2 27z 5 31. Let f (x) = 1 x 2 and g(x) = 1 x. Find f (x) g(x). 32. Let f (x) 1 x 2 and g(x) 1 x . Find f (x) g(x) . 33. Let f (x) 16 x 2 and g(x) 4 x . Find f (g(x)) . of the relation f= {(–1, –5), (–5, –1), (9, –9)}. 35. Find the inverse for the inverse ofthe relation y 5x 3. 36. Find an equation 39. Sketch the graph of the function. Is the inverse of f(x) a function? f (x) 2 x 2 40. Describe how to obtain the graph of y Graph: 41. f(x) = 43. f(x) = x 3 3 from the graph of y x. x 5 3 x3 5 Refer to the function g(x) 2 x 1 . 45. Whatis the domain of g(x)? 46. What is the range of g(x)? Solve the equation. Check for extraneous solutions. 47. x + 2 = x 49. 3 x 5 4 51. 3x 3 4 192 52. 3 y2 5 Chapter 8 59. Graph: f (x) 3x 60. The amount of money, A, accrued at the end of n years when a certain amount, P, is invested at a compound annual rate, r, is given by A P(1 r)n . If a person invests $270 in an account pays 10% interest compounded annually, find the balance after 15 years. that 62. Find the value of $1000 deposited for 10 years in an account paying 7% annual interest compounded yearly. 64. How much money must be deposited now in an account paying 8% annual interest, compounded quarterly, to have a balance of $1000 after 10 years? 68. A piece of equipment costs $85,000 new but depreciates 15% per year in each succeeding year. Find its value after 10 years. 70. Graph f (x) = 3 e x . Simplify the expression. 72. e x 6e 3x 1 5e 2x (e 2x ) 2 73. 4 x e 10e x 74. Is f (x) 13.7e 0.04 t an example of exponential growth or decay? 1 75. Sketch the graph of the function. f (x) e x 2 3 76. Use the formula A Pe rt . If $3900 is deposited in an account at the bank and earns 9% annual interest, compounded continuously, what is the amount in the account, rounded to the nearest dollar, after 7 years? 3 78. Write the equation log 243 27 in exponential form. 5 Find the inverse of the function. 79. y log 8 x Find the inverse of the function. 80. y ln 3x 81. Evaluate without using a calculator. log 7 1 49 83. Graph the function. State the domain and range. y log 2 (x 1) 84. Use the change-of-base formula to evaluate the expression. log 4 7 85. Express as a single logarithm: log a11 log a 35 87. Expand using the properties of logarithms: log 4 3) x(x x5 88. Expand the expression. log 3 (x 2 y 3 ) 1 89. Condense the expression. log 516 3log 5 x 4 log 5 y 2 90. Solve for x correct to four decimal places: e 4 x = 5.8 1 1 91. Solve for x. log 7 3 log 7 x log 7 2 2 1 92. Solve for x. log 10 8 log 10 x log 10 2 3 95. Solve the equation. 35log 6 x 18 96. Solve the equation. 75log 7 x 61 97. Write an exponential function of the form y ab x whose graph passes through the given points. (5, 3), (7, 9) Chapter 9 98. The price per person of renting a bus varies inversely with the number of people renting the bus. It costs $22 per person if 47 people rent the bus. How much will it cost per person if 74 people rent the bus? 99. The variables x and y vary inversely. x = 7 when y = –4. Find an equation that relates the variables. 