The calculus lifesaver

Cover......Page 1
Contents......Page 8
1.1 Functions......Page 26
1.1.1 Interval notation......Page 28
1.1.2 Finding the domain......Page 29
1.1.3 Finding the range using the graph......Page 30
1.1.4 The vertical line test......Page 31
1.2 Inverse Functions......Page 32
1.2.1 The horizontal line test......Page 33
1.2.3 Restricting the domain......Page 34
1.3 Composition of Functions......Page 36
1.4 Odd and Even Functions......Page 39
1.5 Graphs of Linear Functions......Page 42
1.6 Common Functions and Graphs......Page 44
2.1 The Basics......Page 50
2.2 Extending the Domain of Trig Functions......Page 53
2.2.1 The ASTC method......Page 56
2.2.2 Trig functions outside [0; 2]......Page 58
2.3 The Graphs of Trig Functions......Page 60
2.4 Trig Identities ......Page 64
3.1 Limits: The Basic Idea......Page 66
3.2 Left-Hand and Right-Hand Limits......Page 68
3.3 When the Limit Does Not Exist......Page 70
3.4 Limits at 1 and infty......Page 72
3.4.1 Large numbers and small numbers......Page 73
3.5 Two Common Misconceptions about Asymptotes......Page 75
3.6 The Sandwich Principle......Page 76
3.7 Summary of Basic Types of Limits......Page 79
4.1 Limits Involving Rational Functions as x ! a......Page 82
4.3 Limits Involving Rational Functions as x ! 1......Page 86
4.3.1 Method and examples......Page 89
4.4 Limits Involving Poly-type Functions as x ! 1......Page 91
4.5 Limits Involving Rational Functions as x ! infty......Page 95
4.6 Limits Involving Absolute Values......Page 97
5.1 Continuity......Page 100
5.1.1 Continuity at a point......Page 101
5.1.3 Examples of continuous functions......Page 102
5.1.4 The Intermediate Value Theorem......Page 105
5.1.6 Maxima and minima of continuous functions......Page 107
5.2.1 Average speed......Page 109
5.2.2 Displacement and velocity......Page 110
5.2.3 Instantaneous velocity......Page 111
5.2.4 The graphical interpretation of velocity......Page 112
5.2.5 Tangent lines......Page 113
5.2.6 The derivative function......Page 115
5.2.7 The derivative as a limiting ratio......Page 116
5.2.8 The derivative of linear functions......Page 118
5.2.10 When the derivative does not exist......Page 119
5.2.11 Differentiability and continuity......Page 121
6.1 Finding Derivatives Using the Definition......Page 124
6.2 Finding Derivatives (the Nice Way)......Page 127
6.2.2 Sums and differences of functions......Page 128
6.2.3 Products of functions via the product rule......Page 129
6.2.4 Quotients of functions via the quotient rule......Page 130
6.2.5 Composition of functions via the chain rule......Page 132
6.2.6 A nasty example......Page 134
6.2.7 Justification of the product rule and the chain rule......Page 136
6.4 Velocity and Acceleration......Page 139
6.4.1 Constant negative acceleration......Page 140
6.5 Limits Which Are Derivatives in Disguise......Page 142
6.6 Derivatives of Piecewise-Defined Functions......Page 144
6.7 Sketching Derivative Graphs Directly......Page 148
7.1 Limits Involving Trig Functions......Page 152
7.1.1 The small case......Page 153
7.1.2 Solving problems|the small case......Page 154
7.1.3 The large case......Page 159
7.1.5 Proof of an important limit......Page 162
7.2 Derivatives Involving Trig Functions......Page 166
7.2.1 Examples of differentiating trig functions......Page 168
7.2.2 Simple harmonic motion......Page 170
7.2.3 A curious function......Page 171
8.1 Implicit Differentiation......Page 174
8.1.1 Techniques and examples......Page 175
8.1.2 Finding the second derivative implicitly......Page 179
8.2 Related Rates......Page 181
8.2.1 A simple example......Page 182
8.2.2 A slightly harder example......Page 184
8.2.3 A much harder example......Page 185
8.2.4 A really hard example......Page 187
9.1.1 Review of exponentials......Page 192
9.1.2 Review of logarithms......Page 193
9.1.3 Logarithms, exponentials, and inverses......Page 194
9.1.4 Log rules......Page 195
9.2.2 The answer to our question......Page 198
9.2.3 More about e and logs......Page 200
9.3 Differentiation of Logs and Exponentials ......Page 202
