Fluent Essays

Several different questions all about precalculus. Need work to be shown for all questions. pre_calculus_quetions_.docx Unformatted Attachment Preview take for the population of the bacteria to reach 520? 1. To find the distance AB across a river, a distance AC of 783 meters is laid off on one side of the river. It is found that 𝐴 = 108° and 𝐶 = 17°. Find the distance across the river from A to B. 6. Solve the equation 28 = 2𝑒 0.42𝑥 − 7 7. Convert the rectangular coordinate (−2, 2√3) using three different polar representations of the point. 2. Sasha is outside of a building. She sees Thomas on the 4th floor at an angle of elevation (from ground level) of 48°. She then sees Sebastian on the 9th floor at an angle of elevation of 67°. If Thomas is 52 feet from ground level, what is the distance from Thomas to Sebastian? 8. Convert the polar coordinates into rectangular coordinates. 2𝜋 (a) (−8, 5 ) (b) (7, −73°) Given 𝑢 ⃗ = 〈2, −4〉 and 𝑣 = 〈3, 1〉, find: 9. ||𝑢 ⃗ || 3. A baseball player in center field is approximately 330 feet from a television camera that is behind home plate. A batter hits a fly ball that goes to the wall 420 feet from the camera. The camera turns 8° to follow the play. Approximately how far goes the center fielder have to run to make the catch? 10. The direction angle of 𝑢 ⃗ 11. The direction angle of 𝑣 Write the following complex numbers in trig form: 12. −5 + 2𝑖 13. 3 − √3𝑖 4. Tampa is 86 miles from Orlando. Orlando is 244 miles from Miami and Miami is 295 miles from Tampa. At Orlando, what is the angle from Tampa to Miami? 14. −1 − 𝑖 Given 𝑑 = 〈−3, 3〉 and 𝑒 = 〈4, 5〉, find: 15. ⃗⃗⃗⃗ 2𝑑 − 3𝑒 16. 3||𝑑|| 17. ||4𝑒|| 5. The population of a certain bacteria growth over time in minutes can be given by the equation, 𝑃 = 32 ln 𝑡 − 5, where 𝑡 > 0. How long would it Evaluate the following limits: 𝑠𝑖𝑛 𝑥 𝑥→0 𝑥 18. lim 1 𝑙𝑛 (𝑥+1) 𝑥 𝑥→∞ 19. lim −2𝑥 2 +3𝑥−4 3𝑥−5 𝑥→∞ 21. lim+ 20. lim 𝑥→0 |𝑥+2| 𝑥 37. 𝑡𝑎𝑛 3𝑥 = 1 Evaluate each of the following: 1 23. 𝑡𝑎𝑛−1 (−1) 22. 𝑠𝑖𝑛−1 (− ) 38. 𝑠𝑖𝑛 2𝜃 = −𝑐𝑜𝑠 𝜃 2 24. 𝑠𝑒𝑐 −1 (2) 25. 𝑐𝑜𝑡 −1 (−√3) 26. 𝑐𝑠𝑐 −1 (1) 28. 27. 𝑐𝑜𝑠 −1 (− √2 𝑡𝑎𝑛[𝑐𝑜𝑠 −1 ( )] 2 30. Simplify cos (𝜃 − 29. 𝑐𝑜𝑠 −1 39. 2𝑠𝑖𝑛𝜃 − √3 = 0 √2 ) 2 40. Find the general solution of 2𝑠𝑖𝑛2 𝜃 = 𝑐𝑜𝑠 2𝜃. 5𝜋 (𝑐𝑜𝑠 ) 4 3𝜋 ) 2 Given 𝑠𝑖𝑛 𝜃 = 41. 𝑐𝑜𝑠(2𝜃) 11𝜋 12 32. 𝑡𝑎𝑛 7𝜋 12 𝜋 and < 𝜃 < 𝜋, find: 2 42. 𝑠𝑖𝑛(2𝜃) 2 Given 𝑐𝑜𝑠 𝜃 = , find the following: 5 43. 𝑠𝑒𝑐(𝜃) 44. 𝑐𝑜𝑠(−𝜃) Find the exact value of the following: 31. 𝑠𝑖𝑛 4 5 33. 𝑐𝑜𝑠(−75°) 𝜋 45. 𝑠𝑖𝑛 ( 2 − 𝜃) Given 𝑠𝑖𝑛 63° = 𝑎, find the following: 46. 𝑐𝑜𝑠 (27°) 47. 𝑐𝑜𝑠 (−27°) 48. 𝑠𝑖𝑛 (27°) 34. Simplify 𝑐𝑜𝑡𝜃 + 𝑡𝑎𝑛𝜃 35. Simplify 𝑠𝑒𝑐𝜃 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠𝜃 Graph the following polar curves: 49. 𝑟 = 3𝑠𝑖𝑛2𝜃 50. 𝑟 = −2𝑠𝑖𝑛5𝜃 51. 𝑟 = 5𝑐𝑜𝑠4𝜃 Solve each equation over the interval [0,2𝜋) Identify and graph each curve formed by the following: 52. 