AP Calculus AB Course Read through this entire document, then e-mail me, sstirling@palisadessd.org, the student information paper at the end of this document (before June 25). Just provide me with the information via email or type it in Word and send me the attachment. (Please do not send me a picture of the form.) Also, attach your favorite picture of yourself to that email (as a pic or pdf attachment) so I can get to know you better. All of the assignments as well as resources will be in my Canvas Course for completion over the course of the summer. All materials, instructional videos, important dates, announcements, etc. will be done through the Canvas course. All communications and work submission will be done via school email and attachments. If you can t get into the course, email me immediately! Make sure you check your school email periodically (every few weeks) throughout the summer. I would like to start the communication process early and answer questions BEFORE we get back to school. Do not assume you know this stuff, or how to write it correctly! Test yourself and grade your writing style as well!! 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 Copy of Chapter 1 of the Calculus Book: Calculus of a Single Variable, 4 th Edition. PLEASE DO NOT WRITE IN THESE PACKETS! (I would like to use them again.) Graphing Calculator: minimal TI-84 or TI-84+ Check the college board AP Calculus AB website for a full list of calculators you can use on the exam. I will be using a TI-84+. The materials in this packet. All materials can be identified by the document s name located in the header of each document. These materials can also be accessed through the Canvas course. Ms Stirling s Canvas course, AP CALCULUS-18-19. Use it to download materials and to check your understanding of the assignments. I will use this course (as well as your school email) to send reminders and to update you on any information I may need to get out to you. When I enrolled you in my course, you should have received an email. Click Get Started. Go to courses, choose All Courses, and make sure the star next to AP CALCULUS-18-19 is chosen. You must work through the modules in order! If you have any trouble with the course, contact me via phone or e-mail. AP Calculus AB Course Summer Assignments During the summer, you will be reading and studying Chapter 1 from the text packet and my materials and Canvas course. Success in AP Calculus greatly depends upon the concepts and skills you should have learned (and retained) from your pre-requisite mathematics courses. The text s Chapter 1, the materials in this packet, and the materials and videos in Canvas, should help you review (relearn, or learn) the concepts and skills you must be strong in before entering AP Calculus. You should work through ALL of the materials AS listed in the modules and don t skip around. Don t assume you know the material nor the best ways of doing things. Everything in this course is done the way it is done for a reason, so if you need to learn a new way of doing something, please do it! But firstly, a few words of advise S. Stirling Page 1 of 11
Read everything and study everything, including the stuff in the margins and the diagrams. The packet, Ch 1 Notes Preparation for AP Calc, has much of what you will need to know and to be able to do in it. The book s examples are easy to follow so use them as examples as you are working through the exercises. Use my notes, the additional resources and videos I have provided to help you review the major skills and concepts from Algebra, Geometry, Trigonometry and Pre-Calc. In AP Calculus, you are NOT permitted to have notecards, so you will need to MEMORIZE all important facts and procedures. If you know the concepts behind these (the how and why they exist), and use them in the practice exercises, it will be easy to accomplish memorizing. DO IT TO KNOW IT! AND KNOW IT WELL!! Study one section at a time. After you are sure that you completely understand and can remember the material in a section, complete the exercises and/or worksheets for that section (found in the Assignment Check List later in this document). Also, these assignments will be posted in Canvas. You should be testing yourself to see if you really learned the material. And, really test your understanding! Take care in how you show your work! The Free Response section of the AP Calculus AB Exam (which is half of your score on the exam) depends heavily on how well you can communicate the mathematics you are applying. When you check your answers, pay attention to how you write and to the notations you must use to communicate the solutions. For this reason, all work must be shown (in correct mathematical notation) to get homework credit for your summer assignments! Please circle your final answers. Keep assignments clearly labeled with section number, page number, and problem numbers. Leave a little space around each problem in case you need to add some notes or corrections later. Besides the work from the text, I have also have included some worksheets in your packet to give you practice with some of the skills the book doesn t review thoroughly. The following assignments must be complete and corrected by