INCLUDES Course framework Instructional section ample exam S questions AP Calculus ® AB and BC COURSE AND EXAM DESCRIPTION Effective Fall 2020 AP Calculus ® AB and BC COURSE AND EXAM DESCRIPTION Effective Fall 2020 AP COURSE AND EXAM DESCRIPTIONS ARE UPDATED PERIODICALLY Please visit AP Central (apcentral.org) to determine whether a more recent course and exam description is available. About College Board College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, College Board was created to expand access to higher education. Today, the membership association is made up of more than 6,000 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education.
Each year, College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success— including the SAT® and the Advanced Placement® Program. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. For further information, visit collegeboard. AP Equity and Access Policy College Board strongly encourages educators to make equitable access a guiding principle for their AP programs by giving all willing and academically prepared students the opportunity to participate in AP.
We encourage the elimination of barriers that restrict access to AP for students from ethnic, racial, and socioeconomic groups that have been traditionally underrepresented. Schools should make every effort to ensure their AP classes reflect the diversity of their student population. College Board also believes that all students should have access to academically challenging course work before they enroll in AP classes, which can prepare them for AP success. It is only through a commitment to equitable preparation and access that true equity and excellence can be achieved.
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Contents v Acknowledgments 1 About AP 4 AP Resources and Supports 6 Instructional Model 7 About the AP Calculus AB and BC Courses 7 College Course Equivalent 7 Prerequisites COURSE FRAMEWORK 11 Introduction 12 Course Framework Components 13 Mathematical Practices 15 Course Content 20 Course at a Glance 25 Unit Guides 26 Using the Unit Guides 29 UNIT 1: Limits and Continuity 51 UNIT 2: Differentiation: Definition and Fundamental Properties 67 UNIT 3: Differentiation: Composite, Implicit, and Inverse Functions 79 UNIT 4: Contextual Applications of Differentiation 91 UNIT 5: Analytical Applications of Differentiation 109 UNIT 6: Integration and Accumulation of Change 129 UNIT 7: Differential Equations 143 UNIT 8: Applications of Integration 163 UNIT 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions bc only 177 UNIT 10: Infinite Sequences and Series bc only INSTRUCTIONAL APPROACHES 201 Selecting and Using Course Materials 203 Instructional Strategies 214 Developing the Mathematical Practices EXAM INFORMATION 223 Exam Overview 228 Sample AP Calculus AB and BC Exam Questions SCORING GUIDELINES 241 Part A (AB or BC): Graphing Calculator Required 243 Part B (AB or BC): Calculator not Permitted 245 Part A (BC ONLY): Graphing Calculator Required 247 Part B (BC ONLY): Calculator not Permitted THIS PAGE IS INTENTIONALLY LEFT BLANK. Acknowledgments College Board would like to acknowledge the following committee members, consultants, and reviewers for their assistance with and commitment to the development of this course. All individuals and their affiliations were current at the time of contribution. Gabe Anderson, Valor Christian High School, Highlands Ranch, CO Gail Burrill, Michigan State University, East Lansing, MI Stephen Davis, Davidson College, Davidson, NC Theresa Horvath, McKinney High School, McKinney, TX Karen Hyers, Tartan High School, Oakdale, MN Jon Kawamura, West Salem High School, Salem, OR Ben Klein, Davidson College, Davidson, NC Quincy Loney, Cornell University, Ithaca, NY Kathleen Monegan, San Ysidro High School, San Diego, CA Brendan Murphy, John Bapst Memorial High School, Bangor, ME Mike Petrucelli, Mountain Lakes High School, Mountain Lakes, NJ Frances Rouse, UMS Wright Preparatory School, Mobile, AL James Sellers, The Pennsylvania State University, University Park, PA Bob Wilder, Design Thinking Academy, Christiana, DE College Board Staff Sara Hunter, Associate Director, AP Curricular Publications Tiffany Judkins, Director, AP Instructional Design and PD Resource Development Claire Lorenz, Senior Director, AP Instructional Design and PD Resource Development Daniel McDonough, Senior Director, AP Content Integration Stephanie Ogden, Director, AP Mathematics Content Development SPECIAL THANKS Benjamin Hedrick, Brian Robinson, and John R.
Williamson AP Calculus AB and BC Course and Exam Description V.1 | v Return to Table of Contents © 2020 College Board THIS PAGE IS INTENTIONALLY LEFT BLANK. About AP College Board’s Advanced Placement® Program (AP®) assessments—Personal Progress Checks—that enables willing and academically prepared students teachers can assign throughout the year to measure to pursue college-level studies—with the opportunity student progress as they acquire content knowledge to earn college credit, advanced placement, or and develop skills. both—while still in high school. Through AP courses in 38 subjects, each culminating in a challenging exam, students learn to think critically, construct solid Enrolling Students: arguments, and see many sides of an issue—skills Equity and Access that prepare them for college and beyond.
