Summer Assignment Directions:

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1 Name: Block: Date: AP Calculus AB Summer Assignment Mr. Carter Welcome to AP Calculus AB! This fall will begin an exciting, challenging journey through the world of mathematics. You will challenge yourself in ways you never thought possible. The work will be challenging and frustrating, but will also be rewarding. You need to remember that you will not be able to get through Calculus by sitting on the sidelines; you will get out of it what you put into it. The most common mistakes involve not the Calculus, but the Algebra, so it is important for you to review your Algebra skills before coming to class. This packet is designed to help you assess your skills so you can come in at the end of the summer ready to review weak areas and move forward through our Calculus journey. Should you have any questions, please me at Also, I will be using Remind again this coming year; please sign up by to I strongly urge you to sign up now so that I may send updates as needed. Signing up now will also allow you to text me questions over the summer. Although graphing calculators will be available during class, it is strongly recommended that you obtain a calculator for use in and out of class. A list of approved calculators is available on the College Board website, but I recommend a TI-84 family calculator as that is what we will be using in class. I cannot guarantee that I will be able to assist you with another brand or model. Directions: The following problems are all algebra based, but cover topics from pre-algebra through pre-calculus. Feel free to use whatever resources you wish, e.g., siblings, parents, classmates, Internet videos, etc. You may also contact me with questions. Please complete all work in this packet and show all your work in the space provided; if you do not show work, you will not receive credit. All answers must be exact; no approximations (except where approximations are required). This packet is due on the first day of class and will count as your first quiz grade. In addition, there will be a unit circle quiz on the first day of class and a test on the material covered in this packet by the end of the second week of school. However, due to time constraints, we will not have time to review this material during class. You may find the following sites useful: Algebra topics: Trigonometry: Khan Academy (algebra, trigonometry, precalculus):

2 Write an equation of the following lines. 1. Passes through the points (2, 4) and (4, 5) 2. Passes through the point (4, 2) with slope m = 3 3. Passes through the point (1, 2) and is perpendicular to 2x 4y = 8 4. Passes through the point ( 1, 2) and is parallel to y = 3 5 x 1 Simplify the following expressions. 5. x x 3 y2 x 5 y 5 3x 7 7. x 2 4x 5 x 2 +2x+1 8. x 4 4 x 9. (x+1) 3 (x 2)+3(x+1) 2 (x+1) (x 4 y 3 ) x 1 3 x 3 5 Solve the following for all real values of x = x 2 x x 2 9x + 9 = x + x = e x +1 = x 1 5 x 1 = 0 1

3 Determine whether the functions are even, odd, or neither f(x) = x 3 + 3x 22. f(x) = x 4 6x f(x) = x3 x x f(x) = sin 2x Find the average rate of change for the following functions. 25. f(x) = x 3 2x; [0, 4] 26. g(x) = 3 x; [4, 25] 27. A car travels 360 miles in a period of 180 minutes. Find the average velocity of the car, in miles per hour, over this time period. 28. During a recent trip to the store, a car s velocity went from 0 to 60 mph in 20 seconds. What is the average acceleration of the car in miles per hour per hour? 2

4 Factor completely x 4 + 4x 3 + x x + 6x x 2 + 5x sin x + tan x x x 2 + 3x 3xy y x x x x 5 2 2x x x 3 + 4x 2 2x x Rationalize 3 x 1 x 2. 3

5 Sketch the graph of the following functions and/or equations. 41. y = x y = x y = 1 x 44. y = e x 45. y = ln x 46. x 2 + y 2 = x 2 + 4y 2 = 36 4

6 48. Given the function g(x) = 2(x 3) 3 + 4, identify the parent function f(x) and describe the transformations used to obtain the graph of g from f. Then sketch the graph of g. Graph the following trigonometric functions. 49. y = 2 sin 2x 50. y = cos (x π 4 ) On the Unit Circle, the value of the function is the x-coordinate and the value of the function is the y-coordinate. 52. What is the domain and range of sine and cosine? 53. What is the domain and range of tangent? 54. What is the fundamental period of a sinusoidal function? Of a tangent function? Evaluate the following trigonometric expressions exactly. 55. sin 11π 56. cos 3π sec 5π sin ( 17π ) 59. tan 6 (π 11π ) 60. csc arcsin ( 3 2 ) 62. arcsin sin 1 ( 2 2 ) 64. arccos ( 3 ) 65. sin (arccos ( 3 )) 66. tan

7 Complete the following trigonometric identities. 67. sec x = 68. csc x = 69. cot x = 70. sin 2 x + cos 2 x = 71. cos(a ± B) = 72. sin(a ± B) = 73. sin(2x) = 74. cos(2x) = 75. If sin x = 5 and π < x < π, find cos x using a 7 2 Pythagorean identity. 76. If cos θ = and 3π 2 < θ < 2π, find sin(2θ). Solve the following trigonometric equations over the interval 0 θ < 2π cos ( θ 2 ) = sin 2 θ + 2 sin θ + 1 = cos 2 x + sin x + 1 = sec 2 ( θ 2 ) 2 = A 20-foot ladder rests against a building 15 feet from the floor. How far does the ladder extend from the base of the wall? What angle does the ladder make with the ground? 6

8 Complete the following properties of logs. 82. log b (AB) = 83. log b ( A ) = 84. log b (A p ) = B 85. b log b x = 86. log b 1 = 87. log b b x = 88. log b b = Identify all possible zeros, then write as a product of linear factors and identify all real zeros. 89. p(x) = x 3 + 4x 2 + x p(x) = 6x 3 17x 2 16x + 7 Solve the system algebraically then check the solution by using technology to graph the system and check for points of intersection. x y + 1 = { x x { x2 4x + 3 = y = 5 x 2 + 6x 9 = y Write the equation in standard form and identify the conic. 93. x = 4y 2 + 8y x 2 16x + 3y y + 52 = 0 Identify the amplitude, period, phase shift, and vertical displacement of the following functions. 95. y = 2 sin (2x π ) 96. y = π cos 4 (π x + π) Does y = 3x 2 9 have an inverse function? Justify your answer. 7

9 State the domain and range of the following functions using interval notation. 98. f(x) = x g(x) = x 3 27 x y = 3 x y = log(x 3) 102. f(x) = x 4 + x y = 2x y = x y = x+1 (domain only) x 2 1 Let f(x) = x 2 + 3x 2, g(x) = 4x 3, h(x) = ln x, and w(x) = x 4. Find the following inverses and compositions g 1 (x) 107. h 1 (x) 108. w 1 (x), x f(g(x)) 110. h (g(f(1))) 111. w(x) = 5, find x 8

10 Simplify the expressions log log log log ln e 115. ln ln e log w w 45 Solve log 6 (x + 3) + log 6 (x + 4) = log x 2 log 100 = log x+1 = e x2 + 3 = 17 Simplify tan 2 x csc 2 x 1 csc x tan 2 x sin x cos 2 x 124. sec 2 x tan 2 x 125. Verify (1 sin 2 x)(1 + tan 2 x) = 1 Let f(x) = 2x 4 11x 3 x x. Use a graphing calculator to determine the following Find all roots (x-intercepts) Find all local minima and maxima Evaluate: a. f( 1) b. f(2) c. f(0) d. f(0.125) 129. Use a calculator to graph y = x 3 + 5x 2 7x + 2 and y = 0.2x and determine their point(s) of intersection. 9