About our authors 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 widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. 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, 3-dimensional manifolds, applied math, and computational complexity. In his free time Hass enjoys kayaking.Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey articles. He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications. Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis. In his spare time, Heil pursues his hobby of astronomy.The late Maurice D. Weir of the the Naval Postgraduate School in Monterey, California was Professor Emeritus as a member of the Department of Applied Mathematics. He held a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. Weir was awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He co-authored 8 books, including University Calculus and Thomas' Calculus.

• 1. Functions1.1 Functions and Their Graphs1.2 Combining Functions; Shifting and Scaling Graphs1.3 Trigonometric Functions1.4 Graphing with Software1.5 Exponential Functions1.6 Inverse Functions and Logarithms  2. Limits and Continuity2.1 Rates of Change and Tangent Lines to Curves2.2 Limit of a Function and Limit Laws2.3 The Precise Definition of a Limit2.4 One-Sided Limits2.5 Continuity2.6 Limits Involving Infinity; Asymptotes of Graphs  3. Derivatives3.1 Tangent Lines and the Derivative at a Point   3.2 The Derivative as a Function3.3 Differentiation Rules3.4 The Derivative as a Rate of Change3.5 Derivatives of Trigonometric Functions3.6 The Chain Rule3.7 Implicit Differentiation3.8 Derivatives of Inverse Functions and Logarithms  3.9 Inverse Trigonometric Functions   3.10 Related Rates   3.11 Linearization and Differentials  4. Applications of Derivatives4.1 Extreme Values of Functions on Closed Intervals  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. Integrals5.1 Area and Estimating with Finite Sums5.2 Sigma Notation and Limits of Finite Sums5.3 The Definite Integral5.4 The Fundamental Theorem of Calculus5.5 Indefinite Integrals and the Substitution Method5.6 Definite Integral Substitutions and the Area Between Curves  6. Applications of Definite Integrals6.1 Volumes Using Cross-Sections6.2 Volumes Using Cylindrical Shells6.3 Arc Length6.4 Areas of Surfaces of Revolution6.5 Work and Fluid Forces6.6 Moments and Centers of Mass  7. Integrals and Transcendental Functions7.1 The Logarithm Defined as an Integral7.2 Exponential Change and Separable Differential Equations   7.3 Hyperbolic Functions   7.4 Relative Rates of Growth     8. Techniques of Integration8.1 Using Basic Integration Formulas8.2 Integration by Parts8.3 Trigonometric Integrals8.4 Trigonometric Substitutions8.5 Integration of Rational Functions by Partial Fractions8.6 Integral Tables and Computer Algebra Systems8.7 Numerical Integration8.8 Improper Integrals8.9 Probability  9. First-Order Differential Equations9.1 Solutions, Slope Fields, and Euler's Method9.2 First-Order Linear Equations9.3 Applications9.4 Graphical Solutions of Autonomous Equations 9.5 Systems of Equations and Phase Planes 10. Infinite Sequences and Series10.1 Sequences10.2 Infinite Series10.3 The Integral Test10.4 Comparison Tests10.5 Absolute Convergence; The Ratio and Root Tests10.6 Alternating Series and Conditional Convergence10.7 Power Series10.8 Taylor and Maclaurin Series10.9 Convergence of Taylor Series10.10 Applications of Taylor Series  11. Parametric Equations and Polar Coordinates11.1 Parametrizations of Plane Curves11.2 Calculus with Parametric Curves11.3 Polar Coordinates11.4 Graphing Polar Coordinate Equations11.5 Areas and Lengths in Polar Coordinates11.6 Conic Sections11.7 Conics in Polar Coordinates  12. Vectors and the Geometry of Space12.1 Three-Dimensional Coordinate Systems12.2 Vectors12.3 The Dot Product12.4 The Cross Product12.5 Lines and Planes in Space12.6 Cylinders and Quadric Surfaces  13. Vector-Valued Functions and Motion in Space13.1 Curves in Space and Their Tangents13.2 Integrals of Vector Functions; Projectile Motion13.3 Arc Length in Space13.4 Curvature and Normal Vectors of a Curve13.5 Tangential and Normal Components of Acceleration13.6 