# AP Calculus BC

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.

## Content

### Limits and Continuity

• 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.

### Differentiation

• 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.

### Integration

• 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.

### Differential Equations

• 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.

### Infinite Sequences and Series

• 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.
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. The Candidates Test 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 With the Trapezoidal Rule 7.21.5. Left and Right Riemann Sums in Sigma Notation 7.21.6. Midpoint and Trapezoidal Rules in Sigma Notation 7.21.7. Approximating Areas Under Graphs of Composite Functions
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 Piecewise Functions
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
7 topics
11.40. Parametric Equations
 11.40.1. Differentiating Parametric Curves 11.40.2. Calculating Tangent and Normal Lines with Parametric Equations 11.40.3. Second Derivatives of Parametric Equations 11.40.4. Arc Lengths of Parametric Curves
11.41. Vector-Valued Functions
 11.41.1. Defining Vector-Valued Functions 11.41.2. Differentiating Vector-Valued Functions 11.41.3. 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
30 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. Further Properties of Infinite Series 14.48.5. 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