I/. FUNCTIONS AND MODELS 10
1.1 Four Ways to Represent a Function 11
1.2 Mathematical Models: A Catalog of Essential Functions 24
1.3 New Functions from Old Functions 37
1.4 Graphing Calculators and Computers 46
1.5 Exponential Functions 52
1.6 Inverse Functions and Logarithms 59
Principles of Problem Solving 76
II/. LIMITS AND DERIVATIVES 82
2.1 The Tangent and Velocity Problems 83
2.2 The Limit of a Function 88
2.3 Calculating Limits Using the Limit Laws 99
2.4 The Precise Definition of a Limit 109
2.5 Continuity 119
2.6 Limits at Infinity; Horizontal Asymptotes 130
2.7 Derivatives and Rates of Change 143
Writing Project N Early Methods for Finding Tangents 153
2.8 The Derivative as a Function 154
III/. DIFFERENTIATION RULES 172
3.1 Derivatives of Polynomials and Exponential Functions 173
Applied Project N Building a Better Roller Coaster 182
3.2 The Product and Quotient Rules 183
3.3 Derivatives of Trigonometric Functions 189
3.4 The Chain Rule 197
Applied Project N Where Should a Pilot Start Descent? 206
3.5 Implicit Differentiation 207
3.6 Derivatives of Logarithmic Functions 215
3.7 Rates of Change in the Natural and Social Sciences 221
3.8 Exponential Growth and Decay 233
3.9 Related Rates 241
3.10 Linear Approximations and Differentials 247
Laboratory Project N Taylor Polynomials 253
3.11 Hyperbolic Functions 254
IV/. APPLICATIONS OF DIFFERENTIATION 270
4.1 Maximum and Minimum Values 271
Applied Project N The Calculus of Rainbows 279
4.2 The Mean Value Theorem 280
4.3 How Derivatives Affect the Shape of a Graph 287
4.4 Indeterminate Forms and L’Hospital’s Rule 298
Writing Project N The Origins of L’Hospital’s Rule 307
4.5 Summary of Curve Sketching 307
4.6 Graphing with Calculus and Calculators 315
4.7 Optimization Problems 322
Applied Project N The Shape of a Can 333
4.8 Newton’s Method 334
4.9 Antiderivatives 340
V/. INTEGRALS 354
5.1 Areas and Distances 355
5.2 The Definite Integral 366
Discovery Project N Area Functions 379
5.3 The Fundamental Theorem of Calculus 379
5.4 Indefinite Integrals and the Net Change Theorem 391
Writing Project N Newton, Leibniz, and the Invention of Calculus 399
5.5 The Substitution Rule 400
VI/.INTEGRALS 414
6.1 Areas between Curves 415
6.2 Volumes 422
6.3 Volumes by Cylindrical Shells 433
6.4 Work 438
6.5 Average Value of a Function 442
VII/. TECHNIQUES OF INTEGRATION 452
7.1 Integration by Parts 453
7.2 Trigonometric Integrals 460
7.3 Trigonometric Substitution 467
7.4 Integration of Rational Functions by Partial Fractions 473
7.5 Strategy for Integration 483
7.6 Integration Using Tables and Computer Algebra Systems 489
7.7 Approximate Integration 495
7.8 Improper Integrals 508
VIII/.FURTHER APPLICATIONS OF INTEGRATION 524
8.1 Arc Length 525
Discovery Project N Arc Length Contest 532
8.2 Area of a Surface of Revolution 532
Discovery Project N Rotating on a Slant 538
8.3 Applications to Physics and Engineering 539
Discovery Project N Complementary Coffee Cups 550
8.4 Applications to Economics and Biology 550
8.5 Probability 555
IX/. DIFFERENTIAL EQUATIONS 566
9.1 Modeling with Differential Equations 567
9.2 Direction Fields and Euler’s Method 572
9.3 Separable Equations 580
Applied Project N How Fast Does a Tank Drain? 588
Applied Project N Which Is Faster, Going Up or Coming Down? 590
9.4 Models for Population Growth 591
Applied Project N Calculus and Baseball 601
9.5 Linear Equations 602
9.6 Predator-Prey Systems 608
X/.PARAMETRIC EQUATIONS AND POLAR COORDINATES 620
10.1 Curves Defined by Parametric Equations 621
Laboratory Project N Running Circles around Circles 629
10.2 Calculus with Parametric Curves 630
Laboratory Project N Bézier Curves 639
10.3 Polar Coordinates 639
10.4 Areas and Lengths in Polar Coordinates 650
10.5 Conic Sections 654
10.6 Conic Sections in Polar Coordinates 662
XI/. INFINITE SEQUENCES AND SERIES 674
11.1 Sequences 675
Laboratory Project N Logistic Sequences 687
11.2 Series 687
11.3 The Integral Test and Estimates of Sums 697
11.4 The Comparison Tests 705
11.5 Alternating Series 710
11.6 Absolute Convergence and the Ratio and Root Tests 714
11.7 Strategy for Testing Series 721
11.8 Power Series 723
11.9 Representations of Functions as Power Series 728
11.10 Taylor and Maclaurin Series 734
Laboratory Project N An Elusive Limit 748
Writing Project N How Newton Discovered the Binomial Series 748
11.11 Applications of Taylor Polynomials 749
XIII/.VECTOR FUNCTIONS 816
13.1 Vector Functions and Space Curves 817
13.2 Derivatives and Integrals of Vector Functions 824
13.3 Arc Length and Curvature 830
13.4 Motion in Space: Velocity and Acceleration 838
XIV/.PARTIAL DERIVATIVES 854
14.1 Functions of Several Variables 855
14.2 Limits and Continuity 870
14.3 Partial Derivatives 878
14.4 Tangent Planes and Linear Approximations 892
14.5 The Chain Rule 901
14.6 Directional Derivatives and the Gradient Vector 910
14.7 Maximum and Minimum Values 922
14.8 Lagrange Multipliers 934
XV/.MULTIPLE INTEGRALS 950
15.1 Double Integrals over Rectangles 951
15.2 Iterated Integrals 959
15.3 Double Integrals over General Regions 965
15.4 Double Integrals in Polar Coordinates 974
15.5 Applications of Double Integrals 980
15.6 Triple Integrals 990
Discovery Project N Volumes of Hyperspheres 1000
15.7 Triple Integrals in Cylindrical Coordinates 1000
Discovery Project N The Intersection of Three Cylinders 1005
15.8 Triple Integrals in Spherical Coordinates 1005
Applied Project N Roller Derby 1012
15.9 Change of Variables in Multiple Integrals 1012
XVI/.VECTOR CALCULUS 1026
16.1 Vector Fields 1027
16.2 Line Integrals 1034
16.3 The Fundamental Theorem for Line Integrals 1046
16.4 Green’s Theorem 1055
16.5 Curl and Divergence 1061
16.6 Parametric Surfaces and Their Areas 1070
16.7 Surface Integrals 1081
16.8 Stokes’ Theorem 1092
16.9 The Divergence Theorem 1099
16.10 Summary 1105
XVII/.SECOND-ORDER DIFFERENTIAL EQUATIONS 1110
17.1 Second-Order Linear Equations 1111
17.2 Nonhomogeneous Linear Equations 1117
17.3 Applications of Second-Order Differential Equations 1125
17.4 Series Solutions 1133
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