Calculus and Analytic Geometry

by Thomas, George B.; Finney, Ross L.

• ISBN13: 9780201531749
• ISBN10: 0201531747
• Edition: 9th
• Format: Hardcover
• Copyright: 1996-01-01
• Publisher: Addison Wesley

Summary

A calculus textbook containing exercises and problem solving strategies to help the student better grasp the different techniques offered.

Table of Contents

To the Instructor	viii
To the Student	xvii
Preliminaries
Real Numbers and the Real Line	1	(7)
Coordinates, Lines, and Increments	8	(9)
Functions	17	(10)
Shifting Graphs	27	(8)
Trigonometric Functions	35	(16)
Questions to Guide Your Review	47	(1)
Practice Exercises	48	(1)
Additional Exercises---Theory, Examples, Applications	49	(2)
Limits and Continuity
Rates of Change and Limits	51	(10)
Rules for Finding Limits	61	(5)
Target Values and Formal Definitions of Limits	66	(12)
Extensions of the Limit Concept	78	(9)
Continuity	87	(10)
Tangent Lines	97	(12)
Questions to Guide Your Review	103	(1)
Practice Exercises	104	(1)
Additional Exercises---Theory, Examples, Applications	105	(4)
Derivatives
The Derivative of a Function	109	(12)
Differentiation Rules	121	(10)
Rates of Change	131	(12)
Derivatives of Trigonometric Functions	143	(11)
The Chain Rule	154	(10)
Implicit Differentiation and Rational Exponents	164	(8)
Related Rates of Change	172	(17)
Questions to Guide your Review	180	(1)
Practice Exercises	181	(4)
Additional Exercises---Theory, Examples, Applications	185	(4)
Applications of Derivatives
Extreme Values of Functions	189	(7)
The Mean Value Theorem	196	(9)
The First Derivative Test for Local Extreme Values	205	(4)
Graphing with y' and y''	209	(11)
Limits as x → ± ∞, Asymptotes, and Dominant Terms	220	(13)
Optimization	233	(15)
Linearization and Differentials	248	(12)
Newton's Method	260	(15)
Questions to Guide Your Review	268	(1)
Practice Exercises	269	(3)
Additional Exercises---Theory, Examples, Applications	272	(3)
Integration
Indefinite Integrals	275	(7)
Differential Equations, Initial Value Problems, and Mathematical Modeling	282	(8)
Integration by Substitution---Running the Chain Rule Backward	290	(8)
Estimating with Finite Sums	298	(11)
Riemann Sums and Definite Integrals	309	(14)
Properties, Area, and the Mean Value Theorem	323	(9)
The Fundamental Theorem	332	(10)
Substitution in Definite Integrals	342	(4)
Numerical Integration	346	(19)
Questions to Guide Your Review	356	(1)
Practice Exercises	357	(3)
Additional Exercises---Theory, Examples, Applications	360	(5)
Application of Integrals
Areas Between Curves	365	(9)
Finding Volumes by Slicing	374	(5)
Volumes of Solids of Revolution---Disks and Washers	379	(8)
Cylindrical Shells	387	(6)
Lengths of Plane Curves	393	(7)
Areas of Surfaces of Revolution	400	(7)
Moments and Centers of Mass	407	(11)
Work	418	(9)
Fluid Pressures and Forces	427	(7)
The Basic Pattern and Other Modeling Applications	434	(15)
Questions to Guide Your Review	444	(1)
Practice Exercises	444	(3)
Additional Exercises---Theory, Examples, Applications	447	(2)
Transcendental Functions
Inverse Functions and Their Derivatives	449	(9)
Natural Logarithms	458	(9)
The Exponential Function	467	(7)
ax and logax	474	(8)
Growth and Decay	482	(9)
L'Hopital's Rule	491	(7)
Relative Rates of Growth	498	(6)
Inverse Trigonometric Functions	504	(9)
Derivatives of Inverse Trigonometric Functions; Integrals	513	(7)
Hyperbolic Functions	520	(9)
First Order Differential Equations	529	(12)
Euler's Numerical Method; Slope Fields	541	(14)
Questions to Guide Your Review	547	(1)
Practice Exercises	548	(3)
Additional Exercises---Theory, Examples, Applications	551	(4)
Techniques of Integration
Basic Integration Formulas	555	(7)
Integration by Parts	562	(7)
Partial Fractions	569	(9)
Trigonometric Substitutions	578	(5)
Integral Tables and CAS	583	(11)
Improper Integrals	594	(19)
Questions to Guide Your Review	606	(1)
Practice Exercises	606	(3)
Additional Exercises---Theory, Examples, Applications	609	(4)
Infinite Series
Limits of Sequences of Numbers	613	(9)
