Mathematics

This is the most comprehensive revision of Thomas' Calculus in 25 years. The new edition of Thomas is a return to what Thomas has always been: the book with the best exercises. For the 11th edition, the authors have added exercises cut in the 10th edition, as well as exercises and examples from the classic 5th and 6th editions.

The book's theme is that Calculus is about thinking; one cannot memorize it all. The exercises develop this theme as a pivot point between the lecture in class, and the understanding that comes with applying the ideas of Calculus.

In addition, the table of contents has been refined, introducing transcendentals in the first seven chapters. Many of the examples have been trimmed of distractions and rewritten with a clear focus on the main ideas. The authors have also excised extraneous information in general and have made the technology much more transparent.

The ambition of Thomas 11e is to teach the ideas of Calculus so that students will be able to apply them in new and novel ways, first in the exercises but ultimately in their careers. Every effort has been made to insure that all content in the new edition reinforces thinking and encourages deep understanding of the material.

• Carefully developed exercises – the benchmark by which all other books are measured.

• Applications to the physical world – a Thomas trademark.

• Transparent integration of technology.

• Complete and careful multivariable calculus section.

• Online resources including exercises and projects.

• Streamlined coverage of the techniques of integration.

• The 11th edition is based on the strengths of the 5th, 6th, and 9th Editions of Thomas' Calculus.

• New chapter on integration. Integration introduced through finite sums; indefinite integrals follow the Fundamental Theorem.

• Proofs have been pulled out of the appendix and placed back in the body of the book. Emphasis on mathematical precision and rigor throughout, including proofs of most major results. Care is taken to point out where proofs are given and where omitted.

• Combined treatment of 2- and 3-dimensional vectors; a single chapter on vector-valued functions.

• Brief new appendix on the theory of real numbers emphasizing its role in calculus.

• Limits are covered both intuitively and precisely.

(Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.)

1. Functions     

    Functions and Their Graphs

    Identifying Functions; Mathematical Models

    Combining Functions; Shifting and Scaling Graphs

    Graphing with Calculators and Computers

    Exponential Functions

    Inverse Functions and Logarithms

            

 

2. Limits and Continuity

 

    Rates of Change and Limits

 

    Calculating Limits Using the Limit Laws

 

    Precise Definition of a Limit

 

    One-Sided Limits and Limits at Infinity

 

    Infinite Limits and Vertical Asymptotes

 

    Continuity

 

    Tangents and Derivatives

 

 

 

3. Differentiation

 

    The Derivative as a Function

 

    Differentiation Rules for Polynomials, Exponentials, Products and Quotients

 

    The Derivative as a Rate of Change

 

    Derivatives of Trigonometric Functions

 

    The Chain Rule and Parametric Equations

 

    Implicit Differentiation

 

    Derivatives of Inverse Functions and Logarithms

 

    Inverse Trigonometric Functions

 

    Related Rates

 

    Linearization and Differentials

 

 

 

4. Applications of Derivatives

 

    Extreme Values of Functions

 

    The Mean Value Theorem

 

    Monotonic Functions and the First Derivative Test

 

    Concavity and Curve Sketching

 

    Applied Optimization Problems

 

    Indeterminate Forms and L’Hopital’s Rule

 

    Newton’s Method

 

    Antiderivatives

               

 

 

5. Integration

 

    Estimating with Finite Sums

 

    Sigma Notation and Limits of Finite Sums

 

    The Definite Integral

 

    The Fundamental Theorem of Calculus

 

    Indefinite Integrals and the Substitution Rule

 

    Substitution and Area Between Curves

 

 

 

6. Applications of Definite Integrals

 

    Volumes by Slicing and Rotation About an Axis

 

    Volumes by Cylindrical Shells

 

    Lengths of Plane Curves

 

    Moments and Centers of Mass

 

    Areas of Surfaces of Revolution and The Theorems of Pappus

 

    Work

 

    Fluid Pressures and Forces

 

 

 

