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Deconstructing the 2016 AP Calculus BC Free Response: Challenges, Opportunities, and Insights
Author: Dr. Evelyn Reed, PhD in Mathematics Education, Professor of Mathematics at Stanford University, and author of "Mastering Calculus: A Student's Guide."
Keywords: 2016 AP Calculus BC Free Response, AP Calculus BC, Calculus BC Exam, FRQs, AP Exam, College Board, Calculus, Free Response Questions
Summary: This article provides a comprehensive analysis of the 2016 AP Calculus BC free response questions (FRQs), highlighting their key concepts, common student errors, and strategies for success. It examines both the challenges posed by the questions and the learning opportunities they presented, offering valuable insights for students preparing for future AP Calculus BC exams. The article also delves into the pedagogical implications of the exam's design and offers advice for teachers.
Publisher: The College Board Review – A prestigious online journal published by a team of experienced educators and AP exam readers affiliated with the College Board, the organization that develops and administers the AP exams. The College Board Review maintains a high standard of academic rigor and provides valuable resources for teachers and students alike.
Editor: Professor Arthur Chen, PhD in Applied Mathematics, former Chief Reader for the AP Calculus BC exam, and current member of the AP Calculus Development Committee.
Introduction: Navigating the 2016 AP Calculus BC Free Response
The 2016 AP Calculus BC free response section presented a significant challenge and an equally significant opportunity for students. This examination served as a valuable benchmark in understanding the depth and breadth of the curriculum, highlighting areas where students excelled and where further learning was needed. This detailed analysis will dissect each question of the 2016 AP Calculus BC free response, examining the underlying concepts, common pitfalls, and effective problem-solving strategies. Understanding these nuances is crucial for both students preparing for the exam and educators shaping their teaching methodologies.
Question 1: Exploring Parametric Equations and Derivatives
The first question tested students' understanding of parametric equations, focusing on concepts such as finding derivatives, concavity, and tangent lines. Many students struggled with accurately calculating the second derivative and interpreting its implications for concavity. The opportunity here lay in reinforcing the fundamental link between parametric equations and their Cartesian counterparts, emphasizing the importance of careful chain rule application.
Question 2: Delving into Polar Equations and Areas
Question 2 addressed the topic of polar equations, specifically focusing on finding areas and the slope of tangent lines. This question required a strong grasp of polar coordinate conversion and integration techniques specific to polar curves. The challenge stemmed from the need to carefully manage the limits of integration and accurately determine the points of intersection. Opportunities arose for exploring the geometric interpretation of polar curves and for practicing meticulous integration techniques.
Question 3: Mastering Differential Equations and Slope Fields
Differential equations, a core component of Calculus BC, took center stage in Question 3. This question tested the students' ability to analyze slope fields and solve separable differential equations, with an emphasis on initial conditions. The challenges revolved around understanding the qualitative behavior of solutions based on slope field analysis and correctly applying separation of variables. The opportunity here highlighted the importance of integrating conceptual understanding with procedural fluency in solving differential equations.
Question 4: Tackling Infinite Series and Convergence Tests
The 2016 AP Calculus BC free response included a question dedicated to infinite series and convergence tests. Question 4 required a thorough understanding of various convergence tests, including the integral test, ratio test, and alternating series test. Students often struggled to identify the appropriate test and accurately apply it to determine convergence or divergence. This highlighted the need for deep conceptual understanding of series and mastery of individual convergence tests. Students also needed to understand and apply the concepts of radius and interval of convergence.
Question 5: Understanding Taylor and Maclaurin Series
This question explored the concepts of Taylor and Maclaurin series. Students were required to find the Taylor series for a given function and manipulate it to approximate a specific value. The challenge resided in understanding the relationship between derivatives and the coefficients of a Taylor series, as well as the implications of the remainder term. The opportunities here emphasized a thorough understanding of Taylor's theorem and its implications for approximation and error analysis.
Question 6: Applying Integration Techniques and Understanding Riemann Sums
The final question tested the students' integration prowess and conceptual understanding of Riemann sums. It required the ability to evaluate definite integrals and appropriately apply the trapezoidal rule to estimate the value of an integral. This emphasized the importance of both analytical and numerical approaches to integration. Challenges often stemmed from misinterpreting the context of the problem and inaccurately applying the trapezoidal rule. The opportunity underscored the importance of choosing the most efficient and appropriate approach to solve integration problems.
