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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.

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Content

Limits and Continuity

Differentiation

Integration

Differential Equations

Infinite Sequences and Series

1.
Limits
30 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
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. 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
13 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 and Graphs Using the Chain Rule
4.10.6. 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
32 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
7.22. 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
30 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 Over the Real Line
8.30.3. Improper Integrals of the Second Kind
8.30.4. 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 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