Calculus: Special Edition: Chapters 1-5

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Front Cover Summaries - Where to Look for Help Miscellaneous Formulas Contents Preface For the Student For the Instructor Features of this Book Text Content Innovative Presentation Acknowledgments 1 Functions and Graphs 1.1 What is Calculus The Limit: Zeno’s Paradox The Derivative: The Tangent Problem The Integral: The Area Problem Mathematical Modeling 1.1 Problem Set 1.2 Preliminaries Distance on a Number Line Absolute Value Distance in the Plane Trigonometry Solving Trigonometric Equations 1.2 Problem Set 1.3 Lines in the Plane; Parametric Equations Slope of a Line Forms of the Equation of a Line Parametric Form Parallel and Perpendicular Lines 1.3 Problem Set 1.4 Functions and Graphs Definition of a Function Functional Notation Domain and Range of a Function Composition of Functions Graph of a Function Classification of Functions 1.4 Problem Set 1.5 Inverse Functions; Inverse Trigonometric Functions Inverse Functions Criteria for Existence of an Inverse f^{-1} Graph of f^{-1} Inverse Trigonometric Functions Inverse Trigonometric Identities 1.5 Problem Set Chapter 1 Review Book Report: Ethnomathematics by Marcia Ascher Chapter 1 Group Research Project 2 Limits and Continuity 2.1 The Limit of a Function Intuitive Notion of a Limit One-Sided Limits Limits that do not Exist Formal Definition of a Limit 2.1 Problem Set 2.2 Algebraic Computation of Limits Computations with Limits Using Algebra to find Limits Limits of Piecewise-Defined Functions Two Special Trigonometric Limits 2.2 Problem Set 2.3 Continuity Intuitive Notion of Continuity Definition of Continuity Continuity Theorems Continuity on an Interval Intermediate Value Theorem 2.3 Problem Set 2.4 Exponential and Logarithmic Functions Exponential Functions Logarithmic Functions Natural Base e Natural Logarithms Continuous Compounding of Interest 2.4 Problem Set Chapter 2 Review Chapter 2 Group Research Project 3 Differentiation 3.1 An Introduction to the Derivative: Tangents Tangent Lines The Derivative Relationship Between the Graphs of f and f' Existence of Derivatives Continuity and Differentiability Derivative Notation 3.1 Problem Set 3.2 Techniques of Differentiation Derivative of a Constant Function Derivative of a Power Function Procedural Rules for Finding Derivatives Higher-Order Derivatives 3.2 Problem Set 3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions Derivatives of the Sine and Cosine Functions Differentiation of the Other Trigonometric Functions Derivatives of Exponential and Logarithmic Functions 3.3 Problem Set 3.4 Rates of Change: Modeling Rectilinear Motion Average and Instantaneous Rate of Change Introduction to Mathematical Modeling Rectilinear Motion (Modeling in Physics) Falling Body Problems 3.4 Problem Set 3.5 The Chain Rule Introduction to the Chain Rule Extended Derivative Formulas Justification of the Chain Rule 3.5 Problem Set 3.6 Implicit Differentiation General Procedure for Implicit Differentiation Derivative Formulas for the Inverse Trigonometric Functions Logarithmic Differentiation 3.6 Problem Set 3.7 Related Rates and Applications 3.7 Problem Set 3.8 Linear Approximation and Differentials Tangent Line Approximation Differential Error Propagation Marginal Analysis in Economics The Newton-Raphson Method for Approximating Roots 3.8 Problem Set Chapter 3 Review Book Report: Fermat's Enigma by Simon Singh Chapter 3 Group Research Project 4 Additional Applications of the Derivative 4.1 Extreme Values of a Continuous Function Extreme Value Theorem Relative Extrema Absolute Extrema Optimization 4.1 Problem Set 4.2 The Mean Value Theorem Rolle’s