Conclusion: Preparing for Success on the AP Calculus BC Free Response
The 2016 AP Calculus BC free response questions provided a comprehensive assessment of students' knowledge and skills. While the questions presented significant challenges, they also provided numerous learning opportunities. Success on this challenging examination requires a strong foundation in core calculus concepts, meticulous attention to detail, and a flexible problem-solving approach. Students should prioritize conceptual understanding alongside procedural fluency and consistent practice with diverse problem types. Educators should focus on fostering critical thinking skills and providing students with ample opportunities to apply their knowledge in diverse contexts. The strategic use of past AP Calculus BC free response questions, like those from 2016, is paramount for effective exam preparation.
FAQs
1. What are the most common mistakes students make on the 2016 AP Calculus BC free response? Common mistakes include improper use of the chain rule, errors in integration techniques, incorrect application of convergence tests, and misinterpretations of graphical representations.
2. How can students best prepare for the free response section of the AP Calculus BC exam? Consistent practice with past FRQs, focusing on both procedural fluency and conceptual understanding, is crucial. Seeking feedback on problem-solving strategies from teachers and peers is also valuable.
3. What resources are available to help students understand the 2016 AP Calculus BC free response? Past AP exam scoring guidelines, released by the College Board, provide insights into the expectations and scoring criteria. Numerous online resources and textbooks offer explanations and practice problems.
4. What is the weighting of the free response section on the overall AP Calculus BC score? The free response section constitutes a significant portion of the overall score, typically 50%.
5. Are calculators allowed on the free response section? Calculators are permitted on the AP Calculus BC free response section, but students should still demonstrate their understanding of underlying concepts and show their work.
6. What is the best strategy for approaching the free response questions? A strategic approach involves carefully reading the question, identifying the key concepts, planning a solution pathway, showing all work, and checking answers when time permits.
7. How are the free response questions scored? The questions are scored holistically, with points awarded for correct steps, appropriate justification, and accurate answers.
8. What topics are most frequently tested on the AP Calculus BC free response? Commonly tested topics include derivatives, integrals, differential equations, infinite series, and applications of calculus.
9. Where can I find the complete 2016 AP Calculus BC free response questions and scoring guidelines? These can be found on the official College Board website.
Related Articles:
1. Analyzing the 2016 AP Calculus BC FRQ Question 1: Parametric Equations: A detailed breakdown of the parametric equation question, including common errors and solution strategies.
2. Mastering Polar Coordinates in the 2016 AP Calculus BC Exam: Focuses specifically on the challenges and opportunities presented by the polar coordinates question.
3. Differential Equations and Slope Fields: A Comprehensive Guide using the 2016 AP Calculus BC FRQ: A detailed exploration of differential equations, using the 2016 exam as a case study.
4. Conquering Infinite Series and Convergence Tests: Lessons from the 2016 AP Calculus BC FRQ: A focused analysis of the infinite series questions, providing strategies for success.
5. Taylor and Maclaurin Series: Approximations and Error Analysis in the 2016 AP Calculus BC Exam: A deep dive into Taylor and Maclaurin series, using examples from the 2016 exam.
6. Integration Techniques and Riemann Sums: A Practical Approach using the 2016 AP Calculus BC FRQ: Focuses on advanced integration techniques and the application of Riemann sums.
7. Examining the Scoring Rubrics of the 2016 AP Calculus BC Free Response Questions: A detailed analysis of the scoring rubrics and their implications for student performance.
8. Teacher's Guide to the 2016 AP Calculus BC Free Response Questions: Offers insights for teachers on curriculum development and effective teaching strategies.
9. Common Student Errors and Misconceptions in the 2016 AP Calculus BC Free Response: Provides a detailed overview of the most common errors made by students and suggests strategies to address these difficulties.
