Master the material tested on College Board's Advanced Placement Calculus BC exam and earn up to 2 semesters 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 the 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 arc length, area, and volume.
- Interpret accumulation functions geometrically and contextually and compute their derivatives.
- Solve advanced integrals using techniques like substitution, long division, completing the square, partial fractions, and integration by parts.
- Evaluate improper integrals.

- 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.
- Estimate solutions of differential equations using Euler’s method.
- Find solutions of initial value problems using separation of variables.
- Interpret exponential and logistic growth models in context.
- Parametric/Polar Curves and Vector-Valued Functions
- Extend differentiation techniques to parametric and polar curves.
- Solve motion problems by differentiating and integrating vector-valued functions.
- Use integration to calculate the area of a region bounded by polar curves.

- Understand convergence and divergence in the context of sequences and series.
- Apply tests to determine the convergence or divergence of a series.
- Construct Taylor polynomials and use them to approximate values of functions.
- Compute and interpret the radius and interval of convergence of a power series.

1.

Limits
31 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 Inverse Trigonometric Functions | |

1.3.9. | 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 |

2.

Continuity
13 topics

2.6. Continuity

2.6.1. | Determining Continuity from Graphs | |

2.6.2. | Defining Continuity at a Point | |

2.6.3. | Left and Right Continuity | |

2.6.4. | Further Continuity of Piecewise Functions | |

2.6.5. | Point Discontinuities | |

2.6.6. | Jump Discontinuities | |

2.6.7. | Discontinuities Due to Vertical Asymptotes | |

2.6.8. | Continuity Over an Interval | |

2.6.9. | Continuity of Functions | |

2.6.10. | The Intermediate Value Theorem |

2.7. Removing Discontinuities

2.7.1. | Removing Point Discontinuities | |

2.7.2. | Removing Jump Discontinuities | |

2.7.3. | Removing Discontinuities From Rational Functions |

3.

Introduction to Differentiation
19 topics

3.8. Introduction to Differentiation

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

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

3.8.3. | Defining the Derivative Using Derivative Notation | |

3.8.4. | Connecting Differentiability and Continuity | |

3.8.5. | The Power Rule for Differentiation | |

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

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

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

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

3.9. Derivatives of Functions and the Rules of Differentiation

3.9.1. | Differentiating Exponential Functions | |

3.9.2. | Differentiating Logarithmic Functions | |

3.9.3. | Differentiating Trigonometric Functions | |

3.9.4. | Second and Higher Order Derivatives | |

3.9.5. | The Product Rule for Differentiation | |

3.9.6. | The Quotient Rule for Differentiation | |

3.9.7. | Differentiating Reciprocal Trigonometric Functions | |

3.9.8. | Calculating Derivatives From Data and Tables | |

3.9.9. | Calculating Derivatives From Graphs | |

3.9.10. | Recognizing Derivatives in Limits |

4.

Advanced Differentiation
14 topics

4.10. Differentiating Composite Functions

4.10.1. | The Chain Rule for Differentiation | |

4.10.2. | The Chain Rule With Exponential Functions | |

4.10.3. | The Chain Rule With Logarithmic Functions | |

4.10.4. | The Chain Rule With Trigonometric Functions | |

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

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

4.10.7. | Selecting Procedures for Calculating Derivatives |

4.11. Differentiating Implicit and Inverse Functions

4.11.1. | Implicit Differentiation | |

4.11.2. | Calculating Slopes of Circles, Ellipses, and Parabolas | |

4.11.3. | Calculating dy/dx Using dx/dy | |

4.11.4. | Differentiating Inverse Functions | |

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

4.11.6. | Differentiating Inverse Trigonometric Functions | |

4.11.7. | Differentiating Inverse Reciprocal Trigonometric Functions |

5.

