AP Calculus AB

Build the language of limits, connect graphical, numerical, and algebraic representations, and use continuity theorems to justify conclusions about function behavior.

1. 1.1Introducing Calculus: Can Change Occur at an Instant?→
2. 1.2Defining Limits and Using Limit Notation→
3. 1.3Estimating Limit Values from Graphs→
4. 1.4Estimating Limit Values from Tables→
5. 1.5Determining Limits Using Algebraic Properties of Limits→
6. 1.6Determining Limits Using Algebraic Manipulation→
7. 1.7Selecting Procedures for Determining Limits→
8. 1.8Determining Limits Using the Squeeze Theorem→
9. 1.9Connecting Multiple Representations of Limits→
10. 1.10Exploring Types of Discontinuities→
11. 1.11Defining Continuity at a Point→
12. 1.12Confirming Continuity over an Interval→
13. 1.13Removing Discontinuities→
14. 1.14Connecting Infinite Limits and Vertical Asymptotes→
15. 1.15Connecting Limits at Infinity and Horizontal Asymptotes→
16. 1.16Working with the Intermediate Value Theorem (IVT)→

Develop the derivative from rates of change and limits, then apply foundational derivative rules while connecting differentiability with continuity.

1. 2.1Defining Average and Instantaneous Rates of Change at a Point→
2. 2.2Defining the Derivative of a Function and Using Derivative Notation→
3. 2.3Estimating Derivatives of a Function at a Point→
4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist→
5. 2.5Applying the Power Rule→
6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple→
7. 2.7Derivatives of cos x, sin x, eˣ, and ln x→
8. 2.8The Product Rule→
9. 2.9The Quotient Rule→
10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions→

Extend differentiation to composite, implicit, inverse, and inverse-trigonometric functions and choose efficient procedures for higher-order derivatives.

1. 3.1The Chain Rule→
2. 3.2Implicit Differentiation→
3. 3.3Differentiating Inverse Functions→
4. 3.4Differentiating Inverse Trigonometric Functions→
5. 3.5Selecting Procedures for Calculating Derivatives→
6. 3.6Calculating Higher-Order Derivatives→

Interpret derivatives as contextual rates, model motion and related rates, and use local linearity to make and assess approximations.

1. 4.1Interpreting the Meaning of the Derivative in Context→
2. 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration→
3. 4.3Rates of Change in Applied Contexts Other Than Motion→
4. 4.4Introduction to Related Rates→
5. 4.5Solving Related Rates Problems→
6. 4.6Approximating Values of a Function Using Local Linearity and Linearization→

Use derivative tests and major theorems to analyze extrema, monotonicity, concavity, graph behavior, optimization, and implicit relations.

1. 5.1Using the Mean Value Theorem→
2. 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points→
3. 5.3Determining Intervals on Which a Function Is Increasing or Decreasing→
4. 5.4Using the First Derivative Test to Determine Relative (Local) Extrema→
5. 5.5Using the Candidates Test to Determine Absolute (Global) Extrema→
6. 5.6Determining Concavity of Functions over Their Domains→
7. 5.7Using the Second Derivative Test to Determine Extrema→
8. 5.8Sketching Graphs of Functions and Their Derivatives→
9. 5.9Connecting a Function, Its First Derivative, and Its Second Derivative→
10. 5.10Introduction to Optimization Problems→
11. 5.11Solving Optimization Problems→
12. 5.12Exploring Behaviors of Implicit Relations→

Connect accumulation, Riemann sums, definite integrals, and the Fundamental Theorem of Calculus, then select effective antiderivative techniques.

1. 6.1Exploring Accumulations of Change→
2. 6.2Approximating Areas with Riemann Sums→
3. 6.3Riemann Sums, Summation Notation, and Definite Integral Notation→
4. 6.4The Fundamental Theorem of Calculus and Accumulation Functions→
5. 6.5Interpreting the Behavior of Accumulation Functions Involving Area→
6. 6.6Applying Properties of Definite Integrals→
7. 6.7The Fundamental Theorem of Calculus and Definite Integrals→
8. 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation→
9. 6.9Integrating Using Substitution→
10. 6.10Integrating Functions Using Long Division and Completing the Square→
11. 6.14Selecting Techniques for Antidifferentiation→

Model change with differential equations, interpret slope fields, verify and approximate solutions, and solve contextual growth and decay problems.

1. 7.1Modeling Situations with Differential Equations→
2. 7.2Verifying Solutions for Differential Equations→
3. 7.3Sketching Slope Fields→
4. 7.4Reasoning Using Slope Fields→
5. 7.6Finding General Solutions Using Separation of Variables→
6. 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables→
7. 7.8Exponential Models with Differential Equations→

Apply definite integrals to average value, motion, net accumulation, planar area, cross-sectional volume, and solids of revolution.

1. 8.1Finding the Average Value of a Function on an Interval→
2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals→
3. 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts→
4. 8.4Finding the Area Between Curves Expressed as Functions of x→
5. 8.5Finding the Area Between Curves Expressed as Functions of y→
6. 8.6Finding the Area Between Curves That Intersect at More Than Two Points→
7. 8.7Volumes with Cross Sections: Squares and Rectangles→
8. 8.8Volumes with Cross Sections: Triangles and Semicircles→
9. 8.9Volume with Disc Method: Revolving Around the x- or y-Axis→
10. 8.10Volume with Disc Method: Revolving Around Other Axes→
11. 8.11Volume with Washer Method: Revolving Around the x- or y-Axis→
12. 8.12Volume with Washer Method: Revolving Around Other Axes→