p preliminaries

1 lines 1

2 functions and graphs 10

3 exponential functions 24

4 inverse functions and logarithms 31

5 trigonometric functions and their inverses 44

6 parametric equations 60

7 modeling change 67

questions to guide your review 76

practice exercises 77

additional exercises' theory, examples, applications 80

1 limits and continuity

1.1 rates of change and limits 85

1.2 finding limits and one-sided'limits gg

1.3 limits involving infinity 112

1.4 continuity 123

1.5 tangent lines 134

questions to guide your review 141

practice exercises 142

additional exercises: theory, examples, applications 143

.2 derivatives

2.1 the derivative as a function 147

2.2 the derivative as a rate of change 160

2.3 derivatives of products, quotients, and negative powers 173

2.4 derivatives of trigonometric functions 179

2.5 the chain rule and parametric equations 187

2.6 implicit differentiation 198

2.7 related rates 207

questions to guide your review 216

practice exercises 217

addmonal exercises: theory, examples, applications 221

3 applications of derivatives 225

3.1 extreme values of functions 225

3.2 the mean value theorem and differential equations 237

3.3 the shape of a graph 245

3.4 graphical solutions of autonomous differential equations 257

3.5 modeling and optimization 266

3.6 linearization and differentials 283

3.7 newton's method 297

questions to guide your review 305

practice exercises 305

additional exercises: theory, examples, applications 309

4 integration 313

4.1 indefinite integrals, differential equations, and modeling 313

4.:2 integral rules; integration by substitution 322

4.3 estimating with finite sums 329

4.4 riemann sumsand definitelntegrals 340

4.5 the mean value and fundamentaliheorems 351

4.6 substitution in definite integrals 364

4.7 numerical integration 373

ouestions to guide your review 384

practice exercises 385

additional exercises: theory, examples, applications 389

393

5 applications of integrals

5.1 volumes by slicing and rotation about an axis 3a3

5.2 modeling volume using cylindrical shells 406

5.3 lengths of plane curves 413

5.4 springs, pumping, and lifting 421

5.5 fluid forces 432

5.6 moments and centers of mass 439

questions to guide your review 451

practice exercises 451

addmonal exercises: theory, examples, appucations 454

6 transcendental functions and differential equations 457

6.1 logarithms 457

6.2 exponential functions 466

6.3 derivatives of inverse trigonometric functions; integrals 477

6.4 first-order separable differential equations 485

6.5 linear first-order differential equations 499

6.6 euler's method; population models 507

6.7 hyperbolic functions 520

questions to guide your review 530

practice exercises 531

additional exercises: theory, examples, applications 535

7 integration techniques, l'h6pital's rule, and improper integrals 539

7.1 basic integration formulas 539

7.2 integration by parts 546

7.3 partial fractions 555

7.4 trigonometric substitutions 565

7.5 integral tables, computer algebra systems, and

monte carlo integration 570

7.6 l'hopital's rule 578

7.7 improper integrals 586

questions to guide your review 600

practice exercises 601

additional exercises: theory, examples, applications 603

8 infinite series 607

8.1 limits of sequences of numbers 608

8.2 subsequences, bounded sequences, and picard's method 619

8.3 infinite series 627

8.4 series of nonnegative terms 639

8.5 alternating series, absolute and conditional convergence 651

8.6 power series 660

8.7 taylor andmaclaurin series 669

8.8 applications of power series 683

8.9 fourier series 691

8.10 fourier cosine and sine series 698

questions to guide your review 707

practice exercises 708

additional exercises: theory, examples, applications 711

9 vectors in the plane and polar functions 717

9.1 vectors in the plane 717

9.2 dot products 728

9.3 vector-valued functions 738

9.4 modeling project!ic motion 749

9.5 polar coordinates and graphs 761

9.6 calculus of polar curves 770

questions to guide your review 780

practice exercises 780

additional exercises: theory, examples, applications 784

10 vectors and motion in space 787

10.1 cartesian (rectangular) coordinates and vectors in space 787

10.2 dot and cross products 796

10.3 lines and planes:in space 807

10.4 cylinders and quadricsurfaees 816

10.5 vector-valued functions and space curves 825

10.6 arc length and the unit tangent vector t 838

10.7 the tnb frame; tangential and normal components of acceleration 847

10,8 planetary motion and satellites 857

questions to guide your review 866

practice exercises 867

additional exercises: theory, examples, applications 870

11 multivariable functions and their derivatives 873

11.1 functions of several variables 873

11.2 limits and continuity in higher dimensions 882

11.3 partial derivatives 890

11.4 the chain rule 902

11.5 directional derivatives, gradient vectors, and tangent planes 911

11.6 linearization and differentials 925

11.7 extreme values and saddle points 936

11.8 lagrange multipliers 947

11.9 *partial derivatives with constrained variables 958

11.10 taylor's formula for two variables 963

questions to guide your review 968

practice exercises 968

additional exercises: theory,examples, applications 972

12 multiple integrals 975

12.1 double integrals 975

12.2 areas, moments and centers of mass* 987

12.3 double integrals in polar form 1000

12.4 triple integrals in rectangular coordinates 1007

12.5 masses and moments in three dimensions 1017

12.6 triple integrals in cylindrical and spherical coordinates 1024

12.7 substitutions in multiple integrals 1037

questions to guide your review 1046

practice exercises 1047

additional exercises: theory, examples, applications 1049

13 integration in vector fields 1053

13.1 line integrals 1053

13.2 vector fields, work, circulation, and flux 1059

13.3 path independence, potential functions, and conservative fields 1070

13.4 green's theorem in the plane 1080

13.5 surface area and surface integrals 1092

13.6 parametrized surfaces 1103

13.7 stokes' theorem 1113

13.8 divergence theorem and a unified theory 1124

questions to guide your review 1136

practice exercises 1136

additional exercises: theory, examples, applications 1139

14 appendices 1143

a. 1 mathematical induction 1143

a.2 proofs of limit theorems in section 1.2 1146

a.3 proof of the chain rule 1150

a.4 complex numbers 1151

a.5 simpson's one-third rule 1162

a.6 cauchy's mean value theorem and the stronger form of l'hopitars rule 1163

a.7 limits that arise frequently 1164

a.8 proof of taylor's theorem 1166

a.9 the distributive law for vector cross products 1167

a. 10 determinants and cramer's rule 1169

a. 11 the mixed derivative theorem and the increment theorem 1176

a. 12 the area of a parallelogram's projection on a plane 1181

a. 13 conic sections 1183

answers a-1

index i-1

a brief table of integrals t-1