Honors calculus:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Glenview, Ill.
Scott, Foresman
1970
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 889 S. graph. Darst. |
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| 100 | 1 | |a Bartle, Robert G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Honors calculus |c Robert G. Bartle ; Cassius Ionescu Tulcea* |
| 264 | 1 | |a Glenview, Ill. |b Scott, Foresman |c 1970 | |
| 300 | |a XI, 889 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804117250567831552 |
|---|---|
| adam_text | CONTENTS
Chapter 1
SETS AND NUMBERS
1.1 Natural numbers. Integers. Rational Numbers 1
1.2 The Theorem of Pythagoras 3
1.3 Sets. Examples and Notations 6
1.4 Sets. Intersection. Union. Difference 7
1.5 Mathematical Induction 9
1.6 Real Numbers 12
1.7 Various Examples 19
1.8 The Absolute Value of a Real Number 24
Appendix 27
Chapter 2
ELEMENTS OF ANALYTIC GEOMETRY
2.1 Cartesian Products. The distance in R2 30
2.2 The Line 32
2.3 Equations of Lines 39
2.4 Illustrations of R and R2 44
2.5 Circles 47
2.6 Standard Conies 51
2.7 Rectangles. Squares. Disks 53
Chapter 3
FUNCTIONS AND GRAPHS
3.1 Functions 56
3.2 Operations with Functions 61
3.3 Graphs 63
3.4 Intervals 65
3.5 Graphs. Illustrations 67
3.6 The Composition of Functions 71
3.7 Restrictions of Functions 73
3.8 Inverse Functions 74
3.9 Monotone Functions 78
Chapter 4
LIMITS OF SEQUENCES
4.1 Sequences. Examples 80
4.2 Convergent Sequences in R. Limits in R 82
4.3 Bounded Sequences 87
4.4 Properties of Sequences 89
4.5 Monotone Sequences. The Axiom IV 94
4.6 The Sequence a, a2, ...,an, ... (of R) 98
4.7 Consequences of Axiom IV 99
4.8 The Bolzano Weierstrass Theorem 101
4.9 The Axiom of Archimedes 103
Appendix
I The Uniqueness of the Limit 104
II The Proof of Theorem 4.13 104
Chapter 5
LIMITS OF FUNCTIONS
5.1 Limits 109
5.2 Properties of the Limit 115
Appendix
I The Uniqueness of the Limit 121
II The Proof of Theorem 5.13 122
III The Proof of Theorem 5.17 122
Chapter 6
CONTINUOUS FUNCTIONS
6.1 Continuity at a Point 124
6.2 Continuous Functions on a Set 128
6.3 The Minimum maximum Theorem. The Intermediate
Value Theorem 130
6.4 The Continuity of the Composition of Two Functions 133
6.5 The Continuity of the Inverse Function of a Strictly
Monotone Function 134
6.6 The pth Root of a Positive Real Number 136
6.7 Uniform Continuity 140
Appendix
I The Proof of the Minimum maximum Theorem 142
II The Proof of the Intermediate Value Theorem 143
Chapter 7
THE DERIVATIVE
THE MAIN PROPERTIES OF THE DERIVATIVE
7.1 The Definition of the Derivative. Examples 145
7.2 Illustrations and Applications of the Derivative 152
7.3 Properties of Functions Differentiable at a Point 159
7.4 Properties of Functions Differentiable on an Interval 166
7.5 The Derivatives of Compositions and Inverse Functions 168
7.6 Rational Powers 175
7.7 The Derivative of the Restriction 178
ADDITIONAL PROPERTIES AND APPLICATIONS
7.8 The Derivatives of Functions Defined Implicitly 179
7.9 Related Rates of Change 184
7.10 Derivatives of Higher Order 188
7.11 Derivatives of Functions with Arbitrary Domains 192
7.12 Remarks Concerning Notations and Terminology 195
Chapter 8
THE MEAN VALUE THEOREM
8.1 Points of Relative Extremum 199
8.2 Rolle s Theorem 203
8.3 The Mean Value Theorem 204
8.4 Applications of the Mean Value Theorem 207
8.5 Motion of a Particle 211
8.6 A Special Case of Taylor s Theorem 218
Chapter 9
MAXIMA AND MINIMA
9.1 Points of Relative Strict Extremum 220
9.2 The First Derivative Test 222
9.3 The Second Derivative Test 227
9.4 Points of Absolute Extremum 230
9.5 Applications 232
Appendix 239
Chapter 10
CONVEX AND CONCAVE FUNCTIONS
