Calculus: Single Variable, 2nd Edition

Blank and Krantz's Calculus 2e brings together time-tested methods and innovative thinking to address the needs of today's students, who come from a wide range of backgrounds and look ahead to a variety of futures. Using meaningful examples, credible applications, and incisive technology, Blank and Krantz's Calculus 2e strives to empower students, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the major or discipline they ultimately choose to study. Blank and Krantz's engaging style and clear writing make the language of mathematics accessible, understandable and enjoyable, while maintaining high standards for mathematical rigor. Using meaningful examples, credible applications, and incisive technology, Blank and Krantz's Calculus 2e strives to empower students, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the major or discipline they ultimately choose to study. Blank and Krantz's engaging style and clear writing make the language of mathematics accessible, understandable and enjoyable, while maintaining high standards for mathematical rigor. Blank and Krantz's Calculus 2e is available with WileyPLUS , an online teaching and learning environment initially developed for Calculus and Differential Equations courses. WileyPLUS integrates the complete digital textbook with powerful student and instructor resources as well as online auto-graded homework. Cover......Page 1 Title Page......Page 7 Copyright......Page 8 Contents......Page 11 Preface......Page 15 Supplementary Resources......Page 19 Acknowledgments......Page 20 About the Authors......Page 23 Preview......Page 27 1 Number Systems......Page 28 Sets of Real Numbers......Page 29 Intervals......Page 30 Approximation......Page 33 EXERCISES......Page 35 2 Planar Coordinates and Graphing in the Plane......Page 37 The Distance Formula and Circles......Page 38 The Method of Completing the Square......Page 39 Parabolas, Ellipses, and Hyperbolas......Page 42 Regions in the Plane......Page 44 EXERCISES......Page 45 Slopes......Page 47 Equations of Lines......Page 49 Least Squares Lines......Page 54 EXERCISES......Page 57 4 Functions and Their Graphs......Page 60 Examples of Functions of a Real Variable......Page 61 Piecewise-Defined Functions......Page 62 Graphs of Functions......Page 63 Sequences......Page 65 Functions from Data......Page 66 EXERCISES......Page 71 Arithmetic Operations......Page 75 Polynomial Functions......Page 76 Composition of Functions......Page 77 Inverse Functions......Page 78 Vertical and Horizontal Translations......Page 83 Even and Odd Functions......Page 84 Pairing Functions—Parametric Curves......Page 85 Parameterized Curves and Graphs of Functions......Page 87 EXERCISES......Page 88 Sine and Cosine Functions......Page 91 Other Trigonometric Functions......Page 94 Trigonometric Identities......Page 96 Modeling with Trigonometric Functions......Page 97 EXERCISES......Page 98 Summary of Key Topics......Page 101 Review Exercises......Page 104 The Number π......Page 106 Analytic Geometry......Page 107 The Completeness Property of the Real Numbers......Page 108 Preview......Page 109 1 The Concept of Limit......Page 111 One-Sided Limits......Page 112 Specified Degrees of Accuracy......Page 113 Graphical Methods......Page 115 EXERCISES......Page 117 2 Limit Theorems......Page 120 One-Sided Limits......Page 121 Basic Limit Theorems......Page 123 A Rule That Tells When a Limit Does Not Exist......Page 124 The Pinching Theorem......Page 125 Some Important Trigonometric Limits......Page 126 EXERCISES......Page 128 The Definition of Continuity at a Point......Page 131 An Equivalent Formulation of Continuity......Page 132 Continuous Extensions......Page 134 One-Sided Continuity......Page 135 Some Theorems about Continuity—Arithmetic Operations and Composition......Page 136 Advanced Properties of Continuous Functions......Page 137 EXERCISES......Page 141 Infinite-Valued Limits......Page 144 Vertical Asymptotes......Page 145 Limits at Infinity......Page 147 Horizontal Asymptotes......Page 148 EXERCISES......Page 150 5 Limits of Sequences......Page 153 A Precise Discussion of Convergence and Divergence......Page 154 Some Special Sequences......Page 155 Limit Theorems......Page 157 Geometric Series......Page 159 Using Continuous Functions to Calculate Limits......Page 161 EXERCISES......Page 162 The Monotone Convergence Property of the Real Numbers......Page 164 Irrational Exponents......Page 166 Exponential and Logarithmic Functions......Page 168 The Number e......Page 171 An Application: Compound Interest......Page 173 The Natural Logarithm......Page 175 Exponential Decay......Page 176 EXERCISES......Page 178 Summary