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1 | 1.1Overview 1.1.1 An Introduction to Thinkwell's Calculus 1.1.2 The Two Questions of Calculus 1.1.3 Average Rates of Change 1.1.4 How to Do Math 1.2 Precalculus Review 1.2.1 Functions 1.2.2 Graphing Lines 1.2.3 Parabolas 1.2.4 Some Non-Euclidean Geometry 2.1 Concepts of the Limit 2.1.1 Finding Rate of Change over an Interval 2.1.2 Finding Limits Graphically 2.1.3 The Formal Definition of a Limit 2.1.4 The Limit Laws, Part I 2.1.5 The Limit Laws, Part II 2.1.6 One-Sided Limits 2.1.7 The Squeeze Theorem 2.1.8 Continuity and Discontinuity 2.2 Evaluating Limits 2.2.1 Evaluating Limits 2.2.2 Limits and Indeterminate Forms 2.2.3 Two Techniques for Evaluating Limits 2.2.4 An Overview of Limits Supplementary notes Intermediate Theorem Squeeze Theorem | Exercises 1.1.1 through 2.2.4 Discussion 1: Calculus History Chapter 2 Homework |
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2 | 3.1 Understanding the Derivative 3.1.1 Rates of Change, Secants, and Tangents 3.1.2 Finding Instantaneous Velocity 3.1.3 The Derivative 3.1.4 Differentiability 3.2 Using the Derivative 3.2.1 The Slope of a Tangent Line 3.2.2 Instantaneous Rate 3.2.3 The Equation of a Tangent Line 3.2.4 More on Instantaneous Rate 3.3 Some Special Derivatives 3.3.1 The Derivative of the Reciprocal Function 3.3.2 The Derivative of the Square Root Function Supplementary notes Differentiability | Exercises 3.1.1 through 3.3.2 Calculus Lab 1: Average Velocity and Instantaneous Velocity Chapter 3 Homework |
3 | 4 Computational Techniques 4.1 The Power Rule 4.1.1 A Shortcut for Finding Derivatives 4.1.2 A Quick Proof of the Power Rule 4.1.3 Uses of the Power Rule 4.2 The Product and Quotient Rules 4.2.1 The Product Rule 4.2.2 The Quotient Rule 4.3 The Chain Rule 4.3.1 An Introduction to the Chain Rule 4.3.2 Using the Chain Rule 4.3.3 Combining Computational Techniques | Exercises 4.1.1 through 4.3.3 Chapter 4 Homework |
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4 | 5.1 Trigonometric Functions 5.1.1 A Review of Trigonometry 5.1.2 Graphing Trigonometric Functions 5.1.3 The Derivatives of Trigonometric Functions 5.1.4 The Number Pi 5.2 Exponential Functions 5.2.1 Graphing Exponential Functions 5.2.2 Derivatives of Exponential Functions 5.2.3 The Music of Math 5.3 Logarithmic Functions 5.3.1 Evaluating Logarithmic Functions 5.3.2 The Derivative of the Natural Log Function 5.3.3 Using the Derivative Rules with Transcendental Functions Supplementary notes Differentiation Notation | Exercises 5.1.1 through 5.3.3 Discussion 2: Concept of Derivative Chapter 5 Homework |
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5 | 6.1 Implicit Differentiation Basics 6.1.1 An Introduction to Implicit Differentiation 6.1.2 Finding the Derivative Implicitly 6.2 Applying Implicit Differentiation 6.2.1 Using Implicit Differentiation 6.2.2 Applying Implicit Differentiation 6.3 Inverse Functions 6.3.1 The Exponential and Natural Log Functions 6.3.2 Differentiating Logarithmic Functions 6.3.3 Logarithmic Differentiation 6.3.4 The Basics of Inverse Functions 6.3.5 Finding the Inverse of a Function 6.4 The Calculus of Inverse Functions 6.4.1 Derivatives of Inverse Functions 6.5 Inverse Trigonometric Functions 6.5.1 The Inverse Sine, Cosine, and Tangent Functions 6.5.2 The Inverse Secant, Cosecant, and Cotangent Functions 6.5.3 Evaluating Inverse Trigonometric Functions 6.6 The Calculus of Inverse Trigonometric Functions 6.6.1 Derivatives of Inverse Trigonometric Functions 6.7 The Hyperbolic Functions 6.7.1 Defining the Hyperbolic Functions 6.7.2 Hyperbolic Identities 6.7.3 Derivatives of Hyperbolic Functions | Exercises 6.1.1 through through 6.7.3 Whiteboard Lab 1: Calculator Techniques on Function Derivatives and Graphs Chapter 6 Homework |
