Preface.
To the Student.
Diagnostic Tests.
A Preview of Calculus.
1. FUNCTIONS AND MODELS.
Four Ways to Represent a Function. Mathematical Models: A Catalog
of Essential Functions. New Functions from Old Functions.
Exponential Functions. Inverse Functions and Logarithms. Review.
Principles of Problem Solving.
2. LIMITS AND DERIVATIVES.
The Tangent and Velocity Problems.The Limit of a
Function.Calculating Limits Using the Limit Laws. The Precise
Definition of a Limit. Continuity. Limits at Infinity; Horizontal
Asymptotes. Derivatives and Rates of Change. Writing Project: Early
Methods for Finding Tangents.The Derivative as a Function. Review.
Problems Plus.
3. DIFFERENTIATION RULES.
Derivatives of Polynomials and Exponential Functions. Applied
Project: Building a Better Roller Coaster. The Product and Quotient
Rules. Derivatives of Trigonometric Functions. The Chain Rule.
Applied Project: Where Should a Pilot Start Descent? Implicit
Differentiation. Laboratory Project: Families of Implicit Curves.
Derivatives of Logarithmic Functions. Rates of Change in the
Natural and Social Sciences. Exponential Growth and Decay. Applied
Project: Controlling Red Blood Cell Loss During Surgery. Related
Rates. Linear Approximations and Differentials. Laboratory Project:
Taylor Polynomials. Hyperbolic Functions. Review. Problems
Plus.
4. APPLICATIONS OF DIFFERENTIATION.
Maximum and Minimum Values. Applied Project: The Calculus of
Rainbows. The Mean Value Theorem. How Derivatives Affect the Shape
of a Graph. Indeterminate Forms and l’Hospital’s Rule. Writing
Project: The Origins of l’Hospital’s Rule. Summary of Curve
Sketching. Graphing with Calculus and Calculators. Optimization
Problems. Applied Project: The Shape of a Can. Applied Project:
Planes and Birds: Minimizing Energy. Newton’s Method.
Antiderivatives. Review. Problems Plus.
5. INTEGRALS.
Areas and Distances. The Definite Integral. Discovery Project: Area
Functions. The Fundamental Theorem of Calculus. Indefinite
Integrals and the Net Change Theorem. Writing Project: Newton,
Leibniz, and the Invention of Calculus. The Substitution Rule.
Review. Problems Plus.
6. APPLICATIONS OF INTEGRATION.
Areas Between Curves. Applied Project: The Gini Index. Volume.
Volumes by Cylindrical Shells. Work. Average Value of a Function.
Applied Project: Calculus and Baseball. Applied Project: Where to
Sit at the Movies. Review. Problems Plus.
7. TECHNIQUES OF INTEGRATION.
Integration by Parts. Trigonometric Integrals. Trigonometric
Substitution. Integration of Rational Functions by Partial
Fractions. Strategy for Integration. Integration Using Tables and
Computer Algebra Systems. Discovery Project: Patterns in Integrals.
Approximate Integration. Improper Integrals. Review. Problems
Plus.
8. FURTHER APPLICATIONS OF INTEGRATION.
Arc Length. Discovery Project: Arc Length Contest. Area of a
Surface of Revolution. Discovery Project: Rotating on a Slant.
Applications to Physics and Engineering. Discovery Project:
Complementary Coffee Cups. Applications to Economics and Biology.
Probability. Review. Problems Plus.
9. DIFFERENTIAL EQUATIONS.
Modeling with Differential Equations. Direction Fields and Euler’s
Method. Separable Equations. Applied Project: How Fast Does a Tank
Drain? Applied Project: Which is Faster, Going Up or Coming Down?
Models for Population Growth. Linear Equations. Predator-Prey
Systems. Review. Problems Plus.
10. PARAMETRIC EQUATIONS AND POLAR COORDINATES.
Curves Defined by Parametric Equations. Laboratory Project:
Families of Hypocycloids. Calculus with Parametric Curves.
