Thomas´ Calculus Early Transcendentals – George B. Thomas – 13th Edition


Many of today’s students have been exposed to the terminology and computational methods of calculus in high school. Despite this familiarity, their acquired algebra and trigonometry skills sometimes limit their ability to master calculus at the college level. In this text, we seek to balance students’ prior experience in calculus with the algebraic skill development they may still need, without slowing their progress through calculus itself. We have taken care to provide enough review material (in the text and appendices), detailed solutions, and variety of examples and exercises, to support a complete understanding of calculus for students at varying levels.

We present the material in a way to encourage student thinking, going beyond memorizing formulas and routine procedures, and we show students how to generalize key concepts once they are introduced. References are made throughout which tie a new concept to a related one that was studied earlier, or to a generalization they will see later on.

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  • 1. Functions
    1.1 Functions and Their Graphs
    1.2 Combining Functions; Shifting and Scaling Graphs
    1.3 Trigonometric Functions
    1.4 Graphing with Software

    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 Using Basic Integration Formulas
    8.2 Integration by Parts
    8.3 Trigonometric Integrals
    8.4 Trigonometric Substitutions
    8.5 Integration of Rational Functions by Partial Fractions
    8.6 Integral Tables and Computer Algebra Systems
    8.7 Numerical Integration
    8.8 Improper Integrals
    8.9 Probability

    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 Absolute Convergence; The Ratio and Root Tests
    10.6 Alternating Series 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 Polar Coordinate Equations
    11.5 Areas and Lengths in Polar Coordinates
    11.6 Conic Sections
    11.7 Conics in Polar Coordinates

    12. Vectors and the Geometry of Space
    12.1 Three-Dimensional Coordinate Systems
    12.2 Vectors
    12.3 The Dot Product
    12.4 The Cross Product
    12.5 Lines and Planes in Space
    12.6 Cylinders and Quadric Surfaces

    13. Vector-Valued Functions and Motion in Space
    13.1 Curves in Space and Their Tangents
    13.2 Integrals of Vector Functions; Projectile Motion
    13.3 Arc Length in Space
    13.4 Curvature and Normal Vectors of a Curve
    13.5 Tangential and Normal Components of Acceleration
    13.6 Velocity and Acceleration in Polar Coordinates

    14. Partial Derivatives
    14.1 Functions of Several Variables
    14.2 Limits and Continuity in Higher Dimensions
    14.3 Partial Derivatives
    14.4 The Chain Rule
    14.5 Directional Derivatives and Gradient Vectors
    14.6 Tangent Planes and Differentials
    14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers
    14.9 Taylor's Formula for Two Variables
    14.10 Partial Derivatives with Constrained Variables

    15. Multiple Integrals
    15.1 Double and Iterated Integrals over Rectangles
    15.2 Double Integrals over General Regions
    15.3 Area by Double Integration
    15.4 Double Integrals in Polar Form
    15.5 Triple Integrals in Rectangular Coordinates
    15.6 Moments and Centers of Mass
    15.7 Triple Integrals in Cylindrical and Spherical Coordinates
    15.8 Substitutions in Multiple Integrals

    16. Integrals and Vector Fields
    16.1 Line Integrals
    16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
    16.3 Path Independence, Conservative Fields, and Potential Functions
    16.4 Green's Theorem in the Plane
    16.5 Surfaces and Area
    16.6 Surface Integrals
    16.7 Stokes' Theorem
    16.8 The Divergence Theorem and a Unified Theory

    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


    1. Real Numbers and the Real Line
    2. Mathematical Induction
    3. Lines, Circles, and Parabolas
    4. Proofs of Limit Theorems
    5. Commonly Occurring Limits
    6. Theory of the Real Numbers
    7. Complex Numbers
    8. The Distributive Law for Vector Cross Products
    9. The Mixed Derivative Theorem and the Increment Theorem
  • Citation

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