Learn advanced calculus techniques for computing limits, derivatives, and integrals, and apply calculus to solve problems in the context of related rates, optimization, particle motion, and differential equations. Dive deeper into complex numbers, vectors, matrices, parametric and polar curves, probability, and statistics.

- Leverage factoring and graphing as a strategy to solve polynomial inequalities.
- Visualize two-variable linear inequalities in the coordinate plane.
- Apply trigonometric identities to simplify trigonometric expressions.
- Solve trigonometric equations by extending previous knowledge of algebra and leveraging properties of trigonometric functions.

- Compute limits algebraically using advanced techniques such as L'Hopital's rule and conjugate multiplication.
- Understand the relationship between limits and asymptotes.
- Explain the behavior of a function on an interval using the Intermediate Value Theorem.

- Determine whether a sequence is convergent or divergent, and compute the limit of a convergent sequence.
- Express an arithmetic or geometric series using sigma notation.
- Compute the sum of an arithmetic or geometric series.

- Relate the graph of a function to properties of its derivative and understand the relationship between differentiability and continuity.
- Understand that the derivative of position is velocity, and the derivative of velocity is acceleration.
- Calculate derivatives of implicit and inverse functions and use implicit differentiation to solve related rates problems.
- Estimate derivatives numerically and approximate values of functions using linearization.
- Solve optimization problems by using the derivative to find extrema of functions.

- Approximate areas using Riemann sums and compute the definite 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 and area and translate between position, velocity, and acceleration.
- Interpret accumulation functions geometrically 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.

- Verify solutions of differential equations and solve elementary differential equations using direct integration.
- Solve differential equations using separation of variables and fit unknown coefficients to initial conditions.
- Estimate solutions to initial value problems using Euler’s method.

- Translate between graphs and equations of circles, parabolas, ellipses, and hyperbolas.
- Extend prior knowledge of rectangular curves to parametric and polar curves.
- Extend differentiation techniques to parametric and polar curves.
- Use integration to calculate the area of a polar region or the arc length of a polar curve.

- Connect the algebraic and geometric interpretations of complex numbers through Euler’s formula.
- Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
- Compute dot products and cross products of vectors and interpret them geometrically.
- Interpret matrices as linear transformations of points in the coordinate plane.
- Compute inverses and determinants of matrices and interpret them geometrically.

- Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
- Define and perform computations with probability mass functions, cumulative distributions, and expected values of discrete random variables.
- Apply binomial, geometric, and normal distributions in modeling contexts.

1.

Finite Series
12 topics

1.1. Arithmetic Series

1.1.1. | Expressing an Arithmetic Series in Sigma Notation | |

1.1.2. | Finding the Sum of an Arithmetic Series | |

1.1.3. | Finding the First Term of an Arithmetic Series | |

1.1.4. | Calculating the Number of Terms in an Arithmetic Series |

1.2. Finite Geometric Series

1.2.1. | The Sum of a Finite Geometric Series | |

1.2.2. | The Sum of the First N Terms of a Geometric Series | |

1.2.3. | Writing Geometric Series in Sigma Notation | |

1.2.4. | Finding the Sum of a Geometric Series Given in Sigma Notation |

1.3. The Binomial Theorem

1.3.1. | Pascal's Triangle and the Binomial Coefficients | |

1.3.2. | Expanding a Binomial Using Binomial Coefficients | |

1.3.3. | The Special Case of the Binomial Theorem | |

1.3.4. | Approximating Values Using the Binomial Theorem |

2.

Inequalities
19 topics

2.4. Single-Variable Inequalities

2.4.1. | Solving Elementary Quadratic Inequalities | |

2.4.2. | Solving Quadratic Inequalities From Graphs | |

2.4.3. | Solving Quadratic Inequalities Using the Graphical Method | |

2.4.4. | Solving Quadratic Inequalities Using the Sign Table Method | |

2.4.5. | Inequalities Involving Powers of Binomials | |

2.4.6. | Solving Polynomial Inequalities Using a Graphical Method | |

2.4.7. | Solving Polynomial Inequalities Using the Sign Table Method | |

2.4.8. | Solving Rational Inequalities | |

2.4.9. | Solving Inequalities Involving Exponential Functions and Polynomials | |