our first class period. I will put answer keys in Canvas, so check and correct your mistakes as you work through these assignments. Check Canvas frequently for any additional help and new postings. Code items you are having trouble with so that we can quickly go over them with the rest of the class when you return to school. There will be limited time when we return before you have your Chapter 1 Test. Also you may email me your questions, or call, and I will try to get the answers posted on Canvas and/or email you back ASAP. Do make sure you really know your stuff and don t assume you really know your stuff!!! If you don t get something, call me! Seriously!! My number (solely for my AP students) is written at the end of this document and I am providing you with it so that you can use it! Don t be shy. If you are stuck on a concept or have any questions at all, please call me. Many students in the past have found this helpful, and it also gives me an opportunity to get to know you better before class officially begins in the fall. I will check and record grades for the summer assignments during our first few class periods. These assignments will be accepted ONLY on these dates. These assignments will count as homework credit in the 1 st quarter. (Your homework average is 10% of your quarter grade.) When we return to school, we will spend a very short period of time reviewing the summer assignments, and then we will have our Chapter 1 Test. So you really need to know this material really, really well before returning! You should be able to quickly and easily get through this test, with a minimum amount of stress. So practice, study and then study and practice some more. Exams are 90% of your quarter grade, and in the 1 st quarter we will only have 2 tests (approx. 100 points each), one quiz (approx. 40 points) and one AP Calculus Free Response packet (approx. 25 points). So your test scores are very influential in determining your quarter grade. So It is essential that you have a firm grip on these algebraic, geometric and trigonometric topics before we start into Calculus. Lack of understanding and poor skills in algebra, geometry and trigonometry cause more problems in learning Calculus that the Calculus itself. So, if you have questions make note of them, get help right away (call me or go to the resources) and keep reviewing and studying them. Also, work on avoiding the silly mistakes (sign errors, computational errors, etc ). Work on your writing skills (check how you should write procedures and check that you are using the most sophisticated mathematical notations). 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. Any changes (if any) will be made in Canvas. Review your work and notes thoroughly before returning to school. S. Stirling Page 2 of 11
Note: Answer keys (with procedures) will be available in Canvas. Take note of how the work is done as well as the final answers. Also, I may give you some ideas about a better way to approach things and write things. So check the work closely! The other websites listed in the materials can also be helpful! Use them if you need them! One more time DO IT TO KNOW IT AND KNOW IT WELL! Assignment Check List All document titles are in the upper left corner of the document! Read the two articles on What is Calculus? This will give you the big picture about all you will be learning and they will help you put it all together when the time comes. Besides, it s just plain fascinating! Read and Study the Notes General Concepts. Includes some basic things you should always keep in mind. (I will also give you some new ways of looking at procedures and concepts.) It reviews algebraic skills you should know really well. It s the ultimate cheat sheet! Then, for practice of these skills Do the worksheet Getting Warmed Up. You may do this section by section, or all at once. Keep referring to the General Concepts notes and WORK TOWARDS BOTH SPEED AND ACCURACY! Look for the VIPs (very important procedures or problems). They will come back! Review your calculator skills!! Open the document Ch 1 Calculator Notes which can be found on Canvas. You must get good and quick on your calculator, because half of the AP Calculus test uses the graphing calculator. Work through the examples with your calculator and then use your calculator as you work through the summer assignments to check your answers and give you additional practice. Now to the book (well copy of the book) Chapter 1: Preparation for Calculus (You will now be working from the book and the Ch 1 Notes Preparation for AP Calc document.) The notes will provide you with additional materials that address what the book may not have covered completely, and may provide you with a summary of what the book covered. Use both the book and my notes simultaneously. Section 1.1 Graphs and Models You probably know most of this already, but 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 you need to know on the graphing calculators, and as you go through problems analytically (using algebra techniques) use your calculator to check your results both graphically and numerically. Remember to follow the examples to see HOW to write, which is just as important as getting the correct answers. S. Stirling Page 3 of 11
