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Full Description
The Calculus Consortium's focus on the "Rule of Four" (viewing problems graphically, numerically, symbolically, and verbally) has become an integral part of teaching calculus in a way that promotes critical thinking to reveal solutions to mathematical problems. Their approach reinforces the conceptual understanding necessary to reduce complicated problems to simple procedures without losing sight of the practical value of mathematics. In this edition, the authors continue their focus on introducing different perspectives for students with an increased emphasis on active learning in a 'flipped' classroom.
The 8th edition of Calculus: Single and Multivariable features a variety of problems with applications from the physical sciences, health, biology, engineering, and economics, allowing for engagement across multiple majors. The Consortium brings Calculus to (real) life with current, relevant examples and a focus on active learning.
Contents
1 Foundation For Calculus: Functions and Limits 1
1.1 Functions and Change 2
1.2 Exponential Functions 14
1.3 New Functions From Old 26
1.4 Logarithmic Functions 34
1.5 Trigonometric Functions 42
1.6 Powers, Polynomials, and Rational Functions 53
1.7 Introduction To Limits and Continuity 62
1.8 Extending The Idea of A Limit 71
1.9 Further Limit Calculations Using Algebra 80
1.10 Preview of The Formal Definition of A Limit Online
2 Key Concept: The Derivative 87
2.1 How Do We Measure Speed? 88
2.2 The Derivative At A Point 96
2.3 The Derivative Function 105
2.4 Interpretations of The Derivative 113
2.5 The Second Derivative 121
2.6 Differentiability 130
3 Short-Cuts To Differentiation 135
3.1 Powers and Polynomials 136
3.2 The Exponential Function 146
3.3 The Product and Quotient Rules 151
3.4 The Chain Rule 158
3.5 The Trigonometric Functions 165
3.6 The Chain Rule and Inverse Functions 171
3.7 Implicit Functions 178
3.8 Hyperbolic Functions 181
3.9 Linear Approximation and The Derivative 185
3.10 Theorems About Differentiable Functions 193
4 Using The Derivative 199
4.1 Using First and Second Derivatives 200
4.2 Optimization 211
4.3 Optimization and Modeling 220
4.4 Families of Functions and Modeling 234
4.5 Applications To Marginality 244
4.6 Rates and Related Rates 253
4.7 L'hopital's Rule, Growth, and Dominance 264
4.8 Parametric Equations 271
5 Key Concept: The Definite Integral 285
5.1 How Do We Measure Distance Traveled? 286
5.2 The Definite Integral 298
5.3 The Fundamental Theorem and Interpretations 308
5.4 Theorems About Definite Integrals 319
6 Constructing Antiderivatives 333
6.1 Antiderivatives Graphically and Numerically 334
6.2 Constructing Antiderivatives Analytically 341
6.3 Differential Equations and Motion 348
6.4 Second Fundamental Theorem of Calculus 355
7 Integration 361
7.1 Integration By Substitution 362
7.2 Integration By Parts 373
7.3 Tables of Integrals 380
7.4 Algebraic Identities and Trigonometric Substitutions 386
7.5 Numerical Methods For Definite Integrals 398
7.6 Improper Integrals 408
7.7 Comparison of Improper Integrals 417
8 Using The Definite Integral 425
8.1 Areas and Volumes 426
8.2 Applications To Geometry 436
8.3 Area and Arc Length In Polar Coordinates 447
8.4 Density and Center of Mass 456
8.5 Applications To Physics 467
8.6 Applications To Economics 478
8.7 Distribution Functions 489
8.8 Probability, Mean, and Median 497
9 Sequences and Series 507
9.1 Sequences 508
9.2 Geometric Series 514
9.3 Convergence of Series 522
9.4 Tests For Convergence 529
9.5 Power Series and Interval of Convergence 539
10 Approximating Functions Using Series 549
10.1 Taylor Polynomials 550
10.2 Taylor Series 560
10.3 Finding and Using Taylor Series 567
10.4 The Error In Taylor Polynomial Approximations 577
10.5 Fourier Series 584
11 Differential Equations 599
11.1 What is a Differential Equation? 600
11.2 Slope Fields 605
11.3 Euler's Method 614
11.4 Separation of Variables 619
11.5 Growth and Decay 625
11.6 Applications and Modeling 637
11.7 The Logistic Model 647
11.8 Systems of Differential Equations 657
11.9 Analyzing The Phase Plane 667
11.10 Second-Order Differential Equations: Oscillations 674
11.11 Linear Second-Order Differential Equations 682
12 Functions of Several Variables 693
12.1 Functions of Two Variables 694
12.2 Graphs and Surfaces 702
12.3 Contour Diagrams 711
12.4 Linear Functions 725
12.5 Functions of Three Variables 732
12.6 Limits and Continuity 739
13 A Fundamental Tool: Vectors 745
13.1 Displacement Vectors 746
13.2 Vectors In General 755
13.3 The Dot Product 763
13.4 The Cross Product 774
14 Differentiating Functions of Several Variables 785
14.1 The Partial Derivative 786
14.2 Computing Partial Derivatives Algebraically 795
14.3 Local Linearity and The Differential 800
14.4 Gradients and Directional Derivatives In The Plane 809
14.5 Gradients and Directional Derivatives In Space 819
14.6 The Chain Rule 827
14.7 Second-Order Partial Derivatives 838
14.8 Differentiability 847
15 Optimization: Local and Global Extrema 855
15.1 Critical Points: Local Extrema and Saddle Points 856
15.2 Optimization 866
15.3 Constrained Optimization: Lagrange Multipliers 876
16 Integrating Functions of Several Variables 889
16.1 The Definite Integral of A Function of Two Variables 890
16.2 Iterated Integrals 898
16.3 Triple Integrals 908
16.4 Double Integrals In Polar Coordinates 916
16.5 Integrals In Cylindrical and Spherical Coordinates 921
16.6 Applications of Integration To Probability 931
17 Parameterization and Vector Fields 937
17.1 Parameterized Curves 938
17.2 Motion, Velocity, and Acceleration 948
17.3 Vector Fields 958
17.4 The Flow of A Vector Field 966
18 Line Integrals 973
18.1 The Idea of A Line Integral 974
18.2 Computing Line Integrals Over Parameterized Curves 984
18.3 Gradient Fields and Path-Independent Fields 992
18.4 Path-Dependent Vector Fields and Green's Theorem 1003
19 Flux Integrals and Divergence 1017
19.1 The Idea of A Flux Integral 1018
19.2 Flux Integrals For Graphs, Cylinders, and Spheres 1029
19.3 The Divergence of A Vector Field 1039
19.4 The Divergence Theorem 1048
20 The Curl and Stokes' Theorem 1055
20.1 The Curl of A Vector Field 1056
20.2 Stokes' Theorem 1064
20.3 The Three Fundamental Theorems 1071
21 Parameters, Coordinates, and Integrals 1077
21.1 Coordinates and Parameterized Surfaces 1078
21.2 Change of Coordinates In A Multiple Integral 1089
21.3 Flux Integrals Over Parameterized Surfaces 1094
Appendices Online
A Roots, Accuracy, and Bounds Online
B Complex Numbers Online
C Newton's Method Online
D Vectors In The Plane Online
E Determinants Online
Ready Reference 1099
Answers To Odd Numbered Problems 1117
Index 1177
