Calculus: Single and Multivariable

1 Foundation for Calculus: Functions and Limits 1 1.1 Functions and Change 2 1.2 Exponential Functions 13 1.3 New Functions from Old 23 1.4 Logarithmic Functions 32 1.5 Trigonometric Functions 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 6 Constructing Antiderivatives 315 6.1 Antiderivatives Graphically and Numerically 316 6.2 Constructing Antiderivatives Analytically 322 6.3 Differential Equations and Motion 329 6.4 Second Fundamental Theorem of Calculus 335 Review Problems Online Projects Online 7 Integration 341 7.1 Integration by Substitution 342 7.2 Integration by Parts 353 7.3 Tables of Integrals 360 7.4 Algebraic Identities and Trigonometric Substitutions 366 7.5 Numerical Methods for Definite Integrals 376 7.6 Improper Integrals 385 7.7 Comparison of Improper Integrals 394 Review Problems Online Projects Online 8 Using the Definite 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Polynomial Approximations 539 10.5 Fourier Series 546 Review Problems Online Projects Online 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 Separation 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 15 Optimization: Local and Global Extrema 805 15.1 Critical Points: Local Extrema and Saddle Points 806 15.2 Optimization 815 15.3 Constrained Optimization: Lagrange Multipliers 825 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Review Problems Online Projects Online 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 Field 982 19.4 The Divergence Theorem 991 Review Problems Online Projects Online 20 The Curl and Stokes' Theorem 999 20.1 The Curl of a Vector Field 1000 20.2 STOKES' Theorem 1008 20.3 The Three Fundamental Theorems 1015 Review Problems Online Projects Online 21 Parameters, Coordinates, and Integrals 1021 21.1 Coordinates and Parameterized Surfaces 1022 21.2 Change of Coordinates in a Multiple Integral 1033 21.3 Flux Integrals Over Parameterized Surfaces 1038 Review Problems Online Projects Online 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 1043 Answers to Odd-Numbered Problems 1061 Index 1131

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• Cálculo

measure distance traveled; directional derivatives; fundamental theorem; chain rule; definite integral; fundamental tool; arc length; local extrema; line integrals; differential equation; measure speed; formal definition; equations; curl