Calculus: One and Several Variables, 10/e (Hardcover)

Description

For ten editions, readers have turned to Salas to learn the difficult concepts of calculus without sacrificing rigor. The book consistently provides clear calculus content to help them master these concepts and understand its relevance to the real world. Throughout the pages, it offers a perfect balance of theory and applications to elevate their mathematical insights. Readers will also find that the book emphasizes both problem-solving skills and real-world applications.

 

 

Chapter 1. Precalculus Review.
Introduction: What is Calculus?
Review of Elementary Mathematics.
Review of Inequalities.
Coordinate Plane; Analytic Geometry.
Functions.
The Elementary Functions.
Combinations of Functions.
A Note on Mathematical Proof; Mathematical Induction.

Chapter 2. Limits and Continuity.
The Limit Process (An Intuitive Introduction).
Definition of Limit.
Some Limit Theorems.
Continuity.
The Pinching Theorem; Trigonometric Limits.
Two Basic Theorems.

Chapter 3. The Derivative; The Process of Differentiation.
The Derivative.
Some Differentiation Formulas.
The d/dx Notation; Derivatives of Higher Order.
The Derivative as a Rate of Change.
The Chain Rule.
Differentiating the Trigonometric Functions.
Implicit Differentiation; Rational Powers.

Chapter 4. The Mean-Value Theorem; Applications of the First and Second Derivatives.
The Mean-Value Theorem.
Increasing and Decreasing Functions.
Local Extreme Values.
Endpoint Extreme Values; Absolute Extreme Values.
Some Max-Min Problems.
Concavity and Points of Inflection.
Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps.
Some Curve Sketching.
Velocity and Acceleration; Speed.
Related Rates of Change Per Unit Time.
Differentials.
Newton-Raphson Approximations.

Chapter 5.  Integration.
An Area Problem; A Speed-Distance Problem.
The Definite Integral of a Continuous Function.
The Function F (X) = f f(t) dt.
The Fundamental Theorem of Integral Calculus.
Some Area Problems.
Indefinite Integrals.
Working Back from the Chain Rule; The u-Substitution.
Additional Properties of the Definite Integral.
Mean-Value Theorems for Integrals; Average Value of a Function.

Chapter 6.  Some Applications of the Integral.
More on Area.
Volume by Parallel Cross-Sections; Discs and Washers.
Volume by the Shell Method.
The Centroid of a Region; Pappus’s Theorem on Volumes.
The Notion of Work.
Fluid Force.

Chapter 7.  The Transcendental Functions.
One-to-One Functions’ Inverse Functions.
The Logarithm Function, Part I.
The Logarithm Function, Part II.
The Exponential Function.
Arbitrary Powers; Other Bases.
Exponential Growth and Decay.
The Arc Trigonometric Functions.
The Hyperbolic Sine and Cosine.
The Other Hyperbolic Functions.

Chapter 8.  Techniques of Integration.
Review; Integral Tables.
Integration by Parts.
Powers and Products of Trigonometric Functions.
Trigonometric Substitutions.
Rational Functions.
Some Rationalizing Substitutions.
Numerical Integration.

Chapter 9.  Differential Equations.
Introduction.
First Order Linear Equations.
Integral Curves; Separable Equations.
The Equation y + ay = 0.

Chapter 10.  The Conic Sections; Polar Coordinates; Parametric Equations.
Geometry of Parabola, Ellipse Hyperbola.
Polar Coordinates.
Graphing in Polar Coordinates.
Area in Polar Coordinates.
Curves Given Parametrically.
Tangents and Area.
Arc Length and Speed.
The Area of a Surface of Revolution; Pappus’s Theorem on Surface Area.

Chapter 11.  Sequences; Indeterminate Forms; Improper Integrals.
The Least Upper Bound Axiom.
Sequences of Real Numbers.
The Limit of a Sequence.
Some Important Limits.
The Indeterminate Forms (0/0).
The Indeterminate Form; Other Indeterminate Forms.
Improper Integrals.

Chapter 12.  Infinite Series.
Infinite Series.
The Integral Test; Comparison Tests.
The Roof Test; The Ratio Test.
Absolute and Conditional Convergence; Alternating Series.
Taylor Polynomials in x; Taylor Series in x.
Taylor Polynomials in x – a; Taylor Series in x – a.
Power Series.
Differentiation and Integration of Power Series.
The Binomial Series.

Chapter 13.  Vectors.
Cartesian Space Coordinates.
Displacements and Forces.
Vectors.
The Dot Product.
The Cross Product.
Lines.
Planes.

Chapter 14.  Vector Calculus.
Vector Functions.
The Calculus of Vector Functions.
Curves.
Arc Length.
Curvilinear Motion; Curvature.
Vector Calculus in Mechanics.
Planetary Motion.

Chapter 15.  Functions of Several Variables.
Elementary Examples.
A Brief Catalogue of Quadric Surfaces; Projections.
Graphs; Level Curves and Level Surfaces.
Partial Derivatives.
Open Sets and Closed Sets.
Limits and Continuity; Equality of Mixed Partials.

Chapter 16.  Gradients; Extreme Values; Differentials.
Differentiability and Gradient.
Gradients and Directional Derivatives.
Chain Rules.
The Gradient as a Normal; Tangent Lines and Tangent Planes.
Local Extreme Values.
Absolute Extreme Values.
Maxima and Minima with Side Conditions.
Differentials.
Reconstructing A Function From Its Gradient.

Chapter 17.  Multiple Integrals.
Multiple Sigma Notation.
The Double Integral.
The Evaluation of a Double Integral by Repeated Integrals.
Double Integrals in Polar Coordinates.
Some Applications of Double Integration.
Triple Integrals.
Reduction to Repeated Integrals.
Triple Integrals in Cylindrical Coordinates.
The Triple Integral as a Limit of Riemann Sums; Spherical Coordinates.
Jacobians; Changing Variables in Multiple Integration.

Chapter 18.  Line Integrals and Surface Integrals.
Line Integrals.
The Fundamental Theorem for Line Integrals.
Work-Energy Formula; Conservation of Mechanical Energy.
Line Integrals with Respect to Arc Length.
Green’s Theorem.
Parametrized Surfaces; Surface Area..
Surface Integrals.
The Vector Differential Operator.
The Divergence Theorem.
Stokes’ Theorem.

Chapter 19.  Elementary Differential Equations.