The Calculus Lifesaver

作者简介

阿德里安•班纳Adrian Banner

澳大利亚新南威尔士大学数学学士及硕士，普里斯顿大学数学博士。2002年起任职于INTECH公司，现为INTECH公司首席执行官兼首席投资官。同时，他在普林斯顿大学教学数学系任兼职教师。

本书目录

TABLE OF CONTENTS:
Welcome xviii
How to Use This Book to Study for an Exam xix
Two all-purpose study tips xx
Key sections for exam review by topic xx
Acknowledgments xxiii
Chapter 1: Functions, Graphs, and Lines 1
1.1 Functions 1
Interval notation 3
Finding the domain 4
Finding the range using the graph 5
The vertical line test 6
1.2 Inverse Functions 7
The horizontal line test 8
Finding the inverse 9
Restricting the domain 9
Inverses of inverse functions 11
1.3 Composition of Functions 11
1.4 Odd and Even Functions 14
1.5 Graphs of Linear Functions 17
1.6 Common Functions and Graphs 19
Chapter 2: Review of Trigonometry 25
2.1 The Basics 25
2.2 Extending the Domain of Trig Functions 28
The ASTC method 31
Trig functions outside [0; 2π] 33
2.3 The Graphs of Trig Functions 35
2.4 Trig Identities 39
Chapter 3: Introduction to Limits 41
3.1 Limits: The Basic Idea 41
3.2 Left-Hand and Right-Hand Limits 43
3.3 When the Limit Does Not Exist 45
3.4 Limits at 1 and —∞ 47
Large numbers and small numbers 48
3.5 Two Common Misconceptions about Asymptotes 50
3.6 The Sandwich Principle 51
3.7 Summary of Basic Types of Limits 54
Chapter 4: How to Solve Limit Problems Involving Polynomials 57
4.1 Limits Involving Rational Functions as χ → αa 57
4.2 Limits Involving Square Roots as χ → α 61
4.3 Limits Involving Rational Functions as χ → ∞ 61
Method and examples 64
4.4 Limits Involving Poly-type Functions as χ → ∞ 66
4.5 Limits Involving Rational Functions as χ → -∞ 70
4.6 Limits Involving Absolute Values 72
Chapter 5: Continuity and Differentiability 75
5.1 Continuity 75
Continuity at a point 76
Continuity on an interval 77
Examples of continuous functions 77
The Intermediate Value Theorem 80
A harder IVT example 82
Maxima and minima of continuous functions 82
5.2 Differentiability 84
Average speed 84
Displacement and velocity 85
Instantaneous velocity 86
The graphical interpretation of velocity 87
Tangent lines 88
The derivative function 90
The derivative as a limiting ratio 91
The derivative of linear functions 93
Second and higher-order derivatives 94
When the derivative does not exist 94
Differentiability and continuity 96
Chapter 6: How to Solve Differentiation Problems 99
6.1 Finding Derivatives Using the Definition 99
6.2 Finding Derivatives the Nice Way 102
Constant multiples of functions 103
Sums and Differences of functions 103
Products of functions via the product rule 104
Quotients of functions via the quotient rule 105
Composition of functions via the chain rule 107
A nasty example 109
Justification of the product rule and the chain rule 111
6.3 Finding the Equation of a Tangent Line 114
6.4 Velocity and Acceleration 114
Constant negative acceleration 115
6.5 Limits Which Are Derivatives in Disguise 117
6.6 Derivatives of Piecewise-Defined Functions 119
6.7 Sketching Derivative Graphs Directly 123
Chapter 7: Trig Limits and Derivatives 127
7.1 Limits Involving Trig Functions 127
The small case 128
Solving problems|the small case 129
The large case 134
The &34;other&34; case 137
Proof of an important limit 137
7.2 Derivatives Involving Trig Functions 141
Examples of Differentiating trig functions 143
Simple harmonic motion 145
A curious function 146
Chapter 8: Implicit Differentiation and Related Rates 149
8.1 Implicit Differentiation 149
Techniques and examples 150
Finding the second derivative implicitly 154
8.2 Related Rates 156
A simple example 157
A slightly harder example 159
A much harder example 160
A really hard example 162
Chapter 9: Exponentials and Logarithms 167
9.1 The Basics 167
Review of exponentials 167
Review of logarithms 168
Logarithms, exponentials, and inverses 169
Log rules 170
9.2 Definition of e 173
A question about compound interest 173
The answer to our question 173
More about e and logs 175
9.3 Differentiation of Logs and Exponentials 177
