1. Parametric Equations, Polar Coordinates, and Vector-Valued Functions1.1 Defining and Differentiating Parametric Equations0/01.1.1 Introduction to Parametric Equations1.1.2 Finding dy/dx for Parametric Curves1.1.3 Tangent Lines to Parametric Curves1.2 Second Derivatives of Parametric Equations0/01.2.1 The Formula for the Second Derivative of Parametric Equations1.2.2 Analyzing Concavity of Parametric Curves1.3 Finding Arc Lengths of Curves Given by Parametric Equations0/01.3.1 The Arc Length Formula for Parametric Curves1.3.2 Applying the Arc Length Formula1.4 Defining and Differentiating Vector-Valued Functions0/01.4.1 Introduction to Vector-Valued Functions1.4.2 Differentiating Vector-Valued Functions1.5 Integrating Vector-Valued Functions0/01.5.1 Integrating Vector-Valued Functions1.5.2 Initial Value Problems for Vector-Valued Functions1.6 Solving Motion Problems Using Parametric and Vector-Valued Functions0/01.6.1 Velocity and Acceleration Vectors1.6.2 Calculating Speed on a Curve1.6.3 Displacement Using the Velocity Vector1.6.4 Total Distance Traveled Along a Curve1.7 Defining Polar Coordinates and Differentiating in Polar Form0/01.7.1 Introduction to Polar Coordinates1.7.2 Converting Between Polar and Rectangular Coordinates1.7.3 Derivatives of r, x, and y with Respect to θ1.7.4 Finding dy/dx for Polar Curves1.7.5 Second Derivatives and Analysis of Polar Curves1.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve0/01.8.1 The Polar Area Formula1.8.2 Calculating the Area Enclosed by a Polar Curve1.8.3 Areas of Rose Petals and Limaçon Loops1.9 Finding the Area of the Region Bounded by Two Polar Curves0/01.9.1 Finding Intersections of Polar Curves1.9.2 Area Between Two Polar Curves1.9.3 Areas Inside One Curve and Outside Another1. Parametric Equations, Polar Coordinates, and Vector-Valued Functions1.1 Defining and Differentiating Parametric Equations0/01.1.1 Introduction to Parametric Equations1.1.2 Finding dy/dx for Parametric Curves1.1.3 Tangent Lines to Parametric Curves1.2 Second Derivatives of Parametric Equations0/01.2.1 The Formula for the Second Derivative of Parametric Equations1.2.2 Analyzing Concavity of Parametric Curves1.3 Finding Arc Lengths of Curves Given by Parametric Equations0/01.3.1 The Arc Length Formula for Parametric Curves1.3.2 Applying the Arc Length Formula1.4 Defining and Differentiating Vector-Valued Functions0/01.4.1 Introduction to Vector-Valued Functions1.4.2 Differentiating Vector-Valued Functions1.5 Integrating Vector-Valued Functions0/01.5.1 Integrating Vector-Valued Functions1.5.2 Initial Value Problems for Vector-Valued Functions1.6 Solving Motion Problems Using Parametric and Vector-Valued Functions0/01.6.1 Velocity and Acceleration Vectors1.6.2 Calculating Speed on a Curve1.6.3 Displacement Using the Velocity Vector1.6.4 Total Distance Traveled Along a Curve1.7 Defining Polar Coordinates and Differentiating in Polar Form0/01.7.1 Introduction to Polar Coordinates1.7.2 Converting Between Polar and Rectangular Coordinates1.7.3 Derivatives of r, x, and y with Respect to θ1.7.4 Finding dy/dx for Polar Curves1.7.5 Second Derivatives and Analysis of Polar Curves1.8 Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve0/01.8.1 The Polar Area Formula1.8.2 Calculating the Area Enclosed by a Polar Curve1.8.3 Areas of Rose Petals and Limaçon Loops1.9 Finding the Area of the Region Bounded by Two Polar Curves0/01.9.1 Finding Intersections of Polar Curves1.9.2 Area Between Two Polar Curves1.9.3 Areas Inside One Curve and Outside Another2. Infinite Sequences and SeriesPremium2.1 Defining Convergent and Divergent Infinite Series0/02.1.1 Introduction to Infinite Series and Partial Sums2.1.2 Determining Convergence and Divergence from the Sequence of Partial Sums2.2 Working with Geometric Series0/02.2.1 Identifying Geometric Series2.2.2 The Sum of a Convergent Geometric Series2.2.3 Applications of Geometric Series2.3 The nth Term Test for Divergence0/02.3.1 The nth Term Test for Divergence2.3.2 Applying the nth Term Test2.4 Integral Test for Convergence0/02.4.1 Conditions and Statement of the Integral Test2.4.2 Applying the Integral Test2.5 Harmonic Series and p-Series0/02.5.1 The Harmonic Series2.5.2 The Alternating Harmonic Series2.5.3 p-Series and the p-Series Test2.6 Comparison Tests for Convergence0/02.6.1 The Direct Comparison Test2.6.2 The Limit Comparison Test2.6.3 Strategies for Choosing a Comparison Series2.7 Alternating Series Test for Convergence0/02.7.1 Identifying Alternating Series and the