100. The variable z varies jointly with the product of x and y. z = 2.4 when x = 3 and y = 2. Find an equation that relates the variables. 103. Sketch the graph of the function. f (x) x2 x 2 3 104. Locate the asymptotes and graph the rational function f (x) = 2 . x 4 x2 106. Identify all horizontal and vertical asymptotes of the graph of the function. f (x) 2 x 4 5x 107. Identify all horizontal and vertical asymptotes of the graph of the function. f (x) 2 x 1 Simplify the rational expression. n 2 8n 15 108. n 2 25 Divide: x6 x 2 9x 18 110. 2 x6 x 9 x 2 2x 3 112. Simplify the expression. x2 1 2x 3 x2 4 x 3 114. Multiply and simplify. (x 3) 2 4 x2 9 x 2 8x 20 x 2 9x 115. Divide and simplify. 5x 3 50x 2 x 2 7x 18 116. Simplify. (2x 2 y 3 ) 2 (4 x) 2 y 3 3 2 3 (x (xy) 4 y ) 3x 4 2 118. Perform the operations and simplify. 2 x 16 x4 4x 2 2 119. Perform the operations and simplify. 2 x 9 x3 x3 Solve: x x = 7 120. 3 10 121. 4 3 = 0 f 3 f 4 122. r r 1 r 1 r 9 Solve: 124. Is x = –2 a solution of 4x 8 1 ? x 2 x 2 Chapter 13 127. Find x to the nearest hundredth. 10 32° x 129. Evaluate without using a calculator. sec 30 130. Find the missing side lengths for x and y. 30° 8 x y 3 45° x y 131. Find the missing side lengths for x and y. 132. Solve ABC using the diagram and the given measurements. (The triangle is not drawn to scale.) B = 34 , a = 14 A c b C a B Find one positive angle and one negative angle that are coterminal with the given angle. 134. 59 2 135. 3 137. Convert 48 to radian measure. 138. Find the arc length of a sector with a radius of 6 feet and a central angle of 12 . 139. Find the reference angle for 228 . 140. The point (–3, 3) is on the terminal side of an angle . Find cos . 141. Triangle ABC is a right triangle. Find sin A and the measure of A to the nearest degree. A 21 C 20 29 B 142. Use a calculator to evaluate the expression. Round your answer to three significant digits. tan 1 8 Solve the equation for . Round your answer to three significant digits. 143. sin 0.62; 180 270 144. tan 3.2; 180 270 145. You are flying a kite and want to know its angle of elevation. The string on the kite is 41 meters long and the kite is level with the top of a building that you know is 24 meters high. Use an inverse trigonometric function to find the angle of elevation of the kite. 146. Given triangle ABC with a = 7, b = 10, and A = 23 , find c. Round your answer to two decimal places. 148. Solve triangle ABC given that A = 48 , B= 54 , and b = 67. 150. Find the area of ABC . The figure is not drawn to scale. B 3 cm 60° A 10 cm C 152. Given triangle ABC with b = 2, c = 4, and A = 82°, find a. Round the answer to two decimal places. 154. Solve triangle ABC given that a = 15, b = 11, and c = 21. 155. Find the area of ABC using Heron’s formula. B 28 A 21 22 C Chapter 14 156. Sketch one cycle of the graph of the function. x f x 4 cos 3 157. Write two x-values at which the function has a minimum. f (x) 4sin 2x 158. Graph y = 3 cos x on the interval x . 