AP Calculus AB Course
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1 AP Calculus AB Course Course Design and Philosophy It is my belief that students gain a deeper understanding of mathematics if they have a solid conceptual basis before learning the mechanics needed to solve problems. The four major areas of study are limits, derivatives, indefinite integrals, and definite integrals. The concepts in each of these areas are first developed by examining the how and the why behind each idea or theorem, and through examining multiple representations (graphical, numeric, and symbolic) of each. The supporting skills are then discovered and/or explained and connections to and between the concepts are stressed. You will then be able figure out how to apply each concept to a variety of problems. This cycle of learning repeats throughout the course and should help you gain an appreciation for the beauty of calculus. Needed Materials Calculus Book: Calculus of a Single Variable, 4 th Edition Graphing Calculator: TI-84, TI-84+ or TI Inspire. AP Calculus AB Course Summer Assignments Read the article Prologue: What is Calculus? Read and study Chapter 1 from the text a section at a time. Read everything, including the stuff in the margins. The Notes Ch 1 Prerequisites for Calc AP have much of what you need to know, and do, in them. The books examples are easy to follow so use them as examples as you are working through the exercises. Use the notes and other Additional Resources provided to help you review the major skills and concepts from Algebra, Geometry and Trigonometry. On the non-calculator portion of the AP test, you are not permitted notecards so you will need to memorize important facts and procedures. If you know the concepts behind these, and the how and why, it will be easy to accomplish this. After you are sure that you will remember the material in a section, on separate sheets of paper, complete that section from the assignment list from the book. All work must be shown! Please circle your final answers. Keep assignments clearly labeled with section, page number, and problem numbers. Leave a little space around each problem in case we, or I, need to add some notes. I also have included some worksheets in your packet to give you practice with some of the skills the book doesn t review. The following assignments need to be mailed to me by the due dates shown below. I will check, grade and write notes on them for you so that you can correct your mistakes as you review. These assignments will count as homework assignments for the 1 st quarter. I will answer any further questions you may still have when we return to school. When we return to school, we will spend a very short period of time reviewing the summer assignments. Then we will have our Chapter 1 Exam. It is essential that we have a firm grip on these algebraic and trigonometric topics before we start into Calculus. Lack of knowledge and poor skills in algebra, geometry and trig cause more problems in learning Calculus that the Calculus itself. So, if you have questions make note of them and work on avoiding the silly mistakes (sign errors, computational errors, etc ) Use the following as a check list to make sure you are completing the assignments that you need to complete over the summer. Complete them in the order listed.
2 Note: Answers to odd-numbered exercises are in the back of the book, So if you can t get an evennumbered problem, try a similar looking odd-numbered problem first. Since having very good algebra skills is very important to solving calculus problems, these first two additional resources deal with refreshing and refining some of your algebra skills. You may want to work back and forth between the notes and the assignments. DO IT TO KNOW IT AND WELL! Study Notes General Concepts Do worksheet Getting Warmed Up Chapter 1: Preparation for Calculus (You will now be working from the book.) Section 1.1 Graphs and Models You probably know most of this really well already and you need to know this really well!! It is important that you are able to examine all concepts from multiple perspectives (situational, graphical, numeric, and symbolic) so don t skip the stuff in the margins. Also, we will be using the graphing calculators frequently to explore concepts and to solve problems. So review the procedures needed on the graphing calculators as you go through problems analytically (using algebra procedures). Read pages 2 7: Graphs by point plotting. Intercepts. Symmetry (y-axis, x-axis and origin). Know the analytic tests as well as the graphical representation. Sketching using intercepts and symmetry. Check your sketches on the calculator. Finding points of intersections. Math Models from scatterplots. Notes Ch 1 Prerequisites for Calc AP page 1. This lists the calculator procedures you will need. Do Page 8 9. DO NOT use a calculator to complete the problem unless the instructions have the graphing utility symbol next to them. You may use the calculator to check your answers. #1 4 match eqn. to graph. #7, 9, 11, 14 graph by point plotting. You should also use your knowledge about the shape of the functions. #20, 21, x- and y-intercepts. #27, 29, 30, 34, 35, 37 symmetry. Use the tests! #40, 44, 48, 50, 52, 54, 55, 60 sketch graph. Use intercepts, symmetry and your knowledge about the shape. #63, 68, 70, 72 points of intersection. #73, 75 models #77, 81, 83, 84 misc. On #83 & 84, see if you can find a formula for the intercepts. Section 1.2 Linear Models and Rates of Change Again, you probably know most of this and you need to!! Really try to get the concept that slope and rate of change are basically the same thing. Pay close attention to the use of point-slope