Read pages 1 2 of the Ch 1 Notes Preparation for AP Calc. It lists the calculator procedures you will need. If you need more detailed notes, go back to the Canvas course and work through Ch 1 Calculator Notes. It also gives you a summary of the symmetry of functions. Read pages 2 7 from the book. Pay attention to the procedures used in the examples. Also start to connect the form of the equation to the picture of graph and its characteristics. You will need to sketch graphs quickly and easily from the equations. You should know how to Make graphs by point plotting and by checking on the calculator (use [TABLE] to check points). Find intercepts. Determine the symmetry (y-axis, x-axis and origin). Know the analytic tests as well as the graphical representation. DO NOT use numeric values to test for symmetry! Sketch graphs using intercepts and symmetry. Check your sketches on the calculator. Find points of intersections analytically, and find them on the calculator. Make and use Math Models from scatterplots on your calculator. Do Page 8 9. DO NOT use a calculator to complete the problem unless the instructions have the graphing utility symbol next to them. However, you should use the calculator to check your answers. PAY ATTENTION to the procedures and what functions look like in graphical form. #1 4 Match equation to graph. #7, 9, 11, 14 Graph by point plotting., but you should also use your knowledge about the general shapes of the functions. #21, 23, 26 Find x- and y-intercepts. #27, 29, 30, 37 Test symmetry. Use the analytic tests, not specific values! #40, 44, 48, 50, 52, 54, 55, 60 Sketch graphs. Use intercepts, symmetry and your knowledge about the general shape of the graphs. #63, 68, 70, 72 Find points of intersection analytically. #73 Create and analyze models using the graphing calculator. #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 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 in the text. Make sure you are using sophisticated procedures and notations!! Read pages 10 15 from the book. You should know how to Find the slope of a line. Write equations of lines: Point-slope form and slope-intercept form. Know both! AP really likes y m x x y! Use this instead of the other forms. the second form of the Point-slope Form, 1 1 Your book solution may give a different form, but stick with this Point-Slope Form. The General Form and Intercept Form are not used, but occasionally the standard form Ax + By = C, where A and B are not both zero, is used. Apply ratios and rates of change. The units will mean a lot here! Example 2, pay close attention to the units in the applications! You must always write the units!! S. Stirling Page 4 of 11
Write equations for parallel and perpendicular lines. Pay close attention to the STUDY TIP and the TECHNOLOGY PITFALL, use [ZOOM] 5:ZSquare on your calculator! Read page 3 of the Ch 1 Notes Preparation for AP Calc. It gives a good summary of this section and emphasizes the forms AP likes. Do Page 16 18. 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. Note copies of graphs are on the website www.mathgraphs.com. #2, 5 Find the slope from a graph. #11 13 Plot points & find slope. #19, 22 Slope applications. #24, 25 Find slope & y-intercept! #30, 32, 33, 36, 39 Write the equation for the line. Answer in Point-slope Form, y m( x x1) y1!! #44, 46 Derive and use the Intercept Form for a line. #49, 50, 53 Sketch a graph of the line. #59, 62, 64 Find equations of parallel and perpendicular lines. #70 Find the equation through intersection points, found on the calculator. #72 Collinear points? #77, 79, 82 Applications #84 Write the equation for the tangent line to a circle. VIP!!! 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!! You must know how to graph many different types of functions without a calculator, accurately and quickly!! Some of the notations may be new and you will need to read and write with them, so pay very close attention to the notation, especially the interval notation that is in my Ch 1 Notes! AP loves interval notation so learn to use it now! There are many supplements in this section, so be sure to complete the assignments in the order given. There are many assumptions about your prerequisite skills made by the book. The resources I have provided should fill in the holes. It may seem chopped up, but by the time you get to the exercises, your understanding should be coming all together. Start with studying the packet New Functions from Transformations. This is the BEST resource I have ever found to help you understand transformations. Spend some time with this! It is a really good resource. Transformations, and the knowledge of shapes of graphs, are the key to visualization. Very important in AP Calc!!! Read pages 19 21 from the book. You should know how to Understand the concept of a function, including: real numbers, domain & range, implicit & explicit forms, dependent & independent variables, relations, one-to-one functions and piecewise functions. Use function notation to define functions and to evaluate functions. Find the domain & range, with or without a graph. Study the procedures and how they relate to the underlying concepts! S. Stirling Page 5 of 11
