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Thomas' Calculus

Maurice D. Weir and Joel Hass
Publication Date: 
Number of Pages: 
[Reviewed by
Dennis W. Gordon
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The cover image of a tree line on a snow-swept landscape, by the photographer Michael Kenna, was taken in Hokkaido, Japan. The artist was not thinking of calculus when he composed the image, but rather, of a visual haiku consisting of a few elements that would spark a viewer’s imagination. Similarly, the minimal design of this text allows the central ideas of calculus developed in this book to unfold to ignite the learner’s imagination.

Originally written in 1951 and revised, expanded, and modernized over the years, Thomas’s iconic Calculus and Analytic Geometry offered several generations of students a fascinating way to learn this wonderful subject. This edition is presented as “based on the original work by George B. Thomas, Jr. as revised by Maurice D. Weir and Joel Haas.” Comparing it to my thoroughly used copy was interesting.

As a student I heard talk of the alleged beauty of mathematics, but at the time I was too busy to notice, bogged down as I was with the demands of heavy class loads and the drudgery of part time jobs. Then I came across Thomas′s splendid passage on Newton’s famous solution to the brachistochrone problem and the related tautochrone problem. The treatment was so elegant that I experienced the great epiphany of suddenly seeing that Bertrand Russell’s words were true that “[m]athematics, rightly viewed, possess not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture.” In this superb new edition the problem is presented much like in the 1951 edition, but the exposition is charmingly enhanced with additional material on the cycloid curve and color is added for superior clarity. The cycloid was also used by Christian Huygens in his pendulum clock, and is well explained in words and with nice illustrations. Like these examples, the rest of the book delights with its worked-out problems and figures.

In bringing Thomas to the 21st century, the new authors have included a variety of helpful material such as printed material and internet references; in addition, some of the exercises require software programs like Maple or Mathematica.

Did you know that if you cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y = √ (r2 – x2) is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.)

Further examples of the appearance of calculus in other areas are the remarkably accurate sine curve describing the weather over the course of the year in Fairbanks, Alaska, why the trachea constricts when an individual is coughing, the calculation of the optimum angle for kicking a field goal in football, why machines often break down when run at excessive speed, and even art forgery. Sociologists will enjoy the treatment of social diffusion. These are all real-life examples which serve to show the vast usefulness of our beloved subject. Calculus is fun stuff and learning it is all the more enjoyable with this book.

In spite of having studied chemistry (Wayne State University and The University of Kansas) and a professional career in both academic and industrial research, Dennis’ greatest personal interest in science is mathematics. Now retired, he is a voracious reader, and with his wife Sally, they enjoy traveling in their sports car, bluegrass music, and the wonders of Wisconsin. Dennis may be contacted at

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

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 Related Rates

3.9 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 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 Method

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 and Fluid Forces

6.6 Moments and Centers of Mass


7. Transcendental Functions

7.1 Inverse Functions and Their Derivatives

7.2 Natural Logarithms

7.3 Exponential Functions

7.4 Exponential Change and Separable Differential Equations

7.5 Indeterminate Forms and L'Hôpital's Rule

7.6 Inverse Trigonometric Functions

7.7 Hyperbolic Functions

7.8 Relative Rates of Growth


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. First-Order Differential Equations

9.1 Solutions, Slope Fields, and Euler's Method

9.2 First-Order Linear Equations

9.3 Applications

9.4 Graphical Solutions of Autonomous Equations

9.5 Systems of Equations and Phase Planes


10. Infinite Sequences and Series

10.1 Sequences

10.2 Infinite Series

10.3 The Integral Test

10.4 Comparison Tests

10.5 The Ratio and Root Tests

10.6 Alternating Series, Absolute and Conditional Convergence

10.7 Power Series

10.8 Taylor and Maclaurin Series

10.9 Convergence of Taylor Series

10.10 The Binomial Series and Applications of Taylor Series


11. Parametric Equations and Polar Coordinates

11.1 Parametrizations of Plane Curves

11.2 Calculus with Parametric Curves

11.3 Polar Coordinates

11.4 Graphing in Polar Coordinates

11.5 Areas and Lengths in Polar Coordinates

11.6 Conic Sections

11.7 Conics in Polar Coordinates