Laboratory Project: Bézier Curves. Polar Coordinates. Laboratory
Project: Families of Polar Curves. Areas and Lengths in Polar
Coordinates. Conic Sections. Conic Sections in Polar Coordinates.
Review. Problems Plus.
11. INFINITE SEQUENCES AND SERIES.
Sequences. Laboratory Project: Logistic Sequences. Series. The
Integral Test and Estimates of Sums. The Comparison Tests.
Alternating Series. Absolute Convergence and the Ratio and Root
Tests. Strategy for Testing Series. Power Series. Representations
of Functions as Power Series. Taylor and Maclaurin Series.
Laboratory Project: An Elusive Limit. Writing Project: How Newton
Discovered the Binomial Series. Applications of Taylor Polynomials.
Applied Project: Radiation from the Stars. Review. Problems
Plus.
12. VECTORS AND THE GEOMETRY OF SPACE.
Three-Dimensional Coordinate Systems. Vectors. The Dot Product. The
Cross Product. Discovery Project: The Geometry of a Tetrahedron.
Equations of Lines and Planes. Cylinders and Quadric Surfaces.
Review. Problems Plus.
13. VECTOR FUNCTIONS.
Vector Functions and Space Curves. Derivatives and Integrals of
Vector Functions. Arc Length and Curvature. Motion in Space:
Velocity and Acceleration. Applied Project: Kepler’s Laws. Review.
Problems Plus.
14. PARTIAL DERIVATIVES.
Functions of Several Variables. Limits and Continuity. Partial
Derivatives. Tangent Planes and Linear Approximation. Applied
Project: The Speedo LZR Race Suit. The Chain Rule. Directional
Derivatives and the Gradient Vector. Maximum and Minimum Values.
Applied Project: Designing a Dumpster. Discovery Project: Quadratic
Approximations and Critical Points. Lagrange Multipliers. Applied
Project: Rocket Science. Applied Project: Hydro-Turbine
Optimization. Review. Problems Plus.
15. MULTIPLE INTEGRALS.
Double Integrals over Rectangles. Double Integrals over General
Regions. Double Integrals in Polar Coordinates. Applications of
Double Integrals. Surface Area. Triple Integrals. Discovery
Project: Volumes of Hyperspheres. Triple Integrals in Cylindrical
Coordinates. Discovery Project: The Intersection of Three
Cylinders. Triple Integrals in Spherical Coordinates. Applied
Project: Roller Derby. Change of Variables in Multiple Integrals.
Review. Problems Plus.
16. VECTOR CALCULUS.
Vector Fields. Line Integrals. The Fundamental Theorem for Line
Integrals. Green’s Theorem. Curl and Divergence. Parametric
Surfaces and Their Areas. Surface Integrals. Stokes’ Theorem.
Writing Project: Three Men and Two Theorems. The Divergence
Theorem. Summary. Review. Problems Plus.
17. SECOND-ORDER DIFFERENTIAL EQUATIONS.
Second-Order Linear Equations. Nonhomogeneous Linear Equations.
Applications of Second-Order Differential Equations. Series
Solutions. Review. Problems Plus.
APPENDIXES.
A Numbers, Inequalities, and Absolute Values. B Coordinate Geometry
and Lines. C Graphs of Second-Degree Equations. D Trigonometry. E
Sigma Notation. F Proofs of Theorems. G The Logarithm Defined as an
Integral. H Complex Numbers. I Answers to Odd-Numbered
Exercises.
INDEX.
The late James Stewart received his M.S. from Stanford University and his Ph.D. from the University of Toronto. He conducted research at the University of London and was influenced by the famous mathematician George Polya at Stanford University. Dr. Stewart most recently served as a professor of mathematics at McMaster University, and his research focused on harmonic analysis. Dr. Stewart authored a best-selling calculus textbook series, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS and CALCULUS: CONCEPTS AND CONTEXTS as well as a series of successful precalculus texts.
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