University Calculus

Joel Hass received his PhD from the University of California–Berkeley. He is currently a professor of mathematics at the University of California–Davis. He has coauthored six widely used calculus texts as well as two calculus study guides. He is currently on the editorial board of Geometriae Dedicata and Media-Enhanced Mathematics. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass’s current areas of research include the geometry of proteins, three dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking.   Maurice D. Weir holds a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. He is a Professor Emeritus of the Department of Applied Mathematics at the Naval Postgraduate School in Monterey, California. Weir enjoys teaching Mathematical Modeling and Differential Equations. His current areas of research include modeling and simulation as well as mathematics education. Weir has been awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He has coauthored eight books, including the University Calculus series and the twelfth edition of Thomas’ Calculus.   George B. Thomas, Jr. (late) of the Massachusetts Institute of Technology, was a professor of mathematics for thirty-eight years; he served as the executive officer of the department for ten years and as graduate registration officer for five years. Thomas held a spot on the board of governors of the Mathematical Association of America and on the executive committee of the mathematics division of the American Society for Engineering Education. His book, Calculus and Analytic Geometry, was first published in 1951 and has since gone through multiple revisions. The text is now in its twelfth edition and continues to guide students through their calculus courses. He also co-authored monographs on mathematics, including the text Probability and Statistics.

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

1.5 Exponential Functions

1.6 Inverse Functions and Logarithms

 

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

2.5 Continuity

2.6 Limits Involving Infinity; Asymptotes of Graphs

 

3. Differentiation

3.1 Tangents and the Derivative at a Point

3.2 The Derivative as a Function

3.3 Differentiation Rules

3.4 The Derivative as a Rate of Change

3.5 Derivatives of Trigonometric Functions

3.6 The Chain Rule

3.7 Implicit Differentiation

3.8 Derivatives of Inverse Functions and Logarithms

3.9 Inverse Trigonometric Functions

3.10 Related Rates

3.11 Linearization and Differentials

 

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 Indeterminate Forms and L'Hôpital's Rule

4.6 Applied Optimization

4.7 Newton's Method

4.8 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 Using Cross-Sections

6.2 Volumes Using Cylindrical Shells

6.3 Arc Length

6.4 Areas of Surfaces of Revolution

6.5 Work

6.6 Moments and Centers of Mass

 

7. Integrals and Transcendental Functions

7.1 The Logarithm Defined as an Integral

7.2 Exponential Change and Separable Differential Equations

7.3 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 and Applications of Taylor Series

 

10. Parametric Equations and Polar Coordinates

10.1 Parametrizations of Plane Curves

10.2 Calculus with Parametric Curves

10.3 Polar Coordinates

10.4 Graphing in Polar Coordinates

10.5 Areas and Lengths in Polar Coordinates

10.6 Conics in Polar Coordinates

 

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 Curves in Space and Their Tangents

12.2 Integrals of Vector Functions; Projectile Motion

12.3 Arc Length in Space

12.4 Curvature and Normal Vectors 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

 

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 and Line Integrals: Work, Circulation, and Flux

15.3 Path Independence, Conservative Fields, and Potential Functions

15.4 Green's Theorem in the Plane

15.5 Surfaces and Area

15.6 Surface Integrals

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 Euler's Method

16.2 First-Order Linear Equations

16.3 Applications

16.4 Graphical Solutions of Autonomous Equations

16.5 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. Conic Sections

5. Proofs of Limit Theorems

6. Commonly Occurring Limits

7. Theory of the Real Numbers

8. Complex Numbers

9. The Distributive Law for Vector Cross Products

10. The Mixed Derivative Theorem and the Increment Theorem

11. Taylor's Formula for Two Variables