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6 | Chapter 7: Practical Applications of the Derivative 7.1 Position and Velocity 7.1.1 Acceleration and the Derivative 7.1.2 Solving Word Problems Involving Distance and Velocity 7.2 Linear Approximation 7.2.1 Higher-Order Derivatives and Linear Approximation 7.2.2 Using the Tangent Line Approximation Formula 7.2.3 Newton's Method 7.3 Related Rates 7.3.1 The Pebble Problem 7.3.2 The Ladder Problem 7.3.3 The Baseball Problem 7.3.4 The Blimp Problem 7.3.5 Math Anxiety 7.4 Optimization 7.4.1 The Connection Between Slope and Optimization 7.4.2 The Fence Problem 7.4.3 The Box Problem 7.4.4 The Can Problem 7.4.5 The Wire-Cutting Problem Supplementary note Local and Absolute Extreme Values | Exercises 7.1.1 through 7.4.5 Discussion 3: Applications of Derivative Calculus Lab 2: Volume of a Cylinder Chapter 7 Homework |
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8 | 8.1 Introduction 8.1.1 An Introduction to Curve Sketching 8.1.2 Three Big Theorems 8.1.3 Morale Moment 8.2 Critical Points 8.2.1 Critical Points 8.2.2 Maximum and Minimum 8.2.3 Regions Where a Function Increases or Decreases 8.2.4 The First Derivative Test 8.2.5 Math Magic 8.3 Concavity 8.3.1 Concavity and Inflection Points 8.3.2 Using the Second Derivative to Examine Concavity 8.3.3 The Möbius Band 8.4 Graphing Using the Derivative 8.4.1 Graphs of Polynomial Functions 8.4.2 Cusp Points and the Derivative 8.4.3 Domain-Restricted Functions and the Derivative 8.4.4 The Second Derivative Test 8.5 Asymptotes 8.5.1 Vertical Asymptotes 8.5.2 Horizontal Asymptotes and Infinite Limits 8.5.3 Graphing Functions with Asymptotes 8.5.4 Functions with Asymptotes and Holes 8.5.5 Functions with Asymptotes and Critical Points Supplementary notes Mean Value Theorem | Exercises 8.4.1 through 8.5.5 Calculus Lab 3: Curve Sketching Chapter 8 Homework |
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9 | 9.1 Antiderivatives 9.1.1 Antidifferentiation 9.1.2 Antiderivatives of Powers of x 9.1.3 Antiderivatives of Trigonometric and Exponential Functions 9.2 Integration by Substitution 9.2.1 Undoing the Chain Rule 9.2.2 Integrating Polynomials by Substitution 9.3 Illustrating Integration by Substitution 9.3.1 Integrating Composite Trigonometric Functions by Substitution 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution 9.3.3 More Integrating Trigonometric Functions by Substitution 9.3.4 Choosing Effective Function Decompositions | Exercises 9.1.1 through 9.3.4 Calculus Lab 3: Curve Sketching |
10 | 9.4 The Fundamental Theorem of Calculus 9.4.1 Approximating Areas of Plane Regions 9.4.2 Areas, Riemann Sums, and Definite Integrals 9.4.3 The Fundamental Theorem of Calculus, Part I 9.4.4 The Fundamental Theorem of Calculus, Part II 9.4.5 Illustrating the Fundamental Theorem of Calculus 9.4.6 Evaluating Definite Integrals 9.5 Numerical Integration 9.5.1 Deriving the Trapezoidal Rule 9.5.2 An Example of the Trapezoidal Rule | Exercises 9.4.1 through 9.5.2 Chapter 9 Homework |