Velocity and Acceleration in Polar Coordinates  14. Partial Derivatives14.1 Functions of Several Variables14.2 Limits and Continuity in Higher Dimensions14.3 Partial Derivatives14.4 The Chain Rule14.5 Directional Derivatives and Gradient Vectors14.6 Tangent Planes and Differentials14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers14.9 Taylor's Formula for Two Variables14.10 Partial Derivatives with Constrained Variables  15. Multiple Integrals15.1 Double and Iterated Integrals over Rectangles15.2 Double Integrals over General Regions15.3 Area by Double Integration15.4 Double Integrals in Polar Form15.5 Triple Integrals in Rectangular Coordinates15.6 Applications15.7 Triple Integrals in Cylindrical and Spherical Coordinates15.8 Substitutions in Multiple Integrals  16. Integrals and Vector Fields16.1 Line Integrals of Scalar Functions16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux16.3 Path Independence, Conservative Fields, and Potential Functions16.4 Green's Theorem in the Plane16.5 Surfaces and Area16.6 Surface Integrals16.7 Stokes' Theorem16.8 The Divergence Theorem and a Unified Theory  17. Second-Order Differential Equations (Online at www.goo.gl/MgDXPY)17.1 Second-Order Linear Equations17.2 Nonhomogeneous Linear Equations17.3 Applications17.4 Euler Equations17.5 Power-Series Solutions  Appendices1. Real Numbers and the Real Line2. Mathematical Induction3. Lines, Circles, and Parabolas4. Proofs of Limit Theorems5. Commonly Occurring Limits6. Theory of the Real Numbers7. Complex Numbers8. The Distributive Law for Vector Cross Products9. The Mixed Derivative Theorem and the Increment Theorem

About our authors 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 widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. 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, 3-dimensional manifolds, applied math, and computational complexity. In his free time Hass enjoys kayaking.Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey articles. He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications. Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis. In his spare time, Heil pursues his hobby of astronomy.The late Maurice D. Weir of the the Naval Postgraduate School in Monterey, California was Professor Emeritus as a member of the Department of Applied Mathematics. He held a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. Weir was awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He co-authored 8 books, including University Calculus and Thomas' Calculus.

• 1. Functions1.1 Functions and Their Graphs1.2 Combining Functions; Shifting and Scaling Graphs1.3 Trigonometric Functions1.4 Graphing with Software1.5 Exponential Functions1.6 Inverse Functions and Logarithms  2. Limits and Continuity2.1 Rates of Change and Tangent Lines to Curves2.2 Limit of a Function and Limit Laws2.3 The Precise Definition of a Limit2.4 One-Sided Limits2.5 Continuity2.6 Limits Involving Infinity; Asymptotes of Graphs  3. Derivatives3.1 Tangent Lines and the Derivative at a Point   3.2 The Derivative as a Function3.3 Differentiation Rules3.4 The Derivative as a Rate of Change3.5 Derivatives of Trigonometric Functions3.6 The Chain Rule3.7 Implicit Differentiation3.8 Derivatives of Inverse Functions and Logarithms  3.9 Inverse Trigonometric Functions   3.10 Related Rates   3.11 Linearization and Differentials  4. Applications of Derivatives4.1 Extreme Values of Functions on Closed Intervals  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. Integrals5.1 Area and Estimating with Finite Sums5.2 Sigma Notation and Limits of Finite Sums5.3 The Definite Integral5.4 The Fundamental Theorem of Calculus5.5 Indefinite Integrals and the Substitution Method5.6 Definite Integral Substitutions and the Area Between Curves  6. Applications of Definite Integrals6.1 Volumes Using Cross-Sections6.2 Volumes Using Cylindrical Shells6.3 Arc Length6.4 Areas of Surfaces of Revolution6.5 Work and Fluid Forces6.6 Moments and Centers of Mass  7. Integrals and Transcendental Functions7.1 The Logarithm Defined as an Integral7.2 Exponential Change and Separable Differential Equations   7.3 Hyperbolic Functions   7.4 