## Table of Contents

**1. Functions and Limits **

1.1 Functions and Their Graphs

1.2 Combining Functions; Shifting and Scaling Graphs

1.3 Rates of Change and Tangents to Curves

1.4 Limit of a Function and Limit Laws

1.5 Precise Definition of a Limit

1.6 One-Sided Limits

1.7 Continuity

1.8 Limits Involving Infinity

Questions to Guide Your Review

Practice and Additional Exercises

**2. Differentiation**

2.1 Tangents and Derivatives at a Point

2.2 The Derivative as a Function

2.3 Differentiation Rules

2.4 The Derivative as a Rate of Change

2.5 Derivatives of Trigonometric Functions

2.6 Exponential Functions

2.7 The Chain Rule

2.8 Implicit Differentiation

2.9 Inverse Functions and Their Derivatives

2.10 Logarithmic Functions

2.11 Inverse Trigonometric Functions

2.12 Related Rates

2.13 Linearization and Differentials

Questions to Guide Your Review

Practice and Additional Exercises

**3. Applications of Derivatives **

3.1 Extreme Values of Functions

3.2 The Mean Value Theorem

3.3 Monotonic Functions and the First Derivative Test

3.4 Concavity and Curve Sketching

3.5 Parametrizations of Plane Curves

3.6 Applied Optimization

3.7 Indeterminate Forms and L'Hopital's Rule

3.8 Newton's Method

3.9 Hyperbolic Functions

Questions to Guide Your Review

Practice and Additional Exercises

**4. Integration**

4.1 Antiderivatives

4.2 Estimating with Finite Sums

4.3 Sigma Notation and Limits of Finite Sums

4.4 The Definite Integral

4.5 The Fundamental Theorem of Calculus

4.6 Indefinite Integrals and the Substitution Rule

4.7 Substitution and Area Between Curves

Questions to Guide Your Review

Practice and Additional Exercises

**5. Techniques of Integration**

5.1 Integration by Parts

5.2 Trigonometric Integrals

5.3 Trigonometric Substitutions

5.4 Integration of Rational Functions by Partial Fractions

5.5 Integral Tables and Computer Algebra Systems

5.6 Numerical Integration

5.7 Improper Integrals

Questions to Guide Your Review

Practice and Additional Exercises

**6. Applications of Definite Integrals**

6.1 Volumes by Slicing and Rotation About an Axis

6.2 Volumes by Cylindrical Shells

6.3 Lengths of Plane Curves

6.4 Exponential Change and Separable Differential Equations

6.5 Work and Fluid Forces

6.6 Moments and Centers of Mass

Questions to Guide Your Review

Practice and Additional Exercises

**7. Infinite Sequences and Series**

7.1 Sequences

7.2 Infinite Series

7.3 The Integral Test

7.4 Comparison Tests

7.5 The Ratio and Root Tests

7.6 Alternating Series, Absolute and Conditional Convergence

7.7 Power Series

7.8 Taylor and Maclaurin Series

7.9 Convergence of Taylor Series

7.10 The Binomial Series

Questions to Guide Your Review

Practice and Additional Exercises

**8. Polar Coordinates and Conics**

8.1 Polar Coordinates

8.2 Graphing in Polar Coordinates

8.3 Areas and Lengths in Polar Coordinates

8.4 Conics in Polar Coordinates

8.5 Conics and Parametric Equations; The Cycloid

Questions to Guide Your Review

Practice and Additional Exercises

**9. Vectors and the Geometry of Space**

9.1 Three-Dimensional Coordinate Systems

9.2 Vectors

9.3 The Dot Product

9.4 The Cross Product

9.5 Lines and Planes in Space

9.6 Cylinders and Quadric Surfaces

Questions to Guide Your Review

Practice and Additional Exercises

**10. Vector-Valued Functions and Motion in Space**

10.1 Vector Functions and Their Derivatives

10.2 Integrals of Vector Functions

10.3 Arc Length and the Unit Tangent Vector T

10.4 Curvature and the Unit Normal Vector N

10.5 Torsion and the Unit Binormal Vector B

10.6 Planetary Motion

Questions to Guide Your Review

Practice and Additional Exercises

**11. Partial Derivatives**

11.1 Functions of Several Variables

11.2 Limits and Continuity in Higher Dimensions

11.3 Partial Derivatives

11.4 The Chain Rule

11.5 Directional Derivatives and Gradient Vectors

11.6 Tangent Planes and Differentials

11.7 Extreme Values and Saddle Points

11.8 Lagrange Multipliers

Questions to Guide Your Review

Practice and Additional Exercises

**12. Multiple Integrals**

12.1 Double and Iterated Integrals over Rectangles

12.2 Double Integrals over General Regions

12.3 Area by Double Integration

12.4 Double Integrals in Polar Form

12.5 Triple Integrals in Rectangular Coordinates

12.6 Moments and Centers of Mass

12.7 Triple Integrals in Cylindrical and Spherical Coordinates

12.8 Substitutions in Multiple Integrals

Questions to Guide Your Review

Practice and Additional Exercises

**13. Integration in Vector Fields**

13.1 Line Integrals

13.2 Vector Fields, Work, Circulation, and Flux

13.3 Path Independence, Potential Functions, and Conservative Fields

13.4 Green's Theorem in the Plane

13.5 Surface Area and Surface Integrals

13.6 Parametrized Surfaces

13.7 Stokes' Theorem

13.8 The Divergence Theorem and a Unified Theory

Questions to Guide Your Review

Practice and Additional Exercises

**Appendices**

1. Real Numbers and the Real Line

2. Mathematical Induction

3. Lines, Circles, and Parabolas

4. Trigonometric Functions

5. Basic Algebra and Geometry Formulas

6. Proofs of Limit Theorems and L'Hopital's Rule

7. Commonly Occurring Limits

8. Theory of the Real Numbers

9. Convergence of Power Series and Taylor's Theorem

10. The Distributive Law for Vector Cross Products

11. The Mixed Derivative Theorem and the Increment Theorem

12. Taylor's Formula for Two Variables

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