Contextual Applications of Differentiation
15 topics

5.12. Contextual Applications of Differentiation

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

5.12.2. | Rates of Change in Applied Contexts |

5.13. Estimating Derivatives

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

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

5.13.3. | Estimating Derivatives Using a Central Difference Quotient |

5.14. Related Rates of Change

5.14.1. | Introduction to Related Rates | |

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

5.14.3. | Calculating Related Rates With Squares | |

5.14.4. | Calculating Related Rates With Rectangular Solids | |

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

5.14.6. | Calculating Related Rates Using Similar Triangles | |

5.14.7. | Calculating Related Rates Using Trigonometry | |

5.14.8. | Calculating Related Rates With Cones |

5.15. L'Hopital's Rule

5.15.1. | L'Hopital's Rule | |

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

6.

Analytical Applications of Differentiation
26 topics

6.16. Analytical Applications of Differentiation

6.16.1. | The Mean Value Theorem | |

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

6.16.3. | The Extreme Value Theorem | |

6.16.4. | Using Differentiation to Calculate Critical Points | |

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

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

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

6.16.8. | Intervals of Concavity | |

6.16.9. | Relating Concavity to the Second Derivative | |

6.16.10. | Points of Inflection | |

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

6.17. Analysis of Curves

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

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

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

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

6.17.5. | Sketching a Function Given Some Derivative Properties |

6.18. Approximating Values of a Function

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

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

6.18.3. | Approximating Trigonometric Functions Using Local Linearity |

6.19. Optimization

6.19.1. | Optimization Problems Involving Rectangles | |

6.19.2. | Optimization Problems Involving Sectors of Circles | |

6.19.3. | Optimization Problems Involving Boxes and Trays | |

6.19.4. | Optimization Problems Involving Cylinders | |

6.19.5. | Finding Minimum Distances | |

6.19.6. | Optimization Problems With Inscribed Shapes | |

6.19.7. | Optimization Problems in Economics |

7.

Integration
33 topics

7.20. Indefinite Integrals

7.20.1. | The Antiderivative | |

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

7.20.3. | The Sum Rule for Indefinite Integrals | |

7.20.4. | Integrating the Reciprocal Function | |

7.20.5. | Integrating Exponential Functions | |

7.20.6. | Integrating Trigonometric Functions | |

7.20.7. | Integration Using Inverse Trigonometric Functions |

7.21. Approximating Areas with Riemann Sums

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

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

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

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

7.21.5. | Approximating Areas With the Trapezoidal Rule | |

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

7.21.7. | Midpoint and Trapezoidal Rules in Sigma Notation |

7.22. Definite Integrals

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

7.22.2. | The Fundamental Theorem of Calculus | |

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

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

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

7.23. The Area Under a Curve

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

7.23.2. | Evaluating Definite Integrals Using Symmetry | |

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

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

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

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

7.23.7. | Definite Integrals of Functions with Discontinuities |

7.24. Accumulation Functions

7.24.1. | The Integral as an Accumulation Function | |

7.24.2. | The Second Fundamental Theorem of Calculus | |

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

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

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

7.24.6. | Integrating Rates of Change | |

7.24.7. | Integrating Density Functions |

8.