2.4.10. | Solving Radical Inequalities | |

2.4.11. | Solving Inequalities Involving Exponential Functions | |

2.4.12. | Solving Inequalities Involving Logarithmic Functions | |

2.4.13. | Solving Inequalities Involving Geometric Sequences |

2.5. Two-Variable Inequalities

2.5.1. | Graphing Strict Two-Variable Linear Inequalities | |

2.5.2. | Graphing Non-Strict Two-Variable Linear Inequalities | |

2.5.3. | Further Graphing of Two-Variable Linear Inequalities | |

2.5.4. | Solving Systems of Linear Inequalities | |

2.5.5. | Solving Two-Variable Nonlinear Inequalities | |

2.5.6. | Further Solving of Two-Variable Nonlinear Inequalities |

3.

Parametric & Polar Coordinates
15 topics

3.6. Parametric Equations

3.6.1. | Graphing Curves Defined Parametrically | |

3.6.2. | Cartesian Equations of Parametric Curves | |

3.6.3. | Finding Intersections of Parametric Curves and Lines | |

3.6.4. | Differentiating Parametric Curves | |

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

3.6.6. | Second Derivatives of Parametric Equations | |

3.6.7. | The Arc Length of a Parametric Curve |

3.7. Polar Coordinates

3.7.1. | Introduction to Polar Coordinates | |

3.7.2. | Converting from Polar Coordinates to Cartesian Coordinates | |

3.7.3. | Polar Equations of Circles Centered at the Origin | |

3.7.4. | Polar Equations of Radial Lines | |

3.7.5. | Differentiating Curves Given in Polar Form | |

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

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

3.7.8. | The Arc Length of a Polar Curve |

4.

Conic Sections
27 topics

4.8. Circles

4.8.1. | The Center and Radius of a Circle in the Coordinate Plane | |

4.8.2. | Equations of Circles Centered at the Origin | |

4.8.3. | Equations of Circles Centered at a General Point | |

4.8.4. | Finding the Center and Radius of a Circle by Completing the Square | |

4.8.5. | Calculating Intercepts of Circles | |

4.8.6. | Intersections of Circles with Lines | |

4.8.7. | Parametric Equations of Circles |

4.9. Parabolas

4.9.1. | Upward and Downward Opening Parabolas | |

4.9.2. | Left and Right Opening Parabolas | |

4.9.3. | The Vertex of a Parabola | |

4.9.4. | Calculating the Vertex of a Parabola by Completing the Square | |

4.9.5. | Calculating Intercepts of Parabolas | |

4.9.6. | Parametric Equations of Parabolas | |

4.9.7. | Parametric Equations of Parabolas Centered at (h,k) |

4.10. Ellipses

4.10.1. | Introduction to Ellipses | |

4.10.2. | Equations of Ellipses Centered at the Origin | |

4.10.3. | Equations of Ellipses Centered at a General Point | |

4.10.4. | Finding the Center and Axes of Ellipses by Completing the Square | |

4.10.5. | Finding Intercepts of Ellipses | |

4.10.6. | Parametric Equations of Ellipses |

4.11. Hyperbolas

4.11.1. | Equations of Hyperbolas Centered at the Origin | |

4.11.2. | Equations of Hyperbolas Centered at a General Point | |

4.11.3. | Asymptotes of Hyperbolas Centered at the Origin | |

4.11.4. | Asymptotes of Hyperbolas Centered at a General Point | |

4.11.5. | Finding Intercepts and Intersections of Hyperbolas | |

4.11.6. | Parametric Equations of Horizontal Hyperbolas | |

4.11.7. | Parametric Equations of Vertical Hyperbolas |

5.

Trigonometry
26 topics

5.12. The Inverse Trigonometric Functions

5.12.1. | Graphing the Inverse Sine Function | |

5.12.2. | Graphing the Inverse Cosine Function | |

5.12.3. | Graphing the Inverse Tangent Function | |

5.12.4. | Evaluating Expressions Containing Inverse Trigonometric Functions | |

5.12.5. | Limits of Inverse Trigonometric Functions |

5.13. Elementary Trigonometric Equations

5.13.1. | Elementary Trigonometric Equations Containing Sine | |

5.13.2. | Elementary Trigonometric Equations Containing Cosine | |

5.13.3. | Elementary Trigonometric Equations Containing Tangent | |

5.13.4. | Elementary Trigonometric Equations Containing Secant | |

5.13.5. | Elementary Trigonometric Equations Containing Cosecant | |

5.13.6. | Elementary Trigonometric Equations Containing Cotangent | |

5.13.7. | Solving Trigonometric Equations Using the Sin-Cos-Tan Identity | |

5.13.8. | General Solutions of Elementary Trigonometric Equations | |

5.13.9. | General Solutions of Trigonometric Equations With Transformed Functions | |

