Calculus Early Vectors

by Stewart, James

• ISBN13: 9780534493486
• ISBN10: 0534493483
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-05-09
• Publisher: Brooks Cole

Summary

REVIEW AND PREVIEW. Functions and Their Graphs. Types of Functions. Shifting and Scaling. Graphing Calculators and Computers. Principles of Problem Solving. A Preview of Calculus. 1. INTRODUCTION TO VECTORS AND VECTOR FUNCTIONS. Vectors. The Dot Product. Vector Functions. Review. 2. LIMITS AND RATES OF CHANGE. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Law. The Precise Definition of a Limit. Continuity. Limits at Infinity. Horizontal Asymptotes. Tangents, Velocities, and Other Rates of Change. Review. 3. DERIVATIVES. Derivatives. Differentiation Formulas. Rates of Change in the Natural and Social Sciences. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation. Derivatives of Vector Functions. Higher Derivatives. Slopes and Tangents of Parametric Curves. Related Rates. Differentials; Linear and Quadratic Approximations. Newton''s Method. Review. Problems Plus. 4. INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS. Exponential Functions and Their Derivatives. Inverse Functions. Logarithmic Functions. Derivatives of Logarithmic Functions. Exponential Growth and Decay. Inverse Trigonometric Functions. Hyperbolic Functions. Indeterminate Forms and L''Hospital''s Rule. Review. Applications Plus. 5. APPLICATIONS OF DIFFERENTIATION. What does f Say about f?. Maximum and Minimum Values. Derivatives and the Shapes of Curves. Graphing with Calculus and Calculators. Applied Maximum and Minimum Problems. Applications to Economics. Antiderivatives. Review. Problems Plus. 6. INTEGRALS. Sigma Notation. Area. The Definite Integral. The Fundamental Theorem of Calculus. The Substitution Rule. The Logarithm Defined as an Integral. Review. Applications Plus. 7. APPLICATIONS OF INTEGRATION. Areas between Curves. Volume. Volumes by Cylindrical Shells. Work. Average Value of a Function. Review. Problems Plus. 8. TECHNIQUES OF INTEGRATION. Integration by Parts. Trigonometric Integrals. Trigonometric Substitution. Integration of Rational Functions by Partial Fractions. Rationalizing Substitutions. Strategy for Integration. Using Tables of Integrals and Computer Algebra Systems. Approximate Integration. Improper Integrals. Review. Applications Plus. 9. FURTHER APPLICATIONS OF INTEGRATION. Differential Equations. First-Order Linear Equations. Arc Length. Area of Surface of Revolution. Moments and Centers of Mass. Hydrostatic Pressure and Force. Applications to Economics and Biology. Review. Problems Plus. 10. INFINITE SEQUENCES AND SERIES. Sequences. Series. The Integral Test and Comparison Tests. Other Convergence Tests. Power Series. Representation of Functions as Power Series. Taylor and Maclaurin Series. The Binomial Series. Writing Project: How Newton Discovered the Binomial Series. Applications of Taylor Polynomials. Review. Applications Plus. 11. THREE-DIMENSIONAL ANALYTIC GEOMETRY AND VECTORS. Three-Dimensional Coordinate Systems. Vectors and the Dot Product in Three Dimension. The Cross Product. Equations of Lines and Planes. Quadric Surfaces. Vector Functions and Space Curves. Arc Length and Curvature. Motion in Space. Review. Applications Plus. 12. PARTIAL DERIVATIVES. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Tangent Planes and Differentials. The Chain Rule. Directional Derivatives and the Gradient Vector. Maximum and Minimum Values. Lagrange Multipliers. Review. Problems Plus. 13. MULTIPLE INTEGRALS. Double Integrals over Rectangles. Iterated Integrals. Double Integrals over General Regions. Polar Coordinates. Applications of Double Integrals. Surface Area. Triple Integrals. Cylindrical and Spherical Coordinates. Triple Integrals in Cylindrical and Spherical Coordinates. Change of Variables in Multiple Integrals. Review. Applications Plus. 14. VECTOR CALCULUS. Vector Fields. Line Integrals. The Fundamental Theorem for Line Integrals. Green''s Theorem. Curl and Divergence. Parametric Surfaces and their Are

