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 |
1.2.1. | 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 |
1.3.1. | 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 |
2.4.1. | 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 Inequalities Involving Exponential Functions and Polynomials | |
2.4.8. | Solving Inequalities Involving Radical Functions | |
2.4.9. | Solving Inequalities Involving Exponential Functions | |
2.4.10. | Solving Inequalities Involving Logarithmic Functions |
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 |
3.6.1. | The Center and Radius of a Circle in the Coordinate Plane | |
3.6.2. | Equations of Circles Centered at the Origin | |
3.6.3. | Equations of Circles Centered at a General Point | |
3.6.4. | Finding the Center and Radius of a Circle by Completing the Square | |
3.6.5. | Calculating Intercepts of Circles | |
3.6.6. | Intersections of Circles with Lines |
3.7.1. | Upward and Downward Opening Parabolas | |
3.7.2. | Left and Right Opening Parabolas | |
3.7.3. | The Vertex of a Parabola | |
3.7.4. | Calculating the Vertex of a Parabola by Completing the Square |
3.8.1. | Introduction to Ellipses | |
3.8.2. | Equations of Ellipses Centered at the Origin | |
3.8.3. | Equations of Ellipses Centered at a General Point | |
3.8.4. | Finding the Center and Axes of Ellipses by Completing the Square |
3.9.1. | Equations of Hyperbolas Centered at the Origin | |
3.9.2. | Equations of Hyperbolas Centered at a General Point |
4.10.1. | Graphing the Inverse Sine Function | |
4.10.2. | Graphing the Inverse Cosine Function | |
4.10.3. | Graphing the Inverse Tangent Function | |
4.10.4. | Evaluating Expressions Containing Inverse Trigonometric Functions | |
4.10.5. | Limits of Inverse Trigonometric Functions |
4.11.1. | Elementary Trigonometric Equations Containing Sine | |
4.11.2. | Elementary Trigonometric Equations Containing Cosine | |
4.11.3. | Elementary Trigonometric Equations Containing Tangent | |
4.11.4. | Elementary Trigonometric Equations Containing Secant | |
4.11.5. | Elementary Trigonometric Equations Containing Cosecant | |
4.11.6. | Elementary Trigonometric Equations Containing Cotangent | |
4.11.7. | Solving Trigonometric Equations Using the Sin-Cos-Tan Identity | |
4.11.8. | General Solutions of Elementary Trigonometric Equations | |
4.11.9. | General Solutions of Trigonometric Equations With Transformed Functions | |
4.11.10. | Trigonometric Equations Containing Transformed Tangent Functions |
4.12.1. | Simplifying Expressions Using Basic Trigonometric Identities | |
4.12.2. | Simplifying Expressions Using the Pythagorean Identity | |
4.12.3. | Alternate Forms of the Pythagorean Identity | |
4.12.4. | Simplifying Expressions Using the Secant-Tangent Identity | |
4.12.5. | Alternate Forms of the Secant-Tangent Identity | |
4.12.6. | Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity |
4.13.1. | The Sum and Difference Formulas for Sine | |
4.13.2. | The Sum and Difference Formulas for Cosine | |
4.13.3. | Writing Sums of Trigonometric Functions in Amplitude-Phase Form | |
4.13.4. | The Double-Angle Formula for Sine | |
4.13.5. | The Double-Angle Formula for Cosine |
5.14.1. | The Complex Conjugate | |
5.14.2. | Special Properties of the Complex Conjugate | |
5.14.3. | The Complex Conjugate and the Roots of a Quadratic Equation | |
5.14.4. | Dividing Complex Numbers | |
5.14.5. | Solving Equations by Equating Real and Imaginary Parts | |
5.14.6. | Extending Polynomial Identities to the Complex Numbers |
5.15.1. | The Polar Form of a Complex Number | |
5.15.2. | De Moivre's Theorem | |
5.15.3. | Euler's Formula | |
5.15.4. | Roots of Unity | |
