Essential Calculus: Early Transcendental Functions

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 4
A Word from the Authors......Page 9
Integrated Learning System for Calculus......Page 11
Features......Page 16
1.1 Linear Models and Rates of Change......Page 22
1.2 Functions and Their Graphs......Page 30
1.3 Inverse Functions......Page 41
1.4 Exponential and Logarithmic Functions......Page 52
1.5 Finding Limits Graphically and Numerically......Page 59
1.6 Evaluating Limits Analytically......Page 69
1.7 Continuity and One-Sided Limits......Page 79
1.8 Infinite Limits......Page 91
Review Exercises......Page 98
2.1 The Derivative and the Tangent Line Problem......Page 101
2.2 Basic Differentiation Rules and Rates of Change......Page 111
2.3 Product and Quotient Rules and Higher-Order Derivatives......Page 123
2.4 The Chain Rule......Page 133
2.5 Implicit Differentiation......Page 147
2.6 Derivatives of Inverse Functions......Page 155
2.7 Related Rates......Page 161
2.8 Newton's Method......Page 169
Review Exercises......Page 174
3.1 Extrema on an Interval......Page 178
3.2 Rolle's Theorem and the Mean Value Theorem......Page 185
3.3 Increasing and Decreasing Functions and the First Derivative Test......Page 191
3.4 Concavity and the Second Derivative Test......Page 201
3.5 Limits at Infinity......Page 208
3.6 Optimization Problems......Page 218
3.7 Differentials......Page 229
Review Exercises......Page 235
4.1 Antiderivatives and Indefinite Integration......Page 238
4.2 Area......Page 248
4.3 Riemann Sums and Definite Integrals......Page 259
4.4 The Fundamental Theorem of Calculus......Page 269
4.5 Integration by Substitution......Page 281
4.6 Numerical Integration......Page 294
4.7 The Natural Logarithmic Function: Integration......Page 300
4.8 Inverse Trigonometric Functions: Integration......Page 308
4.9 Hyperbolic Functions......Page 315
Review Exercises......Page 325
5.1 Area of a Region Between Two Curves......Page 327
5.2 Volume: The Disk Method......Page 336
5.3 Volume: The Shell Method......Page 346
5.4 Arc Length and Surfaces of Revolution......Page 354
5.5 Applications in Physics and Engineering......Page 364
5.6 Differential Equations: Growth and Decay......Page 380
Review Exercises......Page 387
6.1 Integration by Parts......Page 389
6.2 Trigonometric Integrals......Page 397
6.3 Trigonometric Substitution......Page 405
6.4 Partial Fractions......Page 413
6.5 Integration by Tables and Other Integration Techniques......Page 421
6.6 Indeterminate Forms and L'Hôpital's Rule......Page 426
6.7 Improper Integrals......Page 436
Review Exercises......Page 446
7.1 Sequences......Page 448
7.2 Series and Convergence......Page 459
7.3 The Integral and Comparison Tests......Page 469
7.4 Other Convergence Tests......Page 477
7.5 Taylor Polynomials and Approximations......Page 487
7.6 Power Series......Page 497
7.7 Representation of Functions by Power Series......Page 506
7.8 Taylor and Maclaurin Series......Page 512
Review Exercises......Page 523
8.1 Plane Curves and Parametric Equations......Page 525
8.2 Parametric Equations and Calculus......Page 534
8.3 Polar Coordinates and Polar Graphs......Page 543
8.4 Area and Arc Length in Polar Coordinates......Page 552
8.5 Polar Equations and Conics and Kepler's Laws......Page 560
Review Exercises......Page 567
9.1 Vectors in the Plane......Page 570
9.2 Space Coordinates and Vectors in Space......Page 580
9.3 The Dot Product of Two Vectors......Page 587
9.4 The Cross Product of Two Vectors in Space......Page 595
9.5 Lines and Planes in Space......Page 602
9.6 Surfaces in Space......Page 613
9.7 Cylindrical and Spherical Coordinates......Page 622
Review Exercises......Page 628
10.1 Vector-Valued Functions......Page 630
10.2 Differentiation and Integration of Vector-Valued Functions......Page 637
10.3 Velocity and Acceleration......Page 644
10.4 Tangent Vectors and Normal Vectors......Page 652
10.5 Arc Length and Curvature......Page 661
Review Exercises......Page 672
11.1 Introduction to Functions of Several Variables......Page 674
11.2 Limits and Continuity......Page 685
11.3 Partial Derivatives......Page 694
11.4 Differentials and the Chain Rule......Page 703
11.5 Directional Derivatives and Gradients......Page 715
11.6 Tangent Planes and Normal Lines......Page 725
11.7 Extrema of Functions of Two Variables......Page 733
11.8 Lagrange Multipliers......Page 741
Review Exercises......Page 747
12.1 Iterated Integrals and Area in the Plane......Page 749
12.2 Double Integrals and Volume......Page 756
12.3 Change of Variables: Polar Coordinates......Page 766
12.4 Center of Mass and Moments of Inertia......Page 773
12.5 Surface Area......Page 780
12.6 Triple Integrals and Applications......Page 786
12.7 Triple Integrals in Cylindrical and Spherical Coordinates......Page 796
12.8 Change of Variables: Jacobians......Page 802
Review Exercises......Page 808
13.1 Vector Fields......Page 811
13.2 Line Integrals......Page 821
13.3 Conservative Vector Fields and Independence of Path......Page 834
13.4 Green's Theorem......Page 843
13.5 Parametric Surfaces......Page 851
13.6 Surface Integrals......Page 860
13.7 Divergence Theorem......Page 871
13.8 Stokes's Theorem......Page 879
Review Exercises......Page 885
Appendix A: Proofs of Selected Theorems......Page 888
Appendix B: Integration Tables......Page 904
Appendix C: Business and Economic Applications......Page 909
Answers to Odd-Numbered Exercises......Page 916
Index......Page 996