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Calculus: Single and Multivariable, Enhanced eText, 7th Edition (1119320496) cover image

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Calculus: Single and Multivariable, 7th Edition continues the effort to promote courses in which understanding and computation reinforce each other. The 7th Edition reflects the many voices of users at research universities, four-year colleges, community colleges, and secondary schools. This new edition has been streamlined to create a flexible approach to both theory and modeling. The program includes a variety of problems and examples from the physical, health, and biological sciences, engineering and economics; emphasizing the connection between calculus and other fields.

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Table of Contents

1. Foundations for Calculus: Functions and Limits 1
1.1 Functions and Change 2
1.2 Expotential Functions 13
1.3 New Functions from Old 23
1.4 Logarithmic Functions 32
1.5 Trigonometric Funtions 39
1.6 Powers, Polynomials, and Rational Functions 49
1.7 Introduction to Limits and Continuity 58
1.8 Extending the Idea of a Limit 67
1.9 Further Limit Calculations Using Algebra 75
1.10 Optional Preview of the Formal Definition of a Limit Online
    
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2. Key Concept: The Derivative 83
2.1 How Do We Measure Speed? 84
2.2 The Derivative at a Point 91
2.3 The Derivative Function 99
2.4 Interpretations of the Derivative 108
2.5 The Second Derivative 115
2.6 Differentiability 123
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3. Short-Cuts to Differentiation 129
3.1 Powers and Polynomials 130
3.2 The Exponential Function 140
3.3 The Product and Quotient Rules 144
3.4 The Chain Rule 151
3.5 The Trigonometric Functions 158
3.6 The Chain Rule and Inverse Functions 164
3.7 Implicit Functions 171
3.8 Hyperbolic Functions 174
3.9 Linear Approximation and the Derivative 178
3.10 Theorems About Differentiable Functions 186
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4. Using the Derivative 191
4.1 Using First and Second Derivatives 192
4.2 Optimization 203
4.3 Optimization and Modeling 212
4.4 Families of Functions and Modeling 224
4.5 Applications to Marginality 233
4.6 Rates and Related Rates 243
4.7 L'Hopital's Rule, Growth, and Dominance 252
4.8 Parametric Equations 259
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5. Key Concept: The Definite Integral 271
5.1 How Do We Measure Distance Traveled? 272
5.2 The Definite Integral 283
5.3 The Fundamental Theorem and Interpretations 292
5.4 Theorems About Definite Integrals 302
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6. Constructing Antiderivatives 315
6.1 Antiderivatives Graphically and Numerically
6.2 Constructing Antiderivatives Analytically 322
6.3 Differential Equations and Motion 329
6.4 Second Fundamental Theorem of Calculus 335
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7. Integration 341
7.1 Integration by Substitution 342
7.2 Integration by Parts 353
7.3 Tables of Integrals 360
7.4 Algebraic Methods for Definite Integrals 376
7.5 Numberical Methods for Definite Integrals 376
7.6 Improper Integrals 385
7.7 Comparison of Improper Integrals 397
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8. Using the Definitive Integral 401
8.1 Areas and Volumes 402
8.2 Applications to Geometry 410
8.3 Area and Arc Length in Polar Coordinates 420
8.4 Density and Center of Mass 429
8.5 Applications to Physics 439
8.6 Applications to Economics 450
8.7 Distribution Functions 457
8.8 Probability, Mean, and Median 464
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9. Sequences and Series 473
9.1 Sequences 474
9.2 Geometric Series 480
9.3 Convergence of Series 488
9.4 Tests for Convergence 494
9.5 Power Series and Interval of Convergence 504
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10. Approximating Functions Using Series 513
10.1 Taylor Polynomials 514
10.2 Taylor Series 523
10.3 Finding and Using Taylor Series 530
10.4 The Error in Taylor Palynomial Approximations 539
10.5 Fourier Series 546
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11. Differential Equations 561
11.1 What is a Differential Equation? 562
11.2 Slope Fields 567
11.3 Euler's Method 575
11.4 Serparation of Variables 580
11.5 Growth and Decay 586
11.6 Applications and Modeling 597
11.7 The Logistic Model 606
11.8 Systems of Differential Equations 616
11.9 Analyzing the Phase Plane 626
11.10 Second-Order Differential Equations: Oscillations 632
11.11 Linear Second-Order Differential Equations 640
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12. Functions of Several Variables 651
12.1 Functions of Two Variables 652
12.2 Graphs and Surfaces 660
12.3 Contour Diagrams 668
12.4 Linear Functions 682
12.5 Functions of Three Variables 689
12.6 Limits and Continuity 695
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13. A Fundamental Tool: Vectors 701
13.1 Displacement Vectors 702
13.2 Vectors in General 710
13.3 The Dot Product 718
13.4 The Cross Product 728
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14. Differentiating Functions of Several Variables 739
14.1 The Partial Derivative 740
14.2 Computing Partial Derivatives Algebraically 748
14.3 Local Linearity and the Differential 753
14.4 Gradients and Directional Derivatives in the Plane 762
14.5 Gradients and Directional Derivatives in Space 772
14.6 The Chain Rule 780
14.7 Second-Order Partial Derivatives 790
14.8 Differentiability 799
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15. Optimization: Local and Global Extreme 805
15.1 Critical Points: Local Extreme and Saddle Points 806
15.2 Optimization 815
15.3 Constrained Optimization: Lagrange Multipliers 825
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16. Integrating Functions of Several Variables 839
16.1 The Definite Integral of a Function of Two Variables 840
16.2 Iterated Integrals 847
16.3 Triple Integrals 857
16.4 Double Integrals in Polar Coordinates 864
16.5 Integrals in Cylindrical and Spherical Coordinates 869
16.6 Applications of Integration to Probability 878
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17. Parameterization and Vector Fields 885
17.1 Parameterized Curves 886
17.2 Motion, Velocity, and Acceleration 896
17.3 Vector Fields 905
17.4 The Flow of a Vector Field 913
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18. Line Integrals 921
18.1 The Idea of a Line Integral 922
18.2 Computing Line Integrals Over Parameterized Curves 931
18.3 Gradient Fields and Path-Independent Fields 939
18.4 Path-Dependent Vector Fields and Green's Theorem 949
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19. Flux Integrals and Divergence 961
19.1 The Idea of a Flux Integral 962
19.2 Flux Integrals for Graphs, Cylinders, and Spheres 973
19.3 The Divergence of a Vector Fields 982
19.4 The Divergence Theorem 991
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20. The Curl and Stokes' Theorem 999
20.1 The Curl of a Vector Fields 1000
20.2 Stokes' Theorem 1008
20.3 The Three Fundamental Theorems 1015
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21. Parameters, Coordinates, and Integrals 1021
21.1 Coordinates and Parameterized Surfaces 1022
21.2 Change of Corrdinates in a Multiple Integral 1033
21.3 Flux Integrals Over Parameterized Surfaces 1038
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