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Wheeler High School To: AP Calculus AB Students and Parents From: Lynn Barry, AP Calculus Instructor Re: Summer work Date: May 25, 2007 AP Calculus AB is a college level course covering material traditionally taught in the first semester of college calculus. The course is taught in one semester consisting of 90-minute classes. Students need a strong foundation to be ready for the rigorous work required throughout the term. Completing the review packet before the beginning of the course will ensure a proper background. This packet consists of review material studied during Algebra II and Analysis. Students should expect to work approximately 10 hours on this assignment. The packet will be collected on the first day of class and will be given a grade that will be based on completeness of solutions and accuracy. In preparation for the AP test, students need to begin showing all work with logical steps. You must show your work for problems in the review packet. Do not list only an answer. Students enrolled in AP Calculus AB will be using a graphing calculator throughout the course since one is required to be used for the AP test. Students will be issued a TI 89 calculator for use in class during the semester and, with parental permission, students may take the calculator home to use as well. Students will have the opportunity to continue studying Calculus for a second term through the BC Calculus course. The AB course is a prerequisite for BC Calculus. The BC course will cover material traditionally taught in the second semester of college Calculus. Students taking both the AB and BC courses and passing the AP Calculus BC test could earn up to 6 hours of college credit depending upon the Advanced Placement policy at the college. Some colleges also award credit for a passing score on the AP Calculus AB exam. Since the AP test is only offered in May, students taking only the AB course should be prepared to study for the test independently, although there will be review sessions available. The success of each student in the AP Calculus program depends upon diligent effort and practice of newly learned skills. Although a suggested assignment is given for each lesson, completion of some assignments is optional. The previous night’s assignment will be reviewed in class each day and there will be ample opportunity to ask questions. Calculus is a challenging, stimulating, and dynamic field of mathematics and I look forward to sharing my enthusiasm for the material with you. If you have any questions or need to contact me over the summer, my e-mail address is Lynn.Barry@cobbk12.org . Sincerely, Mrs. Lynn Barry 1 Calculus Summer Packet Work the following problems on your own paper. Show all necessary work. I. Algebra 8 x yz 2x 1 3 3 3 A. Exponents: 1) 1 2 1 3 3 4x yz B. Factor Completely: Hint: Factor as difference of squares first, then as the 2) 9x2 + 3x - 3xy - y (use grouping) 3) 64x 6 - 1 sum and difference of cubes second. 5 3 1 4) 42x 4 + 35x 2 - 28 5) 15 x 2 2x 2 24 x 2 Hint: Factor GCF x 1/2 first. 6) x -1 -3x -2 + 2x -3 Hint: Factor out GCF x-3 first. C. Rationalize denominator / numerator : 1 x x+ 1 + 1 7) 8) 1 x 2 x D. Simplify the rational expression: (x + 1)3 (x - 2) + 3(x + 1 ) 2 9) (x + 1)4 E. Solve algebraic equations and inequalities 10. – 11. Use synthetic division to help factor the following, state all factors and roots. 10) p(x) = x3 + 4x 2 + x - 6 11) p(x) = 6x3 - 17x 2 - 16x + 7 3 12) Explain why cannot be a root of 4x5 + cx3 - dx + 5 = 0, where c and d are integers. 2 (hint: You can look at the possible rational roots.) 