Master the material tested on the College Board's Advanced Placement Calculus AB exam and earn up to 1 semester of university-level calculus credit. Learn the mathematics of change that underlies science and engineering.

- Connect algebraic and graphical interpretations of limits, including relationships to vertical and horizontal asymptotes.
- Estimate limits numerically and compute limits using algebraic manipulation.
- Define continuity in terms of limits, determine intervals over which a function is continuous, and remove discontinuities of functions.
- Explain the behavior of a function on an interval using the Intermediate Value Theorem.

- Interpret the difference quotient geometrically and use it to compute the derivative of a function.
- Estimate derivatives numerically and apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives.
- Understand the relationship between differentiability and continuity.
- Use implicit differentiation to solve related rates problems.
- Approximate values of functions using linearization.
- Use L’Hopital’s Rule to calculate limits of indeterminate forms.
- Relate the graph of a function to properties of its derivative.
- Solve optimization problems by using the derivative to find extrema of functions.

- Evaluate Riemann sums and interpret them geometrically and contextually.
- Compute the integral of a function as the limit of a Riemann sum.
- Relate integrals and antiderivatives through the fundamental theorem of calculus.
- Use integration to compute area and volume.
- Interpret accumulation functions geometrically and contextually and compute their derivatives.
- Solve advanced integrals using techniques like substitution, long division, and completing the square.

- Construct differential equations to model real-world situations involving rates of change.
- Verify solutions of differential equations, sketch slope fields, and relate solutions to slope fields.
- Find solutions of initial value problems using separation of variables.
- Interpret exponential growth models in context.

1.

Limits and Continuity
43 topics

1.1. Estimating Limits from Graphs

1.1.1. | The Finite Limit of a Function | |

1.1.2. | The Left and Right-Sided Limits of a Function | |

1.1.3. | Finding the Existence of a Limit Using One-Sided Limits | |

1.1.4. | Limits at Infinity from Graphs | |

1.1.5. | Infinite Limits from Graphs |

1.2. The Algebra of Limits

1.2.1. | Limits of Power Functions, and the Constant Rule for Limits | |

1.2.2. | The Sum Rule for Limits | |

1.2.3. | The Product and Quotient Rules for Limits | |

1.2.4. | The Power and Root Rules for Limits |

1.3. Limits of Functions

1.3.1. | Limits at Infinity of Polynomials | |

1.3.2. | Limits of Reciprocal Functions | |

1.3.3. | Limits of Exponential Functions | |

1.3.4. | Limits of Logarithmic Functions | |

1.3.5. | Limits of Radical Functions | |

1.3.6. | Limits of Trigonometric Functions | |

1.3.7. | Limits of Reciprocal Trigonometric Functions | |

1.3.8. | Limits of Piecewise Functions |

1.4. Determining Limits Using Algebraic Manipulation

1.4.1. | Calculating Limits of Rational Functions by Factoring | |

1.4.2. | Limits of Absolute Value Functions | |

1.4.3. | Calculating Limits of Radical Functions Using Conjugate Multiplication | |

1.4.4. | Calculating Limits Using Trigonometric Identities | |

1.4.5. | Limits at Infinity and Horizontal Asymptotes of Rational Functions | |

1.4.6. | Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions | |

1.4.7. | Evaluating Limits at Infinity of Radical Functions | |

1.4.8. | Vertical Asymptotes of Rational Functions | |

1.4.9. | Connecting Infinite Limits and Vertical Asymptotes of Rational Functions |

1.5. Special Limits

1.5.1. | The Squeeze Theorem | |

1.5.2. | Special Limits Involving Sine | |

1.5.3. | Evaluating Special Limits Involving Sine Using a Substitution | |

1.5.4. | Special Limits Involving Cosine |

1.6. Continuity

1.6.1. | Determining Continuity from Graphs | |

1.6.2. | Defining Continuity at a Point | |

1.6.3. | Left and Right Continuity | |

1.6.4. | Further Continuity of Piecewise Functions | |

1.6.5. | Point Discontinuities | |

1.6.6. | Jump Discontinuities | |

1.6.7. | Discontinuities Due to Vertical Asymptotes | |

1.6.8. | Continuity Over an Interval | |

1.6.9. | Continuity of Functions | |

1.6.10. | The Intermediate Value Theorem |

1.7. Removing Discontinuities

1.7.1. | Removing Point Discontinuities | |

1.7.2. | Removing Jump Discontinuities | |

1.7.3. | Removing Discontinuities From Rational Functions |

2.