Conditions of the Test2.7.2 Applying the Alternating Series Test2.8 Ratio Test for Convergence0/02.8.1 The Ratio Test for Convergence2.8.2 Applying the Ratio Test2.9 Determining Absolute or Conditional Convergence0/02.9.1 Absolute Convergence, Conditional Convergence, and Divergence2.9.2 Absolute Convergence Implies Convergence2.9.3 Rearranging Terms of an Absolutely Convergent Series2.10 Alternating Series Error Bound0/02.10.1 The Alternating Series Error Bound2.10.2 Using the Alternating Series Error Bound in Approximation2.11 Finding Taylor Polynomial Approximations of Functions0/02.11.1 Constructing Taylor Polynomials Centered at x = a2.11.2 Approximating Function Values with Taylor Polynomials2.11.3 How Taylor Polynomials Approach the Original Function2.12 Lagrange Error Bound0/02.12.1 The Lagrange Error Bound2.12.2 Using the Alternating Series Error Bound with Taylor Polynomials2.13 Radius and Interval of Convergence of Power Series0/02.13.1 Definition of a Power Series2.13.2 Convergence Behavior of a Power Series2.13.3 Finding the Radius of Convergence Using the Ratio Test2.13.4 Testing Endpoints to Determine the Interval of Convergence2.13.5 Power Series as Taylor Series and Term-by-Term Operations2.14 Finding Taylor or Maclaurin Series for a Function0/02.14.1 From Taylor Polynomials to Taylor Series2.14.2 The Maclaurin Series for 1/(1 − x)2.14.3 Maclaurin Series for sin x, cos x, and eË£2.14.4 Building New Maclaurin Series from Known Series2.15 Representing Functions as Power Series0/02.15.1 Deriving Power Series by Substitution and Algebraic Processes2.15.2 Deriving Power Series by Term-by-Term Differentiation2.15.3 Deriving Power Series by Term-by-Term Integration2.15.4 Using Properties of Geometric Series to Represent Functions2. Infinite Sequences and SeriesPremium2.1 Defining Convergent and Divergent Infinite Series0/02.1.1 Introduction to Infinite Series and Partial Sums2.1.2 Determining Convergence and Divergence from the Sequence of Partial Sums2.2 Working with Geometric Series0/02.2.1 Identifying Geometric Series2.2.2 The Sum of a Convergent Geometric Series2.2.3 Applications of Geometric Series2.3 The nth Term Test for Divergence0/02.3.1 The nth Term Test for Divergence2.3.2 Applying the nth Term Test2.4 Integral Test for Convergence0/02.4.1 Conditions and Statement of the Integral Test2.4.2 Applying the Integral Test2.5 Harmonic Series and p-Series0/02.5.1 The Harmonic Series2.5.2 The Alternating Harmonic Series2.5.3 p-Series and the p-Series Test2.6 Comparison Tests for Convergence0/02.6.1 The Direct Comparison Test2.6.2 The Limit Comparison Test2.6.3 Strategies for Choosing a Comparison Series2.7 Alternating Series Test for Convergence0/02.7.1 Identifying Alternating Series and the Conditions of the Test2.7.2 Applying the Alternating Series Test2.8 Ratio Test for Convergence0/02.8.1 The Ratio Test for Convergence2.8.2 Applying the Ratio Test2.9 Determining Absolute or Conditional Convergence0/02.9.1 Absolute Convergence, Conditional Convergence, and Divergence2.9.2 Absolute Convergence Implies Convergence2.9.3 Rearranging Terms of an Absolutely Convergent Series2.10 Alternating Series Error Bound0/02.10.1 The Alternating Series Error Bound2.10.2 Using the Alternating Series Error Bound in Approximation2.11 Finding Taylor Polynomial Approximations of Functions0/02.11.1 Constructing Taylor Polynomials Centered at x = a2.11.2 Approximating Function Values with Taylor Polynomials2.11.3 How Taylor Polynomials Approach the Original Function2.12 Lagrange Error Bound0/02.12.1 The Lagrange Error Bound2.12.2 Using the Alternating Series Error Bound with Taylor Polynomials2.13 Radius and Interval of Convergence of Power Series0/02.13.1 Definition of a Power Series2.13.2 Convergence Behavior of a Power Series2.13.3 Finding the Radius of Convergence Using the Ratio Test2.13.4 Testing Endpoints to Determine the Interval of Convergence2.13.5 Power Series as Taylor Series and Term-by-Term Operations2.14 Finding Taylor or Maclaurin Series for a Function0/02.14.1 From Taylor Polynomials to Taylor Series2.14.2 The Maclaurin Series for 1/(1 − x)2.14.3 Maclaurin Series for sin x, cos x, and eË£2.14.4 Building New Maclaurin Series from Known Series2.15 Representing Functions as Power Series0/02.15.1 Deriving Power Series by Substitution and Algebraic Processes2.15.2 Deriving Power Series by Term-by-Term Differentiation2.15.3 Deriving Power Series by Term-by-Term Integration2.15.4 Using Properties of Geometric Series to Represent Functions