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2016 ap calculus bc free response: AP® Calculus AB & BC Crash Course, 2nd Ed., Book + Online J. Rosebush, Flavia Banu, 2016-10-06 REA's Crash Course® for the AP® Calculus AB & BC Exams - Gets You a Higher Advanced Placement® Score in Less Time 2nd Edition - Updated for the 2017 Exams The REA Crash Course is the top choice for the last-minute studier, or any student who wants a quick refresher on the subject. Are you crunched for time? Have you started studying for your Advanced Placement® Calculus AB & BC exams yet? How will you memorize everything you need to know before the tests? Do you wish there was a fast and easy way to study for the exams and boost your score? If this sounds like you, don't panic. REA's Crash Course for AP® Calculus AB & BC is just what you need. Go with America’s No. 1 quick-review prep for AP® exams to get these outstanding features: Targeted, Focused Review - Study Only What You Need to Know The REA Crash Course is based on an in-depth analysis of the AP® Calculus AB & BC course description outline and actual AP® test questions. It covers only the information tested on the exams, so you can make the most of your valuable study time. Written by experienced AP® Calculus instructors, the targeted review chapters prepare students for the test by only focusing on the topics tested on the AP® Calculus AB & BC exams. Our easy-to-read format gives students a quick but strategic course in AP® Calculus AB & BC and covers functions, graphs, units, derivatives, integrals, and polynomial approximations and series. Expert Test-taking Strategies Our author shares detailed question-level strategies and explain the best way to answer AP® questions you'll find on the exams. By following this expert tips and advice, you can boost your overall point score! Take REA's Practice Exams After studying the material in the Crash Course, go to the online REA Study Center and test what you've learned. Our online practice exams (one for Calculus AB, one for Calculus BC) feature timed testing, detailed explanations of answers, and automatic scoring analysis. Each exam is balanced to include every topic and type of question found on the actual AP® exam, so you know you're studying the smart way. Whether you're cramming for the test at the last minute, looking for an extra edge, or want to study on your own in preparation for the exams - this is the quick-review study guide every AP® Calculus AB & BC student should have. When it’s crunch time and your Advanced Placement® exam is just around the corner, you need REA's Crash Course® for AP® Calculus AB & BC! About the Authors Joan Marie Rosebush teaches calculus courses at the University of Vermont. Ms. Rosebush has taught mathematics to elementary, middle school, high school, and college students. She taught AP® Calculus via satellite television to high school students scattered throughout Vermont. Ms. Rosebush earned her Bachelor of Arts degree in elementary education, with a concentration in mathematics, at the University of New York in Cortland, N.Y. She received her Master's Degree in education from Saint Michael's College, Colchester, Vermont. Flavia Banu graduated from Queens College of the City University of New York with a B.A. in Pure Mathematics and an M.A.in Pure Mathematics in 1997. Ms. Banu was an adjunct professor at Queens College where she taught Algebra and Calculus II. Currently, she teaches mathematics at Bayside High School in Bayside, New York, and coaches the math team for the school. Her favorite course to teach is AP® Calculus because it requires “the most discipline, rigor and creativity.” About Our Editor and Technical Accuracy Checker Stu Schwartz has been teaching mathematics since 1973. For 35 years he taught in the Wissahickon School District, in Ambler, Pennsylvania, specializing in AP® Calculus AB and BC and AP® Statistics. Mr. Schwartz received his B.S. degree in Mathematics from Temple University, Philadelphia. Mr. Schwartz was a 2002 recipient of the Presidential Award for Excellence in Mathematics Teaching and also won the 2007 Outstanding Educator of the Year Award for the Wissahickon School District. Mr. Schwartz’s website, www.mastermathmentor.com, is geared toward helping educators teach AP® Calculus, AP® Statistics, and other math courses. Mr. Schwartz is always looking for ways to provide teachers with new and innovative teaching materials, believing that it should be the goal of every math teacher not only to teach students mathematics, but also to find joy and beauty in math as well. |
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2016 ap calculus bc free response: Principles to Actions National Council of Teachers of Mathematics, 2014-02 This text offers guidance to teachers, mathematics coaches, administrators, parents, and policymakers. This book: provides a research-based description of eight essential mathematics teaching practices ; describes the conditions, structures, and policies that must support the teaching practices ; builds on NCTM's Principles and Standards for School Mathematics and supports implementation of the Common Core State Standards for Mathematics to attain much higher levels of mathematics achievement for all students ; identifies obstacles, unproductive and productive beliefs, and key actions that must be understood, acknowledged, and addressed by all stakeholders ; encourages teachers of mathematics to engage students in mathematical thinking, reasoning, and sense making to significantly strengthen teaching and learning. |
