2023年AP微积分BC考试大纲 AP Calculus BC

Course Content

Unit 1: Limits and Continuity

Unit 2: Differentiation: Definition and Fundamental Properties

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Unit 4: Contextual Applications of Differentiation

Unit 5: Analytical Applications of Differentiation

Unit 6: Integration and Accumulation of Change

Unit 7: Differential Equations

Unit 8: Applications of Integration

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Unit 10: Infinite Sequences and Series

AP Calculus BC Exam Date

Mon, May 9, 2022, 8 AM Local

Unit 1: Limits and Continuity

You’ll start to explore how limits will allow you to solve problems involving change and to better understand mathematical reasoning about functions.

Topics may include:

How limits help us to handle change at an instant

Definition and properties of limits in various representations

Definitions of continuity of a function at a point and over a domain

Asymptotes and limits at infinity

Reasoning using the Squeeze theorem and the Intermediate Value Theorem

On The Exam

4%–7% of exam score

Unit 2: Differentiation: Definition and Fundamental Properties

You’ll apply limits to define the derivative, become skillful at determining

derivatives, and continue to develop mathematical reasoning skills.

Topics may include:

Defining the derivative of a function at a point and as a function

Connecting differentiability and continuity

Determining derivatives for elementary functions

Applying differentiation rules

On The Exam

4%–7% of exam score

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

You’ll master using the chain rule, develop new differentiation techniques, and

be introduced to higher-order derivatives.

Topics may include:

The chain rule for differentiating composite functions

Implicit differentiation

Differentiation of general and particular inverse functions

Determining higher-order derivatives of functions

On The Exam

4%–7% of exam score

Unit 4: Contextual Applications of Differentiation

You’ll apply derivatives to set up and solve real-world problems involving

instantaneous rates of change and use mathematical reasoning to determine limits

of certain indeterminate forms.

Topics may include:

Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change

Applying understandings of differentiation to problems involving motion

Generalizing understandings of motion problems to other situations involving

rates of change

Solving related rates problems

Local linearity and approximation

L’Hospital’s rule

On The Exam

6%–9% of exam score

Unit 5: Analytical Applications of Differentiation

After exploring relationships among the graphs of a function and its derivatives,

you'll learn to apply calculus to solve optimization problems.

Topics may include:

Mean Value Theorem and Extreme Value Theorem

Derivatives and properties of functions

How to use the first derivative test, second derivative test, and candidates test

Sketching graphs of functions and their derivatives

How to solve optimization problems

Behaviors of Implicit relations

On The Exam

8%–11% of exam score

Unit 6: Integration and Accumulation of Change

You’ll learn to apply limits to define definite integrals and how the

Fundamental Theorem connects integration and differentiation. You’ll apply

properties of integrals and practice useful integration techniques.

Topics may include:

Using definite integrals to determine accumulated change over an interval

Approximating integrals with Riemann Sums

Accumulation functions, the Fundamental Theorem of Calculus, and definite

integrals

Antiderivatives and indefinite integrals

Properties of integrals and integration techniques, extended

Determining improper integrals

on The Exam

17%–20% of exam score

Unit 7: Differential Equations

You’ll learn how to solve certain differential equations and apply that

knowledge to deepen your understanding of exponential growth and decay and

logistic models.

Topics may include:

Interpreting verbal descriptions of change as separable differential equations

Sketching slope fields and families of solution curves

Using Euler’s method to approximate values on a particular solution curve

Solving separable differential equations to find general and particular solutions

Deriving and applying exponential and logistic models

On The Exam

6%–9% of exam score

Unit 8: Applications of Integration

You’ll make mathematical connections that will allow you to solve a wide range

of problems involving net change over an interval of time and to find lengths of

curves, areas of regions, or volumes of solids defined using functions.

Topics may include:

Determining the average value of a function using definite integrals

Modeling particle motion

Solving accumulation problems

Finding the area between curves

Determining volume with cross-sections, the disc method, and the washer method

Determining the length of a planar curve using a definite integral

on The Exam

6%–9% of exam score

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

You’ll solve parametrically defined functions, vector-valued functions, and

polar curves using applied knowledge of differentiation and integration. You’ll

also deepen your understanding of straight-line motion to solve problems

involving curves.

Topics may include:

Finding derivatives of parametric functions and vector-valued functions

Calculating the accumulation of change in length over an interval using a

definite integral

Determining the position of a particle moving in a plane

Calculating velocity, speed, and acceleration of a particle moving along a curve

Finding derivatives of functions written in polar coordinates

Finding the area of regions bounded by polar curves

On The Exam

11%–12% of exam score

Unit 10: Infinite Sequences and Series

You’ll explore convergence and divergence behaviors of infinite series and learn

how to represent familiar functions as infinite series. You’ll also learn how to

determine the largest possible error associated with certain approximations

involving series.

Topics may include:

Applying limits to understand convergence of infinite series

Types of series: Geometric, harmonic, and p-series

A test for divergence and several tests for convergence

Approximating sums of convergent infinite series and associated error bounds

Determining the radius and interval of convergence for a series

Representing a function as a Taylor series or a Maclaurin series on an

appropriate interval

On The Exam

17%–18% of exam score

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