Calculus: One and several variables with analytic geometry
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Wiley
1978
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| ISBN: | 0471032859 |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
Chapter 1 Introduction 1
1.1 What is Calculus? 1
1.2 Notions and Formulas from Elementary Mathematics 5
1.3 Some Problems on Lines 12
1.4 Inequalities 17
1.5 Inequalities and Absolute Value 19
1.6 Functions 23
1.7 The Composition of Functions 28
1.8 One-to-one Functions; Inverses 33
Chapter 2 Limits and Continuity 39
2.1 The Idea of Limit 39
2.2 Definition of Limit 45
2.3 Some Limit Theorems 56
2.4 More on Limits 64
2.5 One-Sided Limits 68
2.6 Continuity 73
2.7 Additional Exercises 83
Chapter 3 Differentiation 87
3.1 The Derivative 87
3.2 Some Differentiation Formulas 95
3.3 The d/dx Notation 102
3.4 The Derivative as a Rate of Change 107
3.5 The Chain Rule 110
3.6 Derivatives of Higher Order 118
3.7 Differentiating Inverses; Fractional Exponents 120
ix
Contents
3.8 Additional Practice in Differentiation 125
3.9 Tangent Lines and Normal Lines 125
3.10 Implicit Differentiation; The Angle between Two Curves 129
Chapter 4 The Mean-Value Theorem and Applications 135
4.1 The Mean-Value Theorem 135
4.2 Increasing and Decreasing Functions 139
4.3 Maxima and Minima 146
4.4 More on Maxima and Minima 153
4.5 Some Max-Min Problems 158
4.6 Concavity and Points of Inflection 165
4.7 Some Curve Sketching 167
4.8 Rates of Change Per Unit Time 171
4.9 The Little-o(h) Idea; Differentials 177
4.10 Additional Exercises 183
* Chapter 5 Integration 187
5.1 Motivation: An Area Problem; A Speed-Distance Problem 187
5.2 The Definite Integral of a Continuous Function 191
5.3 The Function F(x) = ( fit) dt 197
J a
5.4 The Fundamental Theorem of Integral Calculus 204
5.5 Some Area Problems; Some Problems of Motion 208
5.6 The Linearity of the Integral 211
5.7 Indefinite Integrals 213
5.8 The w-Substitution; Change of Variables 215
5.9 Some Further Properties of the Definite Integral 218
5.10 More on Area 222
5.11 Additional Exercises 227
5.12 Riemann Sums 229
Chapter 6 The Logarithm and Exponential Functions 233
6.1 In Search of a Notion of Logarithm 233
6.2 The Logarithm Function, Part I 235
6.3 The Logarithm Function, Part II 242
6.4 The Exponential Function 249
6.5 Arbitrary Powers; Other Bases; Estimating e 258
6.6 Exponential Growth and Decline 268
6.7 Integration by Parts 274
6.8 (Optional) The Equation y x) + P(x)y(x) = Q(x) 279
6.9 Additional Exercises 283
Chapter 7 The Trigonometric and Hyperbolic Functions 287
7.1 Differentiating the Trigonometric Functions 287
7.2 Integrating the Trigonometric Functions 299
Contents
7.3 The Inverse Trigonometric Functions 304
7.4 Additional Exercises 311
7.5 Simple Harmonic Motion 312
7.6 (Optional) The Hyperbolic Sine and Cosine 318
7.7 (Optional) Other Hyperbolic Functions 323
Chapter 8 The Technique of Integration 329
8.1 A Short Table of Integrals; Review 329
8.2 Partial Fractions 333
8.3 Powers and Products of Sines and Cosines 342
8.4 Other Trigonometric Powers 345
8.5 Integrals Involving Va2 ± x2 and V*2 ± a2 349
8.6 Rationalizing Expressions in sin x and cos x 351
8.7 Some Rationalizing Substitutions 354
8.8 Approximate Integration 355
8.9 Additional Exercises 361
Chapter 9 The Conic Sections 365
9.1 Introduction 365
9.2 The Distance Between a Point and a Line; Translations 366