of Key Topics......Page 181 Review Exercises......Page 184 Bolzano and the Intermediate Value Theorem......Page 187 Cauchy and Weierstrass......Page 188 Preview......Page 189 1 Rates of Change and Tangent Lines......Page 190 The Definition of Instantaneous Velocity......Page 192 Instantaneous Rate of Change......Page 194 Sums of Functions......Page 196 The Concept of Tangent Line......Page 197 Normal Lines to Curves......Page 199 Corners and Vertical Tangent Lines......Page 200 EXERCISES......Page 201 2 The Derivative......Page 204 Other Notations for the Derivative......Page 206 The Derived Function......Page 207 Differentiability and Continuity......Page 208 Investigating Differentiability Graphically......Page 209 Derivatives of Sine and Cosine......Page 210 Summary of Differentiation Formulas......Page 212 EXERCISES......Page 213 3 Rules for Differentiation......Page 215 Addition, Subtraction, and Multiplication by a Constant......Page 216 Products and Quotients......Page 217 Numeric Differentiation......Page 220 EXERCISES......Page 223 Powers of x......Page 226 Trigonometric Functions......Page 229 The Derivative of the Natural Exponential Function......Page 230 EXERCISES......Page 233 A Rule for Differentiating the Composition of Two Functions......Page 236 Some Examples......Page 237 An Application......Page 240 Derivatives of Exponential Functions......Page 241 EXERCISES......Page 243 Continuity and Differentiability of Inverse Functions......Page 246 Derivatives of Logarithms......Page 249 Logarithmic Differentiation......Page 251 EXERCISES......Page 254 Notation for Higher Derivatives......Page 256 Velocity and Acceleration......Page 257 Approximation of Second Derivatives......Page 258 Leibniz’s Rule......Page 259 EXERCISES......Page 260 The Main Idea of Implicit Differentiation......Page 263 Some Examples......Page 264 Calculating Higher Derivatives......Page 267 Parametric Curves......Page 268 EXERCISES......Page 269 9 Differentials and Approximation of Functions......Page 272 Linearization......Page 274 Important Linearizations......Page 275 Differentials......Page 276 EXERCISES......Page 277 Inverse Trigonometric Functions......Page 279 Inverse Sine and Cosine......Page 280 The Inverse Tangent Function......Page 283 Other Inverse Trigonometric Functions......Page 284 The Hyperbolic Functions......Page 286 Derivatives of the Hyperbolic Functions......Page 288 The Inverse Hyperbolic Functions......Page 289 EXERCISES......Page 293 Summary of Key Topics......Page 294 Review Exercises......Page 298 The Solutions of Fermat and Descartes to the Tangent Problem......Page 301 Newton’s Method of Differentiation......Page 303 Proof of the Chain Rule......Page 304 Preview......Page 307 The Role of Implicit Differentiation......Page 308 Basic Steps for Solving a Related Rates Problem......Page 310 EXERCISES......Page 313 2 The Mean Value Theorem......Page 315 Maxima and Minima......Page 316 Locating Maxima and Minima......Page 317 Rolle’s Theorem and the Mean Value Theorem......Page 319 An Application of the Mean Value Theorem......Page 320 EXERCISES......Page 322 Using the Derivative to Tell When a Function Is Increasing or Decreasing......Page 325 Critical Points and the First Derivative Test for Local Extrema......Page 327 EXERCISES......Page 330 4 Applied Maximum-Minimum Problems......Page 333 Closed Intervals......Page 334 Examples with the Solution at an Endpoint......Page 337 Profit Maximization......Page 339 An Example Involving a Transcendental Function......Page 340 EXERCISES......Page 341 5 Concavity......Page 346 Using the Second Derivative to Test for Concavity......Page 347 Points of Inflection......Page 348 The Second Derivative Test at a Critical Point......Page 349 EXERCISES......Page 351 Basic Strategy of Curve Sketching......Page 353 Periodic Functions......Page 357 Skew-Asymptotes......Page 358 Graphing Calculators/Software......Page 360 EXERCISES......Page 362 l’Hôpital’s Rule for the Indeterminate Form 0/0......Page 364 l’Hôpital’s Rule for the Indeterminate Form ∞/∞......Page 367 The Indeterminate Form 0o......Page 368 The Indeterminate Form ∞o......Page 369 Putting Terms over a Common Denominator......Page 370 EXERCISES......Page 371 The Geometry of the Newton-Raphson Method......Page 374 Calculating with the Newton-Raphson Method......Page 375 Accuracy......Page 376 A Computer Implementation......Page 378 An Application in Economics: Bond Valuation......Page 379 EXERCISES......Page 380 Antidifferentiation......Page 383 Antidifferentiating Powers of x......Page 384 Antidifferentiation of other Functions......Page 386 Velocity and Acceleration......Page 387 EXERCISES......Page 389 Summary of Key