9.3.1 Examples of differentiating exponentials and logs......Page 204
9.4 How to Solve Limit Problems Involving Exponentials or Logs......Page 205
9.4.1 Limits involving the definition of e......Page 206
9.4.2 Behavior of exponentials near 0......Page 207
9.4.3 Behavior of logarithms near 1......Page 208
9.4.4 Behavior of exponentials near 1 or infty......Page 209
9.4.5 Behavior of logs near 1......Page 212
9.4.6 Behavior of logs near 0......Page 213
9.5 Logarithmic Differentiation......Page 214
9.5.1 The derivative of xa......Page 217
9.6 Exponential Growth and Decay......Page 218
9.6.1 Exponential growth......Page 219
9.6.2 Exponential decay......Page 220
9.7 Hyperbolic Functions......Page 223
10.1.1 Using the derivative to show that an inverse exists......Page 226
10.1.2 Derivatives and inverse functions: what can go wrong......Page 228
10.1.3 Finding the derivative of an inverse function......Page 229
10.1.4 A big example......Page 231
10.2.1 Inverse sine......Page 233
10.2.2 Inverse cosine......Page 236
10.2.3 Inverse tangent......Page 238
10.2.4 Inverse secant......Page 241
10.2.5 Inverse cosecant and inverse cotangent......Page 242
10.2.6 Computing inverse trig functions......Page 243
10.3 Inverse Hyperbolic Functions......Page 245
10.3.1 The rest of the inverse hyperbolic functions......Page 247
11.1.1 Global and local extrema......Page 250
11.1.2 The Extreme Value Theorem......Page 252
11.1.3 How to find global maxima and minima......Page 253
11.2 Rolle's Theorem......Page 255
11.3 The Mean Value Theorem......Page 258
11.3.1 Consequences of the Mean Value Theorem......Page 260
11.4 The Second Derivative and Graphs......Page 262
11.4.1 More about points of inflection......Page 263
11.5 Classifying Points Where the Derivative Vanishes......Page 264
11.5.1 Using the first derivative......Page 265
11.5.2 Using the second derivative ......Page 267
12.1 How to Construct a Table of Signs......Page 270
12.1.1 Making a table of signs for the derivative......Page 272
12.1.2 Making a table of signs for the second derivative......Page 273
12.2 The Big Method......Page 275
12.3.1 An example without using derivatives......Page 277
12.3.2 The full method: example 1......Page 279
12.3.3 The full method: example 2......Page 281
12.3.4 The full method: example 3......Page 284
12.3.5 The full method: example 4......Page 287
13.1.1 An easy optimization example......Page 292
13.1.3 An optimization example......Page 294
13.1.4 Another optimization example......Page 296
13.1.5 Using implicit differentiation in optimization......Page 299
13.1.6 A difficult optimization example......Page 300
13.2 Linearization......Page 303
13.2.1 Linearization in general......Page 304
13.2.2 The differential......Page 306
13.2.3 Linearization summary and examples......Page 308
13.2.4 The error in our approximation......Page 310
13.3 Newton's Method......Page 312
14.1 L'H^opital's Rule......Page 318
14.1.1 Type A: 0/0 case......Page 319
14.1.2 Type A: 1=1 case......Page 321
14.1.3 Type B1 (1infty......Page 323
14.1.4 Type B2 (0 1)......Page 324
14.1.5 Type C (11, 00, or 10)......Page 326
14.1.6 Summary of l'H^opital's Rule types......Page 327
14.2 Overview of Limits......Page 328
15.1 Sigma Notation......Page 332
15.1.1 A nice sum......Page 335
15.1.2 Telescoping series......Page 336
15.2.1 Three simple cases......Page 339
15.2.2 A more general journey......Page 342
15.2.3 Signed area......Page 344
15.2.4 Continuous velocity ......Page 345
15.2.5 Two special approximations......Page 348
16.1 The Basic Idea......Page 350
16.1.1 Some easy examples......Page 352
16.2 Definition of the Definite Integral......Page 355
16.2.1 An example of using the definition......Page 356
16.3 Properties of Definite Integrals......Page 359
16.4.1 Finding the unsigned area......Page 364
16.4.2 Finding the area between two curves......Page 367
16.4.3 Finding the area between a curve and the y-axis......Page 369
16.5 Estimating Integrals......Page 371
16.5.1 A simple type of estimation......Page 372
16.6 Averages and the Mean Value Theorem for Integrals......Page 375
