Calculus Companion

Calculus Calculus is advanced math for the high school student, but it's the starting point for math in the most selective colleges and universities. Thinkwell's Calculus course covers both Calculus I and Calculus II, each of which is a one-semester course in college. If you plan to take the AP® Calculus AB or AP® Calculus BC exam, you should consider our Calculus for AP® courses, which have assessments targeted to the AP® exam.   Thinkwell's award-winning math professor, Edward Burger, has a gift for explaining and demonstrating the underlying structure of calculus so that your students will retain what they've learned. It's a great head start for the college-bound math, science, or engineering student. Thinkwell Calculus is approved by the University of California and on their A-G course list. Our complete Calculus package includes: 12-month Online Subscription to our complete Calculus course with video lessons, automatically graded calculus problems, and much more. Printed Notes (optional) are the Calculus course notes from the Online Subscription printed in a color, on-the-go format.   Calculus Materials Online Subscription, 12-month access Access to a complete online package that includes everything you need: High-quality video lessons explain all of the Calculus math topics and concepts Automatically graded calculus problems with immediate feedback allow you to track your progress Printable full-color illustrated notes help you review what you've learned in the video lesson Subscriptions start when you are ready. Buy now and activate your course anytime you like. Wait up to one year to activate your subscription; your 12-month subscription doesn't begin until you say so! Calculus Details Thinkwell's Calculus has all the features your home school needs: More than 270 educational video lessons 2000+ interactive calculus exercises with immediate feedback allow you to track your progress Automatically graded calculus tests for all 21 chapters, as well as practice tests, a midterm, and a final examPrintable illustrated notes for each topic Real-world application examples in both lectures and exercises Interactive animations with audio Glossary of more than 450 mathematical terms Engaging content to help students advance their mathematical knowledge: Limits and derivatives Computational techniques such as the power rule, product rule, quotient rule, and chain rule Differentiation, optimization, and related rates Antiderivatives, integration, and the fundamental theorem of calculus Indeterminate forms and L'Hopital's rule Sequences and series Differential equations Parametric equations and polar coordinates Vector calculus Table of Contents 1. The Basics 1.1 Overview 1.1.1 An Introduction to Thinkwell 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. Limits 2.1 The Concept 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 3. An Introduction to Derivatives 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 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 5. Special Functions 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 6. Implicit Differentiation 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 7. Applications of Differentiation 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 8. Curve Sketching 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 9. The Basics of Integration 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 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 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 11. Calculus I Review 11.1 The Close of Calculus I 11.1.1 A Glimpse Into Calculus II 12. Math Fun 12.1 Paradoxes 12.1.1 An Introduction to Paradoxes 12.1.2 Paradoxes and Air Safety 12.1.3 Newcomb's Paradox 12.1.4 Zeno's Paradox 12.2 Sequences 12.2.1 Fibonacci Numbers 12.2.2 The Golden Ratio 13. An Introduction to Calculus II 13.1 Introduction 13.1.1 Welcome to Calculus II 13.1.2 Review: Calculus I in 20 Minutes 14. L'Hôpital's Rule 14.1 Indeterminate Quotients 14.1.1 Indeterminate Forms 14.1.2 An Introduction to L'Hôpital's Rule 14.1.3 Basic Uses of L'Hôpital's Rule 14.1.4 More Exotic Examples of Indeterminate Forms 14.2 Other Indeterminate Forms 14.2.1 L'Hôpital's Rule and Indeterminate Products 14.2.2 L'Hôpital's Rule and Indeterminate Differences 14.2.3 L'Hôpital's Rule and One to the Infinite Power 14.2.4 Another Example of One to the Infinite Power 15. Elementary Functions and Their Inverses 15.1 Inverse Functions 