Read and study the Ch 1 Notes Preparation for AP Calc, pages 4 6 (top). It provides additional information that the book assumes you know, as well as providing a summary. AP loves using the interval notation, so start using it so that you get used to it. Look at this notation very carefully! If you need more help with domain and range, use the note sheet Mastering Domains, which is on Canvas but not in your packet. It provides additional examples, showing you how to find domains and ranges. Read page 22 from the book. You should know How to determine and sketch the graph of a function simply by reading the equation that defines it. All of the basic shapes and characteristics of the basic functions (asymptotes, zeros, domain, range, increasing, decreasing, etc...) How to use the vertical line test. Study the Ch 1 Notes Preparation for AP Calc pages 7 8. Summarizes the types of functions you should be able to sketch quickly. Know the characteristics of each type really well. The transcendental functions (trigonometric, exponential and logarithmic functions) are covered in a different section of Chapter 1, so they are not studied here. But you will need to know them too. Read page 23 from the book. You should know How to transform any basic function from the function definition, the equation. You should have read and studied the packet New Functions from Transformations before reading this page of the book. Study the Ch 1 Notes Preparation for AP Calc page 6. Gives a quick summary of transformations, again you need to know these well! If needed, do the worksheet Function Frenzy, which is on Canvas but not in your packet. This is additional practice for the material covered in this section. You may test yourself on the functions 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! If needed, do the Transforming Functions Worksheet, which is on Canvas but not in your packet. Read pages 24 25 (stop at composition) then read page 26 from the book. You should know How to classify functions and the 4 basic ways to combine functions. Know the categories: algebraic & transcendental (trigonometric, exponential & logarithmic); leading coefficient test & constant term. How to determine whether a function is even or odd. Do 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 the properties, then try applying it. There should be videos in Canvas for this. S. Stirling Page 6 of 11
Study the Ch 1 Notes Preparation for AP Calc page 9. This gives a purpose for composing functions as well as discussing the importance of the domain and range. Read pages 25 (bottom) from the book. You should know How to form a composite function. How to determine the domain and range of a composition. Do Page 27 30. 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. Note copies of graphs are on the website www.mathgraphs.com. #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 #97 100 true or false (Be sure to fully explain!) 1.4 Fitting Models to Data The most important understanding you need to get from this section is what type of function fits what type of data. You should use your graphing calculator!! Read pages 31 33 from the book. You should know how to Fit a Linear Model, Quadratic Model or Trigonometric Model to Data. Study the Ch 1 Notes Preparation for AP Calc page 10, top, and page 2 for calculator key presses. Do Page 34 36. USE a calculator to complete these problems!! #1 4 choose model from scatterplot #6, 8, 10, 13, 18 Application problems. Don t need an accurate graph on paper here. 1.5 Inverse Functions S. Stirling Page 7 of 11
Everything in this section is important, don t forget the explorations and study the examples closely! Also make sure you are using the inverse notation appropriately! Refer back to the definition of a one-to-one function on page 21 before beginning this. Read pages 37 40 (exclude Ex 4) from the book. You should know how to Verify that functions are inverse functions by showing that both composites yield x. Verify that a function is one-to-one. The idea is to show that if the y-values are the same for two different x s, then the x values are the same, so one-to-one. Write an inverse function. (see the guidelines for finding an inverse function). On Example 3, it is easier to interchange the x and y first and then solve for y. Your book solves for x and then interchanges the x and y. Study the Ch 1 Notes Preparation for AP Calc pages 10 11. The on-line notes in Canvas are color coded for extra understanding. Read pages 40 (start with Ex 4) 43: You should study your Trig Toolkit before you go through this section. And refer to it (over and over again) as you go through the exercises. Again, everything in this section is important! Notations may differ for inverse trig functions so know both (arcsin and sin -1 ). Study the examples very carefully!! REALLY IMPORTANT!! Study the entire Trig Toolkit. I am giving you a Trigonometric Toolkit which contains every thing you will ever need to know about trigonometry for this course. You will need to memorize most of the facts on these sheets, if you haven t already, so start trying 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. AP WILL NOT let you have note cards. The remaining pages of the tool kit are to help you learn and/or to clarify the concepts and to develop good procedures. Look at the version of the Trig Toolkit on Canvas!! It is color coded to help you learn the concepts easier! Do Page 44 48. 