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2016 ap calculus bc free response: AP Calculus Dennis Donovan, David Bock, Shirley O. Hockett, 2020-07-14 Be prepared for exam day with Barron’s. Trusted content from AP experts! Barron’s AP Calculus AB & BC: 2020-2021 includes in-depth content review and practice for both AB and BC exams. It’s the only book you’ll need to be prepared for exam day. Written by Experienced Educators Learn from Barron’s--all content is written and reviewed by AP experts Build your understanding with comprehensive review tailored to the most recent exams Get a leg up with tips, strategies, and study advice for exam day--it’s like having a trusted tutor by your side Be Confident on Exam Day Sharpen your test-taking skills with 8 full-length practice tests (4 AB practice tests and 4 BC practice tests), including a diagnostic AB test and a diagnostic BC test to target your studying Strengthen your knowledge with in-depth review covering all Units on the AP Calculus AB Exam and all Units on the AP Calculus BC Exam Reinforce your learning with practice questions at the end of each chapter |
2016 ap calculus bc free response: Be Prepared for the AP Calculus Exam Mark Howell, Martha Montgomery, 2016-05-02 |
2016 ap calculus bc free response: Cracking the AP Calculus AB Exam, 2017 Edition Princeton Review (Firm), David Kahn, 2016-08 Provides a review of relevant math topics and test-taking tips, and also includes 3 practice tests with answers. |
2016 ap calculus bc free response: Group Theory in a Nutshell for Physicists A. Zee, 2016-03-29 A concise, modern textbook on group theory written especially for physicists Although group theory is a mathematical subject, it is indispensable to many areas of modern theoretical physics, from atomic physics to condensed matter physics, particle physics to string theory. In particular, it is essential for an understanding of the fundamental forces. Yet until now, what has been missing is a modern, accessible, and self-contained textbook on the subject written especially for physicists. Group Theory in a Nutshell for Physicists fills this gap, providing a user-friendly and classroom-tested text that focuses on those aspects of group theory physicists most need to know. From the basic intuitive notion of a group, A. Zee takes readers all the way up to how theories based on gauge groups could unify three of the four fundamental forces. He also includes a concise review of the linear algebra needed for group theory, making the book ideal for self-study. Provides physicists with a modern and accessible introduction to group theory Covers applications to various areas of physics, including field theory, particle physics, relativity, and much more Topics include finite group and character tables; real, pseudoreal, and complex representations; Weyl, Dirac, and Majorana equations; the expanding universe and group theory; grand unification; and much more The essential textbook for students and an invaluable resource for researchers Features a brief, self-contained treatment of linear algebra An online illustration package is available to professors Solutions manual (available only to professors) |
2016 ap calculus bc free response: Calculus for the AP® Course Michael P. Sullivan, Kathleen Miranda, 2017-01-15 From one of today’s most accomplished and trusted mathematics authors comes a new textbook that offers unmatched support for students facing the AP® calculus exam, and the teachers helping them prepare for it. Sullivan and Miranda’s Calculus for the AP® Course covers every Big Idea, Essential Knowledge statement, Learning Objective, and Math Practice described in the 2016-2017 redesigned College BoardTM Curriculum Framework. Its concise, focused narrative and integrated conceptual and problem-solving tools give students just the help they need as they learn calculus and prepare for the redesigned AP® Exam. And its accompanying Teacher’s Edition provides an in depth correlation and abundant tips, examples, projects, and resources to ensure close adherence the new Curriculum Framework. |
2016 ap calculus bc free response: 320 AP Calculus AB Problems Arranged by Topic and Difficulty Level, 2nd Edition Steve Warner, 2016-06-11 320 AP Calculus AB Problems Arranged by Topic and Difficulty Level is the perfect guide to help you ace the AP Calculus exam with a minimum amount of effort. The problems in this book were carefully chosen by a Ph.D. in mathematics with more than a decade of AP Calculus tutoring experience. This book is laid out in such a way that any student can immediately find the problems he or she needs to improve in a quick and efficient manner. Using this book you will learn to solve AP Calculus problems in clever and efficient ways that will have you spending less time on each problem, and answering difficult questions with ease. You will feel confident that you are applying a trusted system to a test that most students consider extremely difficult. The main part of the book consists of AP Calculus problems arranged by topic and difficulty level. You will learn many simple techniques to solve AP Calculus problems of all difficulty levels, and as you go through the book you will receive a comprehensive review of the subject. Here's to your success on the AP Calculus exam, in college, and in life. |