3 form for a line. It will be used more often than the slope-intercept form. Don t forget to carefully examine the margins and explorations. Read pages 10 15: Slope of a line. Equations of Lines: point-slope form and slope-intercept form. Know both! General form not used so much but standard form Ax + By = C where A and B are not both zero is used frequently. Ratios and Rates of Change. The units will means a lot here! Pay close attention to them! Parallel and Perpendicular Lines. Pay close attention to the STUDY TIP and the TECHNOLOGY PITFALL! Notes Ch 1 Prerequisites for Calc AP page 2. Do Page DO NOT use a calculator to complete the problem unless the instructions have Note copies of graphs are on the website #2, 5 slope from graph. #10 13 plot points & find slope. #19, 22 slope applications. #24, 25 slope & y-intercept! #30, 32, 33, 36, 39, 41, 44, 46 write the equation for the line. Answering in Point-slope form is fine. #49, 50, 53, 56 sketch a graph of the line. #59, 62, 64 equations of parallel and perpendicular lines. #70 equation through intersection points. #72 collinear points? #77, 79, 82 applications #84 tangent line to a circle. Remember: tangent is perpendicular to the radius at the point of tangency. Section 1.3 Functions and Their Graphs This is an extremely important section! Know all terms, concepts, procedures and notations!! Also, you must know how to graph many different types of functions with out a calculator, accurately and quickly!! Some of the notation may be new and you will need to read and write with it, so pay close attention to the notation, especially the interval notation that is in the Notes for Ch 1! Read pages 19 26: Function and Function Notation. Including real numbers, domain & range, implicit & explicit, dependent & independent, and evaluate. Finding domain & range. Study procedures and understand the underlying concepts. Includes one-to-one function. The Graph of a Function. Know the vertical line test and all of the basic shapes and characteristics of the basic functions. Transformations of Functions. Know how to transform any basic function to fit. See the packet New Functions from Transformations. Classifications and Combinations of Functions. Know categories (algebraic, trigonometric, exponential & logarithmic), transcendental; leading coefficient test & constant term; composite functions; even & odd functions. Packet New Functions from Transformations. This is the BEST resource I have ever found to help you understand transformations. Spend some time with this!
4 Notes Ch 1 Prerequisites for Calc AP pages 3 8. Do Page DO NOT use a calculator to complete the problem unless the instructions have Note copies of graphs are on the website #2 domain, range and solutions from graph #5, 7, 9, 10, 11 evaluate and simplify #13, 14, 18 find domain & range #20, 21, 24 find domain #26, 28 evaluate, find domain & range (piecewise) #29, 30 write piecewise functions, given the graph #33, 34, 35, 37, 38 sketch graph, find domain & range #41 44, 45, 48 Is the relation a function? #49 54 match function to transformed graph #55, 56 given a graph, sketch the transformed graphs #59, 61, 63, 65, 66 composites #69, 71, 75, 76 even, odd or neither #79, 80 writing functions #81 84 match function to table #86, 87, 95 applications # true or false (Be sure to fully explain!) More Worksheet Function Frenzy This is additional practice for the material covered in this section. There are no additional practice problems for sketching from equations without the calculator, so quiz yourself by making up an equation, graphing it by hand and checking it on your graphing calculator. You should be able to graph, just about anything without the calculator! Notes and worksheet Characteristics of Rational Functions. This will be really important! Make sure you really know this stuff! Study the notes first and memorize it, then try applying it. 1.4 Fitting Models to Data Read pages 31 33: Fitting a Linear Model to Data. Fitting a Quadratic Model to Data. Fitting Trigonometric Model to Data. Notes Ch 1 Prerequisites for Calc AP page 8. Do Page USE a calculator to complete most of the problems. #1 4 choose model from scatterplot #6, 8, 10, 13, 18 application problems 1.5 Inverse Functions Read pages 37 43: Inverse Functions Everything in this section is important, don t forget the Exploration and example here! Existence of an Inverse Function Refer back to the definition of a one-to-one function on page 21 before beginning this. Example 2 part a shows how to verify. The idea is that if the y-values are the same for two different x s, then to be one-to-one, the x values must be the same. Also see the guidelines. On Example 3, it is easier to interchange the x and y first and then solve for y.