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11 | Chapter 10: Applications of Integration 10.1 Motion 10.1.1 Antiderivatives and Motion 10.1.2 Gravity and Vertical Motion 10.1.3 Solving Vertical Motion Problems 10.2 Finding the Area between Two Curves 10.2.1 The Area between Two Curves 10.2.2 Limits of Integration and Area 10.2.3 Common Mistakes to Avoid When Finding Areas 10.2.4 Regions Bound by Several Curves 10.3 Integrating with Respect to y 10.3.1 Finding Areas by Integrating with Respect to y: Part One 10.3.2 Finding Areas by Integrating with Respect to y: Part Two 10.3.3 Area, Integration by Substitution, and Trigonometry 10.4 The Average Value of a Function 10.4.1 Finding the Average Value of a Function 10.5 Finding Volumes Using Cross-Sections 10.5.1 Finding Volumes Using Cross-Sectional Slices 10.5.2 An Example of Finding Cross-Sectional Volumes 10.6 Disks and Washers 10.6.1 Solids of Revolution 10.6.2 The Disk Method along the y-Axis 10.6.3 A Transcendental Example of the Disk Method 10.6.4 The Washer Method across the x-Axis 10.6.5 The Washer Method across the y-Axis | Exercises 10.1.1 through 10.6.5 Whiteboard Lab 2: Calculator Techniques on Area Under the Curve and Numerical Integration |
12 | 10.7 Shells 10.7.1 Introducing the Shell Method 10.7.2 Why Shells Can Be Better Than Washers 10.7.3 The Shell Method: Integrating with Respect to y 10.8 Arc Lengths and Functions 10.8.1 An Introduction to Arc Length 10.8.2 Finding Arc Lengths of Curves Given by Functions 10.9 Work 10.9.1 An Introduction to Work 10.9.2 Calculating Work 10.9.3 Hooke's Law 10.10 Moments and Centers of Mass 10.10.1 Center of Mass 10.10.2 The Center of Mass of a Thin Plate Supplementary notes Pumping Out a Tank | Exercises 10.7.1 through 10.10..2 Chapter 10 Homework |
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13 | 11.1 Separable Differential Equations 11.1.1 An Introduction to Differential Equations 11.1.2 Solving Separable Differential Equations 11.1.3 Finding a Particular Solution 11.1.4 Direction Fields 11.2 Growth and Decay Problems 11.2.1 Exponential Growth 11.2.2 Radioactive Decay | Calculus Lab 4: Slope Field |
14 | 12.1 Indeterminate Quotients 12.1.1 Indeterminate Forms 12.1.2 An Introduction to L'Hôpital's Rule 12.1.3 Basic Uses of L'Hôpital's Rule 12.1.4 More Exotic Examples of Indeterminate Forms 12.2 Other Indeterminate Forms 12.2.1 L'Hôpital's Rule and Indeterminate Products 12.2.2 L'Hôpital's Rule and Indeterminate Differences 12.2.3 L'Hôpital's Rule and One to the Infinite Power 12.2.4 Another Example of One to the Infinite Power 12.3 Improper Integrals 12.3.1 The First Type of Improper Integral 12.3.2 The Second Type of Improper Integral 12.3.3 Infinite Limits of Integration, Convergence, and Divergence Supplementary notes Extra Remarks on Improper Integrals | Exercises 12.1.1 through 12.3.3 Chapter 12 Homework |
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15 | 13.1 The Close of Calculus I 13.1 Paradoxes 13.2 Sequences | Exercises 13.1.1 through 13.1.2 |
