Student's Solution and Survival Manual for Calculus

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Front Cover Title Copyright Contents 1 Functions and Graphs 1.1 What Is Calculus? 1.2 Preliminaries 1.3 Lines in the Plane; Parametric Equations 1.4 Functions and Graphs 1.5 Inverse Functions; Inverse Trigonometric Functions Chapter 1 Review 2 Limits and Continuity 2.1 The Limit of a Function 2.2 Algebraic Computation of Limits 2.3 Continuity 2.4 Exponential and Logarithmic Functions Chapter 2 Review 3 Differentiation 3.1 An Introduction to the Derivative: Tangents 3.2 Techniques of Differentiation 3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions 3.4 Rates of Change: Modeling Rectilinear Motion 3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Related Rates and Applications 3.8 Linear Approximation and Differentials Chapter 3 Review 4 Additional Applications of the Derivative 4.1 Extreme Values of a Continuous Function 4.2 The Mean Value Theorem 4.3 Using Derivatives to Sketch the Graph of a Function 4.4 Curve Sketching with Asymptotes: Limits Involving Infinity 4.5 l'Hôpital's Rule 4.6 Optimization in the Physical Sciences and Engineering 4.7 Optimization in Business, Economics, and the Life Sciences Chapter 4 Review 5 Integration 5.1 Antidifferentiation 5.2 Area as the Limit of a Sum 5.3 Riemann Sums and the Definite Integral 5.4 The Fundamental Theorems of Calculus 5.5 Integration by Substitution 5.6 Introduction to Differential Equations 5.7 The Mean Value Theorem for Integrals; Average Value 5.8 Numerical Integration: The Trapezoidal Rule and Simpson's Rule 5.9 An Alternative Approach: The Logarithm as an Integral Chapter 5 Review Cumulative Review Problems - Chapters 1–5 6 Additional Applications of the Integral 6.1 Area Between Two Curves 6.2 Volume 6.3 Polar Forms and Area 6.4 Arc Length and Surface Area 6.5 Physical Applications: Work, Liquid Force, and Centroids 6.6 Applications to Business, Economics, and Life Sciences Chapter 6 Review 7 Methods of Integration 7.1 Review of Substitution and Integration by Table 7.2 Integration By Parts 7.3 Trigonometric Methods 7.4 Method of Partial Fractions 7.5 Summary of Integration Techniques 7.6 First-Order Differential Equations 7.7 Improper Integrals 7.8 Hyperbolic and Inverse Hyperbolic Functions Chapter 7 Review 8 Infinite Series 8.1 Sequences and Their Limits 8.2 Introduction to Infinite Series; Geometric Series 8.3 The Integral Test; p-series 8.4 Comparison Tests 8.5 The Ratio Test and the Root Test 8.6 Alternating Series; Absolute and Conditional Convergence 8.7 Power Series 8.8 Taylor and Maclaurin Series Chapter 8 Review 9 Vectors in the Plane and in Space 9.1 Vectors in R^2 9.2 Coordinates and Vectors in R^3 9.3 The Dot Product 9.4 The Cross Product 9.5 Lines in R^3 9.6 Planes in R^3 9.7 Quadric Surfaces Chapter 9 Review 10 Vector-Valued Functions 10.1 Introduction to Vector Functions 10.2 Differentiation and Integration of Vector Functions 10.3 Modeling Ballistics and Planetary Motion 10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature 10.5 Tangential and Normal Components of Acceleration Chapter 10 Review Cumulative Review Problems - Chapters 1–10 11 Partial Differentiation 11.1 Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives 11.4 Tangent Planes, Approximations, and Differentiability 11.5 Chain Rules 11.6 Directional Derivatives and the Gradient 11.7 Extrema of Functions of Two Variables 11.8 Lagrange Multipliers Chapter 11 Review 12 Multiple Integration 12.1 Double Integration over Rectangular Regions 12.2 Double Integration over Nonrectangular Regions 12.3 Double Integrals in Polar Coordinates 12.4 Surface Area 12.5 Triple Integrals 12.6 Mass, Moments, and Probability Density Functions 12.7 Cylindrical and Spherical Coordinates 12.8 Jacobians: Change of Variables Chapter 12 Review 13 Vector Analysis 13.1 Properties of a Vector Field: Divergence and Curl 13.2 Line Integrals 13.3 The Fundamental Theorem and Path Independence 13.4 Green's Theorem 13.5 Surface Integrals 13.6 Stokes' Theorem and Applications 13.7 Divergence Theorem and Applications Chapter 13 Review Cumulative Review Problems - Chapters 11–13 14 Introduction to Differential Equations 14.1 First-Order Differential Equations 14.2 Second-Order Homogeneous Linear Differential Equations 14.3 Second-Order Nonhomogeneous Linear Differential Equations Chapter 14 Review Back Cover