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Calculus: Early Transcendental Functions

Chapter 0: Preliminaries

0.1, “The Real Numbers and the Cartesian Plane”

0.2, “Lines and Functions”

0.3, “Graphing Calculators and Computer Algebra Systems”

0.4, “Trigonometric Functions”

0.5, “Transformations of Functions”

Chapter 1: Limits and Continuity

1.1, “A Brief Preview of Calculus: Tangent Lines and the Length of a Curve

1.2, “The Concept of Limit”

1.3, “Computation of Limits”

1.4, “Continuity and Its Consequences”

1.5, “Limits Involving Infinity; Asymptotes”

1.6, “Formal Definition of the Limit”

1.7, “Limits and Loss-of-Significance Errors”

Chapter 2: Differentiation

2.1, “Tangent Lines and Velocity”

2.2, “The Derivative”

2.3, “Computation of Derivatives: The Power Rule”

2.4, “The Product and Quotient Rules”

2.5, “The Chain Rule”

2.6, “Derivatives of Trigonometric Functions”

2.7, “Implicit Differentiation”

2.8, “The Mean Value Theorem”

Chapter 3: Applications of the Derivative

3.1, “Linear Approximations and Newton’s Method””

3.2, “Maximum and Minimum Values”

3.3, “Increasing and Decreasing Functions”

3.4, “Concavity and the Second Derivative Test”

3.5, “Overview of Curve Sketching”

3.6, “Optimization”

3.7, “Related Rates”

3.8, “Rates of Change in Economics and the Sciences”

Chapter 4: Integration

4.1, “Antiderivatives”

4.2, “Sums and Sigma Notation”

4.3, “Area”

4.4, “The Definite Integral”

4.5, “The Fundamental Theorem of Calculus”

4.6, “Integration by Substitution”

4.7, “Numerical Integration”

Chapter 5: Applications of the Definite Integral

5.1, “Area Between Curves”

5.2, “Volume: Slicing, Disks and Washers”

5.3, “Volumes by Cylindrical Shells”

5.4, “Arc Length and Surface Area”

5.5, “Projectile Motion”

5.6, “Applications of Integration to Physics and Engineering”

Chapter 6: Exponentials, Logarithms and Other Transcendental Functions

6.1, “The Natural Logarithm”

6.2, “Inverse Functions”

6.3, “The Exponential Function”

6.4, “The Inverse Trigonometric Functions”

6.5, “The Calculus of the Inverse Trigonometric Functions”

6.6, “The Hyperbolic Functions”

Chapter 7: Integration Techniques

7.1, “Review of Formulas and Techniques”

7.2, “Integration by Parts”

7.3, “Trigonometric Techniques of Integration”

7.4, “Integration of Rational Functions Using Partial Fractions”

7.5, “Integration Tables and Computer Algebra Systems”

7.6, “Indeterminate Forms and L’Hopital’s Rule”

7.7, “Improper Integrals”

7.8, “Probability”

Chapter 8: First-Order Differential Equations

8.1, “Modeling with Differential Equations”

8.2, “Separable Differential Equations”

8.3, “Direction Fields and Euler’s Method”

8.4, “Systems of First-Order Differential Equations”

Chapter 9: Infinite Series

9.1, “Sequences of Real Numbers”

9.2, “Infinite Series”

9.3, “The Integral and Comparison Tests”

9.4, “Alternating Series”

9.5, “Absolute Convergence and the Ratio Test”

9.6, “Power Series”

9.7, “Taylor Series”

9.8, “Applications of Taylor Series”

9.9, “Fourier Series”

Chapter 10: Parametric Equations and Polar Coordinates

10.1, “Plane curves and Parametric Equations”

10.2, “Calculus and Parametric Equations”

10.3, “Arc Length and Surface Area in Parametric Equations”

10.4, “Polar Coordinates”

10.5, “Calculus and Polar Coordinates”

10.6, “Conic Sections”

10.7, “Conic Sections in Polar Coordinates”

Chapter 11: Vectors and the Geometry of Space

11.1, “Vectors in the Plane”

11.2, “Vectors in Space”

11.3, “The Dot Product”

11.4, “The Cross Product”

11.5, “Lines and Planes in Space”

11.6, “Surfaces in Space”

Chapter 12: Vector-Valued Functions

12.1, “Vector-Valued Functions”

12.2, “The Calculus of Vector-Valued Functions”

12.3, “Motion in Space”

12.4, “Curvature”

12.5, “Tangent and Normal Vectors”

12.6, “Parametric Surfaces”

Chapter 13: Functions of Several Variables and Partial Differentiation

13.1, “Functions of Several Variables”

13.2, “Limits and Continuity”

13.3, “Partial Derivatives”

13.4, “Tangent Planes and Linear Approximations”

13.5, “The Chain Rule”

13.6, “The Gradient and Directional Derivatives”

13.7, “Extrema of Functions of Several Variables”

13.8, “Constrained Optimization and and Lagrange Multipliers”

Chapter 14: Multiple Integrals

14.1, “Double Integrals”

14.2, “Area, Volume and Center of Mass”

14.3, “Double Integrals in Polar Coordinates”

14.4, “Surface Area”

14.5, “Triple Integrals”

14.6, “Cylindrical Coordinates”

14.7, “Spherical Coordinates”

14.8, “Change of Variables in Multiple Integrals”

Chapter 15: Vector Calculus

15.1, “Vector Fields”

15.2, “Line Integrals”

15.3, “Independence of Path and Conservative Vector Fields”

15.4, “Green’s Theorem”

15.5, “Curl and Divergence”

15.6, “Surface Integrals”

15.7, “The Divergence Theorem”

15.8, “Stokes’ Theorem”

15.9, “Applications of Vector Calculus”

Chapter 16: Second Order Differential Equations

16.1, Second-Order Equations With Constant Coefficients”

16.2, “Nonhomogeneous Equations: Undetermined Coefficients”

16.3, “Applications of Second-Order Equations”

16.4, “Power Series Solutions of Differential Equations”

Appendix A: Proofs of Selected Theorems

Appendix B: Answers to Odd-Numbered Exercises