10.1 Convex and Concave Functions 242
10.2 Criteria for Convexity and Concavity 244
10.3 More on Convexity and Concavity. Points of Inflection 248
10.4 Graphs 253
Chapter 11
INFINITE LIMITS. LIMITS AT INFINITY
11.1 Infinite Limits 258
11.2 Limits at Infinity 264
11.3 Infinite Limits at Infinity 269
11.4 Examples 273
ASYMPTOTES
11.5 Asymptotes of Type I 280
11.6 Asymptotes of Type II 281
SKETCHING GRAPHS
11.7 The General Method. Examples 285
Chapter 12
PRIMITIVES AND INTEGRALS
12.1 Primitives and their Elementary Properties 292
12.2 First List of Primitives 295
12.3 The Integral of a Continuous Function 297
12.4 Properties of the Integral 301
Chapter 13
INTEGRALS AND AREA
13.1 OrdinateSets 307
13.2 Sequences of Sets 309
13.3 Area 312
13.4 The Fundamental Theorem of Calculus 319
13.5 Examples 325
Chapter 14
THE INTEGRAL AS A LIMIT
14.1 Upper and Lower Riemann Sums 330
14.2 Numerical Integration 336
Appendix
I The Proof of Theorem 14.16 344
II The Proof of Theorem 14.20 345
Chapter 15
COMPLEX NUMBERS
15.1 Complex numbers 348
15.2 The Number i 352
15.3 The Conjugate and Absolute Value 354
15.4 Illustrations 357
15.5 Limits 362
15.6 Continuous Functions 365
15.7 Differentiable Functions 367
Appendix
I Proofs of the Properties 15.17 372
II Conies 373
Chapter 16
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
16.1 The Function E:C~+C 377
16.2 The Real Exponential Function Er 380
16.3 The Logarithmic Function 385
16.4 The Exponential Function x —»ax (a 0) 391
16.5 The Logarithmic Function loga (a 0, o ^ 1) 394
16.6 The Power Function x —? x° (a € R) 396
16.7 Exponential Growth 399
Chapter 17
TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS
17.1 The Cosine and Sine Functions 406
17.2 The Number 7T 411
17.3 The Graphs of cos and sin 415
17.4 The Other Trigonometric Functions 418
17.5 Hyperbolic Functions 424
17.6 Angles and Trigonometric Functions 428
Appendix
I Remarks Concerning the Periodicity of cos, sin, and E 434
II Proof of Theorem 2.27 437
III Angles between Rays 437
Chapter 18
INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS
18.1 The Functions Arcsin and Arccos 444
18.2 The Function Arctan 450
18.3 Inverse Hyperbolic Functions 454
Chapter 19
INTEGRATION BY PARTS. SUBSTITUTION
19.1 A Second List of Primitives 460
19.2 Integration by Parts 464
19.3 The First Substitution Method 469
19.4 The Second Substitution Method 475
19.5 Calculation of Several Areas 478
19.6 Two more Examples 481
19.7 Remarks Concerning Notation 483
Chapter 20
RATIONAL FUNCTIONS. TRIGONOMETRIC INTEGRALS
20.1 Examples of Integrals of Rational Functions 484
20.2 General Method of Integration of Rational Functions 485
20.3 Examples 492
20.4 Trigonometric Integrals 497
20.5 Complex Primitives and Integrals 504
Chapter 21
INTEGRALS HAVING SPECIAL FORMS
ft
21.1 Integrals of the form / f(e?)dx 510
•* a
21.2 Integrals of the form / /(sin x, cos z)dx 512
* o
fb
21.3 Integrals of the form / /(tan x)dx 518
• a
21.4 Integrals of the form / xa(Ax» + B)vdx 520
• a
fb I (Ax + B ll(i
21.5 Integrals of the form I /1 x, I I dx 524
K Cx + Dj I
Chapter 22
APPLICATIONS OF INTEGRATION IN R1
22.1 Curves in/?2 527
22.2 Polar Coordinates 534
22.3 Length and Area in Polar Coordinates 537
22.4 Work 542
22.5 Density of Wires 545
22.6 Moments and Centers of Gravity of a Wire 547
22.7 Center of Gravity of a Lamina 550
Chapter 23
APPLICATIONS OF INTEGRATION IN R*
23.1 The Space/?3 556
23.2 Curves in/?3 564
23.3 Surfaces of Revolution 567
23.4 Solids of Revolution 574
23.5 Cylindrical and Spherical Coordinates 580
Chapter 24
L HOSPITAL S RULES
24.1 Cauchy s Mean Value Theorem 589
24.2 L Hospital s Rules. The Case 0/0 591
24.3 L Hospital s Rules. The Case °o/oo 597
