Wheeler High School To: Honors Calculus and AP Calculus AB Students From: Mrs. Barry, Mrs. Massey, Mrs. Walls, and Mr. Furstein - Calculus Instructors Re: Summer work Honors Calculus and AP Calculus AB are college level courses covering material traditionally taught in the first semester of college calculus. These courses are taught in one semester consisting of daily 90-minute classes. Students need a strong foundation to be ready for the rigorous work required throughout the term. Completing the review packet, which consists of topics studied in pre-requisite courses, will help ensure a proper background. Students should epect to work approimately 0 hours on this assignment. The packet will be given a grade based on completeness and accuracy of solutions. Students are epected to show all work with logical steps. You must show your work for problems on separate paper. Do not list only an answer. The packets are due to your instructor on your first day of class. Students enrolled in Honors Calculus and AP Calculus AB will be using a graphing calculator throughout the course. A TI 89 calculator will be available to AP Calculus students for use during the semester if they do not already have one. AP Calculus AB students will have the opportunity to continue studying Calculus for a second term through the BC Calculus course. The BC course will cover material traditionally taught in the second semester of college Calculus. The AB course is a prerequisite for BC 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 eam. Since the AP test is only offered in May, students taking only the AB course during the fall semester should be prepared to study for the test independently. The success of each student in the Honors Calculus or AP Calculus program depends upon diligent effort, consistent completion of homework assignments, and practice of newly learned skills. Although a suggested assignment is given for each lesson, completion of some assignments may be optional. Calculus is a challenging, stimulating, and dynamic field of mathematics and we look forward to sharing our enthusiasm for the topics with you. If you have any questions or need to contact someone over the summer, please feel free to e-mail me at Lynn.Barry@cobbk.org. Sincerely, Mrs. Lynn Barry Math Department Chair
Calculus Summer Packet Work the following problems on separate paper. Show all necessary work and write your final answer on the answer sheet found at the end of the packet Turn in your work and your answer sheet to your instructor on the first day of class. PART : ALGEBRA Eponents. 8 yz ) Simplify completely: Factor Completely. 4 yz ) 9 + - y - y ) 64 6 4) 4 4 + 5-8 5) 5 5 4 Rationalize the denominator / numerator. 6) 7) Simplify the rational epression. 8) ( ) ( ) ( ) 4 ( ) Solve algebraic equations and inequalities. Use synthetic division to help factor the following, state all factors and roots. 9) p() = + 4 + - 6 0) p() = 6-7 - 6 + 7 4 ) p( ) 5 50 9 8 ) p( ) 4 ) Eplain why cannot be a root of 4 5 + c - d + 5 = 0, where c and d are integers. 4) ) Eplain why 4 + 7 + - 5 = 0 must have a root in the interval [0, ], ( 0 ) Solve. 5) ( + ) > 4 6) 5 0 7) 0 8) 9 0 9) 0 8 0) 5 9
Solve the system. y 0 ) y 5 ) 4 y 6 9 y PART : GRAPHING AND FUNCTIONS Linear Graphs - Write the equation of the line described below. ) Passes through the point (, -) and has slope -/. 4) Passes through the point (4, -) and is perpendicular to y 4. 5) Passes through (-, -) and Is parallel to y. 5 Conic Sections - Write the equation in standard form and identify the conic. 6) = 4y + 8y - 7) y y 4 4 0 8) 4 6 y 4y 5 0 9) y y 4 4 9 0 Functions - Find the domain and range of the following. domain restrictions - denominator 0, argument of a log or ln > 0, radicand of even inde must be 0 range restrictions- reasoning, if all else fails, use graphing calculator 0) y = ) y = log( - ) ) y = 4 + + ) y = 4) y = - 5 5) Given f() below, graph over the domain [ -, ]. What is the range? if 0 f( ) if - < 0 if < - Compositions and Inverses - Find the compositions and inverses as indicated below. Let: f() = + - g() = 4 - h() = ln w() = 4 6) g - () 7) h - () 8) w - (), for 4 9) f(g()) 40) h(g(f())) 4) g(h()) 4) g(w()) 4) Does y = - 9 have an inverse function? Eplain your answer.