Differentiation: Definition and Fundamental Properties
19 topics

2.8. Introduction to Differentiation

2.8.1. | The Average Rate of Change of a Function over a Varying Interval | |

2.8.2. | The Instantaneous Rate of Change of a Function at a Point | |

2.8.3. | Defining the Derivative Using Derivative Notation | |

2.8.4. | Connecting Differentiability and Continuity | |

2.8.5. | The Power Rule for Differentiation | |

2.8.6. | The Sum and Constant Multiple Rules for Differentiation | |

2.8.7. | Calculating the Slope of a Tangent Line Using Differentiation | |

2.8.8. | Calculating the Equation of a Tangent Line Using Differentiation | |

2.8.9. | Calculating the Equation of a Normal Line Using Differentiation |

2.9. Derivatives of Functions and the Rules of Differentiation

2.9.1. | Differentiating Exponential Functions | |

2.9.2. | Differentiating Logarithmic Functions | |

2.9.3. | Differentiating Trigonometric Functions | |

2.9.4. | Second and Higher Order Derivatives | |

2.9.5. | The Product Rule for Differentiation | |

2.9.6. | The Quotient Rule for Differentiation | |

2.9.7. | Differentiating Reciprocal Trigonometric Functions | |

2.9.8. | Calculating Derivatives From Data and Tables | |

2.9.9. | Calculating Derivatives From Graphs | |

2.9.10. | Recognizing Derivatives in Limits |

3.

Differentiation: Composite, Implicit, and Inverse Functions
13 topics

3.10. Differentiating Composite Functions

3.10.1. | The Chain Rule for Differentiation | |

3.10.2. | The Chain Rule With Exponential Functions | |

3.10.3. | The Chain Rule With Logarithmic Functions | |

3.10.4. | The Chain Rule With Trigonometric Functions | |

3.10.5. | Calculating Derivatives From Data Using the Chain Rule | |

3.10.6. | Calculating Derivatives From Graphs Using the Chain Rule | |

3.10.7. | Selecting Procedures for Calculating Derivatives |

3.11. Differentiating Implicit and Inverse Functions

3.11.1. | Implicit Differentiation | |

3.11.2. | Calculating dy/dx Using dx/dy | |

3.11.3. | Differentiating Inverse Functions | |

3.11.4. | Differentiating an Inverse Function at a Point | |

3.11.5. | Differentiating Inverse Trigonometric Functions | |

3.11.6. | Differentiating Inverse Reciprocal Trigonometric Functions |

4.

Contextual Applications of Differentiation
16 topics

4.12. Contextual Applications of Differentiation

4.12.1. | Interpreting the Meaning of the Derivative in Context | |

4.12.2. | Rates of Change in Applied Contexts |

4.13. Estimating Derivatives

4.13.1. | Estimating Derivatives Using a Forward Difference Quotient | |

4.13.2. | Estimating Derivatives Using a Backward Difference Quotient | |

4.13.3. | Estimating Derivatives Using a Central Difference Quotient |

4.14. Displacement, Velocity, and Acceleration

4.14.1. | Calculating Velocity for Straight-Line Motion Using Differentiation | |

4.14.2. | Determining Characteristics of Moving Objects Using Differentiation | |

4.14.3. | Calculating Acceleration for Straight-Line Motion Using Differentiation |

4.15. Related Rates of Change

4.15.1. | Introduction to Related Rates | |

4.15.2. | Calculating Related Rates With Circles and Spheres | |

4.15.3. | Calculating Related Rates With Squares | |

4.15.4. | Calculating Related Rates With Rectangular Solids | |

4.15.5. | Calculating Related Rates Using the Pythagorean Theorem | |

4.15.6. | Calculating Related Rates Using Similar Triangles | |

4.15.7. | Calculating Related Rates Using Trigonometry | |

4.15.8. | Calculating Related Rates With Cones |

5.