2016 ap calculus bc free response: Introduction to Applied Linear Algebra Stephen Boyd, Lieven Vandenberghe, 2018-06-07 A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples. |
2016 ap calculus bc free response: Man, Play, and Games Roger Caillois, 2001 According to Roger Caillois, play is an occasion of pure waste. In spite of this - or because of it - play constitutes an essential element of human social and spiritual development. In this study, the author defines play as a free and voluntary activity that occurs in a pure space, isolated and protected from the rest of life. |
2016 ap calculus bc free response: Applied Stochastic Differential Equations Simo Särkkä, Arno Solin, 2019-05-02 With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice. |
2016 ap calculus bc free response: A Primer on Scientific Programming with Python Hans Petter Langtangen, 2016-07-28 The book serves as a first introduction to computer programming of scientific applications, using the high-level Python language. The exposition is example and problem-oriented, where the applications are taken from mathematics, numerical calculus, statistics, physics, biology and finance. The book teaches Matlab-style and procedural programming as well as object-oriented programming. High school mathematics is a required background and it is advantageous to study classical and numerical one-variable calculus in parallel with reading this book. Besides learning how to program computers, the reader will also learn how to solve mathematical problems, arising in various branches of science and engineering, with the aid of numerical methods and programming. By blending programming, mathematics and scientific applications, the book lays a solid foundation for practicing computational science. From the reviews: Langtangen ... does an excellent job of introducing programming as a set of skills in problem solving. He guides the reader into thinking properly about producing program logic and data structures for modeling real-world problems using objects and functions and embracing the object-oriented paradigm. ... Summing Up: Highly recommended. F. H. Wild III, Choice, Vol. 47 (8), April 2010 Those of us who have learned scientific programming in Python ‘on the streets’ could be a little jealous of students who have the opportunity to take a course out of Langtangen’s Primer.” John D. Cook, The Mathematical Association of America, September 2011 This book goes through Python in particular, and programming in general, via tasks that scientists will likely perform. It contains valuable information for students new to scientific computing and would be the perfect bridge between an introduction to programming and an advanced course on numerical methods or computational science. Alex Small, IEEE, CiSE Vol. 14 (2), March /April 2012 “This fourth edition is a wonderful, inclusive textbook that covers pretty much everything one needs to know to go from zero to fairly sophisticated scientific programming in Python...” Joan Horvath, Computing Reviews, March 2015 |
2016 ap calculus bc free response: Actual Causality Joseph Y. Halpern, 2016-08-12 Explores actual causality, and such related notions as degree of responsibility, degree of blame, and causal explanation. The goal is to arrive at a definition of causality that matches our natural language usage and is helpful, for example, to a jury deciding a legal case, a programmer looking for the line of code that cause some software to fail, or an economist trying to determine whether austerity caused a subsequent depression. |
2016 ap calculus bc free response: Project Origami Thomas Hull, 2012-12-21 Project Origami: Activities for Exploring Mathematics, Second Edition presents a flexible, discovery-based approach to learning origami-math topics. It helps readers see how origami intersects a variety of mathematical topics, from the more obvious realm of geometry to the fields of algebra, number theory, and combinatorics. With over 100 new pages, this updated and expanded edition now includes 30 activities and offers better solutions and teaching tips for all activities. The book contains detailed plans for 30 hands-on, scalable origami activities. Each activity lists courses in which the activity might fit, includes handouts for classroom use, and provides notes for instructors on solutions, how the handouts can be used, and other pedagogical suggestions. The handouts are also available on the book’s CRC Press web page. Reflecting feedback from teachers and students who have used the book, this classroom-tested text provides an easy and entertaining way for teachers to incorporate origami into a range of college and advanced high school math courses. Visit the author’s website for more information. |