Basic Shapes of Curves: Sketch the graphs on the answer sheet. You may use your graphing calculator to verify the 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. 44) y = 45) y = ln 46) y = - 47) y = 48) y = 49) y = 4 50) y = y e 5) y 5) 5 if 0 5 f ( ) if 0, 5 5 0 if 5 Asymptotes Identify any asymptotes. 5) y 54) y 55) y 56) y 57) 5 5 y 58) y 4 Even and Odd Functions - Identify as odd, even, or neither. Show substitutions! Even: f () = f (-) Odd: f (-) = - f () 59) f() = + 60) f() = 4-6 + 6) f( ) 6) f() = sin Test for symmetry. Show substitutions. Symmetric to y ais: replace with - and relation remains the same. Symmetric to ais: replace y with - y and relation remains the same. Origin symmetry: replace with -, y with - y and the relation is equivalent. 4 6) y 64) y = sin() 65) y = cos() 66) y 5 67) = y + 68) y 4
PART : LOGARITHMIC AND EXPONENTIAL FUNCTIONS Simplify Epressions. 69) log 4 6 70) log - log 8 log 4 7 7) log9 7 7) log 5 7) log w w 45 74) ln e 75) ln 76) ln e 5 Solve equations. 77) log 6 ( + ) + log 6 ( + 4) = 78) log - log 00 = log 79) + = 5 PART 4: TRIGONOMETRY Unit Circle find the following without using a calculator. 80) sec 6 8) 9 tan 8) 4 cos 8) sin 4 84) cot 8 85) tan 5 5 86) csc 6 87) sin 7 State the domain, range, and fundamental period for each function. 88) y = sin 89) y = cos 90) y = tan Identities simplify. 9) ( tan )( csc ) 9) - cos 9) sec - tan ( csc)( tan )( sin) Solve equations. 94) cos = cos + 0 π 95) sin() = 0 π 96) 4cos 97) cos + sin + = 0 5
Inverse Trig Functions Evaluate. 98) Arcsin 99) Arcsin 00) Arccos 0) State domain and range for: a) Arcsin() b) Arccos () c) Arctan () 0) in Arccos Graphing State the amplitude and period of the following functions and graph. 0) y sin( ) 04) y / 4cos(4) PART 5: GEOMETRY Right Triangle Trig - Find the value of. 05) 50 06) 0 X 70 o 70o 0 07) H A B C The roller coaster car shown in the diagram above takes.5 sec. to go up the degree incline segment AH and only.8 seconds to go down the drop from H to C. The car covers horizontal distances of 80 feet on the incline and 60 feet on the drop. a) How high is the roller coaster above point B? b) Find the distances AH and HC. c) How fast (in ft/sec) does the car go up the incline? d) What is the approimate average speed of the car as it goes down the drop? (Assume the car travels along HC.) e) Is your approimate answer too big or too small? ( Advanced Mathematics, Richard G. Brown, Houghton Mifflin,994, pg 6) 6
CALCULUS SUMMER PACKET ANSWER SHEET NAME... 4. 5. 6. 7. 8. 9. 0.... 4. 5. 6. 7. 8. 9. 0.... 4. 7
5. 6. 7. 8. 9. 0.... 4. 5. 6. 7. 8. 9. 40. 4. 4. 4. 44. 8
45. 46. 47. 48. 49. 50. 9
5. 5. 5. 54. 55. 56. 57. 58. 59. 60. 6. 6. 6. 64. 65. 66. 67. 68. 0
69. 70. 7. 7. 7. 74. 75. 76. 77. 78. 79. 80. 8. 8. 8. 84. 85. 86. 87. 88. 89. 90. 9. 9. 9. 94.
95. 96. 97. 98. 99. 00. 0. 0. 0. 04. 05. 06. 07.