AP CALCULUS AB Summer Assignment Page 1
Welcome to AP Calculus AB. This will be the toughest class yet in your mathematical careers but the benefit you will receive by having this experience in high school is immense. Because of the unique nature of this class it is very important that you are ready to start working on the first day. We will NOT be reviewing the material in this packet. The reason for this is to make sure we have adequate time for all of the new material that is required for the AP exam. As an instructor of AP Calculus I have extremely high expectations of students taking our course. I expect a certain level of independence to be demonstrated by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness to me is through this summer work packet which will help you maintain/improve your skills. This packet is a requirement for those entering AP Calculus AB and is due on the first day of class. If it is not completed you will not be allowed to enroll in AP Calculus AB. There are certain math skills that have been taught to you over the previous years that are necessary to be successful in calculus. If you do not fully understand the topics in this packet it is possible that you will get calculus problems wrong in the future not because you do not understand the calculus concept but because you do not understand the algebra or trigonometry behind it. Don t fake your way through any of these problems because you will need to understand everything in this very well. You may work with someone else while you do this but copying will not be tolerate Also don t wait until the last minute to do everything in the packet because you may run out of time and rush through them. Likewise do not do all the problems right at the beginning of summer and completely forget how to do all of them by the time school starts again. If you feel that you need help with some of these topics the best resource may be the internet. There are great sites where you can just type in the topics to get help. Using youtube.com khanacademy.org and coolmath.com not only will help you on this packet this summer but will be referenced often throughout the upcoming school year. For this packet you must show all of your work on a separate sheet of paper and circle your answer. If I cannot find each question and answer easily I will assume they are not there or incorrect so make sure your work is neat. Do not rely on a calculator to do all of the work for you. Half of the AP exam does not allow any calculator at all. This packet will be worth a test grade. Page
The following materials are required for AP Calculus AB. Please have these materials by the first day of school so we are able to get started right away! Graphing Calculator (ti-8+ or ti-84+ only) -Ring Binder (at least inches thick) 4-6 Pocket Dividers (plastic ones hold up the best) Pencils Highlighters Your binder should have the following sections Notes Homework Free Response Keep (random things that you will be told are important to keep for review) Enjoy your summer and don t forget about the packet. September will be here before you know it! If you lose your packet you will be able to access the packets on-line at the school website or a copy can be picked up in the main office. Please feel free to email me at acappabianca@pmamethuen.org if you have any questions. Page
This packet contains the topics that you have learned in your previous courses that are most important to calculus. Make sure that you are familiar with all of the topics listed below when you enter Calculus. I. Using Your Graphing Calculator A graphing calculator is required for this course and is not provided by the school. It is recommended that you purchase a TI8 or TI-84 Plus Silver Edition as this is the calculator that will be used by the instructors in each lesson. Be able to graph functions on your calculator. Be able to find key points on your graph like maximum values minimum values points of intersection et Do not depend on the calculator for graphs that should be memorize (See the section on Parent Functions.) Do not depend on your calculator for simple algebra! II. Interval and Set Notation You will be expected to be able to read and write using different notation for example notice the difference between the following solutions and be prepared to turn this in when school begins again III. Cartesian Coordinate System Know the quadrants axes and how to graph Know the different forms of the equation of a line especially Point-Slope Form Slope (incredibly important in calculus) Parallel and perpendicular lines IV. Algebra Basics You must be able to factor polynomials (GCF by grouping trinomials difference of squares and sum & difference of cubes) You must know how to add subtract multiply and divide rational expressions. You must know how to work with complex fractions. Page 4
