Calculus: Early Transcendental Functions

Smith/Minton: Mathematically Precise. Student-Friendly. Superior Technology. Students who have used Smith/Minton's Calculus say it was easier to read than any other math book they've used. That testimony underscores the success of the authors' approach, which combines the best elements of reform with the most reliable aspects of mainstream calculus teaching, resulting in a motivating, challenging book. Smith/Minton also provide exceptional, reality-based applications that appeal to students' interests and demonstrate the elegance of math in the world around us. New features include: / A new organization placing all transcendental functions early in the book and consolidating the introduction to L'Hopital's Rule in a single section. / More concisely written explanations in every chapter. / Many new exercises (for a total of 7,000 throughout the book) that require additional rigor not found in the 2nd Edition. / New exploratory exercises in every section that challenge students to synthesize key concepts to solve intriguing projects. / New commentaries ("Beyond Formulas") that encourage students to think mathematically beyond the procedures they learn./ New counterpoints to the historical notes, "Today in Mathematics," that stress the contemporary dynamism of mathematical research and applications, connecting past contributions to the present. / An enhanced discussion of differential equations and additional applications of vector calculus.

Table of Contents:
0 Preliminaries 0.1 Polynomials and Rational Functions0.2 Graphing Calculators and Computer Algebra Systems0.3 Inverse Functions0.4 Trigonometric and Inverse Trigonometric Functions0.5 Exponential and Logarithmic FunctionsHyperbolic FunctionsFitting a Curve to Data0.6 Transformations of Functions1 Limits and Continuity 1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve1.2 The Concept of Limit 1.3 Computation of Limits 1.4 Continuity and its Consequences The Method of Bisections1.5 Limits Involving Infinity Asymptotes 1.6 Formal Definition of the Limit Exploring the Definition of Limit Graphically1.7 Limits and Loss-of-Significance Errors Computer Representation of Real Numbers2 Differentiation 2.1 Tangent Lines and Velocity 2.2 The Derivative Numerical Differentiation2.3 Computation of Derivatives: The Power Rule Higher Order DerivativesAcceleration2.4 The Product and Quotient Rules2.5 The Chain Rule 2.6 Derivatives of the Trigonometric Functions 2.7 Derivatives of the Exponential and Logarithmic Functions2.8 Implicit Differentiation and Inverse Trigonometric Functions 2.9 The Mean Value Theorem 3 Applications of Differentiation 3.1 Linear Approximations and Newton’s Method 3.2 Indeterminate Forms and L’Hopital’s Rule 3.3 Maximum and Minimum Values3.4 Increasing and Decreasing Functions3.5 Concavity and the Second Derivative Test 3.6 Overview of Curve Sketching3.7 Optimization3.8 Related Rates3.9 Rates of Change in Economics and the Sciences 4 Integration 4.1 Antiderivatives 4.2 Sums and Sigma Notation Principle of Mathematical Induction4.3 Area 4.4 The Definite IntegralAverage Value of a Function 4.5 The Fundamental Theorem of Calculus 4.6 Integration by Substitution 4.7 Numerical Integration Error Bounds for Numerical Integration4.8 The Natural Logarithm as an Integral The Exponential Function as the Inverse of the Natural Logarithm5 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 Integrals Involving Powers of Trigonometric FunctionsTrigonometric Substitution 6.4 Integration of Rational Functions Using Partial Fractions Brief Summary of Integration Techniques 6.5 Integration Tables and Computer Algebra Systems 6.6 Improper Integrals A Comparison Test7 First-Order Differential Equations 7.1 Modeling with Differential Equations Growth and Decay ProblemsCompound Interest7.2 Separable Differential EquationsLogistic Growth7.3 Direction Fields and Euler's Method 7.4 Systems of First-Order Differential EquationsPredator-Prey Systems 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 Estimating the Sum of an Alternating Series8.5 Absolute Convergence and the Ratio TestThe Root Test Summary of Convergence Tests8.6 Power Series 8.7 Taylor Series Representations of Functions as SeriesProof of Taylor’s Theorem8.8 Applications of Taylor SeriesThe Binomial Series 8.9 Fourier Series9 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 Components and Projections10.4 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 Curvature11.5 Tangent and Normal VectorsTangential and Normal Components of Acceleration Kepler’s Laws11.6 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 ApproximationsIncrements and Differentials12.5 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 IntegralsMass and Center of Mass13.6 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 Integrals14.7 The Divergence Theorem 14.8 Stokes' Theorem 14.9 Applications of Vector Calculus 15 Second-Order Differential Equations15.1 Second-Order Equations with Constant Coefficients 15.2 Nonhomogeneous Equations: Undetermined Coefficients15.3 Applications of Second-Order Differential Equations15.4 Power Series Solutions of Differential EquationsAppendix A: Proofs of Selected TheoremsAppendix B: Answers to Odd-Numbered Exercises