Topics......Page 392 Review Exercises......Page 394 Fermat’s Investigation of Points of Inflection......Page 397 Fermat’s Principle of Least Time......Page 398 The Newton-Raphson Method......Page 399 Preview......Page 401 1 Introduction to Integration—The Area Problem......Page 402 Summation Notation......Page 403 Some Special Sums......Page 404 Approximation of Area......Page 405 A Precise Definition of Area......Page 407 Concluding Remarks......Page 410 EXERCISES......Page 411 Riemann Sums......Page 414 The Riemann Integral......Page 418 Calculating Riemann Integrals......Page 420 Using the Fundamental Theorem of Calculus to Compute Areas......Page 422 EXERCISES......Page 423 3 Rules for Integration......Page 425 Reversing the Direction of Integration......Page 428 Order Properties of Integrals......Page 429 The Mean Value Theorem for Integrals......Page 431 EXERCISES......Page 432 4 The Fundamental Theorem of Calculus......Page 434 Examples Illustrating the First Part of the Fundamental Theorem......Page 436 Examples of the Second Part of the Fundamental Theorem......Page 438 EXERCISES......Page 440 5 A Calculus Approach to the Logarithm and Exponential Functions......Page 443 Properties of the Natural Logarithm......Page 444 Graphing the Natural Logarithm Function......Page 446 Properties of the Exponential Function......Page 447 The Number e......Page 448 Logarithms and Powers with Arbitrary Bases......Page 449 Logarithms with Arbitrary Bases......Page 450 EXERCISES......Page 452 6 Integration by Substitution......Page 454 Some Examples of Indefinite Integration by Substitution......Page 455 The Method of Substitution for Definite Integrals......Page 456 The Role of the Chain Rule in the Method of Substitution......Page 458 Integral Tables......Page 459 Integrating Trigonometric Functions......Page 462 EXERCISES......Page 463 7 More on the Calculation of Area......Page 466 The Area between Two Curves......Page 467 Reversing the Roles of the Axes......Page 469 EXERCISES......Page 470 The Midpoint Rule......Page 472 The Trapezoidal Rule......Page 475 Simpson’s Rule......Page 478 Using Simpson’s Rule with Discrete Data......Page 480 EXERCISES......Page 481 Summary of Key Topics......Page 485 Review Exercises......Page 488 The Method......Page 491 Fermat and the Integral Calculus......Page 492 Notation......Page 493 Bernhard Riemann......Page 494 Preview......Page 495 Some Examples......Page 496 Advanced Examples......Page 498 Reduction Formulas......Page 501 EXERCISES......Page 503 2 Powers and Products of Trigonometric Functions......Page 505 Squares of Sine, Cosine, Secant, and Tangent......Page 506 Higher Powers of Sine, Cosine, Secant, and Tangent......Page 507 Odd Powers of Sine and Cosine......Page 508 Integrals That Involve Both Sine and Cosine......Page 509 Converting to Sines and Cosines......Page 511 EXERCISES......Page 512 3 Trigonometric Substitution......Page 514 Trigonometric Substitution......Page 515 General Quadratic Expressions That Appear Under a Radical......Page 518 Quadratic Expressions Not Under a Radical Sign......Page 519 EXERCISES......Page 521 The Method of Partial Fractions for Linear Factors......Page 524 The Method of Partial Fractions for Distinct Linear Factors......Page 525 Heaviside’s Method......Page 527 Repeated Linear Factors......Page 528 Summary of Basic Partial Fraction Forms......Page 529 EXERCISES......Page 530 Rational Functions with Quadratic Terms in the Denominator......Page 532 Checking for Irreducibility......Page 534 Calculating the Coefficients of a Partial Fraction Decomposition......Page 535 EXERCISES......Page 538 Integrals with Infinite Integrands......Page 540 Integrands with Interior Singularities......Page 541 Proving Convergence Without Evaluation......Page 543 EXERCISES......Page 545 The Integral on an Infinite Interval......Page 547 An Application to Finance......Page 549 Integrals Over the Entire Real Line......Page 550 Proving Convergence Without Evaluation......Page 551 EXERCISES......Page 552 Summary of Key Topics......Page 555 Review Exercises......Page 557 Genesis & Development 6......Page 559 Preview......Page 563 Volumes by Slicing—The Method of Disks......Page 564 Solids of Revolution......Page 566 The Method of Washers......Page 568 Rotation about a Line that Is Not a Coordinate Axis......Page 570 The Method of Cylindrical Shells......Page 571 A Final Remark......Page 574 EXERCISES......Page 575 The Basic Method for Calculating Arc Length......Page 578 Some Examples of Arc Length......Page 579 Parametric Curves......Page 580 Surface Area......Page 582 EXERCISES......Page 585 3 The Average Value of a Function......Page 587 The Basic Technique......Page 