AP Calculus Syllabus

Preparation for Calculus

P.1 Graphs and Models

P.2 Linear Models and Rates of Change

P.3 Functions and Their Graphs

P.4 Fitting Models to Data

1 Limits and Their Properties

1.1 A Preview of Calculus

1.2 Finding Limits Graphically and Numerically

1.3 Evaluating Limits Analytically

1.4 Continuity and One-Sided Limits

1.5 Infinite Limits

2 Differentiation

2.1 The Derivative and the Tangent Line Problem

2.2 Basic Differentiation Rules and Rates of Change

2.3 Product and Quotient Rules and Higher-Order Derivatives

2.4 The Chain Rule

2.5 Implicit Differentiation

2.6 Related Rates

3 Applications of Differentiation

3.1 Extrema on an Interval

3.2 Rolle's Theorem and the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

3.4 Concavity and the Second Derivative Test

3.5 Limits at Infinity

3.6 A Summary of Curve Sketching

3.7 Optimization Problems

3.8 Newton's Method

3.9 Differentials

4 Integration

4.1 Antiderivatives and Indefinite Integration

4.2 Area

4.3 Riemann Sums and Definite Integrals

4.4 The Fundamental Theorem of Calculus

4.5 Integration by Substitution

4.6 Numerical Integration

5 Logarithmic, Exponential, and Other Transcendental Functions

5.1 The Natural Logarithmic Function: Differentiation

5.2 The Natural Logarithmic Function: Integration

5.3 Inverse Functions

5.4 Exponential Functions: Differentiation and Integration

5.5 Bases Other Than e and Applications

5.6 Inverse Trigonometric Functions: Differentiation

5.7 Inverse Trigonometric Functions: Integration

5.8 Hyperbolic Functions

6 Differential Equations

6.1 Slope Fields and Euler's Method

6.2 Differential Equations: Growth and Decay

6.3 Separation of Variables and the Logistic Equation

6.4 First-Order Linear Differential Equations

Applications of Integration

7.1 Area of a Region Between Two Curves

7.2 Volume: The Disk Method

7.3 Volume: The Shell Method

7.4 Arc Length and Surfaces of Revolution

7.5 Work

7.6 Moments, Centers of Mass, and Centroids

7.7 Fluid Pressure and Fluid Force

8 Integration Techniques,L Hopital's Rule, and Improper Integrals

8.1 Basic Integration Rules

8.2 Integration by Parts

8.3 Trigonometric Integrals

8.4 Trigonometric Substitution

8.5 Partial Fractions

8.6 Integration by Tables and Other Integration Techniques

8.7 Indeterminate Forms and L' Hopital's Rule

8.8 Improper Integrals

9 Infinite Series

9.1 Sequences

9.2 Series and Convergence

9.3 The Integral Test and p-Series

9.4 Comparisons of Series

9.5 Alternating Series

9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations

9.8 Power Series

9.9 Representation of Functions by Power Series

9.1 Taylor and Maclaurin Series

1O Conics, Parametric Equations, and Polar Coordinates

10.1 Conics and Calculus

10.2 Plane Curves and Parametric Equations

10.3 Parametric Equations and Calculus

10.4 Polar Coordinates and Polar Graphs

10.5 Area and Arc Length in Polar Coordinates

10.6 Polar Equations of Conics and Kepler's Laws

11 Vectors and the Geometry of Space

11.1 Vectors in the Plane

11.2 Space Coordinates and Vectors in Space

11.3 The Dot Product of Two Vectors

11.4 The Cross Product of Two Vectors in Space

11.5 Lines and Planes in Space

11.6 Surfaces in Space

11.7 Cylindrical and Spherical Coordinates

12 Vector-Valued Functions

12.1 Vector-Valued Functions

12.2 Differentiation and Integration of Vector-Valued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors

12.5 Arc Length and Curvature

13 Functions of Several Variables

13.1 Introduction to Functions of Several Variables

13.2 Limits and Continuity

13.3 Partial Derivatives

13.4 Differentials

13.5 Chain Rules for Functions of Several Variables

13.6 Directional Derivatives and Gradients

13.7 Tangent Planes and Normal Lines

13.8 Extrema of Functions of Two Variables

13.9 Applications of Extrema of Functions of Two Variables

13.1 Lagrange Multipliers

14 Multiple Integration

14.1 Iterated Integrals and Area in the Plane

14.2 Double Integrals and Volume

14.3 Change of Variables:Polar Coordinates14.4 Center of Mass and Moments of Inertia

14.5 Surface Area

14.6 Triple Integrals and Applications

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Change of Variables:Jacobians

15 Vector Analysis

15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields and Independence of Path

15.4 Green's Theorem

15.5 Parametric Surfaces

15.6 Surface Integrals

15.7 Divergence Theorem

15.8 Stokes's Theorem

• 本课程适合已完成Algebra2和Pre Calculus的学生

• This course is suitable for students who have completed Algebra2 and Pre Calculus.