Analytical Applications of Differentiation
28 topics

5.16. L'Hopital's Rule

5.16.1. | L'Hopital's Rule | |

5.16.2. | L'Hopital's Rule Applied to Tables |

5.17. Analytical Applications of Differentiation

5.17.1. | The Mean Value Theorem | |

5.17.2. | Global vs. Local Extrema and Critical Points | |

5.17.3. | The Extreme Value Theorem | |

5.17.4. | Using Differentiation to Calculate Critical Points | |

5.17.5. | Determining Intervals on Which a Function Is Increasing or Decreasing | |

5.17.6. | Using the First Derivative Test to Classify Local Extrema | |

5.17.7. | Using the Candidates Test to Determine Global Extrema | |

5.17.8. | Intervals of Concavity | |

5.17.9. | Relating Concavity to the Second Derivative | |

5.17.10. | Points of Inflection | |

5.17.11. | Using the Second Derivative Test to Determine Extrema |

5.18. Analysis of Curves

5.18.1. | Sketching the Derivative of a Function From the Function's Graph | |

5.18.2. | Interpreting the Graph of a Function's Derivative | |

5.18.3. | Interpreting the Graph of a Function's Derivative: Concavity and Points of Inflection | |

5.18.4. | Sketching a Function From the Graph of its Derivative | |

5.18.5. | Sketching a Function Given Some Derivative Properties |

5.19. Approximating Values of a Function

5.19.1. | Approximating Functions Using Local Linearity and Linearization | |

5.19.2. | Approximating the Roots of a Number Using Local Linearity | |

5.19.3. | Approximating Trigonometric Functions Using Local Linearity |

5.20. Optimization

5.20.1. | Optimization Problems Involving Rectangles | |

5.20.2. | Optimization Problems Involving Sectors of Circles | |

5.20.3. | Optimization Problems Involving Boxes and Trays | |

5.20.4. | Optimization Problems Involving Cylinders | |

5.20.5. | Finding Minimum Distances | |

5.20.6. | Optimization Problems With Inscribed Shapes | |

5.20.7. | Optimization Problems in Economics |

6.

Integration
48 topics

6.21. Indefinite Integrals

6.21.1. | The Antiderivative | |

6.21.2. | The Constant Multiple Rule for Indefinite Integrals | |

6.21.3. | The Sum Rule for Indefinite Integrals | |

6.21.4. | Integrating the Reciprocal Function | |

6.21.5. | Integrating Exponential Functions | |

6.21.6. | Integrating Trigonometric Functions | |

6.21.7. | Integration Using Inverse Trigonometric Functions |

6.22. Approximating Areas with Riemann Sums

6.22.1. | Approximating Areas With the Left Riemann Sum | |

6.22.2. | Approximating Areas With the Right Riemann Sum | |

6.22.3. | Approximating Areas With the Midpoint Riemann Sum | |

6.22.4. | Approximating Areas of Composite Functions Using Riemann Sums | |

6.22.5. | Approximating Areas With the Trapezoidal Rule | |

6.22.6. | Left and Right Riemann Sums in Sigma Notation | |

6.22.7. | Midpoint and Trapezoidal Rules in Sigma Notation |

6.23. Definite Integrals

6.23.1. | Defining Definite Integrals Using Left and Right Riemann Sums | |

6.23.2. | The Fundamental Theorem of Calculus | |

6.23.3. | Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions | |

6.23.4. | The Sum and Constant Multiple Rules for Definite Integrals | |

6.23.5. | Properties of Definite Integrals Involving the Limits of Integration |

6.24. The Area Under a Curve

6.24.1. | The Area Bounded by a Curve and the X-Axis | |

6.24.2. | Evaluating Definite Integrals Using Symmetry | |

6.24.3. | Finding the Area Between a Curve and the X-Axis When They Intersect | |

6.24.4. | The Area Bounded by a Curve and the Y-Axis | |

6.24.5. | Calculating the Definite Integral of a Function Given Its Graph | |

6.24.6. | Calculating the Definite Integral of a Function's Derivative Given its Graph | |