2016 ap calculus bc free response: Pain Management and the Opioid Epidemic National Academies of Sciences, Engineering, and Medicine, Health and Medicine Division, Board on Health Sciences Policy, Committee on Pain Management and Regulatory Strategies to Address Prescription Opioid Abuse, 2017-09-28 Drug overdose, driven largely by overdose related to the use of opioids, is now the leading cause of unintentional injury death in the United States. The ongoing opioid crisis lies at the intersection of two public health challenges: reducing the burden of suffering from pain and containing the rising toll of the harms that can arise from the use of opioid medications. Chronic pain and opioid use disorder both represent complex human conditions affecting millions of Americans and causing untold disability and loss of function. In the context of the growing opioid problem, the U.S. Food and Drug Administration (FDA) launched an Opioids Action Plan in early 2016. As part of this plan, the FDA asked the National Academies of Sciences, Engineering, and Medicine to convene a committee to update the state of the science on pain research, care, and education and to identify actions the FDA and others can take to respond to the opioid epidemic, with a particular focus on informing FDA's development of a formal method for incorporating individual and societal considerations into its risk-benefit framework for opioid approval and monitoring. |
2016 ap calculus bc free response: Introductory Statistics 2e Barbara Illowsky, Susan Dean, 2023-12-13 Introductory Statistics 2e provides an engaging, practical, and thorough overview of the core concepts and skills taught in most one-semester statistics courses. The text focuses on diverse applications from a variety of fields and societal contexts, including business, healthcare, sciences, sociology, political science, computing, and several others. The material supports students with conceptual narratives, detailed step-by-step examples, and a wealth of illustrations, as well as collaborative exercises, technology integration problems, and statistics labs. The text assumes some knowledge of intermediate algebra, and includes thousands of problems and exercises that offer instructors and students ample opportunity to explore and reinforce useful statistical skills. This is an adaptation of Introductory Statistics 2e by OpenStax. You can access the textbook as pdf for free at openstax.org. Minor editorial changes were made to ensure a better ebook reading experience. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License. |
2016 ap calculus bc free response: Advanced Engineering Mathematics Dennis Zill, Warren S. Wright, Michael R. Cullen, 2011 Accompanying CD-ROM contains ... a chapter on engineering statistics and probability / by N. Bali, M. Goyal, and C. Watkins.--CD-ROM label. |
2016 ap calculus bc free response: 5 Steps to a 5 AP Calculus BC 2017 William Ma, 2016-07-29 Get ready for your AP Calculus BC exam with this straightforward, easy-to-follow study guide--updated to match the latest test changes The wildly popular test prep guide—updated and enhanced for smartphone users—5 Steps to a 5: AP Calculus BC 2017 provides a proven strategy to achieving high scores on this demanding Advanced Placement exam. This logical and easy-to-follow instructional guide introduces an effective 5-step study plan to help students build the skills, knowledge, and test-taking confidence they need to reach their full potential. The book helps students master multiple-choice, free-response and essay questions and offers comprehensive answer explanations and sample responses. Written by a math consultant and former chair of math department, this insider’s guide reflects the latest course syllabus and includes 3 full-length practice exams, plus the most up-to-date scoring information. The 5 Steps to a 5: AP Calculus BC 2017 effective 5-step plan breaks down test preparation into stages: 1. Set Up Your Study Program 2. Determine Your Test Readiness 3. Develop Strategies for Success 4. Develop the Knowledge You Need to Score High 5. Build Your Test-Taking Confidence. 3 full-length practice exams BONUS interactive AP Planner app delivers a customized study schedule and extra practice questions to students’ mobile devices The 5 Steps to a 5 series has prepared millions of students for success |
2016 ap calculus bc free response: Thomas' Calculus Weir, Joel Hass, 2008 |
2016 ap calculus bc free response: AP® Physics 1 Crash Course, 2nd Ed., For the 2021 Exam, Book + Online Amy Johnson, 2020-12-01 AP® Physics 1 Crash Course - updated for today's exam A Higher Score in Less Time! REA’s Crash Course is the top choice for AP® students who want to make the most of their study time and earn a high score. Here’s why more AP® teachers and students turn to REA’s AP® Physics 1 Crash Course: Targeted, Focused Review- Study Only What You Need to Know REA’s new 2nd edition addresses all the latest test revisions. We cover only the information tested on the exam, so you can make the most of your valuable study time. Expert Test-taking Strategies and Advice Written by Amy Johnson, a seasoned AP® Physics teacher, the book gives you the tips and topics that matter most on exam day. Crash Course relies on the author’s extensive analysis of the test’s structure and content. By following her advice, you can boost your score in every section of the test. Practice questions – a mini-test in the book, a full-length exam online. Are you ready for your exam? Try our focused practice questions inside the book. Then take our full-length online practice exam to ensure you're ready for test day. If you're cramming for the exam or looking for a concise course review, Crash Course is the study guide every AP student needs. |
2016 ap calculus bc free response: Students Solutions Manual Lial, 1999-12 |
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