Calculus Summer Packet Congratulations on reaching this level of mathematics in high school. I know some or all of you are bummed out about having to do a summer math packet; but keep this in mind: we want you to be successful. The most difficult thing outside of the calculus is having to recall and apply all math skills you have developed so far. Many of the calculus rules we will learn this year are straightforward and easy to apply, but it is the reducing of solutions or improper algebra skills that will trip up some of you. It is etremely important that students entering calculus have a strong foundation in algebra, geometry and functions. Most questions in this packet were included because they concern skills and concepts that will be used etensively in calculus. Others have been included not so much because they are skills that are used frequently, but because being able to answer them indicates a strong grasp of important mathematical concepts and more importantly the ability to problem-solve. In this packet, you will find the following: Algebra Geometry Algebraic Functions Transcendental Functions Trigonometric Functions Graphing Skills This packet will be collected on the first full day of school. It is etremely important for all students to review the concepts contained in this packet and to be prepared for an assessment of these prerequisite skills to take place within the first week of school. (Date to be given on first full day of school.) I believe each of you is capable of success, but I want you to be fully certain of the epectations and be ready to give your best effort all year. Whether you are taking AP or not, my goal is to prepare you for college calculus and for you to fully know and be able to eplain all calculus concepts and methods. Please do not hesitate to contact us if you have questions this summer. Coach Swagart jswagart@baysideacademy.org Ms. Rollin hrolin@baysideacademy.org Special note for AP Students: Prerequisites according to College Board for AP Calculus Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students: courses in which they study algebra, geometry, trigonometry, analytic geometry and elementary functions. These functions include those that are linear, polynomial, rational, eponential, logarithmic, trigonometric, inverse trigonometric, and piece-wise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts and so on) and know the values of the trigonometric functions of the numbers 0, π, π, π, π and their multiples. 6,
Algebra Properties: True or False, if false, change the underlined portion to make the statement true. 1. 1 T F. 1 T F. ( ) 9 T F. 1 T F 1 5. (1) 16( ) T F 6. 5 T F Eponents: Simplify each epression. All powers must be positive. 7. y 5 8. yz 9. 5yz y z Factoring: Factor each epression. 10. 8 yz yz 1 1 11. 9 y y 1. 1 9 1. 5 8 1. 15. 15 5 1 6 6 1 16. 1 1 1 1 Rationalize: Rationalize the denominator or numerator. 17. 1 18. 1
Rational Epressions: Simplify each rational epression into a single fraction 19. 1. 1 1 1 1 0. ( 1) ( ) ( 1) 1 Polynomials, Radicals, Rationals, Inequalities, Absolute Value: Solve for, using correct means.. 5 5 0. 6 0. 6 6 10 0 5. 6 6 0 6. 1 0 7. 6 0 8. 9. 6 1 1 1 1 0. 5 0 1.. 5 5 15. 5 1 9 Linear Graphs: Write the equation of the line described below.. Passes through the point (, -1) and has a slope of -1/ 5. Passes through the point (, -) and is perpendicular to + y = 6. Passes through (-1, -) and is parallel to y = /5 1 7. Passes through the points (6, -1) and (0, 5) Systems of Equations: Determine a solution for each system of equations (point of intersection). y1 8. y 5 9. y y 6 9
Geometry Right Triangles: Evaluate for the unknown. 0. 1.. The roller coaster car shown in the diagram above takes.5 seconds 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 180 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 care go up the incline? d) What is the approimate average speed of the car as it goes down the drop? (Assume the care travels along HC). Area: Evaluate the area for each geometric figure...
Algebraic Functions Function Behavior: a) Determine the domain and range for each function. b) Determine the end behavior for each function 5. f ( ) 6. f ( ) 5 5 7. f( ) 8. f ( ) 5 9. f ( ) 5 50. if 0 f ( ) 1 if 1 0 if 1 Compositions and Inverses: Find the compositions and inverses as indicated below. f ( ) g( ) h( ) 51. g 1 ( ) 5. h 1 ( ), for 5. h( g( f (1))) 5. g( h( )) 55. g( f ( )) 56. f ( g( )) Asymptotes: Identify any asymptotes (horizontal, vertical, slant) 57. 1 f( ) 1 58. f( ) 1 59. f( ) 60. f( ) 5 51 1 Even & Odd Functions: Identify as odd, even or neither! Even: f ( ) f ( ) Odd: f ( ) f ( ) 5 61. f ( ) 6. f ( ) 6 6. f ( )
Transcendental Functions Simply Epressions: Evaluate or reduce each logarithmic epression. 6. log 16 65. 1 1 log log 81 log 66. log9 7 7 67. 1 5 log 68. ln eln1 ln e 69. log w w 15 5 Logarithmic Epressions: Epand each condensed logarithm as much as possible and condense the epanded logarithmic epressions. 70. ln y 71. ln y 7. ln y 7. ln 5ln y 7. ln ln ln a 75. ln ln Logarithmic/Eponential Equations: Solve each equation. Give your solution in eact form and round to three decimals places. 76. ln( ) 77. ln ln 1 78. ln ln( ) ln 79. ln( 1) ln( ) ln 5 80. e 1 1 81. 8 ln 8. 100e 50 8. ln e 9
Trigonometric Functions Do your best to answer all without the use of a Unit Circle first! If you must, do use your unit circle. It is advantageous to evaluate all trigonometry without a Unit Circle. Evaluate: Use the following directions for the given angles: a) Convert each angel in degrees to radians, or radians to degrees. b) Determine a positive and negative cotermainal angle. c) Determine the reference angle. 8. 150 85. 150 86. 0 87. 88. 5 89. 7 6 Evaluate: Find the value of each epression in eact form. 90. sin 91. cos 11 9. 6 tan 9. sec 5 9. csc 7 95. cot 5 6 Evaluate: Find the value(s) of in [0, ) which solve each equation. 96. sin 97. cos 1 98. tan 99. sec 100. csc is undefined 101. cot 1 Evaluate: Find the value(s) of in [0, ) which solve each equation. Solve by factoring. 10. cos 5 9cos 10. cos cos = 0 10. sin cos sin 0 105. tan sin tan 0 Evaluate: Graph each function, identifying and y intercepts, if any, and asymptotes, if any. 106. y = -sin () 107. y = + cos 108. y = tan 1 109. y = sec + 1 110. y = csc () 111. y = cot
Graphing Graphing: Use the graph below to complete each task. 11. a) Complete the graph to make f an EVEN function, using RED ink! b) What are the domain and range of feven? c) What is feven(-)? d) Complete the graph to make f and ODD function, using BLUE ink? e) What are the domain and range of fodd? f) What is fodd(-)? 11. Each gridline represents 1 unit. Each gridline represents 1 unit. 1. f -1 (5). f (g(5)). (g f )(). Solve for : f (g ()) = 5 5. Solve for : f () = g () For parts f i, respond in interval notation. 6. For what values of is f () increasing? 7. For what values of is g () positive? 8. Solve for : f () < 9. Solve for : f () g() 11. Given the graph of y = f () (dashed graph), sketch each transformed graph. a) y = f ( + ) b) y = f () c) y = f () d) y = f () + 1