AP Calculus BC includes all topics in AP Calculus AB, as well as additional topics, such as differential and integral calculus (including parametric, polar, and vector functions) and series. It is equivalent to at least one year of calculus at most colleges and universities. AP Calculus BC is an extension of AP Calculus AB, and each course is challenging and demanding and requires a similar depth of understanding of topics.
AP Calculus AB and AP Calculus BC focus on students’ understanding of calculus concepts and provide experience with methods and applications. Through the use of big ideas of calculus (e.g., modeling change, approximation and limits, and analysis of functions), each course becomes a cohesive whole, rather than a collection of unrelated topics. Both courses require students to use definitions and theorems to build arguments and justify conclusions. The courses feature a multi-representational approach to calculus, with concepts, results, and problems expressed graphically, numerically, analytically, and verbally. Exploring connections among these representations builds understanding of how calculus applies limits to develop important ideas, definitions, formulas, and theorems. A sustained emphasis on clear communication of methods, reasoning, justifications, and conclusions is essential. Teachers and students should regularly use technology to reinforce relationships among functions, to confirm written work, to implement experimentation, and to
assist in interpreting results.
AP Calculus BC is designed to be the equivalent to both first and second semester college calculus courses. AP Calculus BC applies the content and skills learned in AP Calculus AB to parametrically defined curves, polar curves, and vector-valued functions; develops additional integration techniques and applications; and introduces the topics of sequences and series.
Study Scope and Sequence
Segment One
Module 01 - Limits and Continuity
Using Limits to Analyze Instantaneous Change
Estimating Limit Values from Graphs and Tables
Determining Limits Using Algebraic Properties and Manipulation
Selecting Procedures for Determining Limits
Squeeze Theorem and Representations of Limits
Determining Continuity and Exploring Discontinuity
Connecting Limits, Infinity, and Asymptotes
The Intermediate Value Theorem (IVT)
Module 02 - Differentiation: Definition and Fundamental Properties
Average and Instantaneous Rates of Change and the Derivative Definition
Determining Differentiability and Estimating Derivatives
Derivative Rules: Constant, Sum, Difference, Constant Multiple, and Power
The Product Rule and the Quotient Rule
Derivatives of Trigonometric Functions
Derivatives of Exponential and Logarithmic Functions
Module 03 - Differentiation: Composite, Implicit, and Inverse Functions
The Chain Rule
Implicit Differentiation
Differentiating Inverse Functions
Differentiating Inverse Trigonometric Functions
Selecting Procedures for Calculating Derivatives
Calculating Higher-Order Derivatives
Module 04 - Contextual Applications of Differentiation
Interpreting and Applying the Derivative in Motion
Rates of Change in Applied Contexts Other Than Motion
Related Rates
Approximating Values of a Function Using Local Linearity and Linearization
L\'Hospital\'s Rule
Module 05 - Analytical Applications of Differentiation
Mean Value and Extreme Value Theorems
Determining Function Behavior and the First Derivative Test
Using the Candidates Test to Determine Absolute Extrema
Determining Concavity of Functions and the Second Derivative Test
Connecting Graphs of Functions and Their Derivatives
Optimization Problems
Exploring Behaviors of Implicit Relations
Segment Two
Module 06 - Integration and Accumulation of Change
Exploring Accumulations of Change
Riemann Sums and the Definite Integral
Accumulation Functions Involving Area and the Fundamental Theorem of Calculus
Applying Properties of Definite Integrals
Finding Antiderivatives and Indefinite Integrals
Integrating Using Substitution
Integrating Using Integration by Parts
Integrating Using Linear Partial Fractions
Evaluating Improper Integrals
Integrating Functions Using Long Division and Completing the Square
Selecting Techniques for Antidifferentiation
Module 07 - Differential Equations
Solutions of Differential Equations
Sketching and Reasoning Using Slope Fields
Approximating Solutions Using Euler\'s Method
Finding Solutions Using Separation of Variables
Exponential Models with Differential Equations
Logistic Models with Differential Equations
Module 08 - Applications of Integration
Average Value and Connecting Position, Velocity, and Acceleration Using Integrals
Using Accumulation Functions and Definite Integrals in Applied Contexts
Finding the Area Between Curves
Finding the Area Between Curves That Intersect at More Than Two Points
Volumes with Discs
Volumes with Washers
Volumes with Cross Sections
The Arc Length of a Smooth, Planar Curve and Distance Traveled
Module 09 - Parametric, Polar, and Vector-Valued Equations
Differentiating Parametric Equations and Finding Arc Length
Differentiating and Integrating Vector-Valued Functions
Solving Motion Problems Using Parametric and Vector-Valued Functions
Defining Polar Coordinates and Differentiating in Polar Form
Finding Area Bounded by Polar Curves
Module 10 - Infinite Sequences and Series
Convergent and Divergent Infinite Series and Geometric Series
Integral Test for Convergence, Harmonic Series, and p-Series
Comparison Tests for Convergence
Additional Tests to Determine Convergence
Alternating Series and Their Error Bound
Taylor Polynomial Approximations of Functions and Evaluating Error
Radius and Interval of Convergence of Power Series
Finding Taylor or Maclaurin Series for a Function
Representing Functions as Power Series
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