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16 | 14.1 Introduction and Review of differential Calculus 15 Techniques of Integration 15.1 Integration Using Tables 15.1.1 An Introduction to the Integral Table 15.1.2 Making u-Substitutions 15.2 Integrals Involving Powers of Sine and Cosine 15.2.1 An Introduction to Integrals with Powers of Sine and Cosine 15.2.2 Integrals with Powers of Sine and Cosine 15.2.3 Integrals with Even and Odd Powers of Sine and Cosine 15.3 Integrals Involving Powers of Other Trigonometric Functions 15.3.1 Integrals of Other Trigonometric Functions 15.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant 15.3.3 Integrals with Even Powers of Secant and Any Power of Tangent 15.4 An Introduction to Integration by Partial Fractions 15.4.1 Finding Partial Fraction Decompositions 15.4.2 Partial Fractions 15.4.3 Long Division 15.5 Integration by Partial Fractions with Repeated Factors 15.5.1 Repeated Linear Factors: Part One 15.5.2 Repeated Linear Factors: Part Two 15.5.3 Distinct and Repeated Quadratic Factors 15.5.4 Partial Fractions of Transcendental Functions | Exercises 1.1.1 through 15.5.4 Discussion 5: Applications of Calculus |
17 | 15.6 Integration by Parts 15.6.1 An Introduction to Integration by Parts 15.6.2 Applying Integration by Parts to the Natural Log Function 15.6.3 Inspirational Examples of Integration by Parts 15.6.4 Repeated Application of Integration by Parts 15.6.5 Algebraic Manipulation and Integration by Parts 15.7 An Introduction to Trigonometric Substitution 15.7.1 Converting Radicals into Trigonometric Expressions 15.7.2 Using Trigonometric Substitution to Integrate Radicals 15.7.3 Trigonometric Substitutions on Rational Powers 15.8 Trigonometric Substitution Strategy 15.8.1 An Overview of Trigonometric Substitution Strategy 15.8.2 Trigonometric Substitution Involving a Definite Integral: Part One 15.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two 15.9 The Calculus of Inverse Trigonometric Functions 15.9.1 More Calculus of Inverse Trigonometric Functions Supplementary notes Trigonometric Substitution The center of Mass: Centroid | Exercises15.6.1 through 15.9.1 Whiteboard Lab 3: Calculator Techniques on Integration Chapter 15 Homework |
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18 | 16.1 Understanding Parametric Equations 16.1.1 An Introduction to Parametric Equations 16.1.2 The Cycloid 16.1.3 Eliminating Parameters 16.2 Calculus and Parametric Equations 16.2.1 Derivatives of Parametric Equations 16.2.2 Graphing the Elliptic Curve 16.2.3 The Arc Length of a Parameterized Curve 16.2.4 Finding Arc Lengths of Curves Given by Parametric Equations 16.3 Understanding Polar Coordinates 16.3.1 The Polar Coordinate System 16.3.2 Converting between Polar and Cartesian Forms 16.3.3 Spirals and Circles 16.3.4 Graphing Some Special Polar Functions 16.4 Polar Functions and Slope 16.4.1 Calculus and the Rose Curve 16.4.2 Finding the Slopes of Tangent Lines in Polar Form | Exercises 16.1.1 through 3.4.2 |
19 | 16.5 Polar Functions and Area 16.5.1 Heading toward the Area of a Polar Region 16.5.2 Finding the Area of a Polar Region: Part One 16.5.3 Finding the Area of a Polar Region: Part Two 16.5.4 The Area of a Region Bounded by Two Polar Curves: Part One 16.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two | Exercises 16.5.1 through 16.5.5 Chapter 16 Homework |
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21 | 17.1 Sequences 17.1.1 The Limit of a Sequence 17.1.2 Determining the Limit of a Sequence 17.1.3 The Squeeze and Absolute Value Theorems 17.2 Monotonic and Bounded Sequences 17.2.1 Monotonic and Bounded Sequences 17.3 Infinite Series 17.3.1 An Introduction to Infinite Series 17.3.2 The Summation of Infinite Series 17.3.3 Geometric Series 17.3.4 Telescoping Series 17.4 Convergence and Divergence 17.4.1 Properties of Convergent Series 17.4.2 The nth-Term Test for Divergence 17.1 Sequences | Exercises 17.1.1 through 17.4.2 Calculus Lab 5: Application of Series and Sequence |