13) Explain why x4 + 7x2 + x - 5 = 0 must have a root in the interval [0, 1], ( 0 ≤ x ≤ 1) (hint: You can use synthetic division and look at the y values.) Solve: You may use your graphing calculator to check solutions. x+ 5 14) (x + 3) 2 > 4 15) 0 16) 3 x3 14 x2 5 x 0 (Factor first) x- 3 1 x2 - 9 1 4 17) x < 18) 0 19) + > 0 x x+ 1 x- 1 x- 6 2 1 20) x 2 < 4 21) | 2x + 1 | < 4 F. Solve the system. Solve the system algebraically and then check the solution by graphing each function and using your calculator to find the points of intersection. 22) x - y + 1 = 0 23) x 2 - 4x + 3 = y y– x 2= - 5 - x 2 + 6x - 9 = y II. Graphing and Functions: A. Linear graphs: Write the equation of the line described below. 1 24) Passes through the point (2, -1) and has slope - . 3 25) Passes through the point (4, - 3) and is perpendicular to 3x + 2y = 4. 3 26) Passes through ( -1, - 2) and is parallel to y = x - 1. 5 B. Conic Sections: Write the equation in standard form and identify the conic. 27) x = 4y 2 + 8y - 3 28) 4x 2 - 16x + 3y 2 + 24y + 52 = 0 C. Functions: Find the domain and range of the following. Note: domain restrictions - denominator ≠ 0, argument of a log or ln > 0, radicand of even index must be ≥ 0 range restrictions- reasoning, if all else fails, use graphing calculator 3 29) y = 30) y = log(x - 3) 31) y = x 4 + x2 + 2 x-2 x 1 32) y = 2x - 3 33) y= | x-5 | 34) domain only: y x 1 2 35) Given f(x) below, graph over the domain [ -3, 3], what is the range? x if x 0 f (x) 1 if -1 x < 0 x 2 if x < -1 3 Find the composition /inverses as indicated below. Let f(x) = x2 + 3x - 2 g(x) = 4x - 3 h(x) = lnx w(x) = x-4 36) g -1 (x) 37) h -1 (x) 38) w -1 (x), for x ≥ 4 39) f(g(x)) 40) h(g(f(1))) 41) Does y = 3x2 - 9 have an inverse function? Explain your answer. Let f(x) = 2x, g(x) = -x, and h(x) = 4, find 42) (f o g)(x) 43) (f o g o h)(x) 44) Let s (x) = 4-x and t(x) = x2, find the domain and range of (s o t )(x). D. Basic Shapes of Curves: Sketch the graphs. You may use your graphing calculator to verify your graph, but you should be able to graph the following by knowledge of the shape of the curve, by plotting a few points, and by your knowledge of transformations. 1 45) y = x 46) y = lnx 47) y = 48) y = | x - 2 | x 1 x π 49) y = 50) y = 51) y = 2 -x 52) y = 3 sin 2 (x - ) x- 2 x2 - 4 6 25 x 2 if x < 0 x 2 25 53) f ( x ) if x 0, x 5 x 5 0 if x = 5 E. Even, Odd, Tests for Symmetry: Identify as odd, even , or neither: Even: f (x) = f (-x) Odd: f (-x) = - f (x) Show substitutions! x3 x 54) f(x) = x3 + 3x 55) f(x) = x 4 - 6x2 + 3 56) f ( x ) x2 57) f(x) = sin 2x 58) f(x) = x 2 + x 59) f(x) = x(x2 - 1) 1 + |x| 60) f(x) = x2 61) What type of function results from the product of two 4 even functions? odd functions? Test for symmetry. Show substiutions. Symmetric to y axis: replace x with - x and relation remains the same. Symmetric to x axis: replace y with - y and relation remains the same. Origin symmetry: replace x with - x, y with - y and the relation is equivalent. 62) y = x 4 + x 2 63) y = sin(x) 64) y = cos(x) |x| 65) x = y2 + 1 66) y = x2 + 1 IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Expressions: 67) log 4 16 log 3 27 3 1 1 68) 3 log 3 3 - log 3 81 1 69) log 9 27 4 3 70) log125 5 1 71) logw w 45 72) ln e 73) ln 1 74) ln e 2 B. Solve equations: 75) log6 (x + 3) + log6 (x + 4) = 1 76) log x2 - log 100 = log 1 77) 3 x+1 = 15 V TRIGONOMETRY A. Unit Circle: Know the unit circle – radian and degree measure. Be prepared for a quiz. 78) State the domain, range and fundamental period for each function? a) y = sin x b) y = cos x c) y = tan x B. Identities: (tan2 x)(csc2 x) - 1 Simplify: 79) 80) 1 - cos 2 x 81) sec2 x - tan2 x (cscx)(tan2 x)(sinx) 82) Verify : (1 - sin2 x)(1 + tan2 x) = 1 C. Solve the Equations 83) cos2x = cos x + 2, 0 ≤ x ≤ 2π 84) 2 sin(2x) = 3, 0 ≤ x ≤ 2π 85) cos2 x + sinx + 1 = 0, 0 ≤ x ≤ 2 5 D. Inverse Trig Functions: Note: Sin -1 x = Arcsin x 2 3 3 86) Arcsin1 87) Arcsin 2 88) Arccos 2 89) sin Arccos 2 90. State domain and range for: Arcsin(x) , Arccos (x), Arctan (x) E. Right Triangle Trig: Find the value of x. 90. o 92 91. 