Table of Contents

REVIEW AND PREVIEW	2	(46)
1 Functions and Their Graphs	2	(16)
2 Types of Functions; Shifting and Scaling	18	(9)
3 Graphing Calculators and Computers	27	(7)
4 Principles of Problem Solving	34	(7)
5 A Preview of Calculus	41	(7)
1 INTRODUCTION TO VECTORS AND VECTOR FUNCTIONS	48	(24)
1.1 Vectors	48	(6)
1.2 The Dot Product	54	(8)
1.3 Vector Functions	62	(8)
Review	70	(2)
2 LIMITS AND RATES OF CHANGE	72	(76)
2.1 The Tangent and Velocity Problems	72	(7)
2.2 The Limit of a Function	79	(12)
2.3 Calculating Limits Using the Limit Laws	91	(10)
2.4 The Precise Definition of a Limit	101	(11)
2.5 Continuity	112	(10)
2.6 Limits at Infinity; Horizontal Asymptotes	122	(12)
2.7 Tangents, Velocities, and Other Rates of Change	134	(11)
Review	145	(3)
3 DERIVATIVES	148	(94)
3.1 Derivatives	148	(12)
3.2 Differentiation Formulas	160	(10)
3.3 Rates of Change in the Natural and Social Sciences	170	(9)
3.4 Derivatives of Trigonometric Functions	179	(7)
3.5 The Chain Rule	186	(8)
3.6 Implicit Differentiation	194	(6)
3.7 Derivatives of Vector Functions	200	(3)
3.8 Higher Derivatives	203	(7)
3.9 Slopes and Tangents of Parametric Curves	210	(5)
3.10 Related Rates	215	(6)
3.11 Differentials; Linear and Quadratic Approximations	221	(8)
3.12 Newton's Method	229	(5)
Review	234	(4)
PROBLEMS PLUS	238	(4)
4 INVERSE FUNCTIONS: Exponential, Logarithmic, and Inverse Trigonometric Functions	242	(60)
4.1 Exponential Functions and Their Derivatives	242	(8)
4.2 Inverse Functions	250	(7)
4.3 Logarithmic Functions	257	(6)
4.4 Derivatives of Logarithmic Functions	263	(7)
4.5 Exponential Growth and Decay	270	(6)
4.6 Inverse Trigonometric Functions	276	(7)
4.7 Hyperbolic Functions	283	(6)
4.8 Indeterminate Forms and L'Hospital's Rule	289	(7)
Review	296	(4)
APPLICATIONS PLUS	300	(2)
5 APPLICATIONS OF DIFFERENTIATION	302	(62)
5.1 What Does f' Say About f?	302	(5)
5.2 Maximum and Minimum Values	307	(7)
5.3 Derivatives and the Shapes of Curves	314	(10)
5.4 Graphing with Calculus and Calculators	324	(7)
5.5 Applied Maximum and Minimum Problems	331	(10)
5.6 Applications to Economics	341	(4)
5.7 Antiderivatives	345	(11)
Review	356	(4)
PROBLEMS PLUS	360	(4)
6 INTEGRALS	364	(58)
6.1 Sigma Notation	364	(6)
6.2 Area	370	(8)
6.3 The Definite Integral	378	(11)
6.4 The Fundamental Theorem of Calculus	389	(12)
6.5 The Substitution Rule	401	(7)
6.6 The Logarithm Defined as an Integral	408	(8)
Review	416	(3)
APPLICATIONS PLUS	419	(3)
7 APPLICATIONS OF INTEGRATION	422	(36)
7.1 Areas Between Curves	422	(7)
7.2 Volume	429	(11)
7.3 Volumes by Cylindrical Shells	440	(5)
7.4 Work	445	(4)
7.5 Average Value of a Function	449	(3)
Review	452	(2)
PROBLEMS PLUS	454	(4)
8 TECHNIQUES OF INTEGRATION	458	(68)
8.1 Integration by Parts	459	(6)
8.2 Trigonometric Integrals	465	(6)
8.3 Trigonometric Substitution	471	(6)
8.4 Integration of Rational Functions by Partial Fractions	477	(9)
8.5 Rationalizing Substitutions	486	(3)
8.6 Strategy for Integration	489	(6)
8.7 Using Tables of Integrals and Computer Algebra Systems	495	(4)
8.8 Approximate Integration	499	(10)
8.9 Improper Integrals	509	(9)
Review	518	(3)
APPLICATIONS PLUS	521	(5)
9 FURTHER APPLICATiONS OF INTEGRATION	526	(50)
9.1 Differential Equations	526	(10)