4 x 159. Write the equation of the resultinggraph when y 6sin is translated up two units. 3 161. Simplify the expression. sin x sec x 2 162. Simplify the expression. sin 2 x tan 2 x 163. Simplify the expression. cosx sin 2 x sec x 164. Verify the identity. sin x 1 cos x 1 cos x sin x 165. Verify the identity. tan 2 x csc 2 x 1 1 166. Solve 6cos x 3 0 in the interval 0 x 360 . 167. Solve 4 sin 2 x 1 0 in the interval 0 x 2 . Write a function for the sinusoidal 168. graph below. (The period is 2 .) y 5 2 2 –5 x 169. Evaluate the following exactly b. tan( 53 4 ) a. cos 512 170. Simplify the following sin 2 x cos2x cos2 x Verify thefollowing: 171. sin tan sin cos 1 tan 172. Given cos A a. cos2B 3 5 with 0 A b. sin B2 2 and sin B 12 13 with B c. tan2A 3 2 find the following: d. sin(A – B) 173. Simplify the following (sec x tan( x))(1 sin( x)) 174. Given that sinu a. cos v2 3 with 0 u 2 and cosv 12 13 with v 5 b. sin( u v) c. tan 2v 175. Solve the following: a. sin 2x sin x 0 b. sin 2x sec x 2cos x 3 2 , find the following d. cos2u Reference: [6.5.1.69] [1] 2x 3 2x 2 8x 8 + 4 x4 Reference: [6.5.1.79] [3] 3, -3 Reference: [6.6.1.81] [5] x = 5, 5, and 1 Reference: [6.7.1.87] 2 3 [7] –1, 1, , 3 2 Reference: [6.7.1.89] y 10 10 x –10 [9] –10 Reference: [6.7.1.92] [10] f (x) x 3 2x 2 15x Reference: [6.7.1.95] [12] (x 2)(x 2)(x 3) Reference: [6.7.2.97] [13] 1, 11 Reference: [6.7.2.98] [14] 4, 4, –2, –1 Reference: [7.1.1.6] [17] 32 Reference: [7.1.1.7] [18] 9 y = x 3 7x 6 Reference: [7.1.1.8] 1 [19] 32 Reference: [7.1.1.15] [21] 6 18 Reference: [7.2.1.25] 1 [22] 5 Reference: [7.2.1.26] [23] 83 25 Reference: [7.2.1.27] 1 [24] 16 5 Reference: [7.2.1.28] [25] 7 Reference: [7.2.1.30] [26] x 7 12 Reference: [7.2.1.32] [27] 3 Reference: [7.2.1.33] 1 [28] 1/6 7 Reference: [7.2.1.36] 4 y 3xz [30] 9z 3 Reference: [7.3.1.40] [31] x 2 x 2 Reference: [7.3.1.39] [32] x 2 x Reference: [7.3.1.41] [33] 8x x 2 Reference: [7.4.1.45] [35] {(–5, –1), (–1, –5), (–9, 9)} Reference: [7.4.1.46] x3 [36] y = 5 Reference: [7.4.2.63] [39] No f(x ) 3 1 –3 1 3 x –2 –3 Reference: [7.5.1.65] [40] Shift the graph of y x left 3 units, and down 3 units. Reference: [7.5.1.66] y 10 10 x –10 [41] –10 Reference: [7.5.1.68] y 10 10 x –10 –10 [43] Reference: [7.5.1.70] [45] x 1 Reference: [7.5.1.71] [46] g(x) 2 Reference: [7.6.1.81] [47] 2 Reference: [7.6.1.83] [49] –59 Reference: [7.6.1.86] [51] x 256 Reference: [7.6.1.87] [52] y = 127 Reference: [8.1.1.1] y 10 10 x –10 [59] –10 Reference: [8.1.2.2] [60] $1128 Reference: [8.1.2.4] [62] $1967.15 Reference: [8.1.2.7] [64] $452.89 Reference: [8.2.2.22] [68] $16,734.32 Reference: [8.3.1.27] y 10 10 x –10 [70] –10 Reference: [8.3.1.29] [72] 6e 4 x 1 Reference: [8.3.1.31] 1 [73] x 2e Reference: [8.3.1.37] [74] Decay Reference: [8.3.1.38] f (x) 4 3 2 1 –3 –2 –1 –1 1 2 3 x –2 [75] Reference: [8.3.2.41] [76] $7323 Reference: [8.4.1.43] [78] 2433 5 = 27 Reference: [8.4.1.45] [79] y 8 x Reference: [8.4.1.47] ex [80] y 3 Reference: [8.4.1.49] [81] –2 Reference: [8.4.2.56] y 10 