Examples of Differentiating exponentials and logs 179
9.4 How to Solve Limit Problems Involving Exponentials or Logs 180
Limits involving the definition of e 181
Behavior of exponentials near 0 182
Behavior of logarithms near 1 183
Behavior of exponentials near ∞ or -∞1 184
Behavior of logs near ∞ 187
Behavior of logs near 0 188
9.5 Logarithmic Differentiation 189
The derivative of χa 192
9.6 Exponential Growth and Decay 193
Exponential growth 194
Exponential decay 195
9.7 Hyperbolic Functions 198
Chapter 10: Inverse Functions and Inverse Trig Functions 201
10.1 The Derivative and Inverse Functions 201
Using the derivative to show that an inverse exists 201
Derivatives and inverse functions: what can go wrong 203
Finding the derivative of an inverse function 204
A big example 206
10.2 Inverse Trig Functions 208
Inverse sine 208
Inverse cosine 211
Inverse tangent 213
Inverse secant 216
Inverse cosecant and inverse cotangent 217
Computing inverse trig functions 218
10.3 Inverse Hyperbolic Functions 220
The rest of the inverse hyperbolic functions 222
Chapter 11: The Derivative and Graphs 225
11.1 Extrema of Functions 225
Global and local extrema 225
The Extreme Value Theorem 227
How to find global maxima and minima 228
11.2 Rolle&39;s Theorem 230
11.3 The Mean Value Theorem 233
Consequences of the Mean Value Theorem 235
11.4 The Second Derivative and Graphs 237
More about points of inection 238
11.5 Classifying Points Where the Derivative Vanishes 239
Using the first derivative 240
Using the second derivative 242
Chapter 12: Sketching Graphs 245
12.1 How to Construct a Table of Signs 245
Making a table of signs for the derivative 247
Making a table of signs for the second derivative 248
12.2 The Big Method 250
12.3 Examples 252
An example without using derivatives 252
The full method: example 1 254
The full method: example 2 256
The full method: example 3 259
The full method: example 4 262
Chapter 13: Optimization and Linearization 267
13.1 Optimization 267
An easy optimization example 267
Optimization problems: the general method 269
An optimization example 269
Another optimization example 271
Using implicit Differentiation in optimization 274
A difficult optimization example 275
13.2 Linearization 278
Linearization in general 279
The Differential 281
Linearization summary and examples 283
The error in our approximation 285
13.3 Newton&39;s Method 287
Chapter 14: L&39;Hôpital&39;s Rule and Overview of Limits 293
14.1 L&39;Hôpital&39;s Rule 293
Type A: 0/0 case 294
Type A: ±∞ / ±∞ case 296
Type B1 ∞ - ∞ 298
Type B2 0 x ±∞ 299
Type C 1±∞, 00, or ∞0 301
Summary of L&39;Hôpital&39;s Rule types 302
14.2 Overview of Limits 303
Chapter 15: Introduction to Integration 307
15.1 Sigma Notation 307
A nice sum 310
Telescoping series 311
15.2 Displacement and Area 314
Three simple cases 314
A more general journey 317
Signed area 319
Continuous velocity 320
Two special approximations 323
Chapter 16: Definite Integrals 325
16.1 The Basic Idea 325
Some easy examples 327
16.2 Definition of the Definite Integral 330
An example of using the definition 331
16.3 Properties of Definite Integrals 334
16.4 Finding Areas 339
Finding the unsigned area 339
Finding the area between two curves 342
Finding the area between a curve and the y-axis 344
16.5 Estimating Integrals 346
A simple type of estimation 347
16.6 Averages and the Mean Value Theorem for Integrals 350
The Mean Value Theorem for integrals 351
16.7 A Nonintegrable Function 353
Chapter 17: The Fundamental Theorems of Calculus 355
17.1 Functions Based on Integrals of Other Functions 355
17.2 The First Fundamental Theorem 358
Introduction to antiderivatives 361
17.3 The Second Fundamental Theorem 362
17.4 Indefinite Integrals 364
17.5 How to Solve Problems: The First Fundamental Theorem 366
Variation 1: variable left-hand limit of integration 367
Variation 2: one tricky limit of integration 367
Variation 3: two tricky limits of integration 369