Relative Rates of Growth     8. Techniques of Integration8.1 Using Basic Integration Formulas8.2 Integration by Parts8.3 Trigonometric Integrals8.4 Trigonometric Substitutions8.5 Integration of Rational Functions by Partial Fractions8.6 Integral Tables and Computer Algebra Systems8.7 Numerical Integration8.8 Improper Integrals8.9 Probability  9. First-Order Differential Equations9.1 Solutions, Slope Fields, and Euler's Method9.2 First-Order Linear Equations9.3 Applications9.4 Graphical Solutions of Autonomous Equations 9.5 Systems of Equations and Phase Planes 10. Infinite Sequences and Series10.1 Sequences10.2 Infinite Series10.3 The Integral Test10.4 Comparison Tests10.5 Absolute Convergence; The Ratio and Root Tests10.6 Alternating Series and Conditional Convergence10.7 Power Series10.8 Taylor and Maclaurin Series10.9 Convergence of Taylor Series10.10 Applications of Taylor Series  11. Parametric Equations and Polar Coordinates11.1 Parametrizations of Plane Curves11.2 Calculus with Parametric Curves11.3 Polar Coordinates11.4 Graphing Polar Coordinate Equations11.5 Areas and Lengths in Polar Coordinates11.6 Conic Sections11.7 Conics in Polar Coordinates  12. Vectors and the Geometry of Space12.1 Three-Dimensional Coordinate Systems12.2 Vectors12.3 The Dot Product12.4 The Cross Product12.5 Lines and Planes in Space12.6 Cylinders and Quadric Surfaces  13. Vector-Valued Functions and Motion in Space13.1 Curves in Space and Their Tangents13.2 Integrals of Vector Functions; Projectile Motion13.3 Arc Length in Space13.4 Curvature and Normal Vectors of a Curve13.5 Tangential and Normal Components of Acceleration13.6 Velocity and Acceleration in Polar Coordinates  14. Partial Derivatives14.1 Functions of Several Variables14.2 Limits and Continuity in Higher Dimensions14.3 Partial Derivatives14.4 The Chain Rule14.5 Directional Derivatives and Gradient Vectors14.6 Tangent Planes and Differentials14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers14.9 Taylor's Formula for Two Variables14.10 Partial Derivatives with Constrained Variables  15. Multiple Integrals15.1 Double and Iterated Integrals over Rectangles15.2 Double Integrals over General Regions15.3 Area by Double Integration15.4 Double Integrals in Polar Form15.5 Triple Integrals in Rectangular Coordinates15.6 Applications15.7 Triple Integrals in Cylindrical and Spherical Coordinates15.8 Substitutions in Multiple Integrals  16. Integrals and Vector Fields16.1 Line Integrals of Scalar Functions16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux16.3 Path Independence, Conservative Fields, and Potential Functions16.4 Green's Theorem in the Plane16.5 Surfaces and Area16.6 Surface Integrals16.7 Stokes' Theorem16.8 The Divergence Theorem and a Unified Theory  17. Second-Order Differential Equations (Online at www.goo.gl/MgDXPY)17.1 Second-Order Linear Equations17.2 Nonhomogeneous Linear Equations17.3 Applications17.4 Euler Equations17.5 Power-Series Solutions  Appendices1. Real Numbers and the Real Line2. Mathematical Induction3. Lines, Circles, and Parabolas4. Proofs of Limit Theorems5. Commonly Occurring Limits6. Theory of the Real Numbers7. Complex Numbers8. The Distributive Law for Vector Cross Products9. The Mixed Derivative Theorem and the Increment Theorem

About our authors 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 widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. 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, 3-dimensional manifolds, applied math, and computational complexity. In his free time Hass enjoys kayaking.Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey articles. He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications. Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis. In his spare time, Heil pursues his hobby of astronomy.The late Maurice D. Weir of the the Naval Postgraduate School in Monterey, California was Professor Emeritus as a member of the Department of Applied Mathematics. He held a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. Weir was awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He co-authored 8 books, including University Calculus and Thomas' Calculus.