10 x –10 –10 [83] Domain: x | x 1; Range: all real numbers Reference: [8.5.1.59] log 7 1.4 [84] log 4 Reference: [8.5.1.62] [85] log a 385 Reference: [8.5.1.64] [87] log 4 x log 4 (x 3) 5log 4 x Reference: [8.5.1.66] [88] 2log 3 x 3log 3 y Reference: [8.5.1.69] 4 y4 [89] log 5 3 x Reference: [8.6.1.76] [90] –0.4395 Reference: [8.6.2.77] [91] x = 36 Reference: [8.6.2.78] [92] x = 64 Reference: [8.6.2.82] [95] x = 2.513 Reference: [8.6.2.83] [96] 4.868 Reference: [8.7.1.86] x [97] y 0.1921.73 Reference: [9.1.1.1] [98] $13.97 Reference: [9.1.1.6] 28 [99] y or xy 28 x Reference: [9.1.2.9] [100] z 0.4 xy Reference: [9.3.1.19] y 10 10 x –10 [104] –10 Reference: [9.3.1.21] [106] x = 2, x = –2, y = 1 Reference: [9.3.1.22] [107] x = 1, x = –1, y = 0 Reference: [9.4.1.27] n3 [108] n5 Reference: [9.4.1.30] [110] x 6 x3 Reference: [9.4.1.32] [112] x 3 x 1 Reference: [9.4.1.36] x 1 [114] (x 3)(2x 3) Reference: [9.4.1.38] 2 [115] (x 2) 5x 3 Reference: [9.4.1.41] y [116] 4x3 Reference: [9.5.1.49] [118] 1 x4 Reference: [9.5.1.51] 4 [119] x3 Reference: [9.6.1.61] [120] 30 Reference: [9.6.1.64] [121] 25 Reference: [9.6.1.65] [122] 3, 3 Reference: [9.6.1.68] [124] It is not. Reference: [13.1.1.3] [127] 8.48 Reference: [13.1.1.12] 2 2 3 [129] 3 3 Reference: [13.1.1.31] [130] x = 4 3 , y = 4 Reference: [13.1.1.32] [131] x = 3, y = 3 2 Reference: [13.1.1.33] A 56, b 9.44, c 16.89 [132] Reference: [13.2.1.37] [134] Positive coterminal angle: 419 Negative coterminal angle: –301 Reference: [13.2.1.39] angle: 8 [135] Positive coterminal 3 4 Negative coterminal angle: – 3 Reference: [13.2.1.48] [137] 4 radians 15 Reference: [13.2.2.54] [138] arc length: 2 feet 5 Reference: [13.3.1.57] [139] 48 Reference: [13.3.1.65] [140] 1 2 Reference: [13.3.1.68] 20 [141] sin A = , mA = 44 29 Reference: [13.4.1.75] [142] 82.9 Reference: [13.4.1.76] [143] 218 Reference: [13.4.1.77] [144] 253 Reference: [13.4.2.79] 35.83 [145] Reference: [13.5.1.80] [146] c = 15.01 or 3.40 Reference: [13.5.1.84] [148] C = 78, a = 61.54, c = 81.01 Reference: [13.5.2.89] 2 [150] 12.99 cm Reference: [13.6.1.95] [152] 4.22 Reference: [13.6.1.97] [154] A = 43.2°, B = 30.1°, C = 106.7° Reference: [13.6.2.102] [155] 228.29 Reference: [14.1.1.9] f(x ) 6 2 1 –2 3 5 6 x –4 –6 [156] Reference: [14.1.1.16] 3 7 , [157] (There are other correct values.) 4 4 Reference: [14.2.1.22] y 5 [158] 2 2 –5 Reference: [14.2.1.25] x [159] y 2 6sin 3 Reference: [14.3.1.38] [161] 1 Reference: [14.3.1.42] [162] cos2 x Reference: [14.3.1.43] [163] 1 Reference: [14.3.1.44] x [164] sin x sin x1 cos x sin x 1 cos xsin x 1 cos x 1 cos x 2 1 cos x 1 cos x 1 cos x 1 cos x sin 2 x sin x Reference: [14.3.1.45] 2 2 2 2 2 [165] tan x csc x tan x sec x tan x 1 Reference: [14.4.1.49] [166] 120.00 , 240.00 Reference: [14.4.1.53] 7 11 , 5 , , [167] 6 6 6 6 Reference: [14.5.1.61] [168] y = 4 sin(x + ) 2 2 3 2 169. a. 2 170. 2cos x 172. a. 7 25 a. 3 1 1 3 c. 24 7 b. 3 13 13 173. cos x 174. b. 26 26 b. d. 56 65 175. a. 90, 210, 330 c. 120 119 3 7 b. , 4 4 16 65 d. 7 25