5.13.10. | Trigonometric Equations Containing Transformed Tangent Functions |

5.14. Trigonometric Identities

5.14.1. | Simplifying Expressions Using Basic Trigonometric Identities | |

5.14.2. | Simplifying Expressions Using the Pythagorean Identity | |

5.14.3. | Alternate Forms of the Pythagorean Identity | |

5.14.4. | Simplifying Expressions Using the Secant-Tangent Identity | |

5.14.5. | Alternate Forms of the Secant-Tangent Identity | |

5.14.6. | Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity |

5.15. The Sum and Difference Formulas

5.15.1. | The Sum and Difference Formulas for Sine | |

5.15.2. | The Sum and Difference Formulas for Cosine | |

5.15.3. | Writing Sums of Trigonometric Functions in Amplitude-Phase Form | |

5.15.4. | The Double-Angle Formula for Sine | |

5.15.5. | The Double-Angle Formula for Cosine |

6.

Complex Numbers
16 topics

6.16. Further Complex Numbers

6.16.1. | The Complex Conjugate | |

6.16.2. | Special Properties of the Complex Conjugate | |

6.16.3. | The Complex Conjugate and the Roots of a Quadratic Equation | |

6.16.4. | Dividing Complex Numbers | |

6.16.5. | Solving Equations by Equating Real and Imaginary Parts | |

6.16.6. | Extending Polynomial Identities to the Complex Numbers |

6.17. Euler's Formula

6.17.1. | The Polar Form of a Complex Number | |

6.17.2. | De Moivre's Theorem | |

6.17.3. | Euler's Formula | |

6.17.4. | The Roots of Unity | |

6.17.5. | Properties of Roots of Unity |

6.18. The Fundamental Theorem of Algebra

6.18.1. | The Fundamental Theorem of Algebra for Quadratic Equations | |

6.18.2. | The Fundamental Theorem of Algebra with Cubic Equations | |

6.18.3. | Solving Cubic Equations With Complex Roots | |

6.18.4. | The Fundamental Theorem of Algebra with Quartic Equations | |

6.18.5. | Solving Quartic Equations With Complex Roots |

7.

Limits & Continuity
10 topics

7.19. Limits

7.19.1. | L'Hopital's Rule | |

7.19.2. | Limits of Sequences | |

7.19.3. | Special Limits Involving Sine | |

7.19.4. | Limits Involving the Exponential Function | |

7.19.5. | Vertical Asymptotes of Rational Functions | |

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

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

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

7.20. Continuity

7.20.1. | Continuity of Functions | |

7.20.2. | The Intermediate Value Theorem |

8.

Differentiation
25 topics

8.21. Differentiating Implicit and Inverse Functions

8.21.1. | Implicit Differentiation | |

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

8.21.3. | Calculating dy/dx Using dx/dy | |

8.21.4. | Differentiating Inverse Functions | |

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

8.21.6. | Differentiating Inverse Trigonometric Functions | |

8.21.7. | Differentiating Inverse Reciprocal Trigonometric Functions | |

8.21.8. | Integration Using Inverse Trigonometric Functions |

8.22. Analytical Applications of Differentiation

8.22.1. | Connecting Differentiability and Continuity | |

8.22.2. | The Mean Value Theorem | |

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

8.22.4. | The Extreme Value Theorem | |

8.22.5. | Using Differentiation to Calculate Critical Points | |

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

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

8.22.8. | The Candidates Test | |

8.22.9. | Intervals of Concavity | |

8.22.10. | Relating Concavity to the Second Derivative | |

8.22.11. | Points of Inflection | |

8.22.12. | The Second Derivative Test | |

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

8.22.14. | Second-Degree Taylor Polynomials |

8.23. Estimating Derivatives

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

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

8.23.3. | Estimating Derivatives Using a Central Difference Quotient |

9.