• 1. Functions1.1 Functions and Their Graphs1.2 Combining Functions; Shifting and Scaling Graphs1.3 Trigonometric Functions1.4 Graphing with Software1.5 Exponential Functions1.6 Inverse Functions and Logarithms  2. Limits and Continuity2.1 Rates of Change and Tangent Lines to Curves2.2 Limit of a Function and Limit Laws2.3 The Precise Definition of a Limit2.4 One-Sided Limits2.5 Continuity2.6 Limits Involving Infinity; Asymptotes of Graphs  3. Derivatives3.1 Tangent Lines and the Derivative at a Point   3.2 The Derivative as a Function3.3 Differentiation Rules3.4 The Derivative as a Rate of Change3.5 Derivatives of Trigonometric Functions3.6 The Chain Rule3.7 Implicit Differentiation3.8 Derivatives of Inverse Functions and Logarithms  3.9 Inverse Trigonometric Functions   3.10 Related Rates   3.11 Linearization and Differentials  4. Applications of Derivatives4.1 Extreme Values of Functions on Closed Intervals  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. Integrals5.1 Area and Estimating with Finite Sums5.2 Sigma Notation and Limits of Finite Sums5.3 The Definite Integral5.4 The Fundamental Theorem of Calculus5.5 Indefinite Integrals and the Substitution Method5.6 Definite Integral Substitutions and the Area Between Curves  6. Applications of Definite Integrals6.1 Volumes Using Cross-Sections6.2 Volumes Using Cylindrical Shells6.3 Arc Length6.4 Areas of Surfaces of Revolution6.5 Work and Fluid Forces6.6 Moments and Centers of Mass  7. Integrals and Transcendental Functions7.1 The Logarithm Defined as an Integral7.2 Exponential Change and Separable Differential Equations   7.3 Hyperbolic Functions   7.4 Relative Rates of Growth     8. Techniques of Integration8.1 Using Basic Integration Formulas8.2 Integration by Parts8.3 Trigonometric Integrals8.4 Trigonometric Substitutions8.5 Integration of Rational Functions by Partial Fractions8.6 Integral Tables and Computer Algebra Systems8.7 Numerical Integration8.8 Improper Integrals8.9 Probability  9. First-Order Differential Equations9.1 Solutions, Slope Fields, and Euler's Method9.2 First-Order Linear Equations9.3 Applications9.4 Graphical Solutions of Autonomous Equations 9.5 Systems of Equations and Phase Planes 10. Infinite Sequences and Series10.1 Sequences10.2 Infinite Series10.3 The Integral Test10.4 Comparison Tests10.5 Absolute Convergence; The Ratio and Root Tests10.6 Alternating Series and Conditional Convergence10.7 Power Series10.8 Taylor and Maclaurin Series10.9 Convergence of Taylor Series10.10 Applications of Taylor Series  11. Parametric Equations and Polar Coordinates11.1 Parametrizations of Plane Curves11.2 Calculus with Parametric Curves11.3 Polar Coordinates11.4 Graphing Polar Coordinate Equations11.5 Areas and Lengths in Polar Coordinates11.6 Conic Sections11.7 Conics in Polar Coordinates  12. Vectors and the Geometry of Space12.1 Three-Dimensional Coordinate Systems12.2 Vectors12.3 The Dot Product12.4 The Cross Product12.5 Lines and Planes in Space12.6 Cylinders and Quadric Surfaces  13. Vector-Valued Functions and Motion in Space13.1 Curves in Space and Their Tangents13.2 Integrals of Vector Functions; Projectile Motion13.3 Arc Length in Space13.4 Curvature and Normal Vectors of a Curve13.5 Tangential and Normal Components of Acceleration13.6 Velocity and Acceleration in Polar Coordinates  14. Partial Derivatives14.1 Functions of Several Variables14.2 Limits and Continuity in Higher Dimensions14.3 Partial Derivatives14.4 The Chain Rule14.5 Directional Derivatives and Gradient Vectors14.6 Tangent Planes and Differentials14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers14.9 Taylor's Formula for Two Variables14.10 Partial Derivatives with Constrained Variables  15. Multiple Integrals15.1 Double and Iterated Integrals over Rectangles15.2 Double Integrals over General Regions15.3 Area by Double Integration15.4 Double Integrals in Polar Form15.5 Triple Integrals in Rectangular Coordinates15.6 Applications15.7 Triple Integrals in Cylindrical and Spherical Coordinates15.8 Substitutions in Multiple Integrals  16. Integrals and Vector Fields16.1 Line Integrals of Scalar Functions16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux16.3 Path Independence, Conservative Fields, and Potential Functions16.4 Green's Theorem in the Plane16.5 Surfaces and Area16.6 Surface Integrals16.7 Stokes' Theorem16.8 The Divergence Theorem and a Unified Theory  17. Second-Order Differential Equations (Online at www.goo.gl/MgDXPY)17.1 Second-Order Linear Equations17.2 Nonhomogeneous Linear Equations17.3 Applications17.4 Euler Equations17.5 Power-Series Solutions  Appendices1. Real Numbers and the Real Line2. Mathematical Induction3. Lines, Circles, and Parabolas4. Proofs of Limit Theorems5. Commonly Occurring Limits6. Theory of the Real Numbers7. Complex Numbers8. The Distributive Law for Vector Cross Products9. The Mixed Derivative Theorem and the Increment Theorem