6.24.7. | Definite Integrals of Functions with Discontinuities |

6.25. Accumulation Functions

6.25.1. | The Integral as an Accumulation Function | |

6.25.2. | The Second Fundamental Theorem of Calculus | |

6.25.3. | Maximizing a Function Using the Graph of Its Derivative | |

6.25.4. | Minimizing a Function Using the Graph of its Derivative | |

6.25.5. | Further Optimizing Functions Using Graphs of Derivatives | |

6.25.6. | Integrating Rates of Change | |

6.25.7. | Integrating Density Functions |

6.26. Integration Using Substitution

6.26.1. | Integrating Algebraic Functions Using Substitution | |

6.26.2. | Integrating Linear Rational Functions Using Substitution | |

6.26.3. | Integration Using Substitution | |

6.26.4. | Calculating Definite Integrals Using Substitution | |

6.26.5. | Further Integration of Algebraic Functions Using Substitution | |

6.26.6. | Integrating Exponential Functions Using Linear Substitution | |

6.26.7. | Integrating Exponential Functions Using Substitution | |

6.26.8. | Integrating Trigonometric Functions Using Substitution | |

6.26.9. | Integrating Logarithmic Functions Using Substitution | |

6.26.10. | Integration by Substitution With Inverse Trigonometric Functions |

6.27. Integration Using Trigonometric Identities

6.27.1. | Integration Using Basic Trigonometric Identities | |

6.27.2. | Integration Using the Pythagorean Identities | |

6.27.3. | Integration Using the Double Angle Formulas |

6.28. Special Techniques for Integration

6.28.1. | Integrating Functions Using Polynomial Division | |

6.28.2. | Integrating Functions by Completing the Square |

7.

Differential Equations
18 topics

7.29. Introduction to Differential Equations

7.29.1. | Introduction to Differential Equations | |

7.29.2. | Verifying Solutions of Differential Equations | |

7.29.3. | Solving Differential Equations Using Direct Integration | |

7.29.4. | Solving First-Order ODEs Using Separation of Variables | |

7.29.5. | Solving Initial Value Problems Using Separation of Variables | |

7.29.6. | Modeling With Differential Equations | |

7.29.7. | Further Modeling With Differential Equations | |

7.29.8. | Exponential Growth and Decay Models With Differential Equations | |

7.29.9. | Exponential Growth and Decay Models With Differential Equations: Calculating Unknown Times and Initial Values | |

7.29.10. | Exponential Growth and Decay Models With Differential Equations: Half-Life Problems |

7.30. Qualitative Techniques for Differential Equations

7.30.1. | Qualitative Analysis of Differential Equations | |

7.30.2. | Equilibrium Solutions of Differential Equations |

7.31. Slope Fields

7.31.1. | Slope Fields for Directly Integrable Differential Equations | |

7.31.2. | Slope Fields for Autonomous Differential Equations | |

7.31.3. | Slope Fields for Nonautonomous Differential Equations | |

7.31.4. | Analyzing Slope Fields for Directly Integrable Differential Equations | |

7.31.5. | Analyzing Slope Fields for Autonomous Differential Equations | |

7.31.6. | Analyzing Slope Fields for Nonautonomous Differential Equations |

8.

Applications of Integration
18 topics

8.32. Applications of Integration

8.32.1. | The Average Value of a Function | |

8.32.2. | The Area Between Curves Expressed as Functions of X | |

8.32.3. | The Area Between Curves Expressed as Functions of Y | |

8.32.4. | Finding Areas Between Curves that Intersect at More Than Two Points |

8.33. Connecting Position, Velocity and Acceleration Using Integrals

8.33.1. | Calculating Velocity Using Integration | |

8.33.2. | Determining Characteristics of Moving Objects Using Integration | |

8.33.3. | Calculating the Position Function of a Particle Using Integration | |

8.33.4. | Calculating the Displacement of a Particle Using Integration | |

8.33.5. | Calculating the Total Distance Traveled by a Particle | |

8.33.6. | Average Position, Velocity, and Acceleration |

8.34. Volumes of Solids With Known Cross Sections

8.34.1. | Volumes of Solids with Square Cross Sections | |

8.34.2. | Volumes of Solids with Rectangular Cross Sections | |

8.34.3. | Volumes of Solids with Triangular Cross Sections | |

8.34.4. | Volumes of Solids with Circular Cross Sections |

8.35. Volumes of Revolution

8.35.1. | Volumes of Revolution Using the Disc Method: Rotation About the Coordinate Axes | |

8.35.2. | Volumes of Revolution Using the Disc Method: Rotation About Other Axes | |

8.35.3. | Volumes of Revolution Using the Washer Method: Rotation About the Coordinate Axes | |

8.35.4. | Volumes of Revolution Using the Washer Method: Rotation About Other Axes |

9.

Applications of Technology
7 topics

9.36. Using Graphing Calculators

9.36.1. | Evaluating Expressions Using a Graphing Calculator | |

9.36.2. | Finding Roots of Functions Using a Graphing Calculator | |

9.36.3. | Finding Intersections of Functions Using a Graphing Calculator | |

9.36.4. | Finding Extrema of Functions Using a Graphing Calculator | |

9.36.5. | Finding Derivatives Using a Graphing Calculator | |

9.36.6. | Finding Definite Integrals Using a Graphing Calculator | |

9.36.7. | Exploring Functions Using Technology |