Definite Integrals
19 topics

9.24. Approximating Areas with Riemann Sums

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

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

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

9.25. Definite Integrals

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

9.25.2. | The Fundamental Theorem of Calculus | |

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

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

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

9.26. The Area Under a Curve

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

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

9.26.3. | Evaluating Definite Integrals Using Symmetry | |

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

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

9.26.6. | Definite Integrals of Piecewise Functions |

9.27. Accumulation Functions

9.27.1. | The Integral as an Accumulation Function | |

9.27.2. | The Second Fundamental Theorem of Calculus |

9.28. Applications of Integration

9.28.1. | The Average Value of a Function | |

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

9.28.3. | The Arc Length of a Planar Curve |

10.

Integration Techniques
29 topics

10.29. Integration Using Substitution

10.29.1. | Integrating Algebraic Functions Using Substitution | |

10.29.2. | Integrating Linear Rational Functions Using Substitution | |

10.29.3. | Integration Using Substitution | |

10.29.4. | Calculating Definite Integrals Using Substitution | |

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

10.29.6. | Integrating Exponential Functions Using Linear Substitution | |

10.29.7. | Integrating Exponential Functions Using Substitution | |

10.29.8. | Integrating Trigonometric Functions Using Substitution | |

10.29.9. | Integrating Logarithmic Functions Using Substitution | |

10.29.10. | Integration by Substitution With Inverse Trigonometric Functions |

10.30. Integration Using Trigonometric Identities

10.30.1. | Integration Using Basic Trigonometric Identities | |

10.30.2. | Integration Using the Pythagorean Identities | |

10.30.3. | Integration Using the Double-Angle Formulas |

10.31. Special Techniques for Integration

10.31.1. | Integrating Functions Using Polynomial Division | |

10.31.2. | Integrating Functions by Completing the Square |

10.32. Integration by Parts

10.32.1. | Introduction to Integration by Parts | |

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

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

10.32.4. | Integration by Parts in Cyclic Cases |

10.33. Integration Using Partial Fractions

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

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

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

10.33.4. | Integrating Rational Functions Using Partial Fractions | |

10.33.5. | Integrating Rational Functions with Repeated Factors | |

10.33.6. | Integrating Rational Functions with Irreducible Quadratic Factors |

10.34. Improper Integrals

10.34.1. | Improper Integrals | |

10.34.2. | Improper Integrals Involving Exponential Functions | |

10.34.3. | Improper Integrals Involving Arctangent | |

10.34.4. | Improper Integrals Over the Real Line |

11.

Contextual Applications of Calculus
21 topics

11.35. Displacement, Velocity, and Acceleration

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

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

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

11.35.4. | Calculating Velocity Using Integration | |

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

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

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

11.36. The Planar Motion of a Particle

11.36.1. | Velocity and Acceleration for Plane Motion | |

11.36.2. | Calculating Displacement for Plane Motion | |

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

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

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

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

11.37. Related Rates and Optimization

11.37.1. | Rates of Change in Applied Contexts | |

11.37.2. | Introduction to Related Rates | |

11.37.3. | Related Rates With Implicit Functions | |

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

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

11.37.6. | Solving Optimization Problems Using Derivatives | |

11.37.7. | Optimizing Distances | |

11.37.8. | Optimizing Distances to Curves |

12.

Differential Equations
9 topics

12.38. Introduction to Differential Equations

12.38.1. | Introduction to Differential Equations | |

12.38.2. | Verifying Solutions of Differential Equations | |

12.38.3. | Solving Differential Equations Using Direct Integration | |

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

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

12.38.6. | Qualitative Analysis of Differential Equations | |

12.38.7. | Modeling With Differential Equations |

12.39. Numerical Solutions of Differential Equations

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

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

13.