5.15.5. | Properties of Roots of Unity |
5.16.1. | The Fundamental Theorem of Algebra for Quadratic Equations | |
5.16.2. | The Fundamental Theorem of Algebra with Cubic Equations | |
5.16.3. | Solving Cubic Equations With Complex Roots |
6.17.1. | L'Hopital's Rule | |
6.17.2. | Limits of Sequences | |
6.17.3. | Special Limits Involving Sine | |
6.17.4. | Vertical Asymptotes of Rational Functions | |
6.17.5. | Limits at Infinity and Horizontal Asymptotes of Rational Functions | |
6.17.6. | Calculating Limits of Radical Functions Using Conjugate Multiplication | |
6.17.7. | Evaluating Limits at Infinity by Comparing Relative Magnitudes of Functions |
6.18.1. | Continuity of Functions | |
6.18.2. | The Intermediate Value Theorem |
7.19.1. | Implicit Differentiation | |
7.19.2. | Calculating Slopes of Circles, Ellipses, and Parabolas | |
7.19.3. | Calculating dy/dx Using dx/dy | |
7.19.4. | Differentiating Inverse Functions | |
7.19.5. | Differentiating Inverse Trigonometric Functions | |
7.19.6. | Differentiating Inverse Reciprocal Trigonometric Functions | |
7.19.7. | Integration Using Inverse Trigonometric Functions |
7.20.1. | Connecting Differentiability and Continuity | |
7.20.2. | The Mean Value Theorem | |
7.20.3. | Global vs. Local Extrema and Critical Points | |
7.20.4. | The Extreme Value Theorem | |
7.20.5. | Using Differentiation to Calculate Critical Points | |
7.20.6. | Determining Intervals on Which a Function Is Increasing or Decreasing | |
7.20.7. | Using the First Derivative Test to Classify Local Extrema | |
7.20.8. | The Candidates Test | |
7.20.9. | Intervals of Concavity | |
7.20.10. | Relating Concavity to the Second Derivative | |
7.20.11. | Points of Inflection | |
7.20.12. | Using the Second Derivative Test to Determine Extrema | |
7.20.13. | Approximating Functions Using Local Linearity and Linearization |
7.21.1. | Estimating Derivatives Using a Forward Difference Quotient | |
7.21.2. | Estimating Derivatives Using a Backward Difference Quotient | |
7.21.3. | Estimating Derivatives Using a Central Difference Quotient |
8.22.1. | Approximating Areas With the Left Riemann Sum | |
8.22.2. | Approximating Areas With the Right Riemann Sum | |
8.22.3. | Left and Right Riemann Sums in Sigma Notation |
8.23.1. | Defining Definite Integrals Using Left and Right Riemann Sums | |
8.23.2. | The Fundamental Theorem of Calculus | |
8.23.3. | Applying the Fundamental Theorem of Calculus to Exponential and Trigonometric Functions | |
8.23.4. | The Sum and Constant Multiple Rules for Definite Integrals | |
8.23.5. | Properties of Definite Integrals Involving the Limits of Integration |
8.24.1. | The Area Bounded by a Curve and the X-Axis | |
8.24.2. | Evaluating Definite Integrals Using Symmetry | |
8.24.3. | Finding the Area Between a Curve and the X-Axis When They Intersect | |
8.24.4. | Calculating the Definite Integral of a Function Given Its Graph | |
8.24.5. | Definite Integrals of Piecewise Functions |
8.25.1. | The Integral as an Accumulation Function | |
8.25.2. | The Second Fundamental Theorem of Calculus |
8.26.1. | The Average Value of a Function | |
8.26.2. | The Area Between Curves Expressed as Functions of X | |
8.26.3. | The Arc Length of a Smooth Planar Curve |
9.27.1. | Integrating Algebraic Functions Using Substitution | |
9.27.2. | Integrating Linear Rational Functions Using Substitution | |
9.27.3. | Integration Using Substitution | |
9.27.4. | Calculating Definite Integrals Using Substitution | |
9.27.5. | Further Integration of Algebraic Functions Using Substitution | |
9.27.6. | Integrating Exponential Functions Using Linear Substitution | |
9.27.7. | Integrating Exponential Functions Using Substitution | |