588 Random Variables......Page 591 Average Values in Probability Theory......Page 593 Population Density Functions......Page 594 EXERCISES......Page 595 Moments (Two Point Systems)......Page 598 Center of Mass......Page 600 EXERCISES......Page 604 Using Integrals to Calculate Work......Page 606 Examples with Weights That Vary......Page 607 An Example Involving a Spring......Page 609 Examples that Involve Pumping a Fluid from a Reservoir......Page 610 EXERCISES......Page 611 6 First Order Differential Equations—Separable Equations......Page 614 Slope Fields......Page 615 Initial Value Problems......Page 616 Separable Equations......Page 617 Equations of the Form dy/dx = g(x)......Page 618 Examples from the Physical Sciences......Page 619 Logistic Growth......Page 622 EXERCISES......Page 625 Solving Linear Differential Equations......Page 632 An Application: Mixing Problems......Page 635 An Application: Electric Circuits......Page 636 Linear Equations with Constant Coefficients......Page 637 Newton’s Law for Temperature Change......Page 638 The Linear Drag Law......Page 639 EXERCISES......Page 641 Summary of Key Topics......Page 645 Review Exercises......Page 647 Arc Length......Page 651 The Catenary......Page 652 The Catenoid......Page 654 Preview......Page 655 Limits of Infinite Sequences—A Review......Page 657 The Definition of an Infinite Series......Page 659 Convergence of Infinite Series......Page 660 A Telescoping Series......Page 661 The Harmonic Series......Page 662 Basic Properties of Series......Page 663 Series of Powers (Geometric Series)......Page 664 EXERCISES......Page 666 The Divergence Test......Page 669 Series with Nonnegative Terms......Page 670 The Tail End of a Series......Page 671 The Integral Test......Page 672 p-Series......Page 674 An Extension......Page 675 EXERCISES......Page 676 The Comparison Test for Convergence......Page 679 The Comparison Test for Divergence......Page 681 An Advanced Example......Page 682 The Limit Comparison Test......Page 683 EXERCISES......Page 685 The Alternating Series Test......Page 687 Some Examples......Page 688 Absolute Convergence......Page 689 EXERCISES......Page 691 The Ratio Test......Page 693 Examples......Page 696 EXERCISES......Page 698 Radius and Interval of Convergence......Page 700 Power Series about an Arbitrary Base Point......Page 705 Addition and Scalar Multiplication of Power Series......Page 707 Differentiation and Antidifferentiation of Power Series......Page 708 EXERCISES......Page 709 Power Series Expansions of Some Standard Functions......Page 712 The Relationship between the Coefficients and Derivatives of a Power Series......Page 716 An Application to Differential Equations......Page 718 Taylor Series and Polynomials......Page 719 EXERCISES......Page 721 Taylor’s Theorem......Page 724 Estimating the Error Term......Page 727 Achieving a Desired Degree of Accuracy......Page 728 Taylor Series Expansions of the Common Transcendental Functions......Page 730 The Binomial Series......Page 733 Using Taylor Polynomials to Evaluate Indeterminate Forms......Page 734 EXERCISES......Page 735 Summary of Key Topics......Page 738 Review Exercises......Page 741 Infinite Series in the 17th Century......Page 744 James Gregory and Sir Isaac Newton......Page 745 Colin Maclaurin (1698–1746)......Page 746 Table of Integrals......Page 747 Answers to Selected Exercises......Page 761 Index......Page 817 "Calculus is one of the milestones of human thought. Every well-educated person should be acquainted with the basic ideas of the subject. As our world has become more quantified and technological, knowledge of calculus has become essential to a broader cross-section of the population." "This Debut Edition of Calculus by Brian E. Blank and Steven G. Krantz, both of Washington University in St. Louis, is published in two volumes, Single Variable and Multivariable. These respected authors bring together time tested as well as innovative pedagogy and exposition in this book, focusing on today's best practices in calculus teaching. Book jacket." -- BOOK JACKET Blank and Krantz's Calculus brings together time-tested methods and innovative thinking to address the needs of today's readers, who come from a wide range of backgrounds and look ahead to a variety of futures. Some study the subject because it is required, others because it will widen their career options. Mathematics majors go into law, medicine, genome research, the technology sector, and many other professions. Blank and Krantz's Calculus strives to empower these readers, enhance their critical thinking skills, and equip them with the knowledge and skills to succeed in the discipline they ultimately choose to study.