This table of contents outlines the key topics covered in a typical Advanced Placement (AP) Calculus course, whether AB or BC. The depth of coverage for each topic may vary slightly depending on the specific curriculum and teacher. This guide provides a roadmap for students to anticipate the material they'll encounter.
Part 1: Limits and Continuity (Fundamentals)
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Chapter 1: Introduction to Limits
- Intuitive understanding of limits
- Graphical, numerical, and analytical approaches to evaluating limits
- One-sided limits
- Limits at infinity and infinite limits
- Properties of limits
- Indeterminate forms (0/0, ∞/∞) and L'Hôpital's Rule (brief introduction – more in later chapters)
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Chapter 2: Continuity
- Definition of continuity
- Types of discontinuities (removable, jump, infinite)
- Intermediate Value Theorem
- Properties of continuous functions
- Exploring continuity on closed intervals
Part 2: Differentiation (The Core of Calculus)
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Chapter 3: Derivatives and Rates of Change
- Definition of the derivative as a limit
- Geometric interpretation of the derivative (slope of the tangent line)
- Differentiability vs. continuity
- Power rule, sum/difference rule, constant multiple rule, product rule, quotient rule, chain rule
- Higher-order derivatives
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Chapter 4: Applications of Derivatives
- Related rates problems
- Optimization problems (finding maximum and minimum values)
- Mean Value Theorem
- Rolle's Theorem
- Increasing/decreasing functions and concavity
- First and Second Derivative Tests
- Curve sketching and graphing
Part 3: Integration (The Inverse of Differentiation)
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Chapter 5: Introduction to Integrals
- Definite integrals as areas under curves
- Riemann sums (left, right, midpoint)
- Fundamental Theorem of Calculus (Part 1 and Part 2)
- Indefinite integrals
- Power rule for integration
- Basic integration techniques (u-substitution)
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Chapter 6: Applications of Integrals
- Area between curves
- Volumes of solids of revolution (disk/washer and shell methods)
- Average value of a function
Part 4: Advanced Topics (AP Calculus BC Only)
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Chapter 7: Transcendental Functions
- Derivatives and integrals of exponential and logarithmic functions
- Derivatives and integrals of inverse trigonometric functions
- L'Hôpital's Rule (in-depth coverage)
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Chapter 8: Sequences and Series
- Sequences and their limits
- Infinite series (convergence and divergence tests)
- Taylor and Maclaurin series
- Power series
Part 5: Differential Equations (Introduction)
- Chapter 9: Introduction to Differential Equations
- Basic differential equations and their solutions
- Slope fields
- Separable differential equations
Part 6: Practice and Review
- Chapter 10: Review of Key Concepts and Formulas
- Comprehensive summary of important definitions, theorems, and techniques.
- Chapter 11: Practice Problems and Exams
- A wide range of practice problems, including multiple-choice and free-response questions, to prepare for the AP exam.
This table of contents serves as a general guide. The specific order and depth of coverage might differ based on the textbook and instructor's preferences. Students should consult their syllabus and course materials for a detailed outline of their specific AP Calculus curriculum.
