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 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
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