5 Inverse Trigonometric Functions Again, everything in this section is important! Look at the last two pages of the Trig Toolkit for a review and summary of the functions and their graphs. Look at the first page to review function values. Notations differ for inverse trig functions so know both (arcsin and sin -1 ). Study the examples carefully!! Notes Ch 1 Prerequisites for Calc AP page 8. Trig Toolkit pages 1, 3 & 4 I am giving you a Trigonometric Toolkit which contains every thing you will ever need to know about trigonometry. (The boxes that have dimmed headings are not frequently used, page 2.) You will need to memorize most of the facts on these sheets, if you haven t already, so start to try to recall facts before you look them up. I tried to organize the information so that you can start seeing patterns and connections between the concepts. This should help you with the memorization process. Do Page DO NOT use a calculator to complete the problem unless the instructions have #1, 4, 5, 8 use composites (show g o f = f o g = x ) and rough sketch the graph #9 12 just match #14, 15 graph carefully #18, 20, 22 rough sketch graph with the equation #23, 25, 28 try to picture the graph to determine one-to-one #29, 32, 36 good graphs are needed here, label key points & watch limited domains! #39, 41, 42 remember to switch x and y and solve for y for the inverse #43 table for inverse, find table for function 1 st #46 application #53, 55, 56, 58 you may graph and use the horizontal line test on these #59, 61 use the horizontal line test on these and don t forget to find the inverse function #67, 69, 70, 71 DON T find the inverse function, just use the values #77 80 show how you got these #81 #84 graph carefully #87 use trig toolkit to help if needed #89, 91, 94, 96 use Example 5 and trig toolkit to help #98, 99 make sure you are in radian mode! #104, 105 fully explain (you may use examples to help) #108 # , 114, 116 see Example 7 for help. #117, 120 hint: take the sine or cosine of both sides #122 hint: point of intersection is given by arcsin x arcos x = 0, then solve #123, 124, 126, 130, 131 hint use right triangles! #137, 140 use basic shape and transformations #145, 148, 149, Exponential and Logarithmic Functions Read pages 49 54: Exponential Functions Know your properties and know key characteristics of the graphs of these. Know that it is not nice to leave negative exponents in your answers, so make sure your answers have positive exponents only. The number e Get the general idea in Example 3, don t stress over it. Know key characteristics of the graphs of e x. The Natural Logarithmic Function Focus on the inverse idea and know key characteristics of the graphs of ln x.
6 Properties of Logarithms Understand the connections between the exponent properties and the logarithmic properties. Follow the examples, but if you need more help, look at the Ch 1 Notes and do the Exponential and Logs Worksheet first. Notes Ch 1 Prerequisites for Calc AP page Great property summary sheet and many additional examples for working with exponents and logs. Exponential and Logs Worksheet It includes all bases so it is good extra practice! Do Page DO NOT use a calculator to complete the problem unless the instructions have #2, 3, 6 use exponent properties #8, 9, 12, 14, 15 solve by using exponent properties #20, 21, 26, 27 use basic characteristics and transformations #31 38 match graphs to the equations #39 use the given points to help you & test it on the calculator #41, 43 re-write #46, 49, 50 use basic characteristics and transformations #55, 58 show all steps and sketch graphs #59, 61, 62 apply inverse properties #65 you may only use the decimal approximations given #68 numerous explanations are possible #71, 72, 73, 75, 78, 79 expand logs using properties #81, 82, 84, 85 condense logs using properties #88, 89, 92, 93 solve using techniques from the book & notes #95 confirm this for themselves What is due and When? Getting Warmed Up and Sections 1.1 through 1.3 problems and worksheets need to reach me by July 28 th. Sections 1.4 through 1.6 problems and worksheets need to reach me by August 21 th. If you have any problems meeting these due dates, please contact me ASAP! Please do not send these assignments to the school. My summer address is S. Stirling 2038 Susquehanna Road Abington, PA Call my cell phone: if you need to reach me. sstirling@palisadessd.org Please do not share this personal information with any other students. I am giving it to you so that you can contact me over the summer. This is only for you, my AP students! Have a great summer and please do not hesitate to call if you need me, believe it or not, I tutor well over the phone. Thanks for taking on the AP challenge. I m looking forward to a great class with all of you. Thanks! Ms. Stirling
7 Please fill out this student information paper, and return it to me before you leave for the summer so that I can reach you if I need to. AP Calculus Name Book #: Summer Address Last math class taken: Teacher: Approximate final grade: Phone number(s) Please indicate home or cell. address: (Very Important!! So I can send you stuff if you need it!!) Concerns, if any:
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