AP Calculus Summer Packet Going into AP Calculus, there are certain skills that have been taught to you over the previous tears that we assume you have. If you do not have these skills, you will find that you will consistently get problems incorrect next year, even though you understand the Calculus concepts. It is frustrating for students when they are tripped up by the Algebra and not the Calculus. This summer packet is intended for you to brush up and possible relearn these topics. We assume that you have basic skills in Algebra. Being able to solve equations, work with algebraic expressions, and basic factoring, for example should now be a part of you. If not, you would not be going into AP Calculus. So, only the topics I see that students constantly do not have down in their basic skills set are included here. These are skills that are used continually in AP Calculus. Each problem in this packet should be done in pencil on the space provided. Rather than give you a textbook to remind you of the techniques necessary to solve the problem, I have given you several websites that have full instructions on techniques. If and when you are unsure of how to attempt these problems, examine these websites. Don t fake your way through these problems. As stated, students are notoriously weak in them, even students who have achieved well prior to AP Calculus. Use the websites. Realize also that certain concepts are interrelated. Domain, for example, may require you to be expert at working with inequalities. Solving quadratic equations may involve techniques used in solving fractional equations. On the first day of school, August 8 th, you will show me the work you have done over the summer. I understand that even with the use of the provided websites, you may have questions. After I see that you put in the effort during the summer, I will return the packet to you. You will have the next week to ask me questions either before or after class or in the mornings. Work needs to be shown when needed. Also do not rely on the calculator. Half of your AP exam next year is taken without the calculator. So paper and pencil techniques only. This packet will be graded for accuracy and is due at the start of class on August 15 th. Do not wait to do them at the very last minute. These take time. If you do two concepts a day, the whole packet will take you about a week to complete. If you have questions about any of these problems or techniques used in solving them, contact me via e-mail: meredith.alvarez@stpsb.org. Have a great summer and see you in the fall! The topics listed are below. You can certainly do Google searches for any of these topics. But I have given you several sites that will cover pretty much all of these topics. Algebra topics: http://www.purplemath.com/modules/index.htm Beginning algebra topics 1. Exponents - Negative and fractional exponents Intermediate algebra topics 2. Domain 3. Solving inequalities: absolute value 4. Solving inequalities: quadratic 5. Special Factoring formulas 6. Function transformation 7. Factor theorem (p over q method) 8. Even and odd functions 9. Solving quadratic equations and quadratic formula Advanced algebra topics 10. Asymptotes 11. Complex fractions 12. Composition of functions 13. Solving Rational (fractional) equations Trig Information: (Once in the site, go to lecture notes.) http://www.mathematicshelpcentral.com/index.html 14. Basic right angle trig 15. Trig equations
Topic 1: Fractional & Negative Exponents Simplify using only positive exponents. You are allowed to have fractional exponents in the denominator. 1. 3 2. 10 (4 9) ( 9) 3. 2 () 4. (16 ) / 5. 6. 6 () ()() () 7. ()/ 8. + + /
Topic 2: Domain Find the domain of the following functions. Write your answers in either set or interval notation. 9. = 10. = 11. = 12. = 13. = 14. = 15. = 5 14 16. =
17. =log( 4) 18. = Topic 3: Solving Inequalities (Absolute Value) Write the following absolute value expressions as piecewise expressions. 19. = 2 8 20. = 2+10 +1 Solve the following absolute value inequalities. 21. 3 >12 22. 3 5 23. 10+8 >2 24. 3 4 > 8
25. 6 >8 26. 3 18 < 10 Topic 4: Solving Inequalities (quadratic) Write the following absolute value expressions as piecewise expressions. 27. 16 28. + 30 Solve the following by factoring and making appropriate sign charts. 29. 25>0 30. +6 16>0 31. 2 +4 3 32. +4 4
Topic 5: Special Factorization Factor Completely. 33. +27 34. 8 35. 27 125 36. +11 80 37. + 38. 2 +50 20 39. + 40. ( 3) (2+1) +( 3) (2+1)
Topic 6: Function Transformation If ()= 4, describe in words what the following would do to the graph of (): 41. () 4 42. ( 4) 43. (+2) 44. 5()+3 45. (2) 46. () For questions #47-52, here is a graph of =(). Sketch the following graphs. 47. =2() 48. = 2()
49. = () 50. =( 1) 51. =(+3) 52. = () Topic 7: Factor Theorem ( over Method/Synthetic Division) Use the over method and synthetic division to factor the polynomial (). Then solve ()=0. 53. ()= +4 + 6 54. ()= +5 2 24
55. ()= 6 +3 10 56. ()= +2 19 20 57. ()= +5 +6 4 8 58. ()= +11 +41 +61+30 Topic 8: Even and Odd Functions Show all work to determine if the relation is even, odd, or neither. 59. ()=2 7 60. ()= 4 2 61. ()=4 4+4 62. ()=
63. 5 6=1 64. = Solve each equation using the Quadratic Formula. Topic 9: Solving Quadratic Equations and Quadratic Formula 65. 7 3=0 66. 4( 2) 5( 1)=2 67. +6+10=0 68. 2 3+3=0 69. 2 (+2)( 3)=12 70. + =
71. 9 +8=0 72. 10 +9=0 73. =12 Topic 10: Asymptotes For each function, find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they exist). 74. = 75. = 76. = 77. =
78. = 79. = 80. = 81. = Topic 11: Complex Fractions Simplify the following: 82. 83. 84. 85.
86. 87. 88. 89. Topic 12: Composition of Functions If ()=, ()=2 1, and h()=3, find the following. 90. (2) 91. (2) 92. h( 1) 93. h( 1) 94. h 95. () 96. () 97. () 98. (h())
Topic 13: Solving Rational (Fractional) Equations 99. + =5 100. =1 101. = 102. = 103. + = 104. + = 105. + = 106. = +
Topic 14: Solving Rational (Fractional) Equations Solve the following problems. If point is on the terminal side of, find all 6 trig function of. Draw a picture. 107. ( 2,4) 108. ( 5, 2) 109. If cos=,in quadrant I, find sin and tan. 110. If cot=3, in quadrant III, find sin and cos. Find the exact value of the following without calculators (Hint: Use the Unit Circle). 111. sin 225 cos 300 112. (6sec180 4cot90 ) 113. (4cos30 6sin120 ) Solve the following triangles (round to 3 decimal places where needed.) 114. A= a=21.7 115. A= a=6 feet B=16 b= B= b= C=90 c= C=90 c=95 inches
Topic 15: Solving Trigonometric Equations 116. sin= 117. cos =cos 118. 2cos+ 3=0 119. 4sin =1 120. 2sin +sin=1 121. cos +2cos=3 122. 2sincos+sin=0 123. 8cos 2cos=1