Vectors
17 topics

13.40. Vectors in 3D Cartesian Coordinates

13.40.1. | Three-Dimensional Vectors in Component Form | |

13.40.2. | Addition and Scalar Multiplication of Cartesian Vectors in 3D | |

13.40.3. | Calculating the Magnitude of Cartesian Vectors in 3D |

13.41. The Dot Product

13.41.1. | Calculating the Dot Product Using Angle and Magnitude | |

13.41.2. | Calculating the Dot Product Using Components | |

13.41.3. | The Angle Between Two Vectors | |

13.41.4. | Calculating a Scalar Projection | |

13.41.5. | Calculating a Vector Projection |

13.42. The Cross Product

13.42.1. | The Cross Product of Two Vectors | |

13.42.2. | Properties of the Cross Product | |

13.42.3. | Calculating the Cross Product Using Determinants | |

13.42.4. | Finding Areas Using the Cross Product | |

13.42.5. | The Scalar Triple Product | |

13.42.6. | Volumes of Parallelepipeds |

13.43. Vector-Valued Functions

13.43.1. | Defining Vector-Valued Functions | |

13.43.2. | Differentiating Vector-Valued Functions | |

13.43.3. | Integrating Vector-Valued Functions |

14.

Linear Algebra
37 topics

14.44. Introduction to Matrices

14.44.1. | Introduction to Matrices | |

14.44.2. | Index Notation for Matrices | |

14.44.3. | Adding and Subtracting Matrices | |

14.44.4. | Properties of Matrix Addition | |

14.44.5. | Scalar Multiplication of Matrices | |

14.44.6. | Zero, Square, Diagonal and Identity Matrices | |

14.44.7. | The Transpose of a Matrix |

14.45. Matrix Multiplication

14.45.1. | Multiplying a Matrix by a Column Vector | |

14.45.2. | Multiplying Square Matrices | |

14.45.3. | Conformability for Matrix Multiplication | |

14.45.4. | Multiplying Matrices | |

14.45.5. | Powers of Matrices | |

14.45.6. | Multiplying a Matrix by the Identity Matrix | |

14.45.7. | Properties of Matrix Multiplication | |

14.45.8. | Representing 2x2 Systems of Equations Using a Matrix Product | |

14.45.9. | Representing 3x3 Systems of Equations Using a Matrix Product |

14.46. Determinants

14.46.1. | The Determinant of a 2x2 Matrix | |

14.46.2. | The Geometric Interpretation of the 2x2 Determinant | |

14.46.3. | The Minors of a 3x3 Matrix | |

14.46.4. | The Determinant of a 3x3 Matrix |

14.47. The Inverse of a Matrix

14.47.1. | Introduction to the Inverse of a Matrix | |

14.47.2. | Inverses of 2x2 Matrices | |

14.47.3. | Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method | |

14.47.4. | Solving 2x2 Systems of Equations Using Inverse Matrices | |

14.47.5. | Solving Systems of Equations Using Inverse Matrices |

14.48. Linear Transformations

14.48.1. | Introduction to Linear Transformations | |

14.48.2. | The Standard Matrix of a Linear Transformation | |

14.48.3. | Linear Transformations of Points and Lines in the Plane | |

14.48.4. | Linear Transformations of Objects in the Plane | |

14.48.5. | Dilations and Reflections as Linear Transformations | |

14.48.6. | Shear and Stretch as Linear Transformations | |

14.48.7. | Right-Angle Rotations as Linear Transformations | |

14.48.8. | Rotations as Linear Transformations | |

14.48.9. | Combining Linear Transformations Using 2x2 Matrices | |

14.48.10. | Inverting Linear Transformations | |

14.48.11. | Area Scale Factors of Linear Transformations | |

14.48.12. | Singular Linear Transformations in the Plane |

15.

Probability
17 topics

15.49. Conditional Probability

15.49.1. | Conditional Probabilities From Venn Diagrams | |

15.49.2. | The Multiplication Law for Conditional Probability | |

15.49.3. | The Law of Total Probability | |

15.49.4. | Independent Events |

15.50. Discrete Random Variables

15.50.1. | Probability Mass Functions of Discrete Random Variables | |

15.50.2. | Cumulative Distribution Functions for Discrete Random Variables | |

15.50.3. | Expected Values of Discrete Random Variables | |

15.50.4. | The Binomial Distribution | |

15.50.5. | Modeling With the Binomial Distribution | |

15.50.6. | The Geometric Distribution | |

15.50.7. | Modeling With the Geometric Distribution |

15.51. The Normal Distribution

15.51.1. | The Standard Normal Distribution | |

15.51.2. | Symmetry Properties of the Standard Normal Distribution | |

15.51.3. | The Normal Distribution | |

15.51.4. | Mean and Variance of the Normal Distribution | |

15.51.5. | Percentage Points of the Standard Normal Distribution | |

15.51.6. | Modeling With the Normal Distribution |