Variation 4: limit is a derivative in disguise 370
17.6 How to Solve Problems: The Second Fundamental Theorem 371
Finding indefinite integrals 371
Finding definite integrals 374
Unsigned areas and absolute values 376
17.7 A Technical Point 380
17.8 Proof of the First Fundamental Theorem 381
Chapter 18: Techniques of Integration, Part One 383
18.1 Substitution 383
Substitution and definite integrals 386
How to decide what to substitute 389
Theoretical justification of the substitution method 392
18.2 Integration by Parts 393
Some variations 394
18.3 Partial Fractions 397
The algebra of partial fractions 398
Integrating the pieces 401
The method and a big example 404
Chapter 19: Techniques of Integration, Part Two 409
19.1 Integrals Involving Trig Identities 409
19.2 Integrals Involving Powers of Trig Functions 413
Powers of sin and/or cos 413
Powers of tan 415
Powers of sec 416
Powers of cot 418
Powers of csc 418
Reduction formulas 419
19.3 Integrals Involving Trig Substitutions 421
Type 1: 421
Type 2: 423
Type 3: 424
Completing the square and trig substitutions 426
Summary of trig substitutions 426
Technicalities of square roots and trig substitutions 427
19.4 Overview of Techniques of Integration 429
Chapter 20: Improper Integrals: Basic Concepts 431
20.1 Convergence and Divergence 431
Some examples of improper integrals 433
Other blow-up points 435
20.2 Integrals over Unbounded Regions 437
20.3 The Comparison Test Theory 439
20.4 The Limit Comparison Test Theory 441
Functions asymptotic to each other 441
The statement of the test 443
20.5 The p-test Theory 444
20.6 The Absolute Convergence Test 447
Chapter 21: Improper Integrals: How to Solve Problems 451
21.1 How to Get Started 451
Splitting up the integral 452
How to deal with negative function values 453
21.2 Summary of Integral Tests 454
21.3 Behavior of Common Functions near ∞ and -∞ 456
Polynomials and poly-type functions near ∞ and -∞ 456
Trig functions near ∞ and -∞ 459
Exponentials near ∞ and -∞ 461
Logarithms near ∞ 465
21.4 Behavior of Common Functions near 0 469
Polynomials and poly-type functions near 0 469
Trig functions near 0 470
Exponentials near 0 472
Logarithms near 0 473
The behavior of more general functions near 0 474
21.5 How to Deal with Problem Spots Not at 0 or ∞ 475
Chapter 22: Sequences and Series: Basic Concepts 477
22.1 Convergence and Divergence of Sequences 477
The connection between sequences and functions 478
Two important sequences 480
22.2 Convergence and Divergence of Series 481
Geometric series theory 484
22.3 The nth Term Test Theory 486
22.4 Properties of Both Infinite Series and Improper Integrals 487
The comparison test theory 487
The limit comparison test theory 488
The p-test theory 489
The absolute convergence test 490
22.5 New Tests for Series 491
The ratio test theory 492
The root test theory 493
The integral test theory 494
The alternating series test theory 497
Chapter 23: How to Solve Series Problems 501
23.1 How to Evaluate Geometric Series 502
23.2 How to Use the nth Term Test 503
23.3 How to Use the Ratio Test 504
23.4 How to Use the Root Test 508
23.5 How to Use the Integral Test 509
23.6 Comparison Test, Limit Comparison Test, and p-test 510
23.7 How to Deal with Series with Negative Terms 515
Chapter 24: Taylor Polynomials, Taylor Series, and Power Series 519
24.1 Approximations and Taylor Polynomials 519
Linearization revisited 520
Quadratic approximations 521
Higher-degree approximations 522
Taylor&39;s Theorem 523
24.2 Power Series and Taylor Series 526
Power series in general 527
Taylor series and Maclaurin series 529
Convergence of Taylor series 530
24.3 A Useful Limit 534
Chapter 25: How to Solve Estimation Problems 535
25.1 Summary of Taylor Polynomials and Series 535
25.2 Finding Taylor Polynomials and Series 537
25.3 Estimation Problems Using the Error Term 540
First example 541
Second example 543
Third example 544
Fourth example 546
Fifth example 547