9.3 The Parabola 369
9.4 The Ellipse 377
9.5 The Hyperbola 384
9.6 Additional Exercises 389
9.7 Rotations; Eliminating the xy-Term 389
Chapter 10 Volume, Work, and other Applications of the Integral 395
10.1 The Average Value of a Continuous Function 395
10.2 Volume: Parallel Cross Sections 399
10.3 Volume: The Shell Method 409
10.4 The Notion of Work 414
10.5 Fluid Pressure 419
10.6 Revenue Streams 423
10.7 Additional Exercises 426
Chapter 11 Polar Coordinates; Parametric Equations 429
11.1 Polar Coordinates 429
11.2 Graphing in Polar Coordinates 436
11.3 The Intersection of Polar Curves 441
11.4 Area in Polar Coordinates 442
11.5 Curves Given Parametrically 446
11.6 (Optional) The Cycloid; A Note on Area 454
11.7 The Least Upper Bound Axiom; Arc Length 456
11.8 Area of a Surface Revolution 465
11.9 Some Additional Exercises 471
Contents
Chapter 12 Sequences; Indeterminate Forms; Improper Integrals 473
12.1 Sequences of Real Numbers 473
12.2 The Limit of a Sequence 478
12.3 Some Important Limits 488
12.4 Limits asx^±=« 493
12.5 The Indeterminate Form (0/0) 496
12.6 Infinite Limits; The Indeterminate Form (cc/cc) 501
12.7 Improper Integrals 507
Chapter 13 Infinite Series 515
13.1 Sigma Notation 515
13.2 Infinite Series 518
13.3 The Integral Test; Comparison Theorems 525
13.4 The Root Test; The Ratio Test 531
13.5 Absolute and Conditional Convergence; Alternating Series 535
13.6 Taylor Polynomials in x Taylor Series in Powers of x 540
13.7 Taylor Polynomials in x - a; Taylor Series in Powers of x — a 549
13.8 The Logarithm and the Arc Tangent; Computing tt
13.9 Power Series, Part I 556
13.10 Power Series, Part II 562
13.11 Problems on the Binomial Series 570
Chapter 14 Vectors 575
14.1 Cartesian Space Coordinates 575
14.2 Displacements and Forces 577
14.3 Vectors 581
14.4 The Dot Product 589
14.5 Lines 596
14.6 Planes 603
14.7 The Cross Product 609
14.8 Some Geometry by Vector Methods 616
14.9 Additional Exercises 618
Chapter 15 Vector Calculus 621
15.1 Vector Functions 621
15.2 Differentiation Formulas 626
15.3 Curves and Tangents 631
15.4 Velocity and Acceleration 636
15.5 Arc Length and Speed 644
15.6 (Optional) The Curvature of a Plane Curve 651
Chapter 16 Functions of Several Variables 659
16.1 What Are They? 659
16.2 A Brief Catalog of the Quadric Surfaces 663
16.3 Graphs; Level Curves and Level Surfaces 669
16.4 Partial Derivatives 675
Contents
16.5 Open Sets and Closed Sets 682
16.6 Limits and Continuity; Equality of Mixed Partials 685
Chapter 17 Gradients; Extreme Values; Differentials 693
17.1 Differentiability and Gradient 693
17.2 Some Simple Properties of Gradients 699
17.3 The Mean-Value Theorem and Chain Rule 709
17.4 The Gradient as a Normal; Tangent Lines and Tangent Planes 718
17.5 Maximum and Minimum Values 726
17.6 Second-Partials Test 732
17.7 Maxima and Minima with Side Conditions 737
17.8 Differentials 744
17.9 Reconstructing a Function from its Gradient 749
Chapter 18 Double and Triple Integrals 757
18.1 Multiple-Sigma Notation 757
18.2 The Double Integral over a Rectangle 760
18.3 The Double Integral over More General Regions 770
18.4 The Evaluation of Double Integrals by Repeated Integrals 772
18.5 Double Integrals in Polar Coordinates 782
18.6 Triple Integrals 791
18.7 Reduction to Repeated Integrals 794
18.8 Averages and Centroids 801
18.9 Integration in Cylindrical Coordinates 810
