University Calculus

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 6 widely used calculus texts as well as 2 calculus study guides. He is currently on the editorial board of Geometriae Dedicata and Media-Enhanced Mathematics. 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.Maurice D. Weir holds a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. He is a Professor Emeritus of the Department of Applied Mathematics at the Naval Postgraduate School in Monterey, California. Weir enjoys teaching Mathematical Modeling and Differential Equations. His current areas of research include modeling and simulation as well as mathematics education. Weir has been awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He has coauthored 8 books, including the University Calculus series and Thomas' Calculus.Przemyslaw Bogacki is an Associate Professor of Mathematics and Statistics and a University Professor at Old Dominion University. He received his PhD in 1990 from Southern Methodist University. He is the author of a text on linear algebra. He is actively involved in applications of technology in collegiate mathematics. His areas of research include computer aided geometric design and numerical solution of initial value problems for ordinary differential equations.

• Functions 1.1 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 Limits and Continuity 2.1 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 Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Derivatives 3.1 Tangent Lines and the Derivative at a Point3.2 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 Logarithms3.9 Inverse Trigonometric Functions3.10 Related Rates3.11 Linearization and Differentials Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Applications of Derivatives 4.1 Extreme Values of Functions on Closed Intervals4.2 The Mean Value Theorem4.3 Monotonic Functions and the First Derivative Test4.4 Concavity and Curve Sketching4.5 Indeterminate Forms and L’Hôpital’s Rule4.6 Applied Optimization4.7 Newton’s Method4.8 Antiderivatives Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Integrals 5.1 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 Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Applications of Definite Integrals 6.1 Volumes Using Cross-Sections6.2 Volumes Using Cylindrical Shells6.3 Arc Length6.4 Areas of Surfaces of Revolution6.5 Work6.6 Moments and Centers of Mass Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Integrals and Transcendental Functions 7.1 The Logarithm Defined as an Integral7.2 Exponential Change and Separable Differential Equations7.3 Hyperbolic Functions Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Techniques of Integration 8.1 Integration by Parts8.2 Trigonometric Integrals8.3 Trigonometric Substitutions8.4 Integration of Rational Functions by Partial Fractions8.5 Integral Tables and Computer Algebra Systems8.6 Numerical Integration8.7 Improper Integrals Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Infinite Sequences and Series 9.1 Sequences9.2 Infinite Series9.3 The Integral Test9.4 Comparison Tests9.5 Absolute Convergence; The Ratio and Root Tests9.6 Alternating Series and Conditional Convergence9.7 Power Series9.8 Taylor and Maclaurin Series9.9 Convergence of Taylor Series9.10 Applications of Taylor Series Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Parametric Equations and Polar Coordinates 10.1 Parametrizations of Plane Curves10.2 Calculus with Parametric Curves10.3 Polar Coordinates10.4 Graphing Polar Coordinate Equations10.5 Areas and Lengths in Polar Coordinates Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Vectors and the Geometry of Space 11.1 Three-Dimensional Coordinate Systems11.2 Vectors11.3 The Dot Product11.4 The Cross Product11.5 Lines and Planes in Space11.6 Cylinders and Quadric Surfaces Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Vector-Valued Functions and Motion in Space 12.1 Curves in Space and Their Tangents12.2 Integrals of Vector Functions; Projectile Motion12.3 Arc Length in Space12.4 Curvature and Normal Vectors of a Curve12.5 Tangential and Normal Components of Acceleration12.6 Velocity and Acceleration in Polar Coordinates Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Partial Derivatives 13.1 Functions of Several Variables13.2 Limits and Continuity in Higher Dimensions13.3 Partial Derivatives13.4 The Chain Rule13.5 Directional Derivatives and Gradient Vectors13.6 Tangent Planes and Differentials13.7 Extreme Values and Saddle Points13.8 Lagrange Multiplier Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Multiple Integrals 14.1 Double and Iterated Integrals over Rectangles14.2 Double Integrals over General Regions14.3 Area by Double Integration14.4 Double Integrals in Polar Form14.5 Triple Integrals in Rectangular Coordinates14.6 Applications14.7 Triple Integrals in Cylindrical and Spherical Coordinates14.8 Substitutions in Multiple Integrals Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises Integrals and Vector Fields 15.1 Line Integrals of Scalar Functions15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux15.3 Path Independence, Conservative Fields, and Potential Functions15.4 Green’s Theorem in the Plane15.5 Surfaces and Area15.6 Surface Integrals15.7 Stokes’ Theorem15.8 The Divergence Theorem and a Unified Theory Questions to Guide Your ReviewPractice ExercisesAdditional and Advanced Exercises First-Order Differential Equations (online at bit.ly/2pzYlEq) 16.1 Solutions, Slope Fields, and Euler’s Method16.2 First-Order Linear Equations16.3 Applications16.4 Graphical Solutions of Autonomous Equations16.5 Systems of Equations and Phase Planes Second-Order Differential Equations (online at bit.ly/2IHCJyE) 17.1 Second-Order Linear Equations17.2 Non-homogeneous Linear Equations17.3 Applications17.4 Euler Equations17.5 Power-Series Solutions Appendix A.1 Real Numbers and the Real LineA.2 Mathematical InductionA.3 Lines and CirclesA.4 Conic SectionsA.5 Proofs of Limit TheoremsA.6 Commonly Occurring LimitsA.7 Theory of the Real NumbersA.8 Complex NumbersA.9 The Distributive Law for Vector Cross ProductsA.10 The Mixed Derivative Theorem and the increment TheoremAdditional Topics (online) B.1 Relative Rates of GrowthB.2 ProbabilityB.3 Conics in Polar CoordinatesB.4 Taylor’s Formula for Two VariablesB.5 Partial Derivatives with Constrained VariablesOdd Answers