AP Calculus AB Summer Assignment

Transcription

1 AP Calculus AB Summer Assignment Congratulations! You have been accepted into Advanced Placement Calculus AB for the next school year. This course will count as a math credit at Freedom High School and you may also earn college credit if you pass the national AP Calculus AB exam in May 018. Remember that by enrolling in this course, you are making a commitment to excellence in daily work. Successful students in AP Calculus possess the following characteristics: daily review of new content material taught in class diligent completion of homework on a daily basis participation in study groups or working with a study buddy understanding concepts vs. cramming details organizing notes and materials, grouping of similar concepts, discerning differences between concepts, and knowing and understanding major theorems and concepts asking questions in class and out of class before the next concept is introduced proficient in translating mathematical expressions into necessary and sufficient English justifications staying consistently diligent the entire year It is a requirement to have a graphing calculator for this course and the exam. A TI-84 is recommended. The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in AP Calculus AB. Guidelines: o Do NOT use a calculator on ANY of the problems. o You may use previous notes or online tools to help you. o Complete the packet without the help from another person. o It is best to complete one to two weeks before school begins so the concepts are fresh in your mind. o No class time will be used to work on this packet. DUE: Wednesday August 14 th, 017 Summer workshops will be available during preplanning to help assist in preparing for the year. Please see the reverse side for dates, times, and locations. I look forward to an exciting year of Calculus! Cassidy Fuller cassidy.fuller@ocps.net

2 Summer 017 Pre-AP Calculus AB workshops Wednesday August 9 th :0-5:0PM & Thursday August 10 th :0-5:0PM Report to room 56 and be ON TIME! Please bring your summer packet, pencil, paper, and a graphing calculator (if you have one). Topics to include but not limited to: Parent functions and transformations Factoring Solving quadratic equations Solving systems of equations analytically and graphically Function notation and operations Horizontal and vertical asymptotes Linear functions and slope Graphing calculator functions Helpful Videos Below are links to videos that would be helpful when completing the summer assignment. Please utilize them as the material in this packet WILL NOT BE REVIEWED OR TAUGHT IN CLASS. Factoring: Solving Quadratic equations: Logs and Exponentials: Rational and Radical Expressions and Equations Function Notation and Operations Evaluating Trig and Solving Trig Equations Additional Helpful videos that include linear functions, absolute value, piecewise functions, asymptotes, etc. Limits and Continuity

3 AP Calculus AB Summer Assignment DO NOT begin until the beginning of August!!! Due: Wednesday August 16 th, 017 Do NOT use a calculator! Name: PD: SHOW ALL WORK ON THIS PACKET! Simplify #1-7 completely x+x 5 x+5x x 1 y 1 x +1 y x y xy + x z xz z y yz 5. 1 x x+ 6. x+5 x x a b 8c 4 d 75a 4 b 8c d 4 7. x 64 x 4 1. sin x+cos x sec x 8. 7 x x cot x sec x 15. (x + y)

4 16. x 1 ( x + x x) a xy 6x 4 y x x+1 5. x 1 + x a 4 a 1 6. ( x1 4y 1 4 ) 1. (4a 7 ) 7. 6 x x 5 Consider the following to answer #8-7. x 4 7 f(x) 5 h(x) {(,1), (,), (7,4)} g(x) x + 1 k(x) x + 8. (f + h)() f() + h() 0. (f h)(7) 9. (k g)() 1. g(k(7))

5 . k(g(x + )) 5. (g k)(x). g 1 (x) 6. g(k(x )) 4. f(4) g() 7. f 1 (x) For # 8-4, write the equation for the line given the information below. Leave your answer in point-slope form (when possible). y y 1 m(x x 1 ) 8. slope containing the point (, ). 9. containing the points (1, ) and ( 5, ). 40. slope 0 and containing the point (4, ). 41. perpendicular to the line in #8 and containing the point (, 4). 4. passing through the point ( 4,) with an undefined slope. 4. passing through the point (, 8) and parallel to the line y 5 x 1. 6 For #44-46, evaluate the expression f(x+h) f(x) h given f(x) below. Simplify as much as possible. 44. f(x) 9x +

6 45. f(x) 5 x 46. f(x) x + x For #47-51, determine all the points of intersection by solving the system of equations. Write your answer as a coordinate point x y 5 x + y 48. y 4x x + y 49. y x y x 50. xy 4 x + y 8

7 51. y x y x + 1 For #5-59, solve for x. Show all work that leads to your solution. 5. x + 9x x x 5. x x (x + ) x 5x 55. ax + 6ab 7ax + ab 59. a+x b c

8 For #60-6, complete the table using the appropriate notation or graphical representation For #6-68, describe the transformation from the parent function and state the domain and range. Function Transformation Domain Range 6. y (x + 1) 64. y x 65. y x y ln x 67. y e x y x + 7 For #69-76, sketch the function. Label important points (x & y intercepts, vertex, etc). NO Calculator!!!!! 69. y sin x 7. y x 70. y x + 7. y 1 x 71. y x y x 1

9 75. y e x y ln x For #77-79, determine the vertical asymptote(s) for the function. The vertical asymptote occurs where the function is undefined (where the denominator is equal to zero). Your answer should be in the form x#. 77. f(x) 1 x 78. f(x) x x f(x) +x x (1 x) For #80-8, determine the horizontal asymptote using the three cases below. Case 1: The degree of the numerator is less than the degree of the denominator. The function has a horizontal asymptote at y 0. Case : The degree of the numerator is equal to the degree of the denominator. The function has a horizontal asymptote equal to the ratio of the leading coefficients. Case : The degree of the numerator is greater than the degree of the denominator. The function has no horizontal asymptote. 80. f(x) x x+1 x +x f(x) 5x x +8 4x x f(x) 4x5 x 7

10 For #8-96, without a calculator, determine the exact value of each expression For # 97-10, solve each of the equations for 0 x π. Isolate the variable and find all of the solutions within the given domain. (EXACT SOLUTIONS NO DECIMALS!) sin x 100. cos x 98. sin x 101. sin x cos x cos x 1

11 For #10-116, solve for x (without a calculator). Show ALL steps that lead to your answer. Some answers may contain exponential and logarithmic expressions. 10. x log(7x + 1) log(x ) x ln x x ln(x 1) 106. x ln x log(x + 1) e x log(x + 1) e x log x + log x e x+1

12 For #117-10, determine if each function is continuous. If the function is not continuous, find the x value of the discontinuity AND classify the type of discontinuity (removable, jump, or asymptotic) f(x) x x f(x) x+1 x +x f(x) x+1 x x 10. f(x) For #11-1, find the intervals on which each function is continuous. Use interval notation when writing your answer Give an example of a function with a removable discontinuity at x 0 and an asymptotic discontinuity at x Of the six basic trigonometric functions, (a) Which are continuous over all real numbers? (b) Which are not? What type of discontinuity occurs for those functions?

13 For #15-140, evaluate each limit (without a calculator). Show your work! 15. lim x lim x+ x x +x lim x x + x x 4x lim 17. lim x π sin x 4x+ 16. lim x 7x lim 1 x 0 + x 8, x lim f(x), f(x) { x x 1 x 4x 4, x > 1 x+ x x +x 19. lim 10. lim x 7x+1 x x x, x < lim f(x), f(x) { x 1 x + x, x 1 x 1 x x lim 19. Find lim x 1 (f(x) + g(x)) given 1. lim x 1 x 1 x+4 1. lim x 5 x Find lim x 1 g(x) if lim x 1 (f(x) + g(x)) 1, lim x 1 f(x) 4