Calculus: Special Edition: Chapters 1-5

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