9.2 First-Order Linear Equations	536	(5)
9.3 Arc Length	541	(7)
9.4 Area of a Surface of Revolution	548	(6)
9.5 Moments and Centers of Mass	554	(7)
9.6 Hydrostatic Pressure and Force	561	(3)
9.7 Applications to Economics and Biology	564	(6)
Review	570	(2)
PROBLEMS PLUS	572	(4)
10 INFINITE SEQUENCES AND SERIES	576	(76)
10.1 Sequences	576	(10)
10.2 Series	586	(9)
10.3 The Integral Test and Comparison Tests; Estimating Sums	595	(10)
10.4 Other Convergence Tests	605	(7)
10.5 Power Series	612	(6)
10.6 Representation of Functions as Power Series	618	(5)
10.7 Taylor and Maclaurin Series	623	(11)
10.8 The Binomial Series	634	(3)
10.9 Applications of Taylor Polynomials	637	(9)
Review	646	(2)
PROBLEMS PLUS	648	(4)
11 THREE-DIMENSIONAL ANALYTIC GEOMETRY AND VECTORS	652	(70)
11.1 Three-Dimensional Coordinate Systems	652	(5)
11.2 Vectors and the Dot Product in Three Dimensions	657	(9)
11.3 The Cross Product	666	(8)
11.4 Equations of Lines and Planes	674	(9)
11.5 Quadric Surfaces	683	(6)
11.6 Vector Functions and Space Curves	689	(9)
11.7 Arc Length and Curvature	698	(8)
11.8 Motion in Space: Velocity and Acceleration	706	(8)
Review	714	(4)
APPLICATIONS PLUS	718	(4)
12 PARTIAL DERIVATIVES	722	(72)
12.1 Functions of Several Variables	722	(10)
12.2 Limits and Continuity	732	(8)
12.3 Partial Derivatives	740	(9)
12.4 Tangent Planes and Differentials	749	(7)
12.5 The Chain Rule	756	(8)
12.6 Directional Derivatives and the Gradient Vector	764	(10)
12.7 Maximum and Minimum Values	774	(9)
12.8 Lagrange Multipliers	783	(6)
Review	789	(3)
PROBLEMS PLUS	792	(2)
13 MULTIPLE INTEGRALS	794	(74)
13.1 Double Integrals over Rectangles	794	(5)
13.2 Iterated Integrals	799	(6)
13.3 Double Integrals over General Regions	805	(8)
13.4 Polar Coordinates	813	(8)
13.5 Double Integrals in Polar Coordinates	821	(6)
13.6 Applications of Double Integrals	827	(6)
13.7 Surface Area	833	(2)
13.8 Triple Integrals	835	(9)
13.9 Cylindrical and Spherical Coordinates	844	(5)
13.10 Triple Integrals in Cylindrical and Spherical Coordinates	849	(6)
13.11 Change of Variables in Multiple Integrals	855	(7)
Review	862	(4)
APPLICATIONS PLUS	866	(2)
14 VECTOR CALCULUS	868	(76)
14.1 Vector Fields	868	(4)
14.2 Line Integrals	872	(11)
14.3 The Fundamental Theorem for Line Integrals	883	(9)
14.4 Green's Theorem	892	(7)
14.5 Curl and Divergence	899	(8)
14.6 Parametric Surfaces and Their Areas	907	(7)
14.7 Surface Integrals	914	(12)
14.8 Stokes' Theorem	926	(6)
14.9 The Divergence Theorem	932	(5)
14.10 Summary	937	(1)
Review	938	(3)
PROBLEMS PLUS	941	(3)
15 SECOND-ORDER DIFFERENTIAL EQUATIONS	944
15.1 Second-Order Linear Equations	944	(7)
15.2 Nonhomogeneous Linear Equations	951	(6)
15.3 Applications of Second-Order Differential Equations	957	(7)
15.4 Using Series to Solve Differential Equations	964	(5)
Review	969
APPENDIXES	A1
A Numbers, Inequalities, and Absolute Values	A2
B Coordinate Geometry and Lines	A11
C Graphs of Second-Degree Equations	A17
D Trigonometry	A23
E Mathematical Induction	A32
F Proofs of Theorems	A34
G Lies My Calculator and Computer Told Me	A47
H Complex Numbers	A51
I Conic Sections	A58
J Conic Sections in Polar Coordinates	A66
K Table of Integrals	A70
L Answers to Odd-Numbered Exercises	A76
INDEX	A125