1 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 One-Sided 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 Theorem245
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 Three-Dimensional 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 Vector-Valued 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 First-Order Differential Equations (online)
15.1 Solutions, Slope Fields, and Picard’s Theorem
15.2 First-Order 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 Second-Order Differential Equations (online)
16.1 Second-Order Linear Equations
16.2 Nonhomogeneous Linear Equations
16.3 Applications
16.4 Euler Equations
16.5 Power Series Solutions
Appendices AP-1
A.1 Real Numbers and the Real Line AP-1
A.2 Mathematical Induction AP-7
A.3 Lines, Circles, and Parabolas AP-10
A.4 Trigonometry Formulas AP-19
A.5 Proofs of Limit Theorems AP-21
A.6 Commonly Occurring Limits AP-25
A.7 Theory of the Real Numbers AP-26
A.8 The Distributive Law for Vector Cross Products AP-29
A.9 The Mixed Derivative Theorem and the Increment Theorem AP-30
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