Taking AP College Board strongly encourages educators to courses demonstrates to college admission officers make equitable access a guiding principle for their that students have sought the most challenging AP programs by giving all willing and academically curriculum available to them, and research indicates prepared students the opportunity to participate that students who score a 3 or higher on an AP Exam in AP. We encourage the elimination of barriers typically experience greater academic success in that restrict access to AP for students from ethnic, college and are more likely to earn a college degree racial, and socioeconomic groups that have been than non-AP students. Each AP teacher’s syllabus traditionally underserved. College Board also believes is evaluated and approved by faculty from some of that all students should have access to academically the nation’s leading colleges and universities, and AP challenging coursework before they enroll in AP Exams are developed and scored by college faculty and classes, which can prepare them for AP success.
It is experienced AP teachers. Most four-year colleges and only through a commitment to equitable preparation universities in the United States grant credit, advanced and access that true equity and excellence can be placement, or both on the basis of successful AP achieved. Exam scores—more than 3,300 institutions worldwide annually receive AP scores. Offering AP Courses: AP Course Development The AP Course Audit The AP Program unequivocally supports the principle In an ongoing effort to maintain alignment with best that each school implements its own curriculum that will practices in college-level learning, AP courses and enable students to develop the content understandings exams emphasize challenging, research-based and skills described in the course framework.
curricula aligned with higher education expectations. While the unit sequence represented in this publication Individual teachers are responsible for designing their is optional, the AP Program does have a short list of own curriculum for AP courses, selecting appropriate curricular and resource requirements that must be college-level readings, assignments, and resources. fulfilled before a school can label a course “Advanced This course and exam description presents the content Placement” or “AP.” Schools wishing to offer AP and skills that are the focus of the corresponding courses must participate in the AP Course Audit, a college course and that appear on the AP Exam. It also process through which AP teachers’ course materials organizes the content and skills into a series of units are reviewed by college faculty.
The AP Course Audit that represent a sequence found in widely adopted was created to provide teachers and administrators college textbooks and that many AP teachers have with clear guidelines on curricular and resource told us they follow in order to focus their instruction. requirements for AP courses and to help colleges and The intention of this publication is to respect teachers’ universities validate courses marked “AP” on students’ time and expertise by providing a roadmap that they transcripts. This process ensures that AP teachers’ can modify and adapt to their local priorities and courses meet or exceed the curricular and resource preferences. Moreover, by organizing the AP course expectations that college and secondary school faculty content and skills into units, the AP Program is able have established for college-level courses.
to provide teachers and students with free formative AP Calculus AB and BC Course and Exam Description V.1 | 1 Return to Table of Contents © 2020 College Board The AP Course Audit form is submitted by the AP teacher response questions and through-course performance and the school principal (or designated administrator) to assessments, as applicable, are scored by thousands confirm awareness and understanding of the curricular of college faculty and expert AP teachers. Most are and resource requirements. A syllabus or course outline, scored at the annual AP Reading, while a small portion detailing how course requirements are met, is submitted is scored online. All AP Readers are thoroughly trained, by the AP teacher for review by college faculty.
and their work is monitored throughout the Reading for fairness and consistency. In each subject, a highly Please visit collegeboard.org/apcourseaudit for more respected college faculty member serves as Chief information to support the preparation and submission Faculty Consultant and, with the help of AP Readers of materials for the AP Course Audit. in leadership positions, maintains the accuracy of the scoring standards. Scores on the free-response How the AP Program questions and performance assessments are weighted Is Developed and combined with the results of the computer-scored multiple-choice questions, and this raw score is The scope of content for an AP course and exam is converted into a composite AP score on a 1–5 scale.
derived from an analysis of hundreds of syllabi and course offerings of colleges and universities. Using AP Exams are not norm-referenced or graded on a curve. this research and data, a committee of college faculty Instead, they are criterion-referenced, which means that and expert AP teachers work within the scope of every student who meets the criteria for an AP score of the corresponding college course to articulate what 2, 3, 4, or 5 will receive that score, no matter how many students should know and be able to do upon the students that is. The criteria for the number of points completion of the AP course.
The resulting course students must earn on the AP Exam to receive scores framework is the heart of this course and exam of 3, 4, or 5—the scores that research consistently description and serves as a blueprint of the content and validates for credit and placement purposes—include: skills that can appear on an AP Exam. §§ The number of points successful college students The AP Test Development Committees are responsible earn when their professors administer AP Exam for developing each AP Exam, ensuring the exam questions to them. questions are aligned to the course framework. The AP §§ The number of points researchers have found to Exam development process is a multiyear endeavor; all be predictive that an AP student will succeed when AP Exams undergo extensive review, revision, piloting, placed into a subsequent higher-level college course.
and analysis to ensure that questions are accurate, fair, §§ Achievement-level descriptions formulated by and valid, and that there is an appropriate spread of college faculty who review each AP Exam question. difficulty across the questions. Committee members are selected to represent a variety Using and Interpreting AP Scores of perspectives and institutions (public and private, The extensive work done by college faculty and small and large schools and colleges), and a range of AP teachers in the development of the course and gender, racial/ethnic, and regional groups.