| 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 |
| 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. | 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 |
| 5.15.1. | 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. | 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 |
| 6.17.1. | 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. | 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 |
| 7.20.1. | 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 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 |
| 7.22.1. | 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 Functions with Discontinuities |
| 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 Smooth 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. | 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 |
| 11.41.1. | Defining and Differentiating Vector-Valued Functions | |
| 11.41.2. | 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. | 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 |
