AP Calculus Summer 017 Welcome to AP Calculus. I hope this course proves to be a challenging, yet rewarding endeavor for you. Calculus will be unlike any other math class you may have taken, filled with new ideas, concepts, and applications. While the material may be new, solving the problems that are presented in this course will require a strong foundation in problem solving. This summer assignment focuses on the skills that will be necessary to be successful in Calculus. All of the problems in this packet require the tools that you have learned in Algebra and Precalculus. Please read all directions carefully. All answers are to be placed on the last pages in their designated spots. You are free to write on the packet, but please make sure your work is neat and organized. On some sections of this assignment, it is recommended that you not use a calculator. More than half of the AP Calculus Eam does not allow a calculator, so it would be a good idea to adhere to these recommendations. This entire packet will be turned in on September 5, 017, the first day of class and will be counted for a grade. If you would like to use scrap paper, you may attach all loose-leaf paper to the assignment and include it when you turn in your work. If there are any questions or concerns that you may have during the summer please feel free to e-mail me at twaltrich@gmahs.org. I would recommend that you do not do the entire packet at one time, but rather, spread out the assignment over the entire summer. And do not what until the last minute to begin your work. Best of luck to you this summer and I look forward to a great year. Mr. Waltrich
Simplify the following statements. 1 1 + y 1.. y 1 + - 1. + - 1 + 6-16. - 7-8 - 8 - + 16 5. 6. 1 æ ö ç15 ø è 7. e +ln 8. ln e Epand the following using the Laws of Logs. 9. a log 10. b log y 11. log ab 1. log 100 Find the following products. 1. ( +)( 5) 1. ( 1)( +1) 15. (5 )(5 + ) 16. (a + b)(a b) 17. ( 5)( 5) 18. ( + 7)( + 7) 19. ( + )( + ) 0. ( + 1)( + 1)
1. ( 5). ( + ). ( 1). ( + y) 5. (p q) 6. (s + t) 7. (a + b) 8. (a b) 9. (a + b) 0. (a + b) Factor the following statements completely. 1. + 15. 1 19 18. 8 6 + 17 + 6. 8 + 18 7 Find the eact values of the following without using a calculator. 5. sin p 7 6. cos p 6 7. tan p 8. cos p 9. 5 tan p 0. 5 sec p 1. sin 60 o. cos 0 o
. sin 60 o cos 0 o + cos 60 o sin 0 o. (1 + csc 5 o ) 5. tan 60 o + sec 60 o 6. sec 0 csc 0 o o Verify the following identities. 7. (sin + cos ) 1 cos = 1 + sin cos 8. sin - = 9. cos = 1- tan 1 + tan 50. cot = cot - tan Solve the following equations. Give eact answers only. 51. 6 17 + 7 = 0 5. 0 + 6 15 = 9 5. 10 6 9 = 7 5. 1 60 = 0 55. 6 17 5 + 6 = 0 56. + + 6 + + 1 = 0
Solve the following Trig equations. Give eact answers in radians on the interval [0, π). 57. tan = 1 58. sin + cos = 0 59. cos () = 0 60. sin + = -sin Solve the following equations. Round answers to decimal places. 61. log ( - ) = 1 6. + 5ln() = 17 6. ln() + ln() = 6. ln(+1) ln( +8) = 0 65. - = 7 66. e 11e = 0 Answer the following word problems. 67. A wire inches long is cut into pieces to form a rectangle whose shortest side has length. Epress the area of the rectangle as a function of. Determine the Domain and Range of the function.
68. The distance a ball travels when thrown vertically upward from ground level is d(t) = -16t + 8t where t is the time in seconds and d is the distance in feet. a) Find the time when the ball reaches its maimum. b) What is the ball s maimum height? c) How long is the ball in the air? 69. From a point 00 ft from a flagpole, the angle of elevation top of the pole is 18 o while the angle of depression to the bottom is 18 o 5. Approimate the height of the flagpole to the nearest foot. 70. Suppose you start driving away from Gwynedd Valley in a car at 1:00 noon. At any time hours after your trip starts, your distance away from Gwynedd Valley (in miles) is given by the function f() = - 5 + 1-5 + 95. You drive for 5 hours. a) Draw a graph of this function. Use window 0 < < 6 and -50 < y < 00. b) Create a table, including all integer values of from 0 to 6. c) Find f(). Give units for the answer and interpret in terms of the problem. d) Find f(.5). Give units for the answer and interpret in terms of the problem. e) Find the average rate of change of f() from 1 to.5. Give units for the answer and interpret in terms of the problem. f) Find the average rate of change of f() from 1.5 to.5. Give units for the answer and interpret in terms of the problem. g) Find the average rate of change of f() from 1.9 to.1. Give units for the answer and interpret in terms of the problem. h) Find an estimate for the instantaneous rate of change of f() at =. Give units for the answer and interpret in terms of the problem.
Answers: 1.... 5. 6. 7. 8. 9. 10. 11. 1. 1. 1. 15. 16. 17. 18. 19. 0. 1.... 5. 6. 7. 8. 9. 0. 1.... 5. 6. 7. 8. 9. 0. 1.... 5. 6.
7. 8. 9. 50. 51. 5. 5. 5. 55. 56. 57. 58. 59. 60. 61. 6. 6. 6. 65. 66.
Word Problems: 67. 68. a) b) c) 00 69. 70. a) b) 175 150 15 100 75 50 5-5 -50 1 5 6 0 1 5 6 f() c) d) e) f) g) h)