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University Calculusby George B. Thomas
Synopses & ReviewsPublisher Comments:This streamlined version of ‘Thomas’ Calculus’ provides a fasterpaced, precise and accurate presentation of calculus for a collegelevel calculus course. ‘University Calculus’ covers both single variable and multivariable calculus and is appropriate for a three semester or four quarter course.
About the AuthorJoel 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 six widely used calculus texts as well as two calculus study guides. He is currently on the editorial board of Geometriae Dedicata and MediaEnhanced 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, three dimensional manifolds, applied math, and computational complexity. In his free time, Hass enjoys kayaking.
Maurice D. Weir holds a DA and MS from CarnegieMellon 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 eight books, including the University Calculus series and the twelfth edition of Thomas’ Calculus. George B. Thomas, Jr. (late) of the Massachusetts Institute of Technology, was a professor of mathematics for thirtyeight years; he served as the executive officer of the department for ten years and as graduate registration officer for five years. Thomas held a spot on the board of governors of the Mathematical Association of America and on the executive committee of the mathematics division of the American Society for Engineering Education. His book, Calculus and Analytic Geometry, was first published in 1951 and has since gone through multiple revisions. The text is now in its twelfth edition and continues to guide students through their calculus courses. He also coauthored monographs on mathematics, including the text Probability and Statistics. Table of Contents1 Functions 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 14 1.3 Trigonometric Functions 22 1.4 Exponential Functions 30 1.5 Inverse Functions and Logarithms 36 1.6 Graphing with Calculators and Computers 50
2 Limits and Continuity 2.1 Rates of Change and Tangents to Curves 55 2.2 Limit of a Function and Limit Laws 62 2.3 The Precise Definition of a Limit 74 2.4 OneSided Limits and Limits at Infinity 84 2.5 Infinite Limits and Vertical Asymptotes 97 2.6 Continuity 103 2.7 Tangents and Derivatives at a Point 115 QUESTIONS TO GUIDE YOUR REVIEW 119 PRACTICE EXERCISES 120 ADDITIONAL AND ADVANCED EXERCISES 122
3 Differentiation 3.1 The Derivative as a Function 125 3.2 Differentiation Rules for Polynomials, Exponentials, Products, and Quotients 134 3.3 The Derivative as a Rate of Change 146 3.4 Derivatives of Trigonometric Functions 157 3.5 The Chain Rule and Parametric Equations 164 3.6 Implicit Differentiation 177 3.7 Derivatives of Inverse Functions and Logarithms 183 3.8 Inverse Trigonometric Functions 194 3.9 Related Rates 201 3.10 Linearization and Differentials 209 3.11 Hyperbolic Functions 221 QUESTIONS TO GUIDE YOUR REVIEW 227 PRACTICE EXERCISES 228 ADDITIONAL AND ADVANCED EXERCISES 234
4 Applications of Derivatives 4.1 Extreme Values of Functions 237 4.2 The Mean Value Theorem 245 4.3 Monotonic Functions and the First Derivative Test 254 4.4 Concavity and Curve Sketching 260 4.5 Applied Optimization 271 4.6 Indeterminate Forms and L’Hôpital’s Rule 283 4.7 Newton’s Method 291 4.8 Antiderivatives 296 QUESTIONS TO GUIDE YOUR REVIEW 306 PRACTICE EXERCISES 307 ADDITIONAL AND ADVANCED EXERCISES 311
5 Integration 5.1 Estimating with Finite Sums 315 5.2 Sigma Notation and Limits of Finite Sums 325 5.3 The Definite Integral 332 5.4 The Fundamental Theorem of Calculus 345 5.5 Indefinite Integrals and the Substitution Rule 354 5.6 Substitution and Area Between Curves 360 5.7 The Logarithm Defined as an Integral 370 QUESTIONS TO GUIDE YOUR REVIEW 381 PRACTICE EXERCISES 382 ADDITIONAL AND ADVANCED EXERCISES 386
6 Applications of Definite Integrals 6.1 Volumes by Slicing and Rotation About an Axis 391 6.2 Volumes by Cylindrical Shells 401 6.3 Lengths of Plane Curves 408 6.4 Areas of Surfaces of Revolution 415 6.5 Exponential Change and Separable Differential Equations 421 6.6 Work 430 6.7 Moments and Centers of Mass 437 QUESTIONS TO GUIDE YOUR REVIEW 444 PRACTICE EXERCISES 444 ADDITIONAL AND ADVANCED EXERCISES 446