16.6.1 The Mean Value Theorem for integrals......Page 376
16.7 A Nonintegrable Function......Page 378
17.1 Functions Based on Integrals of Other Functions......Page 380
17.2 The First Fundamental Theorem......Page 383
17.2.1 Introduction to antiderivatives......Page 386
17.3 The Second Fundamental Theorem......Page 387
17.4 Indefinite Integrals......Page 389
17.5 How to Solve Problems: The First Fundamental Theorem......Page 391
17.5.2 Variation 2: one tricky limit of integration......Page 392
17.5.3 Variation 3: two tricky limits of integration......Page 394
17.5.4 Variation 4: limit is a derivative in disguise......Page 395
17.6.1 Finding indefinite integrals......Page 396
17.6.2 Finding definite integrals......Page 399
17.6.3 Unsigned areas and absolute values......Page 401
17.7 A Technical Point......Page 405
17.8 Proof of the First Fundamental Theorem......Page 406
18.1 Substitution......Page 408
18.1.1 Substitution and definite integrals......Page 411
18.1.2 How to decide what to substitute......Page 414
18.1.3 Theoretical justification of the substitution method......Page 417
18.2 Integration by Parts......Page 418
18.2.1 Some variations......Page 419
18.3 Partial Fractions ......Page 422
18.3.1 The algebra of partial fractions......Page 423
18.3.2 Integrating the pieces......Page 426
18.3.3 The method and a big example......Page 429
19.1 Integrals Involving Trig Identities......Page 434
19.2.1 Powers of sin and/or cos......Page 438
19.2.2 Powers of tan......Page 440
19.2.3 Powers of sec......Page 441
19.2.5 Powers of csc......Page 443
19.2.6 Reduction formulas......Page 444
19.3.1 Type 1: p a2 infty......Page 446
19.3.2 Type 2: p x2 + a2......Page 448
19.3.3 Type 3: p x2 infty......Page 449
19.3.5 Summary of trig substitutions......Page 451
19.3.6 Technicalities of square roots and trig substitutions......Page 452
19.4 Overview of Techniques of Integration......Page 454
20.1 Convergence and Divergence......Page 456
20.1.1 Some examples of improper integrals......Page 458
20.1.2 Other blow-up points......Page 460
20.2 Integrals over Unbounded Regions......Page 462
20.3 The Comparison Test (Theory)......Page 464
20.4.1 Functions asymptotic to each other......Page 466
20.4.2 The statement of the test......Page 468
20.5 The p-test (Theory)......Page 469
20.6 The Absolute Convergence Test......Page 472
21.1 How to Get Started......Page 476
21.1.1 Splitting up the integral......Page 477
21.1.2 How to deal with negative function values......Page 478
21.2 Summary of Integral Tests......Page 479
21.3.1 Polynomials and poly-type functions near 1 and infty......Page 481
21.3.2 Trig functions near 1 and infty......Page 484
21.3.3 Exponentials near 1 and infty......Page 486
21.3.4 Logarithms near 1......Page 490
21.4.1 Polynomials and poly-type functions near 0......Page 494
21.4.2 Trig functions near 0......Page 495
21.4.3 Exponentials near 0......Page 497
21.4.4 Logarithms near 0......Page 498
21.4.5 The behavior of more general functions near 0......Page 499
21.5 How to Deal with Problem Spots Not at 0 or 1......Page 500
22.1 Convergence and Divergence of Sequences......Page 502
22.1.1 The connection between sequences and functions......Page 503
22.1.2 Two important sequences......Page 505
22.2 Convergence and Divergence of Series......Page 506
22.2.1 Geometric series (theory)......Page 509
22.3 The nth Term Test (Theory)......Page 511
22.4.1 The comparison test (theory)......Page 512
22.4.2 The limit comparison test (theory)......Page 513
22.4.3 The p-test (theory)......Page 514
22.4.4 The absolute convergence test......Page 515
22.5 New Tests for Series......Page 516
22.5.1 The ratio test (theory)......Page 517
22.5.2 The root test (theory)......Page 518
22.5.3 The integral test (theory)......Page 519
22.5.4 The alternating series test (theory)......Page 522
23 How to Solve Series Problems......Page 526
23.1 How to Evaluate Geometric Series......Page 527
23.2 How to Use the nth Term Test......Page 528
23.3 How to Use the Ratio Test......Page 529
23.4 How to Use the Root Test......Page 533
