University Calculus: Early Transcendentals – George B. Thomas – 3rd Edition


University Calculus, Early Transcendentals, Third Edition helps students generalize and apply the key ideas of calculus through clear and precise explanations, thoughtfully chosen examples, meticulously crafted figures, and superior exercise sets. This text offers the right mix of basic, conceptual, and challenging exercises, along with meaningful applications. This revision features more examples, more mid-level exercises, more figures, improved conceptual flow, and the best in technology for learning and teaching.

This third edition of University Calculus provides a streamlined treatment of the material in a standard three-semester or four-quarter course taught at the university level. As the title suggests, the book aims to go beyond what many students may have seen at the high school level. By emphasizing rigor and mathematical precision, supported with examples and exercises, this book encourages students to think more clearly than if they were using rote procedures.

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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 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


    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
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