Techniques of Integration
32 topics

8.25. Integration Using Substitution

8.25.1. | Integrating Algebraic Functions Using Substitution | |

8.25.2. | Integrating Linear Rational Functions Using Substitution | |

8.25.3. | Integration Using Substitution | |

8.25.4. | Calculating Definite Integrals Using Substitution | |

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

8.25.6. | Integrating Exponential Functions Using Linear Substitution | |

8.25.7. | Integrating Exponential Functions Using Substitution | |

8.25.8. | Integrating Trigonometric Functions Using Substitution | |

8.25.9. | Integrating Logarithmic Functions Using Substitution | |

8.25.10. | Integration by Substitution With Inverse Trigonometric Functions |

8.26. Integration Using Trigonometric Identities

8.26.1. | Integration Using Basic Trigonometric Identities | |

8.26.2. | Integration Using the Pythagorean Identities | |

8.26.3. | Integration Using the Double Angle Formulas |

8.27. Special Techniques for Integration

8.27.1. | Integrating Functions Using Polynomial Division | |

8.27.2. | Integrating Functions by Completing the Square |

8.28. Integration by Parts

8.28.1. | Introduction to Integration by Parts | |

8.28.2. | Using Integration by Parts to Calculate Integrals With Logarithms | |

8.28.3. | Applying the Integration By Parts Formula Twice | |

8.28.4. | The Tabular Method of Integration by Parts | |

8.28.5. | Integration by Parts in Cyclic Cases |

8.29. Integration Using Partial Fractions

8.29.1. | Expressing Rational Functions as Sums of Partial Fractions | |

8.29.2. | Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions | |

8.29.3. | Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions | |

8.29.4. | Integrating Rational Functions Using Partial Fractions | |

8.29.5. | Integrating Rational Functions with Repeated Factors | |

8.29.6. | Integrating Rational Functions with Irreducible Quadratic Factors |

8.30. Improper Integrals

8.30.1. | Improper Integrals | |

8.30.2. | Improper Integrals Involving Exponential Functions | |

8.30.3. | Improper Integrals Involving Arctangent | |

8.30.4. | Improper Integrals Over the Real Line | |

8.30.5. | Improper Integrals of the Second Kind | |

8.30.6. | Improper Integrals of the Second Kind: Discontinuities at Interior Points |

9.

Differential Equations
23 topics

9.31. Introduction to Differential Equations

9.31.1. | Introduction to Differential Equations | |

9.31.2. | Verifying Solutions of Differential Equations | |

9.31.3. | Solving Differential Equations Using Direct Integration | |

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

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

9.31.6. | Modeling With Differential Equations | |

9.31.7. | Further Modeling With Differential Equations |

9.32. Qualitative Techniques for Differential Equations

9.32.1. | Qualitative Analysis of Differential Equations | |

9.32.2. | Equilibrium Solutions of Differential Equations |

9.33. Modeling Exponential Growth and Decay With Differential Equations

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

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

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

9.34. Modeling Logistic Growth With Differential Equations

9.34.1. | Logistic Growth Models With Differential Equations | |

9.34.2. | Qualitative Analysis of the Logistic Growth Equation | |

9.34.3. | Solving the Logistic Growth Equation |

9.35. Slope Fields

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

9.35.2. | Slope Fields for Autonomous Differential Equations | |

9.35.3. | Slope Fields for Nonautonomous Differential Equations | |

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

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

9.35.6. | Analyzing Slope Fields for Nonautonomous Differential Equations |

9.36. Numerical Solutions of Differential Equations

9.36.1. | Euler's Method: Calculating One Step | |

9.36.2. | Euler's Method: Calculating Multiple Steps |

10.

Applications of Integration
13 topics

10.37. Applications of Integration

10.37.1. | The Average Value of a Function | |

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

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

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

10.37.5. | The Arc Length of a Smooth Planar Curve |

10.38. Volumes of Solids With Known Cross Sections

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

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

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

10.38.4. | Volumes of Solids with Circular Cross Sections |

10.39. Volumes of Revolution

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

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

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

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

11.

Parametric Equations
6 topics

11.40. Parametric Equations

11.40.1. | Defining and Differentiating Parametric Equations | |

11.40.2. | Calculating Tangent and Normal Lines with Parametric Equations | |

11.40.3. | Second Derivatives of Parametric Equations | |

11.40.4. | Finding the Arc Lengths of Curves Given by Parametric Equations |

11.41. Vector-Valued Functions

11.41.1. | Defining and Differentiating Vector-Valued Functions | |

11.41.2. | Integrating Vector-Valued Functions |

12.

Polar Equations
10 topics

12.42. Polar Coordinates

12.42.1. | Differentiating Curves Given in Polar Form | |

12.42.2. | Further Differentiation of Curves Given in Polar Form | |

12.42.3. | Horizontal and Vertical Tangents to Polar Curves | |

12.42.4. | Horizontal and Vertical Tangents to Polar Curves in Non-Differentiable Cases | |

12.42.5. | Tangent and Normal Lines to Polar Curves | |

12.42.6. | Finding the Area of a Polar Region | |

12.42.7. | Finding the Limits of Integration For a Given Polar Region | |

12.42.8. | The Total Area Bounded by a Single Polar Curve | |

12.42.9. | The Area Bounded by Two Polar Curves | |

12.42.10. | The Arc Length of a Polar Curve |

13.