General techniques for estimating the error term 548
25.4 Another Technique for Estimating the Error 548
Chapter 26: Taylor and Power Series: How to Solve Problems 551
26.1 Convergence of Power Series 551
Radius of convergence 551
How to find the radius and region of convergence 554
26.2 Getting New Taylor Series from Old Ones 558
Substitution and Taylor series 560
Differentiating Taylor series 562
Integrating Taylor series 563
Adding and subtracting Taylor series 565
Multiplying Taylor series 566
Dividing Taylor series 567
26.3 Using Power and Taylor Series to Find Derivatives 568
26.4 Using Maclaurin Series to Find Limits 570
Chapter 27: Parametric Equations and Polar Coordinates 575
27.1 Parametric Equations 575
Derivatives of parametric equations 578
27.2 Polar Coordinates 581
Converting to and from polar coordinates 582
Sketching curves in polar coordinates 585
Finding tangents to polar curves 590
Finding areas enclosed by polar curves 591
Chapter 28: Complex Numbers 595
28.1 The Basics 595
Complex exponentials 598
28.2 The Complex Plane 599
Converting to and from polar form 601
28.3 Taking Large Powers of Complex Numbers 603
28.4 Solving zn = w 604
Some variations 608
28.5 Solving ez = w 610
28.6 Some Trigonometric Series 612
28.7 Euler&39;s Identity and Power Series 615
Chapter 29: Volumes, Arc Lengths, and Surface Areas 617
29.1 Volumes of Solids of Revolution 617
The disc method 619
The shell method 620
Summary . . . and variations 622
Variation 1: regions between a curve and the y-axis 623
Variation 2: regions between two curves 625
Variation 3: axes parallel to the coordinate axes 628
29.2 Volumes of General Solids 631
29.3 Arc Lengths 637
Parametrization and speed 639
29.4 Surface Areas of Solids of Revolution 640
Chapter 30: Differential Equations 645
30.1 Introduction to Differential Equations 645
30.2 Separable First-order Differential Equations 646
30.3 First-order Linear Equations 648
Why the integrating factor works 652
30.4 Constant-coefficient Differential Equations 653
Solving first-order homogeneous equations 654
Solving second-order homogeneous equations 654
Why the characteristic quadratic method works 655
Nonhomogeneous equations and particular solutions 656
Finding a particular solution 658
Examples of finding particular solutions 660
Resolving conicts between yP and yH 662
Initial value problems constant-coefficient linear 663
30.5 Modeling Using Differential Equations 665
Appendix A Limits and Proofs 669
A.1 Formal Definition of a Limit 669
A little game 670
The actual definition 672
Examples of using the definition 672
A.2 Making New Limits from Old Ones 674
Sums and Differences of limits|proofs 674
Products of limits|proof 675
Quotients of limits|proof 676
The sandwich principle|proof 678
A.3 Other Varieties of Limits 678
Inffinite limits 679
Left-hand and right-hand limits 680
Limits at ∞ and -∞ 680
Two examples involving trig 682
A.4 Continuity and Limits 684
Composition of continuous functions 684
Proof of the Intermediate Value Theorem 686
Proof of the Max-Min Theorem 687
A.5 Exponentials and Logarithms Revisited 689
A.6 Differentiation and Limits 691
Constant multiples of functions 691
Sums and Differences of functions 691
Proof of the product rule 692
Proof of the quotient rule 693
Proof of the chain rule 693
Proof of the Extreme Value Theorem 694
Proof of Rolle&39;s Theorem 695
Proof of the Mean Value Theorem 695
The error in linearization 696
Derivatives of piecewise-defined functions 697
Proof of L&39;Hôpital&39;s Rule 698
A.7 Proof of the Taylor Approximation Theorem 700
Appendix B Estimating Integrals 703
B.1 Estimating Integrals Using Strips 703
Evenly spaced partitions 705
B.2 The Trapezoidal Rule 706
B.3 Simpson&39;s Rule 709
Proof of Simpson&39;s rule 710
B.4 The Error in Our Approximations 711
Examples of estimating the error 712
Proof of an error term inequality 714
List of Symbols 717
Index 719
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