18.10 Integration in Spherical Coordinates 816
Chapter 19 Line Integrals and Surface Integrals 821
19.1 Work and Line Integrals 821
19.2 The Fundamental Theorem for Line Integrals 828
19.3 Green s Theorem 834
19.4 Multiple Riemann Sums; Surface Area 837
19.5 The Mass of a Material Surface; Surface Integrals 844
19.6 The Divergence Theorem 851
19.7 Stokes s Theorem 855
Appendix A. Some Elementary Topics 863
A.I Sets 863
A.2 Induction 867
Appendix B. Some Additional Proofs 871
B.I The Intermediate-Value Theorem 871
B.2 The Maximum-Minimum Theorem 872
B.3 Inverses 873
B.4 The Integrability of Continuous Functions 874
Contents
B.5 The Integral as the Limit of Riemann Sums 878
B.6 A Variant of Bliss s Theorem 878
Appendix C. Tables 881
Table 1 Natural Logs 881
Table 2 Exponentials (0.01 to 0.99) 882
Table 3 Exponentials (1.0 to 4.9) 883
Table 4 Sines, Cosines, Tangents (Radian Measure) 884
Table 5 Sines, Cosines, Tangents (Degree Measure) 886
Table 6 Integrals 887
Answers to Starred Exercises A1
Index 11
|
| any_adam_object | 1 |
| author | Salas, Saturnino L. 1930-2012 Hille, Einar 1894-1980 |
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| discipline | Mathematik |
| edition | 3. ed. |
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| spelling | Salas, Saturnino L. 1930-2012 Verfasser (DE-588)1058200674 aut Calculus One and several variables with analytic geometry Saturnino L. Salas ; Einar Hille* 3. ed. New York Wiley 1978 txt rdacontent n rdamedia nc rdacarrier Calculus Analytische Geometrie (DE-588)4001867-2 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Infinitesimalrechnung (DE-588)4072798-1 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Analytische Geometrie (DE-588)4001867-2 s 1\p DE-604 Infinitesimalrechnung (DE-588)4072798-1 s 2\p DE-604 Hille, Einar 1894-1980 Verfasser (DE-588)11909116X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003311724&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Salas, Saturnino L. 1930-2012 Hille, Einar 1894-1980 Calculus One and several variables with analytic geometry Calculus Analytische Geometrie (DE-588)4001867-2 gnd Analysis (DE-588)4001865-9 gnd Infinitesimalrechnung (DE-588)4072798-1 gnd |
| subject_GND | (DE-588)4001867-2 (DE-588)4001865-9 (DE-588)4072798-1 |
| title | Calculus One and several variables with analytic geometry |
| title_auth | Calculus One and several variables with analytic geometry |
| title_exact_search | Calculus One and several variables with analytic geometry |
| title_full | Calculus One and several variables with analytic geometry Saturnino L. Salas ; Einar Hille* |
| title_fullStr | Calculus One and several variables with analytic geometry Saturnino L. Salas ; Einar Hille* |
| title_full_unstemmed | Calculus One and several variables with analytic geometry Saturnino L. Salas ; Einar Hille* |
| title_short | Calculus |
| title_sort | calculus one and several variables with analytic geometry |
| title_sub | One and several variables with analytic geometry |
| topic | Calculus Analytische Geometrie (DE-588)4001867-2 gnd Analysis (DE-588)4001865-9 gnd Infinitesimalrechnung (DE-588)4072798-1 gnd |
| topic_facet | Calculus Analytische Geometrie Analysis Infinitesimalrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003311724&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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