University Calculus: Alternate Edition Book

1. Functions

1.1 Functions and Their Graphs

1.2 Combining Functions; Shifting and Scaling Graphs

1.3 Trigonometric Functions

1.4 Graphing with Calculators and Computers

2. Limits and Continuity

2.1 Rates of Change and Tangents to Curves

2.2 Limit of a Function and Limit Laws

2.3 The Precise Definition of a Limit

2.4 One-Sided Limits and Limits at Infinity

2.5 Infinite Limits and Vertical Asymptotes

2.6 Continuity

2.7 Tangents and Derivatives at a Point

3. Differentiation

3.1 The Derivative as a Function

3.2 Differentiation Rules

3.3 The Derivative as a Rate of Change

3.4 Derivatives of Trigonometric Functions

3.5 The Chain Rule

3.6 Implicit Differentiation

3.7 Related Rates

3.8 Linearization and Differentials

3.9 Parametrizations of Plane Curves

4. Applications of Derivatives

4.1 Extreme Values of Functions

4.2 The Mean Value Theorem

4.3 Monotonic Functions and the First Derivative Test

4.4 Concavity and Curve Sketching

4.5 Applied Optimization

4.6 Newton's Method

4.7 Antiderivatives

5. Integration

5.1 Area and Estimating with Finite Sums

5.2 Sigma Notation and Limits of Finite Sums

5.3 The Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Indefinite Integrals and the Substitution Rule

5.6 Substitution and Area Between Curves

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 Areas of Surfaces of Revolution

6.5 Work

6.6 Moments and Centers of Mass

6.7 Fluid Pressures and Forces

7. Transcendental Functions

7.1 Inverse Functions and Their Derivatives

7.2 Natural Logarithms

7.3 Exponential Functions

7.4 Inverse Trigonometric Functions

7.5 Exponential Change and Separable Differential Equations

7.6 Indeterminate Forms and L'Hopital's Rule

7.7 Hyperbolic Functions

8. Techniques of Integration

8.1 Integration by Parts

8.2 Trigonometric Integrals

8.3 Trigonometric Substitutions

8.4 Integration of Rational Functions by Partial Fractions

8.5 Integral Tables and Computer Algebra Systems

8.6 Numerical Integration

8.7 Improper Integrals

9. Infinite Sequences and Series

9.1 Sequences

9.2 Infinite Series

9.3 The Integral Test

9.4 Comparison Tests

9.5 The Ratio and Root Tests

9.6 Alternating Series, Absolute and Conditional Convergence

9.7 Power Series

9.8 Taylor and Maclaurin Series

9.9 Convergence of Taylor Series

9.10 The Binomial Series

10. Polar Coordinates and Conics

10.1 Polar Coordinates

10.2 Graphing in Polar Coordinates

10.3 Areas and Lengths in Polar Coordinates

10.4 Conic Sections

10.5 Conics in Polar Coordinates

10.6 Conics and Parametric Equations; The Cycloid

11. Vectors and the Geometry of Space

11.1 Three-Dimensional Coordinate Systems

11.2 Vectors

11.3 The Dot Product

11.4 The Cross Product

11.5 Lines and Planes in Space

11.6 Cylinders and Quadric Surfaces

12. Vector-Valued Functions and Motion in Space

12.1 Vector Functions and Their Derivatives

12.2 Integrals of Vector Functions

12.3 Arc Length in Space

12.4 Curvature of a Curve

12.5 Tangential and Normal Components of Acceleration

12.6 Velocity and Acceleration in Polar Coordinates

13. Partial Derivatives

13.1 Functions of Several Variables

13.2 Limits and Continuity in Higher Dimensions

13.3 Partial Derivatives

13.4 The Chain Rule

13.5 Directional Derivatives and Gradient Vectors

13.6 Tangent Planes and Differentials

13.7 Extreme Values and Saddle Points

13.8 Lagrange Multipliers

13.9 Taylor's Formula for Two Variables

14. Multiple Integrals

14.1 Double and Iterated Integrals over Rectangles

14.2 Double Integrals over General Regions

14.3 Area by Double Integration

14.4 Double Integrals in Polar Form

14.5 Triple Integrals in Rectangular Coordinates

14.6 Moments and Centers of Mass

14.7 Triple Integrals in Cylindrical and Spherical Coordinates

14.8 Substitutions in Multiple Integrals

15. Integration in Vector Fields

15.1 Line Integrals

15.2 Vector Fields, Work, Circulation, and Flux

15.3 Path Independence, Potential Functions, and Conservative Fields

15.4 Green's Theorem in the Plane

15.5 Surfaces and Area

15.6 Surface Integrals and Flux

15.7 Stokes' Theorem

15.8 The Divergence Theorem and a Unified Theory

16. First-Order Differential Equations (online)

16.1 Solutions, Slope Fields, and Picard's Theorem

16.2 First-Order Linear Equations

16.3 Applications

16.4 Euler's Method

16.5 Graphical Solutions of Autonomous Equations

16.6 Systems of Equations and Phase Planes

17. Second-Order Differential Equations (online)

17.1 Second-Order Linear Equations

17.2 Nonhomogeneous Linear Equations

17.3 Applications

17.4 Euler Equations

17.5 Power Series Solutions

Appendices

1 Real Numbers and the Real Line

2 Mathematical Induction

3 Lines, Circles, and Parabolas

4 Trigonometry Formulas

5 Proofs of Limit Theorems

6 Commonly Occurring Limits

7 Theory of the Real Numbers

8 The Distributive Law for Vector Cross Products

9 The Mixed Derivative Theorem and the Increment Theorem