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, 5 Use composites (show g f f g x ) and rough sketch the graph (from calculator). #9 12 Just match graph to inverse. #14, 15 Graph these carefully. #18 Make a rough sketch of graph (from the calculator). #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, 42 Remember to switch x and y and solve for y to form the inverse #43 Table for inverse? Find the table for the function 1 st. #46 Application #53, 55, 56, 58 You may graph (by hand) and use the horizontal line test on these. #61 Use the horizontal line test on this and don t forget to find the inverse function #67, 69, 70, 71 DON T find the inverse function, just use the values. Remember, the IN for the inverse is the OUT of the original function. #77 80 Show how you got these. S. Stirling Page 8 of 11
#81 Remember that with inverses the x and y are flipped. #84 Sketch carefully #87 Only use trig toolkit to help if needed. Practice memorizing your values. #89, 91, 94, 96 Use the book s Example 5 and trig toolkit to help #98, 99 Make sure you are in radian mode! #104, 105 Fully explain. (You may use examples if you want.) #108 Calculator. Use [ZOOM] 5: so that your graph is not distorted. #109 112, 114, 116 See book s Example 7 for help. #117, 120 Hint: take the sine or cosine of both sides #122 Hint: point of intersection is when arcsin x = arcos x, then solve #123, 124, 130, 131 Hint use right triangles! #137, 140 Use basic shape and transformations. Graph carefully. #145, 148, 149, 150 Give good explanations! 1.6 Exponential and Logarithmic Functions Again everything in this section is important!! Many of you may not have spent a great deal of time with exponential and logarithmic functions. We utilize them a great deal in AP, especially e x and ln x. Get to know their properties as well as the visual representations of these functions. Read pages 49 54 from the book. You should know Exponential Functions. Know your properties and know key characteristics of the graphs of these, as in Example 2. 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. e is a number just like π and you need to get used to looking at 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 graph of ln x. Switching from logarithmic to exponential form can be very helpful here. Properties of Logarithms. Understand the connections between the exponent properties and the logarithmic properties. You must memorize these ln (log) properties. Follow the examples, but if you need more help, look at the Ch 1 Notes and do the Exponential and Logs Worksheet first, before doing the book s examples. Study the Ch 1 Notes Preparation for AP Calc page 12 15. Great property summary sheet and many additional examples for working with exponents and logs. The book really doesn t do enough. Do the Exponential and Logarithms Worksheet It includes all bases so it is really good extra practice! Do Page 54 56. 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. #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 S. Stirling Page 9 of 11
#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 yourselves Pace your self and do not let this go until the last minute!! You should be done with Getting Warmed Up and Sections 1.1 through 1.3 problems and related worksheets by the end of July. You might be submitting these assignments electronically, so check your email for dates to submit your work (scan in or take a good photo of it). Sections 1.4 through 1.6 problems and worksheets need to be done by the end of August. You might be submitting these assignments electronically, so check your email for due dates. As I stated before, we will be having a test on this material shortly after we return from school. If you have any problems completing this work, please contact me ASAP! There will be very limited time when we return from summer vacation. Call my cell phone: if you need to reach me. E-mail: 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 wonderful 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 year with all of you. Thanks! Ms. Stirling Please fill out this student information paper electronically (the form is on the next page) and return it to me via email before you get too far into the summer so that I can reach you if I need to. S. Stirling Page 10 of 11
AP Calculus Name School Email address: (I will use this one.) Additional E-mail address if you use a personal email instead: (I will only use this one if you are not contacting me through the school email.) Your Grade level for the 2017-2018 school year (i.e. 11) Last math class completed: (i.e. A A Trig. Honors) When did you take it? (i.e. Spring 2017) Last math Teacher (name): (i.e. Ms. Colyer) Approximate final grade: (i.e. A - ) Phone number(s) Please indicate home or cell. Dates that you will be on vacation (or unavailable): Why are you taking this course? Concerns, if any: Send this form via email and remember to also attach a picture of yourself so I can get to know your name & face! S. Stirling Page 11 of 11