This comprehensive course is designed to deliver a thorough, engaging, and effective learning experience. It covers the full range of topics one would expect to find in a complete calculus course, including limits and continuity, differentiation, indefinite and definite integrals, applications of differentiation and integration, differential equations, parametric and polar equations, particle dynamics, sequences and series, power series, and applications of technology.
By completing this course, students will have everything they need to achieve the highest mark on the AP exam. In addition, they will be equipped with the necessary tools to study more advanced mathematics, such as multivariable calculus, linear algebra, and probability and statistics.
| 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 |
| 1.2.1. | 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 |
| 1.3.1. | 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 |
| 1.4.1. | 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 |
| 1.5.1. | 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 | |
| 1.5.5. | Limits Involving the Exponential Function |
| 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 |
| 2.7.1. | Removing Point Discontinuities | |
| 2.7.2. | Removing Jump Discontinuities | |
| 2.7.3. | Removing Discontinuities From Rational Functions |
| 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 |
| 3.9.1. | 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 |
| 4.10.1. | 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 |
| 4.11.1. | 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 |
| 5.12.1. | Interpreting the Meaning of the Derivative in Context | |
| 5.12.2. | Rates of Change in Applied Contexts |
| 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 |
| 5.14.1. | Introduction to Related Rates | |
| 5.14.2. | Related Rates With Implicit Functions | |
| 5.14.3. | Calculating Related Rates With Circles and Spheres | |
| 5.14.4. | Calculating Related Rates With Squares | |
| 5.14.5. | Calculating Related Rates With Rectangular Solids | |
| 5.14.6. | Calculating Related Rates Using the Pythagorean Theorem | |
| 5.14.7. | Calculating Related Rates Using Similar Triangles | |
| 5.14.8. | Calculating Related Rates Using Trigonometry | |
| 5.14.9. | Calculating Related Rates With Cones |
| 5.15.1. | L'Hopital's Rule | |
| 5.15.2. | L'Hopital's Rule Applied to Tables |
| 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. | The Second Derivative Test |
| 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 |
| 6.18.1. | 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 |
| 6.19.1. | Solving Optimization Problems Using Derivatives | |
| 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. | Optimizing Distances | |
| 6.19.6. | Optimizing Distances to Curves | |
| 6.19.7. | Optimization Problems With Inscribed Shapes | |
| 6.19.8. | Optimization Problems in Economics |
| 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 |
| 7.21.1. | 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 |
| 7.22.1. | 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 |
| 7.23.1. | 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 |
| 7.24.1. | 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 |
| 8.25.1. | 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 |
| 8.26.1. | Integration Using Basic Trigonometric Identities | |
| 8.26.2. | Integration Using the Pythagorean Identities | |
| 8.26.3. | Integration Using the Double-Angle Formulas |
| 8.27.1. | Integrating Functions Using Polynomial Division | |
| 8.27.2. | Integrating Functions by Completing the Square |
| 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 |
| 8.29.1. | 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 |
| 8.30.1. | 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 |
| 9.31.1. | 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 |
| 9.32.1. | Qualitative Analysis of Differential Equations | |
| 9.32.2. | Equilibrium Solutions of 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 |
| 9.34.1. | Logistic Growth Models With Differential Equations | |
| 9.34.2. | Qualitative Analysis of the Logistic Growth Equation | |
| 9.34.3. | Solving the Logistic Growth Equation |
| 9.35.1. | 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 |
| 9.36.1. | Euler's Method: Calculating One Step | |
| 9.36.2. | Euler's Method: Calculating Multiple Steps |
| 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 Planar Curve |
| 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 |
| 10.39.1. | 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 |
| 11.40.1. | 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. | The Arc Length of a Parametric Curve |
| 11.41.1. | Defining Vector-Valued Functions | |
| 11.41.2. | Differentiating Vector-Valued Functions | |
| 11.41.3. | Integrating Vector-Valued Functions |
| 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 |
| 13.43.1. | 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 |
| 13.44.1. | 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 |
| 13.45.1. | 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 |
| 14.46.1. | 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 |
| 14.47.1. | Monotonic Sequences | |
| 14.47.2. | Identifying Monotonic Sequences Using Differentiation | |
| 14.47.3. | Identifying Monotonic Sequences Using Ratios |
| 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 |
| 14.49.1. | 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 |
| 14.50.1. | 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 |
| 15.51.1. | 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 |
| 15.52.1. | 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 |
| 16.53.1. | 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 |