𝑟 = 1 − 𝑐𝑜𝑠𝜃 36. cos 2𝜃 + 𝑐𝑜𝑠𝜃 + 2 = 0 53. 𝑟 = 3 + 2𝑠𝑖𝑛𝜃 2 54. 𝑟 2 = 9𝑠𝑖𝑛2𝜃 63. Write the equation of an ellipse in standard form with foci at (5, 2) and (5, -10) and a minor axis of length 6. Convert the following polar equations to rectangular: 55. 𝑟 = −3𝑐𝑜𝑠𝜃 56. 𝑟 = 2 −4𝑠𝑖𝑛𝜃+5𝑐𝑜𝑠𝜃 64. Write the equation of a hyperbola in standard form with vertices at (1, -4) and (7, -4) and foci at (-1, -4) and (9, -4). 57. 𝑟 = 4 58. 𝑟 = 6𝑠𝑖𝑛𝜃 65. Find the vertex, focus, and directrix of a parabola with the equation 𝑦 + 2 = 3(𝑥 − 3)2 . Convert the following rectangular equations to polar: 59. 5𝑥 − 𝑦 + 8 = 0 66. Write the equation of a hyperbola in standard form with vertices at (-2, 1) and (-2, 9) and a conjugate axis of length 4. 60. 𝑥 .2 + (𝑦 + 3)2 = 9 61. 𝑥 2 + 𝑦 2 + 2𝑥 + 5𝑦 = 0 67. Write the equation of a parabola with a vertex at (6, 2), a horizontal line of symmetry, and a distance from the vertex to the focus of 4 units. 62. Rewrite in standard form, graph and list all characteristics of each conic section. a. 4𝑥 2 + 𝑦 2 − 8𝑥 + 6𝑦 + 9 = 0 68. Evaluate (−2 − 𝑖)(5 + 3𝑖) 69. Find the quotient 4−3𝑖 1+𝑖 b. 9𝑥 2 − 4𝑦 2 − 18𝑥 − 8𝑦 − 31 = 0 70. Find the modulus of −4 − 𝑖. 71. Find the component form of 𝑑 given ‖𝑑‖ = 100 and 𝜃 = 135°. c. 𝑥 = −𝑦 2 + 2𝑦 − 6 72. Let 𝑓 be the function defined by (𝑥 + 2)(𝑥 − 1) 𝑥 < −5 𝑓(𝑥) = { 𝑥 2 + 𝑥 − 1 −5≤𝑥 ≤0 𝑥2 − 1 𝑥>0 𝑥 -0.1 𝑓(𝑥) 2.218 –0.01 2.220 –0.001 2.222 0 error 0.001 2.226 0.01 2.228 0.1 2.230 For what values of 𝑥 is 𝑓 not continuous? 73. Given the following table 3 what is the value of lim 𝑓(𝑥) ? 𝑥→0 𝑥 2 −36 𝑥→6 𝑥−6 77. Find lim 𝑒 76. Find lim 𝑥→𝜋 74. Given the graph of 𝑓(𝑥) 2𝑥 3 +3𝑥−18 𝑥→∞ 8𝑥−4𝑥 4 +𝑥 2 𝑥 3 +7𝑥 2 +10𝑥 𝑥→0 𝑥 3 −𝑥 2 −6𝑥 78. Find lim 79. Find lim 80. Which function is NOT continuous at 𝑥 = 2? A) 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 4 a. For what value(s) of 𝑥 is 𝑓(𝑥) B) 𝑔(𝑥) = 𝑥 2 −4𝑥+4 𝑥+2 C) 𝑔(𝑥) = 𝑥 2 −4 𝑥+2 D) 𝑔(𝑥) = 𝑥 2 −4 𝑥−2 81. Given that 𝑔 is a continuous function on the interval [1,5] and 𝑔(1) = −1 and 𝑔(5) = 7, what does the IVT (Intermediate Value Theorem) guarantee for the function 𝑔? discontinuous? b. What is lim 𝑓(𝑥) ? 𝑥→3 82. Using the IVT (Intermediate Value Theorem) given 𝑓(𝑥) = 2𝑥 2 − 𝑥 − 5 is a continuous function, does 𝑓(𝑥) have a zero over the interval [−3, 1]? Why or why not? c. What is lim 𝑓(𝑥) ? 𝑥→0 75. Given the graph of 𝑓(𝑥) 83. Use the DEFINITION of continuity to determine if 𝑓(𝑥) is continuous at 𝑥 = 2. Explain. For what value(s) of 𝑥 is 𝑓(𝑥) discontinuous? A) −2 B) 3 C) −2, 3 D) no points of discontinuity 84. If lim 𝑓(𝑥) = 4 and lim 𝑔(𝑥) = 7, 𝑥→𝑎 𝑥→𝑎 Find: 𝑓(𝑥) a. lim 𝑔(𝑥) 𝑥→𝑎 4 b. lim 𝑓(𝑥) ∙ 𝑔(𝑥) 𝑥→𝑎 c. lim 5𝑓(𝑥) 𝑥→𝑎 85. What domain restrictions must be placed on all six trigonometric functions so that their inverses can be defined. 86. Write the equation of the following polar graphs: a. b. 5 ... Purchase answer to see full attachment