Theorems for Calculating Limits of Sequences	622	(8)
Infinite Series	630	(10)
The Integral Test for Series of Nonnegative Terms	640	(4)
Comparison Tests for Series of Nonnegative Terms	644	(5)
The Ratio and Root Tests for Series of Nonnegative Terms	649	(6)
Alternating Series, Absolute and Conditional Convergence	655	(8)
Power Series	663	(9)
Taylor and Maclaurin Series	672	(6)
Convergence of Taylor Series; Error Estimates	678	(10)
Applications of Power Series	688	(21)
Questions to Guide Your Review	699	(1)
Practice Exercises	700	(3)
Additional Exercises---Theory, Examples, Applications	703	(6)
Conic Sections, Parametrized Curves, and Polar Coordinates
Conic Sections and Quadratic Equations	709	(14)
Classifying Conic Sections by Eccentricity	723	(5)
Quadratic Equations and Rotations	728	(6)
Parametrizations of Plane Curves	734	(10)
Calculus with Parametrized Curves	744	(7)
Polar Coordinates	751	(5)
Graphing in Polar Coordinates	756	(8)
Polar Equations for Conic Sections	764	(6)
Integration in Polar Coordinates	770	(17)
Questions to Guide Your Review	777	(1)
Practice Exercises	778	(5)
Additional Exercises---Theory, Examples, Applications	783	(4)
Vectors and Analytic Geometry in Space
Vectors in the Plane	787	(8)
Cartesian (Rectangular) Coordinates and Vectors in Space	795	(11)
Dot Products	806	(9)
Cross Products	815	(7)
Lines and Planes in Space	822	(7)
Cylinders and Quadric Surfaces	829	(12)
Cylindrical and Spherical Coordinates	841	(14)
Questions to Guide Your Review	847	(1)
Practice Exercises	848	(3)
Additional Exercises---Theory, Examples, Applications	851	(4)
Vector-Valued Functions and Motion in Space
Vector-Valued Functions and Space Curves	855	(13)
Modeling Projectile Motion	868	(8)
Arc Length and the Unit Tangent Vector T	876	(5)
Curvature, Torsion, and the TNB Frame	881	(12)
Planetary Motion and Satellites	893	(16)
Questions to Guide Your Review	902	(1)
Practice Exercises	902	(3)
Additional Exercises---Theory, Examples, Applications	905	(4)
Multivariable Functions and Partial Derivatives
Functions of Several Variables	909	(8)
Limits and Continuity	917	(7)
Partial Derivatives	924	(9)
Differentiability, Linearization, and Differentials	933	(11)
The Chain Rule	944	(8)
Partial Derivatives with Constrained Variables	952	(5)
Directional Derivatives, Gradient Vectors, and Tangent Planes	957	(13)
Extreme Values and Saddle Points	970	(10)
Lagrange Multipliers	980	(9)
Taylor's Formula	989	(12)
Questions to Guide Your Review	993	(1)
Practice Exercises	994	(4)
Additional Exercises---Theory, Examples, Applications	998	(3)
Multiple Integrals
Double Integrals	1001	(11)
Areas, Moments, and Centers of Mass	1012	(8)
Double Integrals in Polar Form	1020	(6)
Triple Integrals in Rectangular Coordinates	1026	(8)
Masses and Moments in Three Dimensions	1034	(5)
Triple Integrals in Cylindrical and Spherical Coordinates	1039	(9)
Substitutions in Multiple Integrals	1048	(13)
Questions to Guide Your Review	1055	(1)
Practice Exercises	1056	(2)
Additional Exercises---Theory, Examples, Applications	1058	(3)
Integration in Vector Fields
Line Integrals	1061	(6)
Vector Fields, Work, Circulation, and Flux	1067	(9)
Path Independence, Potential Functions, and Conservative Fields	1076	(8)
Green's Theorem in the Plane	1084	(12)
Surface Area and Surface Integrals	1096	(10)
Parametrized Surfaces	1106	(8)
Stokes's Theorem	1114	(9)
The Divergence Theorem and a Unified Theory	1123
Questions to Guide Your Review	1134	(1)
Practice Exercises	1134	(3)
Additional Exercises---Theory, Examples, Applications	1137
Appendices
A.1 Mathematical Induction	A-1
A.2 Proofs of Limit Theorems in Section 1.2	A-4
A.3 Complex Numbers	A-7
A.4 Simpson's One-Third Rule	A-17
A.5 Cauchy's Mean Value Theorem and the Stronger Form of l'Hopital's Rule	A-18
A.6 Limits That Arise Frequently	A-20
A.7 The Distributive Law for Vector Cross Products	A-21
A.8 Determinants and Cramer's Rule	A-22
A.9 Euler's Theorem and the Increment Theorem	A-29
Answers	A-35
Index	I-1
A Brief Table of Integrals	T-1