Calculus: Early Transcendental Functions

Title
Table of Contents
0 Preliminaries
	0.1 Polynomials and Rational Functions
	0.2 Graphing Calculators and Computer Algebra Systems
	0.3 Inverse Functions
	0.4 Trigonometric and Inverse Trigonometric Functions
	0.5 Exponential and Logarithmic Functions
	0.6 Transformations of Functions
1 Limits and Continuity
	1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
	1.2 The Concept of Limit
	1.3 Computation of Limits
	1.4 Continuity and Its Consequences
	1.5 Limits Involving Infinity; Asymptotes
	1.6 Formal Definition of the Limit
	1.7 Limits and Loss-of-Significance Errors
2 Differentiation
	2.1 Tangent Lines and Velocity
	2.2 The Derivative
	2.3 Computation of Derivatives: The Power Rule
	2.4 The Product and Quotient Rules
	2.5 The Chain Rule
	2.6 Derivatives of Trigonometric Functions
	2.7 Derivatives of Exponential and Logarithmic Functions
	2.8 Implicit Differentiation and Inverse Trigonometric Functions
	2.9 The Hyperbolic Functions
	2.10 The Mean Value Theorem
3 Applications of Differentiation
	3.1 Linear Approximations and Newton’s Method
	3.2 Indeterminate Forms and l’Hˆopital’s Rule
	3.3 Maximum and Minimum Values
	3.4 Increasing and Decreasing Functions
	3.5 Concavity and the Second Derivative Test
	3.6 Overview of Curve Sketching
	3.7 Optimization
	3.8 Related Rates
	3.9 Rates of Change in Economics and the Sciences
4 Integration
	4.1 Antiderivatives
	4.2 Sums and Sigma Notation
	4.3 Area
	4.4 The Definite Integral
	4.5 The Fundamental Theorem of Calculus
	4.6 Integration by Substitution
	4.7 Numerical Integration
	4.8 The Natural Logarithm as an Integral
5  Applications of the Definite Integral
	5.1 Area Between Curves
	5.2 Volume: Slicing, Disks and Washers
	5.3 Volumes by Cylindrical Shells
	5.4 Arc Length and Surface Area
	5.5 Projectile Motion
	5.6 Applications of Integration to Physics and Engineering
	5.7 Probability
6 Integration Techniques
	6.1 Review of Formulas and Techniques
	6.2 Integration by Parts
	6.3 Trigonometric Techniques of Integration
	6.4 Integration of Rational Functions Using Partial Fractions
	6.5 Integration Tables and Computer Algebra Systems
	6.6 Improper Integrals
7 First-Order Differential Equations
	7.1 Modeling with Differential Equations
	7.2 Separable Differential Equations
	7.3 Direction Fields and Euler’s Method
	7.4 Systems of First-Order Differential Equations
8 Infinite Series
	8.1 Sequences of Real Numbers
	8.2 Infinite Series
	8.3 The Integral Test and Comparison Tests
	8.4 Alternating Series
	8.5 Absolute Convergence and the Ratio Test
	8.6 Power Series
	8.7 Taylor Series
	8.8 Applications of Taylor Series
	8.9 Fourier Series
9 Parametric Equations and Polar Coordinates
	9.1 Plane Curves and Parametric Equations
	9.2 Calculus and Parametric Equations
	9.3 Arc Length and Surface Area in Parametric Equations
	9.4 Polar Coordinates
	9.5 Calculus and Polar Coordinates
	9.6 Conic Sections
	9.7 Conic Sections in Polar Coordinates
10 Vectors and the Geometry of Space
	10.1 Vectors in the Plane
	10.2 Vectors in Space
	10.3 The Dot Product
	10.4 The Cross Product
	10.5 Lines and Planes in Space
	10.6 Surfaces in Space
11 Vector-Valued Functions
	11.1 Vector-Valued Functions
	11.2 The Calculus of Vector-Valued Functions
	11.3 Motion in Space
	11.4 Curvature
	11.5 Tangent and Normal Vectors
	11.6 Parametric Surfaces
12 Functions of Several Variables and Partial Differentiation
	12.1 Functions of Several Variables
	12.2 Limits and Continuity
	12.3 Partial Derivatives
	12.4 Tangent Planes and Linear Approximations
	12.5 The Chain Rule
	12.6 The Gradient and Directional Derivatives
	12.7 Extrema of Functions of Several Variables
	12.8 Constrained Optimization and Lagrange Multipliers
13 Multiple Integrals
	13.1 Double Integrals
	13.2 Area, Volume and Center of Mass
	13.3 Double Integrals in Polar Coordinates
	13.4 Surface Area
	13.5 Triple Integrals
	13.6 Cylindrical Coordinates
	13.7 Spherical Coordinates
	13.8 Change of Variables in Multiple Integrals
14 Vector Calculus
	14.1 Vector Fields
	14.2 Line Integrals
	14.3 Independence of Path and Conservative Vector Fields
	14.4 Green’s Theorem
	14.5 Curl and Divergence
	14.6 Surface Integrals
	14.7 The Divergence Theorem
	14.8 Stokes’ Theorem
	14.9 Applications of Vector Calculus
15 Second-Order Differential Equations
	15.1 Second-Order Equations with Constant Coefficients
	15.2 Nonhomogeneous Equations: Undetermined Coefficients
	15.3 Applications of Second-Order Equations
	15.4 Power Series Solutions of Differential Equations
Appendix A: Proofs of Selected Theorems
Appendix B: Answers to Odd-Numbered Exercises
Credits
Index