22 | 17.5 The Integral Test 17.5.1 An Introduction to the Integral Test 17.5.2 Examples of the Integral Test 17.5.3 Using the Integral Test 17.5.4 Defining p-Series 17.6 The Direct Comparison Test 17.6.1 An Introduction to the Direct Comparison Test 17.6.2 Using the Comparison Test 17.7 The Limit Comparison Test 17.7.1 An Introduction to the Limit Comparison Test 17.7.2 Using the Limit Comparison Test 17.7.3 Inverting the Series in the Limit Comparison Test | Exercises 17.5.1 through 17.7.3 |
23 | 17.8 The Alternating Series 17.8.1 Alternating Series 17.8.2 The Alternating Series Test 17.8.3 Estimating the Sum of an Alternating Series 17.9 Absolute and Conditional Convergences 17.9.1 Absolute and Conditional Convergence 17.10 The Ratio and Root Tests 17.10.1 The Ratio Test 17.10.2 Examples of the Ratio Test 17.10.3 The Root Test 17.11 Polynomial Approximations of Elementary Functions 17.11.1 Polynomial Approximation of Elementary Functions 17.11.2 Higher-Degree Approximations | Discussion 6: Sequence and Series (History and the concept of infinity) Exercises 17.8.1 through 17.11.2 |
24 | 17.12 Taylor and Maclaurin Polynomials 17.12.1 Taylor Polynomials 17.12.2 Maclaurin Polynomials 17.12.3 The Remainder of a Taylor Polynomial 17.12.4 Approximating the Value of a Function 17.13 Taylor and Maclaurin Series 17.13.1 Taylor Series 17.13.2 Examples of the Taylor and Maclaurin Series 17.13.3 New Taylor Series 17.13.4 The Convergence of Taylor Series Supplementary note Taylor and Maclauring Polynomial and Series and Lagrange Formula for the Remainder 17.14 Power Series 17.14.1 The Definition of Power Series 17.14.2 The Interval and Radius of Convergence 17.14.3 Finding the Interval and Radius of Convergence: Part One 17.14.4 Finding the Interval and Radius of Convergence: Part Two 17.14.5 Finding the Interval and Radius of Convergence: Part Three 17.15 Power Series Representations of Functions 17.15.1 Differentiation and Integration of Power Series 17.15.2 Finding Power Series Representations by Differentiation 17.15.3 Finding Power Series Representations by Integration 17.15.4 Integrating Functions Using Power Series | Exercises 17.12.1 through 17.13.4 Exercises 17.14.1 through 17.15.4 Chapter 17 Homework |
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25 | 18.1 Solving a Homogeneous Differential Equation 18.1.1 Separating Homogeneous Differential Equations 18.1.2 Change of Variables 18.2 Solving First-Order Linear Differential Equations 18.2.1 First-Order Linear Differential Equations 18.2.2 Using Integrating Factors Supplementary Note: Euler’s Method 19.1 Vectors and the Geometry of R2 and R3 19.1.1 Coordinate Geometry in Three Dimensional Space 19.1.2 Introduction to Vectors 19.1.3 Vectors in R2 and R3 19.1.4 An Introduction to the Dot Product 19.1.5 Orthogonal Projections 19.1.6 An Introduction to the Cross Product 19.1.7 Geometry of the Cross Product 19.1.8 Equations of Lines and Planes in R3 19.2 Vector Functions 19.2.1 Introduction to Vector Functions 19.2.2 Derivatives of Vector Functions 19.2.3 Vector Functions: Smooth Curves 19.2.4 Vector Functions: Velocity and Acceleration | Exercises 18.1.1 through 18.2.2 Discussion 7: Differential Equations Whiteboard Lab 4: Calculator Techniques on Differential Equations Chapter 18 Homework Exercises 19.1.1 through 19.2.4 |
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