AP Calculus AB Summer Packet V. Function Basics We use functions almost every day in calculus. It is critical that you are able to identify and do the following: VI. Identify functions from equations graphs sets Use function notation Find the domain and range of a function Identify functions as even odd or neither. You should also be able to identify what lines of symmetry go with even and odd functions. Perform operations on functions. Compose functions. Graph and write the equations of piecewise functions. Identify discontinuities. Find inverse functions algebraically and graphically. Use long and synthetic division Find the left hand and right hand behavior of a graph. Find the zeros of a polynomial function algebraically and graphically. Find vertical horizontal and slant asymptotes and any holes in the graph of a function. Logarithmic and Exponential Functions Know the properties of exponents. Know the properties of logarithms. Natural logs will be used a lot in Calculus so be familiar with them. Solve exponential and logarithmic equations VII. Trigonometry The Precalculus course that you just took was your formal trig course. However we use the trig functions and a few of the identities regularly in Calculus. The following are the most important: Calculus is always in radians never degrees. Know the special right triangles! You need to know without a calculator following values: for the. Be able to identify the graph of sine cosine and tangent. The Pythagorean Identities and Double Angle Identities are the most use Be able to work with arcsin arccos and arctan. Page 5
Name: AP Calculus AB Summer Packet Answer Sheet 1... 4. 5. 6. 7. 8. 9. 10. 11. 1. 1. 14. 15. 16. 17. 18. 19. 0. 1... 4. 5. 6. 7. 8. 9. 0. 1... 4. 5. 6. 7. 8. 9. 40. 41. 4. 4. 44. 45. 46. 47. 48. 49. 50. 51. 5. 5. 54. 55. 56. 57. 58. 59. 60. Page 6
Directions: Identify the choice that best completes the statement or answers the question. Write your answers on the answer sheet that has been provided in this packet. 1. Simplify: AP Calculus AB Summer Packet. Simplify:. Simplify: 4. Simplify: + Page 7
5. Simplify: 6. Factor Completely: 7. Factor Completely: 8. Simplify: Assume that no denominator is equal to 0. Page 8
AP Calculus AB Summer Packet For the given graph describe the end behavior determine whether it represents an odd-degree or even-degree polynomial function and state the number of real zeros. 9. f( x) 5 0 15 10 5 5 4 1 5 1 4 5 x 10 15 0 5 The end behavior of the graph is. It is an odd-degree polynomial function. The function has five real zeros. The end behavior of the graph is. It is an odd-degree polynomial function. The function has five real zeros. The end behavior of the graph is. It is an odd-degree polynomial function. The function has four real zeros. The end behavior of the graph is. It is an even-degree polynomial function. The function has five real zeros. as and as as and as as and as as and as Page 9
10. For the given values determine consecutive values of x between which each real zero is locate There is a zero between x = 1 and x =. There are zeros between x = and x = x = 1 and x = 0 x = and x =. There are zeros between x = 1 and x = x = 1 and x =. There is a zero between x = 1 and x =. 11. Estimate the x-coordinares at which the relative maxima and relative minima occur for the function. The relative maximum is at and the relative minimum is at. The relative maximum is at and the relative minimum is at. The relative maximum is at and the relative minimum is at. The relative maximum is at and the relative minimum is at. 1. Find all of the zeros of the function. 6 i 6 + i 6 i 6 i 6 + i 6 i 6 + i 1. Find the inverse of the given function: Page 10
AP Calculus AB Summer Packet 14. Graph the given function. State its domain and range. y 5 4 50 50 40 40 0 0 0 0 10 10 1 10 1 4 5 x 5 1 10 0 0 0 40 40 50 50 5 and the range is 6 The domain is x y 6. 1 4 5 x 5 and the range is 6 y 6. y 4 y 50 50 40 40 0 0 0 0 10 10 1 10 1 4 5 x 5 4 1 10 0 0 0 0 40 40 50 50 The domain is x y 6. 4 0 The domain is x 5 y 5 and the range is 6 The domain is x 1 4 5 x 5 and the range is 6 y 6. Page 11