9.27.8. | Integrating Trigonometric Functions Using Substitution | |
9.27.9. | Integrating Logarithmic Functions Using Substitution | |
9.27.10. | Integration by Substitution With Inverse Trigonometric Functions |
9.28.1. | Integration Using Basic Trigonometric Identities | |
9.28.2. | Integration Using the Pythagorean Identities | |
9.28.3. | Integration Using the Double Angle Formulas |
9.29.1. | Integrating Functions Using Polynomial Division | |
9.29.2. | Integrating Functions by Completing the Square |
9.30.1. | Introduction to Integration by Parts | |
9.30.2. | Using Integration by Parts to Calculate Integrals With Logarithms | |
9.30.3. | Applying the Integration By Parts Formula Twice | |
9.30.4. | Integration by Parts in Cyclic Cases |
9.31.1. | Expressing Rational Functions as Sums of Partial Fractions | |
9.31.2. | Expressing Rational Functions with Repeated Factors as Sums of Partial Fractions | |
9.31.3. | Expressing Rational Functions with Irreducible Quadratic Factors as Sums of Partial Fractions | |
9.31.4. | Integrating Rational Functions Using Partial Fractions | |
9.31.5. | Integrating Rational Functions with Repeated Factors | |
9.31.6. | Integrating Rational Functions with Irreducible Quadratic Factors |
9.32.1. | Improper Integrals | |
9.32.2. | Improper Integrals Involving Exponential Functions | |
9.32.3. | Improper Integrals Involving Arctangent | |
9.32.4. | Improper Integrals Over the Real Line |
10.33.1. | Distance-Time Graphs | |
10.33.2. | Calculating Acceleration From a Speed-Time Graph | |
10.33.3. | Calculating Distance From a Speed-Time Graph | |
10.33.4. | Calculating Velocity for Straight-Line Motion Using Differentiation | |
10.33.5. | Calculating Acceleration for Straight-Line Motion Using Differentiation | |
10.33.6. | Calculating Velocity Using Integration | |
10.33.7. | Calculating the Position Function of a Particle Using Integration | |
10.33.8. | Calculating the Displacement of a Particle Using Integration |
10.34.1. | Rates of Change in Applied Contexts | |
10.34.2. | Introduction to Related Rates | |
10.34.3. | Calculating Related Rates With Circles and Spheres | |
10.34.4. | Calculating Related Rates Using the Pythagorean Theorem | |
10.34.5. | Optimization Problems Involving Rectangles | |
10.34.6. | Finding Minimum Distances |
11.35.1. | Introduction to Differential Equations | |
11.35.2. | Verifying Solutions of Differential Equations | |
11.35.3. | Solving Differential Equations Using Direct Integration | |
11.35.4. | Solving First-Order ODEs Using Separation of Variables | |
11.35.5. | Solving Initial Value Problems Using Separation of Variables | |
11.35.6. | Qualitative Analysis of Differential Equations |
11.36.1. | Euler's Method: Calculating One Step | |
11.36.2. | Euler's Method: Calculating Multiple Steps |
12.37.1. | Graphing Curves Defined Parametrically | |
12.37.2. | Cartesian Equations of Parametric Curves | |
12.37.3. | Finding Intersections of Parametric Curves and Lines | |
12.37.4. | Differentiating Parametric Curves | |
12.37.5. | Calculating Tangent and Normal Lines with Parametric Equations | |
12.37.6. | Second Derivatives of Parametric Equations | |
12.37.7. | Arc Lengths of Parametric Curves |
12.38.1. | Introduction to Polar Coordinates | |
12.38.2. | Converting from Polar Coordinates to Cartesian Coordinates | |
12.38.3. | Polar Equations of Circles Centered at the Origin | |
12.38.4. | Polar Equations of Radial Lines | |
12.38.5. | Differentiating Curves Given in Polar Form | |
12.38.6. | Further Differentiation of Curves Given in Polar Form | |
12.38.7. | Finding the Area of a Polar Region | |
12.38.8. | The Arc Length of a Polar Curve |