24.4 Remarks and Examples 603
Chapter 25
IMPROPER INTEGRALS
25.1 Improper Integrals over Intervals of the Form [a, /3) 607
25.2 Improper Integrals over Intervals of the Form (a, 6] 613
25.3 Improper Integrals over Intervals of the Form (a, j8) 616
25.4 Improper Integrals of Positive Functions 620
25.5 Absolutely Convergent Integrals 625
Chapter 26
TAYLOR S THEOREM
26.1 Taylor s Theorem 630
26.2 The Remainder in Taylor s Formula 636
26.3 The Exponential and Logarithm 637
26.4 The Functions sin and cos 640
26.5 The Binomial Functions 643
Chapter 27
SERIES
27.1 Series. Definition and Examples 648
27.2 Properties of Series 654
27.3 Series with Positive Terms 657
27.4 Absolute Convergence 665
27.5 Two Tests for Absolute Convergence 668
27.6 Alternating Series 673
27.7 Complex Series 675
Chapter 28
SERIES OF FUNCTIONS. POWER SERIES
28.1 Series of Functions. Definitions and Examples 678
28.2 Normally Convergent Series 681
28.3 General Remarks 684
28.4 Some Properties of Normally Convergent Series 685
28.5 Power Series 689
28.6 Supplementary Results 697
28.7 Expansions in Power Series 699
28.8 Series of Complex Functions and Complex Power Series 705
Chapter 29
FUNCTIONS OF SEVERAL VARIABLES.
PARTIAL DIFFERENTIATION
29.1 Functions on R2 to R 707
29.2 Partial Derivatives. Tangent Plane 714
29.3 Partial Derivatives on a Set 721
29.4 Differentiable Functions. The Differential 725
29.5 Partial Derivatives of Composite Functions 730
29.6 Partial Derivatives of Higher Order 739
29.7 Maxima and Minima 745
29.8 Tangent lines. Tangent Planes 749
29.9 Functions on R* to R 751
29.10 Three dimensional Vectors 755
29.11 Gradient, Divergence, Curl 768
Appendix 774
Chapter 30
DOUBLE AND TRIPLE INTEGRALS
30.1 Preliminary Definitions 781
30.2 The Double Integral 783
30.3 The Double Integral as a Limit 788
30.4 Theorems of Fubini Type for Double Integrals 790
30.5 Additional Remarks on the Calculation of Double Integrals 800
30.6 Volume 802
30.7 Substitution Formulas for Double Integrals 805
30.8 Triple Integrals 812
30.9 The Triple Integral as a Limit 812
30.10 Theorems of Fubini Type for Triple Integrals 814
30.11 Substitution Formulas for Triple Integrals 821
APPENDIX I Two Uniqueness Theorems 835
APPENDIX II The Existence and Uniqueness of the Exponential Function 838
APPENDIX III Suprema and Infima. Cauchy Sequences. Intervals. 843
APPENDIX IV Closed Bounded Sets are Borel Sets 853
ANSWERS AND HINTS TO SELECTED EXERCISES 859
INDEX 883
|
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| spelling | Bartle, Robert G. Verfasser aut Honors calculus Robert G. Bartle ; Cassius Ionescu Tulcea* Glenview, Ill. Scott, Foresman 1970 XI, 889 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Calcul infinitésimal Calculus Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Ionescu Tulcea, Cassius Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001852569&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bartle, Robert G. Ionescu Tulcea, Cassius Honors calculus Calcul infinitésimal Calculus Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Honors calculus |
| title_auth | Honors calculus |
| title_exact_search | Honors calculus |
| title_full | Honors calculus Robert G. Bartle ; Cassius Ionescu Tulcea* |
| title_fullStr | Honors calculus Robert G. Bartle ; Cassius Ionescu Tulcea* |
| title_full_unstemmed | Honors calculus Robert G. Bartle ; Cassius Ionescu Tulcea* |
| title_short | Honors calculus |
| title_sort | honors calculus |
| topic | Calcul infinitésimal Calculus Analysis (DE-588)4001865-9 gnd |
| topic_facet | Calcul infinitésimal Calculus Analysis |
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