Course Information

Course Title: Advanced Placement Calculus AB

Grade Level: 11-12

Prerequisites: Pre-Calculus (Honors or College Prep) with a grade of B- or better

Course Duration: Full Year

Credits: 1.0

Course Description

Advanced Placement Calculus AB is a rigorous, college-level mathematics course that follows the curriculum prescribed by the College Board. This course introduces students to the major concepts and tools for studying functions using calculus. Students will explore the concepts of limits, derivatives, integrals, and the Fundamental Theorem of Calculus. They will develop an understanding of these concepts through multiple representations-graphical, numerical, analytical, and verbal.

Belmont High School's AP Calculus AB program is designed to provide students with a solid foundation in differential and integral calculus, preparing them for success in college mathematics courses and potentially enabling them to earn college credit through the Advanced Placement Examination.

Course Objectives

Upon completion of this course, students will be able to:

• Understand and apply the concept of limits, including one-sided limits, limits at infinity, and continuity
• Compute derivatives using rules including the Power Rule, Product Rule, Quotient Rule, and Chain Rule
• Apply derivatives to solve optimization and related rates problems
• Understand and interpret the meaning of the derivative in various contexts
• Graph functions and their derivatives to analyze behavior, extrema, and points of inflection
• Understand and apply the concept of the definite integral
• Calculate integrals using various techniques including substitution
• Understand the relationship between derivatives and integrals as expressed in the Fundamental Theorem of Calculus
• Apply integration to solve area and volume problems
• Model real-world situations with differential equations and solve elementary differential equations
• Convey mathematical ideas clearly and effectively in both written and oral form
• Use technology appropriately as a tool for investigation and problem-solving

Units of Study

Unit 1: Limits and Continuity (approximately 2-3 weeks)

• Definition and intuitive understanding of limits
• Evaluating limits algebraically, graphically, and numerically
• Properties of limits
• One-sided limits and limits at infinity
• Continuity
• Intermediate Value Theorem
• Indeterminate forms and L'Hpital's Rule

Unit 2: Differentiation: Definition and Fundamental Properties (approximately 3 weeks)

• Definition of derivative as a limit
• Tangent lines and rates of change
• Derivative rules: Power Rule, Constant Multiple Rule, Sum Rule
• Product Rule and Quotient Rule
• Chain Rule
• Derivatives of trigonometric functions
• Implicit differentiation

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (approximately 2-3 weeks)

• Chain Rule applications
• Implicit differentiation applications
• Differentiation of inverse functions
• Differentiation of exponential and logarithmic functions
• Differentiation of inverse trigonometric functions

Unit 4: Contextual Applications of Differentiation (approximately 3-4 weeks)

• Related rates problems
• Linearization and differentials
• L'Hpital's Rule extended
• Curve sketching using derivatives
• Optimization problems
• Mean Value Theorem

Unit 5: Analytical Applications of Differentiation (approximately 3 weeks)

• Increasing and decreasing functions
• First and Second Derivative Tests
• Concavity and points of inflection
• Analyzing graphs of functions and derivatives
• Optimization problems
• Behaviors of functions at extreme values

Unit 6: Integration and Accumulation of Change (approximately 4 weeks)

• Antiderivatives and indefinite integrals
• Riemann sums
• Definite integrals
• Fundamental Theorem of Calculus
• Integration by substitution
• Properties of definite integrals
• Approximation methods

Unit 7: Differential Equations (approximately 2-3 weeks)

• Modeling with differential equations
• Verifying solutions to differential equations
• Slope fields
• Separable differential equations
• Exponential growth and decay models

Unit 8: Applications of Integration (approximately 3-4 weeks)

• Finding the area between curves
• Volumes of solids with known cross-sections
• Volumes of revolution (disk and washer methods)
• Arc length
• Applications to physics and economics

Unit 9: AP Review and Preparation (approximately 3-4 weeks)

• Comprehensive review of all calculus topics
• Practice with released AP exam questions
• Test-taking strategies
• Timed practice tests

Assessment Methods

Required Materials

• Textbook: Larson, Ron, and Bruce H. Edwards. Calculus of a Single Variable. 11th ed., Cengage Learning, 2017.
• Graphing calculator (TI-84 Plus, TI-84 Plus CE, or TI-89 recommended)
• Graph paper
• Notebook or binder with dividers
• Writing utensils (pencils, pens, highlighters)

Academic Integrity

Students are expected to maintain the highest standards of academic integrity. Cheating, plagiarism, or any form of academic dishonesty will not be tolerated and will result in appropriate disciplinary action according to Belmont High School's academic honesty policy. Students are encouraged to work collaboratively on many assignments, but must ensure that all written work submitted for grading represents their own understanding and effort.

Extra Help and Resources

• Teacher assistance: Available Tuesdays and Thursdays from 2:30-3:15 PM in Room 312
• Peer tutoring: Available through Belmont High School's Math Honor Society
• Online resources: Khan Academy, AP Classroom, and other supplementary materials will be provided on Google Classroom
• Review sessions: Scheduled prior to major tests and during AP exam preparation

Summer Assignment

Students enrolled in AP Calculus AB are required to complete a summer assignment that reviews essential pre-calculus concepts needed for success in calculus. This assignment will be distributed in June and must be submitted on the first day of classes. The assignment includes problems on functions, trigonometry, algebra, and analytic geometry.

AP Exam Information

The AP Calculus AB examination is scheduled by the College Board for early May. Students are strongly encouraged to take the exam, which consists of two sections:

1. Multiple Choice: 45 questions in 105 minutes (50% of exam score)
   • 30 questions without calculator (60 minutes)
   • 15 questions with calculator (45 minutes)
2. Free Response: 6 questions in 90 minutes (50% of exam score)
   • 2 questions with calculator (30 minutes)
   • 4 questions without calculator (60 minutes)

Colleges may grant credit and/or advanced placement to students who earn a score of 3, 4, or 5 on the AP Calculus AB examination. Students should check with prospective colleges for their specific AP credit policies.

Conclusion

AP Calculus AB at Belmont High School is designed to provide a comprehensive, college-level calculus experience that challenges students to develop deep mathematical understanding and problem-solving skills. Success in this course requires dedication, consistent effort, and active participation. The mathematical thinking developed in calculus will serve students well not only in future mathematics courses but also in fields such as science, engineering, economics, and medicine.