Calculus: Early Transcendentals

Contents......Page 15
Preface......Page 20
Preliminaries......Page 31
0.1 Lines......Page 32
0.2 Functions and Their Graphs......Page 46
0.3 The Trigonometric Functions......Page 57
0.4 Combining Functions......Page 69
0.5 Graphing Calculators and Computers......Page 82
0.6 Mathematical Models......Page 87
0.7 Inverse Functions......Page 103
0.8 Exponential and Logarithmic Functions......Page 114
Limits......Page 129
1.1 An Intuitive Introduction to Limits......Page 130
1.2 Techniques for Finding Limits......Page 142
1.3 A Precise Definition of a Limit......Page 156
1.4 Continuous Functions......Page 164
1.5 Tangent Lines and Rates of Change......Page 179
The Derivative......Page 193
2.1 The Derivative......Page 194
2.2 Basic Rules of Differentiation......Page 206
2.3 The Product and Quotient Rules......Page 216
2.4 The Role of the Derivative in the Real World......Page 226
2.5 Derivatives of Trigonometric Functions......Page 238
2.6 The Chain Rule......Page 245
2.7 Implicit Differentiation......Page 261
2.8 Derivatives of Logarithmic Functions......Page 275
2.9 Related Rates......Page 281
2.10 Differentials and Linear Approximations......Page 291
Applications of the Derivative......Page 311
3.1 Extrema of Functions......Page 312
3.2 The Mean Value Theorem......Page 326
3.3 Increasing and Decreasing Functions and the First Derivative Test......Page 335
3.4 Concavity and Inflection Points......Page 344
3.5 Limits Involving Infinity; Asymptotes......Page 359
3.6 Curve Sketching......Page 377
3.7 Optimization Problems......Page 391
3.8 Indeterminate Forms and l’Hôpital’s Rule......Page 408
3.9 Newton’s Method......Page 419
Integration......Page 433
4.1 Indefinite Integrals......Page 434
4.2 Integration by Substitution......Page 445
4.3 Area......Page 456
4.4 The Definite Integral......Page 474
4.5 The Fundamental Theorem of Calculus......Page 491
4.6 Numerical Integration......Page 509
Applications of the Definite Integral......Page 526
5.1 Areas Between Curves......Page 527
5.2 Volumes: Disks, Washers, and Cross Sections......Page 539
5.3 Volumes Using Cylindrical Shells......Page 555
5.4 Arc Length and Areas of Surfaces of Revolution......Page 564
5.5 Work......Page 577
5.6 Fluid Pressure and Force......Page 586
5.7 Moments and Center of Mass......Page 594
5.8 Hyperbolic Functions......Page 605
Techniques of Integration......Page 622
6.1 Integration by Parts......Page 623
6.2 Trigonometric Integrals......Page 633
6.3 Trigonometric Substitutions......Page 642
6.4 The Method of Partial Fractions......Page 650
6.5 Integration Using Tables of Integrals and a CAS; a Summary of Techniques......Page 661
6.6 Improper Integrals......Page 670
Differential Equations......Page 692
7.1 Differential Equations: Separable Equations......Page 693
7.2 Direction Fields and Euler’s Method......Page 708
7.3 The Logistic Equation......Page 719
7.4 First-Order Linear Differential Equations......Page 729
7.5 Predator-Prey Models......Page 741
Infinite Sequences and Series......Page 754
8.1 Sequences......Page 755
8.2 Series......Page 772
8.3 The Integral Test......Page 782
8.4 The Comparison Tests......Page 788
8.5 Alternating Series......Page 795
8.6 Absolute Convergence; the Ratio and Root Tests......Page 800
8.7 Power Series......Page 810
8.8 Taylor and Maclaurin Series......Page 819
8.9 Approximation by Taylor Polynomials......Page 835
Conic Sections, Plane Curves, and Polar Coordinates......Page 853
9.1 Conic Sections......Page 854
9.2 Plane Curves and Parametric Equations......Page 874
9.3 The Calculus of Parametric Equations......Page 883
9.4 Polar Coordinates......Page 892
9.5 Areas and Arc Lengths in Polar Coordinates......Page 904
9.6 Conic Sections in Polar Coordinates......Page 914
Vectors and the Geometry of Space......Page 927
10.1 Vectors in the Plane......Page 928
10.2 Coordinate Systems and Vectors in 3-Space......Page 939
10.3 The Dot Product......Page 950
10.4 The Cross Product......Page 960
10.5 Lines and Planes in Space......Page 971
10.6 Surfaces in Space......Page 982
10.7 Cylindrical and Spherical Coordinates......Page 997
Vector-Valued Functions......Page 1009
11.1 Vector-Valued Functions and Space Curves......Page 1010
11.2 Differentiation and Integration of Vector-Valued Functions......Page 1018
11.3 Arc Length and Curvature......Page 1026
11.4 Velocity and Acceleration......Page 1036
11.5 Tangential and Normal Components of Acceleration......Page 1044
Functions of Several Variables......Page 1059
12.1 Functions of Two or More Variables......Page 1060
12.2 Limits and Continuity......Page 1074
12.3 Partial Derivatives......Page 1085
12.4 Differentials......Page 1099
12.5 The Chain Rule......Page 1110
12.6 Directional Derivatives and Gradient Vectors......Page 1122
12.7 Tangent Planes and Normal Lines......Page 1134
12.8 Extrema of Functions of Two Variables......Page 1142
12.9 Lagrange Multipliers......Page 1153
Multiple Integrals......Page 1173
13.1 Double Integrals......Page 1174
13.2 Iterated Integrals......Page 1183
13.3 Double Integrals in Polar Coordinates......Page 1194
13.4 Applications of Double Integrals......Page 1201
13.5 Surface Area......Page 1208
13.6 Triple Integrals......Page 1214
13.7 Triple Integrals in Cylindrical and Spherical Coordinates......Page 1226
13.8 Change of Variables in Multiple Integrals......Page 1234
Vector Analysis......Page 1249
14.1 Vector Fields......Page 1250
14.2 Divergence and Curl......Page 1257
14.3 Line Integrals......Page 1267
14.4 Independence of Path and Conservative Vector Fields......Page 1282
14.5 Green’s Theorem......Page 1294
14.6 Parametric Surfaces......Page 1305
14.7 Surface Integrals......Page 1316
14.8 The Divergence Theorem......Page 1330
14.9 Stokes’ Theorem......Page 1337
APPENDIX A The Real Number Line, Inequalities, and Absolute Value......Page 1351
APPENDIX B Proofs of Theorems......Page 1357
APPENDIX C The Definition of the Logarithm
as an Integral......Page 1369
ANSWERS TO SELECTED EXERCISES......Page 1377
INDEX......Page 1455
INDEX OF APPLICATIONS......Page 1475