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AP Calculus Summer Review Packet TO: All 2011/2012 AP Calculus Students FROM: AP Calculus Teacher Ms. Kane—East Room 239, Ms. Radek-Carreon—West We are pleased that you have chosen to complete your math sequence by enrolling in AP Calculus for next year. To help ensure your success in AP Calculus next year, we have created a summer review work packet. This packet contains material that you must have knowledge of on the 1 st day of the course. The packet has questions from Algebra, Advanced Algebra, and Pre-Calculus. This packet is to be completed in the following manner: Work must be clearly enumerated and in consecutive order on separate paper. Each problem with work counts toward the total grade. Each problem must have WORK! NO WORK! NO CREDIT! This packet is worth the equivalent of one test. The packet with work is due on the first FULL day of school. The packet is due whether or not you are physically present in school. Resources you may want to use: Your Algebra, Advanced Algebra, or Pre-Calculus notes Calculus: Graphical, Numerical, Algebraic, Third Edition, Finney, Demana, Waits & Kennedy www.interactmath.com college.hmco.com/mathematics/larson/calculus_analytic/7e/students/index.html college.hmco.com/mathematics/resources/students/ace/college_algebra/ace.html mathforum.org Below is a list of supplies you will need for AP Calculus. Shop for the items when they are on sale and be prepared the first day of school. AP Calculus is a college-level course and you are expected to be prepared with your materials each day. AP Calculus Supplies Needed: Pencils/Erasers Binder Paper Graphing Calculator (TI-83, TI-83 Plus Silver Edition, TI-84, or TI-84 Plus) Calculus Summer Review Packet This packet is a review of information you learned in Algebra, Advanced Algebra & PreCalculus. You need to know this information to be successful in AP Calculus. Therefore, this packet is due on your FIRST DAY IN CALCULUS. It is to be completed CORRECTLY, NEATLY, and on a SEPARATE sheet of paper. Your Calculus teacher will collect your work on your FIRST DAY IN CALCULUS. Failure to turn in your completed work on your FIRST DAY IN CALCULUS may jeopardize your ability to remain in the course. I. Simplify. x6 1. 2 x 5x 6 5. 2x 6 2. 2 x 6x 9 cos3 x cos2 x 1 sin 2 x 6. tan 3 x sec2 x 1 II. Trigonometric Identities 9. Pythagorean Identities (there are 3) _________________________ _________________________ _________________________ 11. sin 2x = 7. sin 2 x 2 sin x cos x 8. sec x tan x 10. cos 2x = ? (there are 3) _________________________ _________________________ _________________________ _________________________ III. Simplify each expression 3 2 2 1 1 12. x 13. x 4 14. 12 10 x 5 x 7 2 x x 16 IV. Solve for a. 17. 15ab 3 4 3 22. n n 1 1 1 15. x 3 3 x 16. 19. c2 2ac 7a 5b 0 18. 7c 3ab 0 V. Expand and simplify n2 21. n0 3 x 2 x 30 4. x 2 25 4 x 3. 2 x 16 1 3 n 2 4x 2 x 6x 9 x 2 x 3 2 20. 2 cos a 1, 0 a 2 VI. Simplify x 23. x 24. eln8 F1I 28. log GJ H5K 29. log 3 9 33. 82 / 3 34. (8a 2 /5 )(2a 5/ 2 ) 5 25. e( 2ln x ) F1 I 30. lnGJ Ha K 26. ln 1 31. e 4ln x 5 27. ln e 4 xy 5 32. 1 4 10 x y 2 35. (4a 3/5 )5/ 2 VII. Using the point-slope form: y y1 m( x x1 ) , write an equation for the line: 36. with slope –4, containing the point (3,2) 37. containing the points (1,-4) and (-5,7) 38. parallel to 2x – 5y =7 and passing through (6,1) 39. perpendicular to the line 2x – 5y = 7 and passing through (6,1) VIII. Without using a calculator, determine the exact value of each expression. 40. sin 0 45. tan 2 41. cos 2 46. tan 42. cos 47. tan 3 4 4 43. sin 44. sin 6 1 48. sin(cos1 ) 2 IX. Solve for x, where x is a real number. Show the work that leads to your solution. 49. x 6x 2 18 x4 4 50. 0 x3 51. x 3 16 53. ( x 4)( x 4) 0 54. x 2 2 x 35 0 55. sin 2 x sin x, 0 x 2 56. ( x 4) 2 ( x 3) ( x 4)( x 3) 2 0 59. log x log( x 9) 1 b g 2 57. 82 x 4 x 3 60. ln x 5 X. Miscellaneous: Follow the directions for each problem. 61. Find the x-intercept(s): 5x 2 7 y 2 3xy 20 0 62. Find all points of intersection of the graphs of x 2 x y 6 and x y 2 . 1 63. If the point ( 1, ) lies on the graph of the equation 2 x ky 11, find k. 4 52. 2 x 2 5x 12 58. e5 x 3 64. Find the equation of the vertical line that passes through the point (5,4). 65. A business had annual sales of $120,000 in 2008 and $264,000 in 2011. Assuming that the annual increase in sales followed by a linear pattern, what were the retail sales in 2010? 66. Given f ( x) x 3 5 , find f (2) f (7) . 67. Given f ( x) x 2 3x 4 , find f ( x 1) f (1) . 68. Find f ( x h) for f ( x) x 2 5x 6 . 69. Let f ( x) 1 2 x, R S T x , 2 x 1 x 1 . Find f(4) and find f(-5). 70. Given f ( x ) 4 x 3 and g( x) x 2 9 , find the product of f ( x ) g ( x ) . 71. Given f ( x ) 3x 6 and g( x) x 2 4 , find f ( x) . g( x) 72. Given f ( x) 2 x 2 5 and g( x) x 3 , find ( f g )( x ) . 73. Given f ( x ) 3x 2 , find f 1 ( x ) . XI. Graph each function, identify the x- & y-intercepts and identify the domain and range. 74. y sin x 75. y cos x 76. y tan x 77. y x 3 x 2 6x 78. y x 2 6 x 8 79. y e x 80. y ln x 81. y x 82. y 3 x 83. y x 3 2 84. y R x | x2 85. y S |T5 2 1 x if x 0 if 0 x 4 if x 4 X. Use the drawing to the right to find the limit of each. 86. lim f ( x ) 87. lim f ( x ) 88. lim f ( x ) x 3 89. lim f ( x ) x 1 x 3 90. lim f ( x) x 1 x 3 91. lim f ( x ) x 1