Theorem Proof of the Mean Value Theorem The Zero-Derivative Theorem 4.2 Problem Set 4.3 Using Derivatives to Sketch the Graph of a Function Increasing and Decreasing Functions The First-Derivative Test Concavity and Inflection Points The Second-Derivative Test Curve Sketching Using the First and Second Derivatives 4.3 Problem Set 4.4 Curve Sketching with Asymptotes: Limits Involving Infinity Limits at Infinity Infinite Limits Graphs with Asymptotes Vertical Tangents and Cusps A General Graphing Strategy 4.4 Problem Set 4.5 l'Hôpital's Rule A Rule to Evaluate Indeterminate Forms Indeterminate Forms 0/0 and ∞∕∞ Other Indeterminate Forms Special Limits Involving e^x and ln(x) 4.5 Problem Set 4.6 Optimization in the Physical Sciences and Engineering Optimization Procedure Fermat’s Principle of Optics and Snell’s Law 4.6 Problem Set 4.7 Optimization in Business, Economics, and the Life Sciences Economics Business Management Physiology 4.7 Problem Set Chapter 4 Review Chapter 4 Group Research Project 5 Integration 5.1 Antidifferentiation Reversing Differentiation Antiderivative Notation Antidifferentiation Formulas Applications Area as an Antiderivative 5.1 Problem Set 5.2 Area as the Limit of a Sum Area as the Limit of a Sum The General Approximation Scheme Summation Notation Area Using Summation Formulas 5.2 Problem Set 5.3 Riemann Sums and the Definite Integral Riemann Sums The Definite Integral Area as an Integral Properties of the Definite Integral Distance as an Integral 5.3 Problem Set 5.4 The Fundamental Theorems of Calculus The First Fundamental Theorem of Calculus The Second Fundamental Theorem of Calculus 5.4 Problem Set 5.5 Integration by Substitution Substitution with Indefinite Integration Substitution with Definite Integration 5.5 Problem Set 5.6 Introduction to Differential Equations Introduction and Terminology Direction Fields Separable Differential Equations Modeling Exponential Growth and Decay Orthogonal Trajectories Modeling Fluid Flow Through an Orifice Modeling the Motion of a Projectile: Escape Velocity 5.6 Problem Set 5.7 The Mean Value Theorem for Integrals; Average Value Mean Value Theorem for Integrals Modeling Average Value of a Function 5.7 Problem Set 5.8 Numerical Integration: The Trapezoidal Rule and Simpson’s Rule Approximation by Rectangles Trapezoidal Rule Simpson’s Rule Error Estimation Summary of Numerical Integration Techniques 5.8 Problem Set 5.9 An Alternative Approach: The Logarithm as an Integral Natural Logarithm as an Integral Geometric Interpretation The Natural Exponential Function 5.9 Problem Set Chapter 5 Review Chapter 5 Group Research Project Cumulative Review Problems - Chapters 1-5 Appendices A: Introduction to the Theory of Limits The Believer/Doubter Format Selected Theorems with Formal Proofs B: Selected Proofs Chain Rule (Section 3.5) Cauchy’s Generalized Mean Value Theorem (Section 4.2) L'Hôpital's Rule* (Section 4.5) Limit Comparison Test (Section 8.4) Taylor’s Theorem (Section 8.8) Sufficient Condition for Differentiability (Section 11.4) Change of Variables Formula for Multiple Integration (Section 12.8) Stokes' Theorem (Section 13.6) Divergence Theorem (Section 13.7) Significant Digits C: Significant Digits Rounding and Rules of Computations Used in this Book Calculator Experiments Trigonometric Evaluations Graphing Blunders D: Short Table of Integrals E: Trigonometry Trigonometric Functions Radians and Degrees Inverse Trigonometric Functions Evaluating Trigonometric Functions Trigonometric Graphs Trigonometric Identities F: Determinants Determinants Properties of Determinants G: Answers to Selected Problems Index Integration Formulas Differentiation Formulas Back Cover