Calculus: Early Transcendental Single Variable (Loose Leaf, 11)

INTRODUCTION: The Roots of Calculus

1 LIMITS AND CONTINUITY

1.1 Limits (An Intuitive Approach)

1.2 Computing Limits

1.3 Limits at Infinity; End Behavior of a Function

1.4 Limits (Discussed More Rigorously)

1.5 Continuity

1.6 Continuity of Trigonometric Functions

1.7 Inverse Trigonometric Functions

1.8 Exponential and Logarithmic Functions

2 THE DERIVATIVE

2.1 Tangent Lines and Rates of Change

2.2 The Derivative Function

2.3 Introduction to Techniques of Differentiation

2.4 The Product and Quotient Rules

2.5 Derivatives of Trigonometric Functions

2.6 The Chain Rule

3 TOPICS IN DIFFERENTIATION

3.1 Implicit Differentiation

3.2 Derivatives of Logarithmic Functions

3.3 Derivatives of Exponential and Inverse Trigonometric Functions

3.4 Related Rates

3.5 Local Linear Approximation; Differentials

3.6 L’Hôpital’s Rule; Indeterminate Forms

4 THE DERIVATIVE IN GRAPHING AND APPLICATIONS

4.1 Analysis of Functions I: Increase, Decrease, and Concavity

4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials

4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents

4.4 Absolute Maxima and Minima

4.5 Applied Maximum and Minimum Problems

4.6 Rectilinear Motion

4.7 Newton’s Method

4.8 Rolle’s Theorem; Mean-Value Theorem

5 INTEGRATION

5.1 An Overview of the Area Problem

5.2 The Indefinite Integral

5.3 Integration by Substitution

5.4 The Definition of Area as a Limit; Sigma Notation

5.5 The Definite Integral

5.6 The Fundamental Theorem of Calculus

5.7 Rectilinear Motion Revisited Using Integration

5.8 Average Value of a Function and its Applications

5.9 Evaluating Definite Integrals by Substitution

5.10 Logarithmic and Other Functions Defined by Integrals

6 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

6.1 Area Between Two Curves

6.2 Volumes by Slicing; Disks and Washers

6.3 Volumes by Cylindrical Shells

6.4 Length of a Plane Curve

6.5 Area of a Surface of Revolution

6.6 Work

6.7 Moments, Centers of Gravity, and Centroids

6.8 Fluid Pressure and Force

6.9 Hyperbolic Functions and Hanging Cables

7 PRINCIPLES OF INTEGRAL EVALUATION

7.1 An Overview of Integration Methods

7.2 Integration by Parts

7.3 Integrating Trigonometric Functions

7.4 Trigonometric Substitutions

7.5 Integrating Rational Functions by Partial Fractions

7.6 Using Computer Algebra Systems and Tables of Integrals

7.7 Numerical Integration; Simpson’s Rule

7.8 Improper Integrals

8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS

8.1 Modeling with Differential Equations

8.2 Separation of Variables

8.3 Slope Fields; Euler’s Method

8.4 First-Order Differential Equations and Applications

9 INFINITE SERIES

9.1 Sequences

9.2 Monotone Sequences

9.3 Infinite Series

9.4 Convergence Tests

9.5 The Comparison, Ratio, and Root Tests

9.6 Alternating Series; Absolute and Conditional Convergence

9.7 Maclaurin and Taylor Polynomials

9.8 Maclaurin and Taylor Series; Power Series

9.9 Convergence of Taylor Series

9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series

10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS

10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves

10.2 Polar Coordinates

10.3 Tangent Lines, Arc Length, and Area for Polar Curves

10.4 Conic Sections

10.5 Rotation of Axes; Second-Degree Equations

10.6 Conic Sections in Polar Coordinates

APPENDICES

A TRIGONOMETRY SUMMARY

B FUNCTIONS (SUMMARY)

C NEW FUNCTIONS FROM OLD (SUMMARY)

D FAMILIES OF FUNCTIONS (SUMMARY)