13.39.1. | Three-Dimensional Vectors in Component Form | |
13.39.2. | Addition and Scalar Multiplication of Cartesian Vectors in 3D | |
13.39.3. | Calculating the Magnitude of Cartesian Vectors in 3D |
13.40.1. | Calculating the Dot Product Using Angle and Magnitude | |
13.40.2. | Calculating the Dot Product Using Components | |
13.40.3. | The Angle Between Two Vectors | |
13.40.4. | Calculating a Scalar Projection | |
13.40.5. | Calculating a Vector Projection |
13.41.1. | Calculating the Cross Product of Two Vectors Using the Definition | |
13.41.2. | Calculating the Cross Product Using Determinants | |
13.41.3. | Finding Areas Using the Cross Product | |
13.41.4. | The Scalar Triple Product | |
13.41.5. | Volumes of Parallelepipeds |
13.42.1. | Defining Vector-Valued Functions | |
13.42.2. | Differentiating Vector-Valued Functions | |
13.42.3. | Integrating Vector-Valued Functions |
14.43.1. | Introduction to Matrices | |
14.43.2. | Index Notation for Matrices | |
14.43.3. | Adding and Subtracting Matrices | |
14.43.4. | Properties of Matrix Addition | |
14.43.5. | Scalar Multiplication of Matrices | |
14.43.6. | Zero, Square, Diagonal and Identity Matrices | |
14.43.7. | The Transpose of a Matrix |
14.44.1. | Multiplying a Matrix by a Column Vector | |
14.44.2. | Multiplying Square Matrices | |
14.44.3. | Conformability for Matrix Multiplication | |
14.44.4. | Multiplying Matrices | |
14.44.5. | Powers of Matrices | |
14.44.6. | Multiplying a Matrix by the Identity Matrix | |
14.44.7. | Properties of Matrix Multiplication | |
14.44.8. | Representing 2x2 Systems of Equations Using a Matrix Product | |
14.44.9. | Representing 3x3 Systems of Equations Using a Matrix Product |
14.45.1. | The Determinant of a 2x2 Matrix | |
14.45.2. | The Geometric Interpretation of the 2x2 Determinant | |
14.45.3. | The Minors of a 3x3 Matrix | |
14.45.4. | The Determinant of a 3x3 Matrix |
14.46.1. | Introduction to the Inverse of a Matrix | |
14.46.2. | Inverses of 2x2 Matrices | |
14.46.3. | Solving 2x2 Systems of Equations Using Inverse Matrices |
14.47.1. | Introduction to Linear Transformations | |
14.47.2. | The Standard Matrix of a Linear Transformation | |
14.47.3. | Linear Transformations of Points and Lines in the Plane | |
14.47.4. | Linear Transformations of Objects in the Plane | |
14.47.5. | Dilations and Reflections as Linear Transformations | |
14.47.6. | Shear and Stretch as Linear Transformations | |
14.47.7. | Right-Angle Rotations as Linear Transformations | |
14.47.8. | Rotations as Linear Transformations | |
14.47.9. | Combining Linear Transformations Using 2x2 Matrices | |
14.47.10. | Inverting Linear Transformations | |
14.47.11. | Area Scale Factors of Linear Transformations | |
14.47.12. | Singular Linear Transformations in the Plane |
15.48.1. | Conditional Probabilities From Venn Diagrams | |
15.48.2. | The Multiplication Law for Conditional Probability | |
15.48.3. | The Law of Total Probability | |
15.48.4. | Independent Events |
15.49.1. | Probability Mass Functions of Discrete Random Variables | |
15.49.2. | Cumulative Distribution Functions for Discrete Random Variables | |
15.49.3. | Expected Values of Discrete Random Variables | |
15.49.4. | The Binomial Distribution | |
15.49.5. | Modeling With the Binomial Distribution | |
15.49.6. | The Geometric Distribution | |
15.49.7. | Modeling With the Geometric Distribution |
15.50.1. | The Standard Normal Distribution | |
15.50.2. | The Normal Distribution | |
15.50.3. | Mean and Variance of the Normal Distribution | |
15.50.4. | Percentage Points of the Standard Normal Distribution | |
15.50.5. | Modeling With the Normal Distribution |