7 Techniques of Integration 7.1 Integration by Parts 448 7.2 Trigonometric Integrals 455 7.3 Trigonometric Substitutions 461 7.4 Integration of Rational Functions by Partial Fractions 464 7.5 Integral Tables and Computer Algebra Systems 471 7.6 Numerical Integration 477 7.7 Improper Integrals 487 QUESTIONS TO GUIDE YOUR REVIEW 497 PRACTICE EXERCISES 497 ADDITIONAL AND ADVANCED EXERCISES 500
8 Infinite Sequences and Series 8.1 Sequences 502 8.2 Infinite Series 515 8.3 The Integral Test 523 8.4 Comparison Tests 529 8.5 The Ratio and Root Tests 533 8.6 Alternating Series, Absolute and Conditional Convergence 537 8.7 Power Series 543 8.8 Taylor and Maclaurin Series 553 8.9 Convergence of Taylor Series 559 8.10 The Binomial Series 569 QUESTIONS TO GUIDE YOUR REVIEW 572 PRACTICE EXERCISES 573 ADDITIONAL AND ADVANCED EXERCISES 575
9 Polar Coordinates and Conics 9.1 Polar Coordinates 577 9.2 Graphing in Polar Coordinates 582 9.3 Areas and Lengths in Polar Coordinates 586 9.4 Conic Sections 590 9.5 Conics in Polar Coordinates 599 9.6 Conics and Parametric Equations; The Cycloid 606 QUESTIONS TO GUIDE YOUR REVIEW 610 PRACTICE EXERCISES 610 ADDITIONAL AND ADVANCED EXERCISES 612
10 Vectors and the Geometry of Space 10.1 ThreeDimensional Coordinate Systems 614 10.2 Vectors 619 10.3 The Dot Product 628 10.4 The Cross Product 636 10.5 Lines and Planes in Space 642 10.6 Cylinders and Quadric Surfaces 652 QUESTIONS TO GUIDE YOUR REVIEW 657 PRACTICE EXERCISES 658 ADDITIONAL AND ADVANCED EXERCISES 660
11 VectorValued Functions and Motion in Space 11.1 Vector Functions and Their Derivatives 663 11.2 Integrals of Vector Functions 672 11.3 Arc Length in Space 678 11.4 Curvature of a Curve 683 11.5 Tangential and Normal Components of Acceleration 689 11.6 Velocity and Acceleration in Polar Coordinates 694 QUESTIONS TO GUIDE YOUR REVIEW 698 PRACTICE EXERCISES 698 ADDITIONAL AND ADVANCED EXERCISES 700
12 Partial Derivatives 12.1 Functions of Several Variables 702 12.2 Limits and Continuity in Higher Dimensions 711 12.3 Partial Derivatives 719 12.4 The Chain Rule 731 12.5 Directional Derivatives and Gradient Vectors 739 12.6 Tangent Planes and Differentials 747 12.7 Extreme Values and Saddle Points 756 12.8 Lagrange Multipliers 765 12.9 Taylor’s Formula for Two Variables 775 QUESTIONS TO GUIDE YOUR REVIEW 779 PRACTICE EXERCISES 780 ADDITIONAL AND ADVANCED EXERCISES 783
13 Multiple Integrals 13.1 Double and Iterated Integrals over Rectangles 785 13.2 Double Integrals over General Regions 790 13.3 Area by Double Integration 799 13.4 Double Integrals in Polar Form 802 13.5 Triple Integrals in Rectangular Coordinates 807 13.6 Moments and Centers of Mass 816 13.7 Triple Integrals in Cylindrical and Spherical Coordinates 825 13.8 Substitutions in Multiple Integrals 837 QUESTIONS TO GUIDE YOUR REVIEW 846 PRACTICE EXERCISES 846 ADDITIONAL AND ADVANCED EXERCISES 848
14 Integration in Vector Fields 14.1 Line Integrals 851 14.2 Vector Fields, Work, Circulation, and Flux 856 14.3 Path Independence, Potential Functions, and Conservative Fields 867 14.4 Green’s Theorem in the Plane 877 14.5 Surfaces and Area 887 14.6 Surface Integrals and Flux 896 14.7 Stokes’Theorem 905 14.8 The Divergence Theorem and a Unified Theory 914 QUESTIONS TO GUIDE YOUR REVIEW 925 PRACTICE EXERCISES 925 ADDITIONAL AND ADVANCED EXERCISES 928
15 FirstOrder Differential Equations (online) 15.1 Solutions, Slope Fields, and Picard’s Theorem 15.2 FirstOrder Linear Equations 15.3 Applications 15.4 Euler’s Method 15.5 Graphical Solutions of Autonomous Equations 15.6 Systems of Equations and Phase Planes
16 SecondOrder Differential Equations (online) 16.1 SecondOrder Linear Equations 16.2 Nonhomogeneous Linear Equations 16.3 Applications 16.4 Euler Equations 16.5 Power Series Solutions
Appendices AP1 A.1 Real Numbers and the Real Line AP1 A.2 Mathematical Induction AP7 A.3 Lines, Circles, and Parabolas AP10 A.4 Trigonometry Formulas AP19 A.5 Proofs of Limit Theorems AP21 A.6 Commonly Occurring Limits AP25 A.7 Theory of the Real Numbers AP26 A.8 The Distributive Law for Vector Cross Products AP29 A.9 The Mixed Derivative Theorem and the Increment Theorem AP30
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