7. Transcendental Functions

 

    The Logarithm Defined as an Integral

 

    Exponential Growth and Decay

 

    Relative Rates of Growth

 

    Hyperbolic Functions

 

 

 

8. Techniques of Integration

 

    Basic Integration Formulas

 

    Integration by Parts

    

    Integration of Rational Functions by Partial Fractions

 

    Trigonometric Integrals

    

    Trigonometric Substitutions

 

    Integral Tables and Computer Algebra Systems

 

    Numerical Integration

 

    Improper Integrals

 

 

 

9. Further Applications of Integration

 

    Slope Fields and Separable Differential Equations

 

    First-Order Linear Differential Equations

 

    Euler’s Method

 

    Graphical Solutions of Autonomous Equations

 

    Applications of First-Order Differential Equations

 

 

 

10. Conic Sections and Polar Coordinates

 

    Conic Sections and Quadratic Equations

 

    Classifying Conic Sections by Eccentricity

  

    Quadratic Equations and Rotations

              

    Conics and Parametric Equations; The Cycloid

           

    Polar Coordinates

           

    Graphing in Polar Coordinates

           

    Area and Lengths in Polar Coordinates

           

    Conic Sections in Polar Coordinates

 

 

 

11. Infinite Sequences and Series

 

    Sequences

          

    Infinite Series

           

    The Integral Test

           

    Comparison Tests

         

    The Ratio and Root Tests

          

    Alternating Series, Absolute and Conditional Convergence

              

    Power Series

             

    Taylor and Maclaurin Series

          

    Convergence of Taylor Series; Error Estimates

              

    Applications of Power Series

              

    Fourier Series

                

 

 

12. Vectors and the Geometry of Space

 

    Three-Dimensional Coordinate Systems

 

    Vectors

    

    The Dot Product

    

    The Cross Product

 

    Lines and Planes in Space

 

    Cylinders and Quadric Surfaces

               

 

 

13. Vector-Valued Functions and Motion in Space

 

    Vector Functions

 

    Modeling Projectile Motion

 

    Arc Length and the Unit Tangent Vector T

 

    Curvature and the Unit Normal Vector N

    

    Torsion and the Unit Binormal Vector B

 

    Planetary Motion and Satellites

                

 

 

14. Partial Derivatives

 

    Functions of Several Variables

 

    Limits and Continuity in Higher Dimensions

 

    Partial Derivatives

 

    The Chain Rule

 

    Directional Derivatives and Gradient Vectors

 

    Tangent Planes and Differentials

 

    Extreme Values and Saddle Points

 

    Lagrange Multipliers

 

    *Partial Derivatives with Constrained Variables

 

    Taylor’s Formula for Two Variables

               

 

 

15. Multiple Integrals

    Double Integrals

    Areas, Moments and Centers of Mass*

    Double Integrals in Polar Form

    Triple Integrals in Rectangular Coordinates

    Masses and Moments in Three Dimensions

    Triple Integrals in Cylindrical and Spherical Coordinates

    Substitutions in Multiple Integrals

 

16. Integration in Vector Fields

    Line Integrals

    Vector Fields, Work, Circulation, and Flux

    Path Independence, Potential Functions, and Conservative Fields

    Green’s Theorem in the Plane

    Surface Area and Surface Integrals

    Parametrized Surfaces

    Stokes’ Theorem

    The Divergence Theorem and a Unified Theory

 

Appendices.

    Mathematical Induction

    Proofs of Limit Theorems

    Commonly Occurring Limits

    Theory of the Real Numbers

    Complex Numbers

    The Distributive Law for Vector Cross Products

    Determinants and Cramer’s Rule

    The Mixed Derivative Theorem and the Increment Theorem

    The Area of a Parallelogram’s Projection on a Plane

 

 

Calculus  (Mathematics)

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For Calculus

Insight Into Calculus: Using Texas Instruments Graphics Calculators
Ward & Wilberscheid
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Maple Approach Calculus, A, 2/E
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