15.1.1 The Exponential and Natural Log Functions 15.1.2 Differentiating Logarithmic Functions 15.1.3 Logarithmic Differentiation 15.1.4 The Basics of Inverse Functions 15.1.5 Finding the Inverse of a Function 15.2 The Calculus of Inverse Functions 15.2.1 Derivatives of Inverse Functions 15.3 Inverse Trigonometric Functions 15.3.1 The Inverse Sine, Cosine, and Tangent Functions 15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions 15.3.3 Evaluating Inverse Trigonometric Functions 15.4 The Calculus of Inverse Trigonometric Functions 15.4.1 Derivatives of Inverse Trigonometric Functions 15.4.2 More Calculus of Inverse Trigonometric Functions 15.5 The Hyperbolic Functions 15.5.1 Defining the Hyperbolic Functions 15.5.2 Hyperbolic Identities 15.5.3 Derivatives of Hyperbolic Functions 16. Techniques of Integration 16.1 Integration Using Tables 16.1.1 An Introduction to the Integral Table 16.1.2 Making u-Substitutions 16.2 Integrals Involving Powers of Sine and Cosine 16.2.1 An Introduction to Integrals with Powers of Sine and Cosine 16.2.2 Integrals with Powers of Sine and Cosine 16.2.3 Integrals with Even and Odd Powers of Sine and Cosine 16.3 Integrals Involving Powers of Other Trigonometric Functions 16.3.1 Integrals of Other Trigonometric Functions 16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant 16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent 16.4 An Introduction to Integration by Partial Fractions 16.4.1 Finding Partial Fraction Decompositions 16.4.2 Partial Fractions 16.4.3 Long Division 16.5 Integration by Partial Fractions with Repeated Factors 16.5.1 Repeated Linear Factors: Part One 16.5.2 Repeated Linear Factors: Part Two 16.5.3 Distinct and Repeated Quadratic Factors 16.5.4 Partial Fractions of Transcendental Functions 16.6 Integration by Parts 16.6.1 An Introduction to Integration by Parts 16.6.2 Applying Integration by Parts to the Natural Log Function 16.6.3 Inspirational Examples of Integration by Parts 16.6.4 Repeated Application of Integration by Parts 16.6.5 Algebraic Manipulation and Integration by Parts 16.7 An Introduction to Trigonometric Substitution 16.7.1 Converting Radicals into Trigonometric Expressions 16.7.2 Using Trigonometric Substitution to Integrate Radicals 16.7.3 Trigonometric Substitutions on Rational Powers 16.8 Trigonometric Substitution Strategy 16.8.1 An Overview of Trigonometric Substitution Strategy 16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One 16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two 16.9 Numerical Integration 16.9.1 Deriving the Trapezoidal Rule 16.9.2 An Example of the Trapezoidal Rule 17. Improper Integrals 17.1 Improper Integrals 17.1.1 The First Type of Improper Integral 17.1.2 The Second Type of Improper Integral 17.1.3 Infinite Limits of Integration, Convergence, and Divergence 18. Applications of Integral Calculus 18.1 The Average Value of a Function 18.1.1 Finding the Average Value of a Function 18.2 Finding Volumes Using Cross-Sections 18.2.1 Finding Volumes Using Cross-Sectional Slices 18.2.2 An Example of Finding Cross-Sectional Volumes 18.3 Disks and Washers 18.3.1 Solids of Revolution 18.3.2 The Disk Method along the y-Axis 18.3.3 A Transcendental Example of the Disk Method 18.3.4 The Washer Method across the x-Axis 18.3.5 The Washer Method across the y-Axis 18.4 Shells 18.4.1 Introducing the Shell Method 18.4.2 Why Shells Can Be Better Than Washers 18.4.3 The Shell Method: Integrating with Respect to y 18.5 Arc Lengths and Functions 18.5.1 An Introduction to Arc Length 18.5.2 Finding Arc Lengths of Curves Given by Functions 18.6 Work 18.6.1 An Introduction to Work 18.6.2 Calculating Work 18.6.3 Hooke's Law 18.7 Moments and Centers of Mass 18.7.1 Center of Mass 18.7.2 The Center of Mass of a Thin Plate 19. Sequences and Series 19.1 Sequences 19.1.1 The Limit of a Sequence 19.1.2 Determining the Limit of a Sequence 19.1.3 The Squeeze and Absolute Value Theorems 19.2 Monotonic and Bounded Sequences 19.2.1 Monotonic and Bounded Sequences 19.3 Infinite Series 19.3.1 An Introduction to Infinite Series 19.3.2 The Summation of Infinite Series 19.3.3 Geometric Series 19.3.4 Telescoping Series 19.4 Convergence and Divergence 19.4.1 Properties of Convergent Series 19.4.2 The nth-Term Test for Divergence 19.5 The Integral Test 19.5.1 An Introduction to the Integral Test 19.5.2 Examples of the Integral Test 19.5.3 Using the Integral Test 19.5.4 Defining p-Series 19.6 The Direct Comparison Test 19.6