Particle Dynamics
13 topics

13.43. Displacement, Velocity, and Acceleration

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

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

13.43.3. | Determining Characteristics of Moving Objects Using Differentiation |

13.44. Connecting Position, Velocity and Acceleration Using Integrals

13.44.1. | Calculating Velocity Using Integration | |

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

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

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

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

13.44.6. | Average Position, Velocity, and Acceleration |

13.45. The Planar Motion of a Particle

13.45.1. | Calculating Velocity for Plane Motion Using Differentiation | |

13.45.2. | Calculating Acceleration for Plane Motion Using Differentiation | |

13.45.3. | Finding Velocity Vectors in Two Dimensions Using Integration | |

13.45.4. | Finding Displacement Vectors in Two Dimensions Using Integration |

14.

Sequences & Series
29 topics

14.46. Sequences

14.46.1. | Limits of Sequences | |

14.46.2. | Convergence of Geometric Sequences | |

14.46.3. | Further Convergence of Geometric Sequences | |

14.46.4. | Limits of Sequences With Factorials | |

14.46.5. | Determining Limits of Sequences Using Relative Magnitudes | |

14.46.6. | Further Determining Limits of Sequences Using Relative Magnitudes |

14.47. Monotonic Sequences

14.47.1. | Monotonic Sequences | |

14.47.2. | Identifying Monotonic Sequences Using Differentiation | |

14.47.3. | Identifying Monotonic Sequences Using Ratios |

14.48. Infinite Series

14.48.1. | Infinite Series and Partial Sums | |

14.48.2. | Convergent and Divergent Infinite Series | |

14.48.3. | Properties of Infinite Series | |

14.48.4. | Telescoping Series |

14.49. Geometric Series

14.49.1. | Finding the Sum of an Infinite Geometric Series | |

14.49.2. | Writing an Infinite Geometric Series in Sigma Notation | |

14.49.3. | Finding the Sum of an Infinite Geometric Series Expressed in Sigma Notation | |

14.49.4. | Convergence of Geometric Series | |

14.49.5. | Repeating Decimals as Infinite Geometric Series |

14.50. Infinite Series Convergence Tests

14.50.1. | The Nth Term Test for Divergence | |

14.50.2. | The Integral Test | |

14.50.3. | Harmonic Series and p-Series | |

14.50.4. | The Comparison Test | |

14.50.5. | The Limit Comparison Test | |

14.50.6. | The Alternating Series Test | |

14.50.7. | The Ratio Test | |

14.50.8. | Absolute and Conditional Convergence | |

14.50.9. | The Alternating Series Error Bound | |

14.50.10. | Determining Convergence Parameters for Infinite Series | |

14.50.11. | Selecting Procedures for Analyzing Infinite Series |

15.

Power Series
14 topics

15.51. Taylor Polynomials

15.51.1. | Second-Degree Taylor Polynomials | |

15.51.2. | Analyzing Second-Degree Taylor Polynomials | |

15.51.3. | Third-Degree Taylor Polynomials | |

15.51.4. | Higher-Degree Taylor Polynomials | |

15.51.5. | The Lagrange Error Bound |

15.52. Taylor Series

15.52.1. | Radius of Convergence of Power Series Centered at the Origin | |

15.52.2. | Radius of Convergence of Power Series | |

15.52.3. | Maclaurin Series | |

15.52.4. | Taylor Series | |

15.52.5. | Representing Functions as Power Series | |

15.52.6. | Recognizing Standard Maclaurin Series | |

15.52.7. | Recognizing Standard Maclaurin Series for Trigonometric Functions | |

15.52.8. | Differentiating Taylor Series | |

15.52.9. | Approximating Integrals Using Taylor Series |

16.

Applications of Technology
9 topics

16.53. Using Graphing Calculators

16.53.1. | Evaluating Expressions Using a Graphing Calculator | |

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

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

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

16.53.5. | Finding Derivatives Using a Graphing Calculator | |

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

16.53.7. | Finding Improper Integrals Using a Graphing Calculator | |

16.53.8. | Exploring Functions Using Technology | |

16.53.9. | Plotting Parametric and Polar Curves Using Technology |