Calculus

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.

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