Calculus

Preface / v
Cas / vi
Calculus Map / xv
Sage Code / xvi

Part I Single Variable Calculus
Chapter 1. Functions
1.1 Functions and Graph 4
1.2 Symmetry 10
1.3 Common Functions 14
1.4 Translation, Stretching and Rotation of Functions 22

Chapter 2. Limits and Continuity
2.1 Limits of Functions 34
2.2 Continuity 53

Chapter 3. Derivatives
3.1 Definition of Derivatives, Differentiation 68
3.2 Derivatives of Functions, The Product and Quotient Rule 71
3.3 The Chain Rule and Inverse Functions 87
*3.4 Approximation and Related Rates 113

Chapter 4. Applications of Derivatives
4.1 Extreme values of a function 124
4.2 Shapes of Curves 138
4.3 The Limit of Indeterminate Forms and L’Hospital’s Rule 150
4.4 Mathematical Optimization Problems 158
4.5 Newton’s Method 167

Chapter 5. Integrals
5.1 Areas and Distances 174
5.2 The Definite Integral 178
5.3 The Fundamental Theorem of Calculus 193
5.4 Indefinite Integrals and the Net Change Theorem 202
5.5 The Substitution Rule 210
5.6 The Logarithm Defined as an Integral 220

Chapter 6. Applications of Integration
6.1 Areas between Curves 232
6.2 Volumes 242
6.3 Volumes by Cylindrical Shells 260
*6.4 Work 269
6.5 Average Value of a Function 276

Chapter 7. Techniques of Integration
7.1 Integration by Parts 286
7.2 Trigonometric Integrals 297
7.3 Trigonometric Substitution 306
7.4 Integration of Rational Functions and CAS 315
7.5 Formulas for Integration 331
*7.6 Integration Using Tables 339
*7.7 Approximate Integration and CAS 344
7.8 Improper Integrals 356

Chapter 8. Further Applications of Integration
8.1 Arc Length 372
8.2 Area of a Surface of Revolution 380
8.3 Center of Mass 390
*8.4 Differential Equations 395

Chapter 9. Infinite Sequences and Infinite Series
9.1 Sequences and Series 406
9.2 Tests for Convergence of Series 427
9.3 Alternating Series and Absolute Convergence 437
9.4 Power Series 446
9.5 Taylor, Maclaurin, and Binomial Series 456

Part II Multivariate Calculus
Chapter 10. Parametric Equations and Polar Coordinates
10.1 Parametric Equations 478
10.2 Calculus with Parametric Curves 489
10.3 Polar Coordinates 501
10.4 Areas and Lengths in Polar Coordinates 515
10.5 Conic Sections 528

Chapter 11. Vectors and the Geometry of Space
11.1 Three-Dimensional Coordinate Systems 550
11.2 Vectors 556
11.3 The Dot Product 565
11.4 The Cross Product 575
11.5 Equations of Lines and Planes 585
11.6 Cylinders and Quadric Surfaces 603

Chapter 12. Vector Valued Functions
12.1 Vector-Valued Functions and Space Curves 618
12.2 Calculus of Vector Functions 628
12.3 Arc Length and Curvature 642
*12.4 Motion Along A Space Curve: Velocity and Acceleration 661

Chapter 13. Partial Derivatives
13.1 Multivariate Functions 674
13.2 Limits and Continuity of Multivariate Functions 684
13.3 Partial Derivatives 699
13.4 Differentiability and Total Differential 707
13.5 The Chain Rule 714
13.6 Directional Derivatives and Gradient 728
13.7 Tangent Plane 740
13.8 Extrema of Multivariate Functions 749
13.9 Lagrange Multiplier 769

Chapter 14. Multiple Integrals
14.1 Double Integrals 786
14.2 Double Integrals in Polar Coordinates 806
14.3 Surface Area 813
14.4 Cylindrical Coordinates and Spherical Coordinates 821
14.5 Triple Integrals 830
14.6 Triple Integrals in Cylindrical and Spherical Coordinates 836
14.7 The Substitution Method 842

Chapter 15. Vector Calculus
15.1 Vector Differentiation 854
15.2 Line Integrals 860
15.3 Potential Function and Independence of Path 865
15.4 Green’s Theorem in the Plane 876
15.5 Curl and Divergence 888
15.6 Surface Area 899
15.7 Surface Integrals 908
15.8 Stokes’Theorem 918
15.9 Divergence Theorem 925

Table / 937
References / 943
Index / 945