15. Find the values of the six trigonometric functions for angle when and. 5 5 4 4 sin = cos = csc = sec = tan = and cot =. 4 5 5 4 4 5 5 4 sin = cos = csc = sec = tan = and cot =. 5 5 4 4 5 4 4 5 sin = cos = csc = sec = tan = and cot =. 4 5 5 4 4 5 5 4 4 sin = cos = csc = sec = tan = and cot =. 5 5 4 16. Simplify. 17. Simplify. Page 1
AP Calculus AB Summer Packet bg 18. Find the domain of the function: f x x 9 x 0 x 4x All real numbers x 1 All real numbers x 1 All real numbers x 5 4 All real numbers x 5 4 bg 19. Find the domain of the function: f x x 4 x 0 x x x 0 bg 0. Find the domain of the function graphed below: f x x 4 y 10 10 10 x 10 b g g [B] b g; b g b g b [D] b g; b 4 g [A] ; 4 [C] ; 4 bg 1. Find the vertical asymptote(s) if any for f x x 7. x 5x 6 [A] x 7 x [C] x x [B] x x x 7 [D] No vertical asymptotes Page 1
AP Calculus AB Summer Packet. Determine if the graph of the rational function has a slant asymptote. If it does find the equation of the slant asymptote. x x 6x f x x4 x bg y x y x y x No slant asymptote bg. Find the horizontal asymptotes if any of the graph of f x y x 8. x 4 x 1 y 0 No horizontal asymptotes y 8 4. Sketch the graph of the rational function. Find any vertical and horizontal asymptotes. x 5x f x x 4 bg y y x x Asymptotes: x x y Asymptotes: x x y y x Asymptotes: x x y y x Asymptotes: x 1 x y Page 14
AP Calculus AB Summer Packet b g bg f x h f x. for the function f x x 1 h b g b g bg 5. Find f x h f x h f x and xh x h 1 x 1 xh 1 x h 1 x 1 xh x h 1 x x h 1 x h 1 x 1 h x h 1 x xh 1 x h 1 x 1 h x h 1 x 1 x h 1 x h 1 x 1 b gb g b b b 6. gb g b gb g b b gb g b Evaluate the expression: gb g gb g gb g gb g. e 7. Evaluate the expression. 14 e 8. bg Solve the given equation. Round to the nearest ten-thousandth if necessary. 1.4 0.5666 0.659 0.168 Page 15
AP Calculus AB Summer Packet 9.Solve the exponential equation algebraically: 5e 0.04 x 67 97 0557. 150.000 44.794 19.454 0. Solve the equation algebraically: 600 575 1 e x 15..06 0.715 0.67 1. Find the value of x. b g ln 4 x 1 19.049 17. 1.08 0.67. What is written as a single logarithm?. What is the solution of? Page 16
4. Find the coordinates of the point of intersection of a 40 angle and the unit circle. F HG F HG 1 1 I KJ I KJ AP Calculus AB Summer Packet F HG F HG I KJ I KJ 5. Use the period of the trigonometric function sin to evaluate the function. 1 0 6. If possible evaluate the expression without the aid of a calculator. 1 sin 1 b g 4 Not possible 7. Evaluate the expression without the aid of a calculator. arcsin 0 0 4 Page 17
8. Use a calculator to approximate the expression. b arctan 0. 67 g 0590. 16. 8551. 00117. 9. Simplify: F tan arcsin 1 I HG K J 40. Factor the expression and use the fundamental identities to simplify. cos x sec x cos x cos x cot x 1 cos x sin x 41.Find the expression that completes the identity: 1 cos u sin u sin u 1 cos u csc u sin u 0 cos u Page 18
4. Find the x-values that are solutions of the equation: 5cot x 15 0 4 5 5 7 4 4 4 4 5 4 6 5 7 11 6 6 6 6 4. Find the x-values that are solutions of the equation: 8cos x 4 0 6 11 6 7 6 11 6 5 6 7 6 6 7 6 44. Find all the solution of the equation in the interval [0 ): 4cosx 0 11 1 5 18 18 18 18 18 11 1 5 5 18 18 18 18 18 18 11 1 5 5 18 18 18 18 18 18 18 1 18 Page 19
AP Calculus AB Summer Packet 45. Find all solutions of sin x 8 8 n 0 n 8 n n 8 n n 46. Classify the conic represented by the equation x 11xy 19 y x 8 y 0. parabola ellipse or circle hyperbola none of these 47. Find an equation that represents a parabol 5y 6x y 11 5x 6x 10 y y x 6x y 5y 0 y 6 y x x 5 0 48. Divide by using synthetic division. 49. Which equation represents a line through (-1 1) with a slope of? Page 0
50. Which of the following equations is shown in the graph below? 51. Which of the following systems of equations has the solution (4-1)? { { { { 5. What function has a graph with a removable discontinuity at )? Page 1
5. Which expression equals 54. Write in simplest form. 55. What is in simplest form? 56. What is the value of Page
57. What is the value of x to the nearest degree? 58. Factor completely: 59. Factor completely: 60. Factor completely: Page
Directions: Match the functions with the graphs and then write the domain and range under each graph. 1... 4. 5. 6. 7. 8. 9. 10. 11. 1. 1. 14. A. C. B. D. Page 4
E. H. F. I. G. J. Page 5
K. M. N. L. Page 6