91. 50 . 93. x 10 X H o o 70 70 10 A 180 ft 60 ft C B 93) The roller coaster car shown in the diagram above takes 23.5 sec. to go up the 23 degree incline segment AH and only 2.8 seconds to go down the drop from H to C. The car covers horizontal distances of 180 feet on the incline and 60 feet on the drop. How high is the roller coaster above point B? Find the distances AH and HC. How fast (in ft/sec) does the car go up the incline? What is the approximate average speed of the car as it goes down the drop? (Assume the car travels along HC. Is your approximate answer too big or too small? ( Advanced M athematics, Richard G. Brown, Houghton M ifflin,1994, pg 336) F. Graphs: Identify the amplitude and period of these functions 94) y = - 2sin(2x) 2 95) y cos x G. Be able to do the following on your graphing caluculator: Be familiar with the the CALC commands; value, root, minimum, maximum, intersect. You may need to zoom in on areas of your graph to find the information. Answers should be accurate to 3 decimal places. Sketch graph. 6 96. – 99. Given the following function f(x) = 2x4 - 11x3 - x2 + 30x 96. Find all roots. Note: Window x min: -10 x max: 10 scale 1 y min: - 100 y max: 60 scale 0 97. Find all local maxima. A local maximu m or local min imu m is a 98. Find all local minima. point on the graph where there is a highest or lowest point within an interval such as the vertex of a parabola. 99. Find the following values: f (-1), f (2), f (0), f (.125) 100. Graph the following two functions and find their points of intersection using the intersect command on your calculator. y = x3 + 5x2 - 7x + 2 and y = .2x2 + 10 Window: x min : -10 x max: 10 scale 1 y min: - 10 y max: 50 scale 0 Neatly graph and label the following. 1 101. y ln x 102. y e x 3. y x 4. y = x 5. y = x Answers: (Remember – you must show all of your work!) 1. 4x11/ 3 y4/ 3 z 2. (3x + 1)(3x - y) 3. (2x - 1)(4x2 + 2x + 1)(2x + 1)(4x2 - 2x +1) 4. 7(2x 2 -1)(3x 2 +4) 5. x1/ 2(5x+6)(3x-4) 1-x 1 + x-2 1 x2 - x + 1 6. x 3 x 2 x 1 7. 8. 9. 3 x x 1 1 (x + 1)2 7 1 10. (x –1)(x + 2)(x + 3); 1, -2, -3 11. (x + 1)(2x – 7)(3x – 1); -1, , 12. not a possible 2 3 rational root 13. f(0) = neg and f(1) = positive 14. x> -1 or x < -5 15. -5 x < 3 1 16. x or 0 x 5 17. 0 < x < 1 or x < -1 18. [ -3, -1) U (3, ∞) 19. x > 6 or 1 < x < 2 3 5 3 20. –2 < x < 2 21. - <x<- 22. (3, 4), (-2, -1) 23. (2, -1) , ( 3, 0) 8 8 7 1 1 2 17 3 7 24. y x 25. y x 26. y x 27. x = 4(y + 1)2 – 7 parabola 3 3 3 3 5 5 x 2 y 4 2 2 28. 1 ellipse 29. D: x ≠ 2 R: y ≠ 0 30. D: x 3 R: all reals 3 4 31. D: all reals R: y ≥ 2 32. D: x > 3/2 R: y > 0 33. D: all reals R: y ≥ 0 x3 34. x >-1 and x ≠ 1 35. R: 0 ≤ y ≤ 3 or - 5 < y < - 3 36. f 1 ( x) 37. h -1 (x) = e x 4 38. y = x2 + 4 x > 0, 39. f(g(x)) = 16x2 – 12x – 2 40. ln 5 41. no, explain 42. –2x 43. - 8 44. D: 0 y 2 R: 2 x 2 47. 49. 53. 45. 51. You must show work on these! 54. odd 55. even 56. odd 57. odd 58. neither 59. odd 60. even 61. even, even 62. symm y axis 63. origin 64. y axis 65. x axis 66. y axis 67. - 2 68. –1 69. 3/2 70. – 1/3 71. 45 72. 1 73. 0 74. 2 75. - 1 log 15 76. x = 10, -10 77. - 1 78. a) D: all reals, R: -1 < x < 1 , 2 л log 3 b) D: all reals, R: -1 < x < 1 , 2 л c) D: x ≠ π/2 R: all reals , π 79. 1 80. sin2 x 5 7 11 3 81. 1 82. yes 83. π 84. , , , 85. 86. 6 6 6 6 2 2 1 87. 88. 89. 90. Arcsin(x) D: [-1, 1] Range: , , 4 6 2 2 2 Arc cos (x) D: [-1, 1]. R: [0, π] Arctan (x) D: all reals R: , 2 2 91. 10 sin 50 92. draw altitude, tan 70 = x/5 94. A: 2, P: π π 95. A = π , Per. 4, y = - π cos ( x + 2π) 96. - 1.5, 0 , 2 , 5 2 97. rel max. ( 1.07, 20.1) 98. rel. min ( - .89, - 18. 48), (3.94, - 88) 99. f( -1) = - 18 100. 3 points of intersection - one is ( -5.77, 16.66) 100. Do not use trace to find points. Use CALC commands. 8