23.5 How to Use the Integral Test......Page 534
23.6 Comparison Test, Limit Comparison Test, and p-test......Page 535
23.7 How to Deal with Series with Negative Terms......Page 540
24.1 Approximations and Taylor Polynomials......Page 544
24.1.1 Linearization revisited......Page 545
24.1.2 Quadratic approximations......Page 546
24.1.3 Higher-degree approximations......Page 547
24.1.4 Taylor's Theorem......Page 548
24.2 Power Series and Taylor Series......Page 551
24.2.1 Power series in general......Page 552
24.2.2 Taylor series and Maclaurin series......Page 554
24.2.3 Convergence of Taylor series......Page 555
24.3 A Useful Limit ......Page 559
25.1 Summary of Taylor Polynomials and Series......Page 560
25.2 Finding Taylor Polynomials and Series......Page 562
25.3 Estimation Problems Using the Error Term......Page 565
25.3.1 First example......Page 566
25.3.2 Second example......Page 568
25.3.3 Third example......Page 569
25.3.4 Fourth example......Page 571
25.3.5 Fifth example......Page 572
25.4 Another Technique for Estimating the Error......Page 573
26.1.1 Radius of convergence......Page 576
26.1.2 How to find the radius and region of convergence......Page 579
26.2 Getting New Taylor Series from Old Ones......Page 583
26.2.1 Substitution and Taylor series......Page 585
26.2.2 Differentiating Taylor series......Page 587
26.2.3 Integrating Taylor series......Page 588
26.2.4 Adding and subtracting Taylor series......Page 590
26.2.5 Multiplying Taylor series......Page 591
26.2.6 Dividing Taylor series......Page 592
26.3 Using Power and Taylor Series to Find Derivatives......Page 593
26.4 Using Maclaurin Series to Find Limits......Page 595
27.1 Parametric Equations......Page 600
27.1.1 Derivatives of parametric equations......Page 603
27.2 Polar Coordinates......Page 606
27.2.1 Converting to and from polar coordinates......Page 607
27.2.2 Sketching curves in polar coordinates......Page 610
27.2.3 Finding tangents to polar curves......Page 615
27.2.4 Finding areas enclosed by polar curves......Page 616
28.1 The Basics......Page 620
28.1.1 Complex exponentials......Page 623
28.2 The Complex Plane......Page 624
28.2.1 Converting to and from polar form......Page 626
28.3 Taking Large Powers of Complex Numbers......Page 628
28.4 Solving zn = w......Page 629
28.4.1 Some variations......Page 633
28.5 Solving ez = w......Page 635
28.6 Some Trigonometric Series ......Page 637
28.7 Euler's Identity and Power Series......Page 640
29.1 Volumes of Solids of Revolution......Page 642
29.1.1 The disc method......Page 644
29.1.2 The shell method......Page 645
29.1.3 Summary......Page 647
29.1.4 Variation 1: regions between a curve and the y-axis......Page 648
29.1.5 Variation 2: regions between two curves......Page 650
29.1.6 Variation 3: axes parallel to the coordinate axes......Page 653
29.2 Volumes of General Solids......Page 656
29.3 Arc Lengths......Page 662
29.3.1 Parametrization and speed......Page 664
29.4 Surface Areas of Solids of Revolution......Page 665
30.1 Introduction to Differential Equations......Page 670
30.2 Separable First-order Differential Equations......Page 671
30.3 First-order Linear Equations......Page 673
30.3.1 Why the integrating factor works......Page 677
30.4 Constant-coefficient Differential Equations......Page 678
30.4.2 Solving second-order homogeneous equations......Page 679
30.4.3 Why the characteristic quadratic method works......Page 680
30.4.4 Nonhomogeneous equations and particular solutions......Page 681
30.4.5 Finding a particular solution......Page 683
30.4.6 Examples of finding particular solutions......Page 685
30.4.7 Resolving conflicts between yP and yH......Page 687
30.4.8 Initial value problems (constant-coefficient linear)......Page 688
30.5 Modeling Using Differential Equations......Page 690
A.1 Formal Definition of a Limit......Page 694
A.1.1 A little game......Page 695
A.1.3 Examples of using the definition......Page 697
A.2.1 Sums and differences of limits|proofs......Page 699
A.2.2 Products of limits|proof......Page 700
A.2.3 Quotients of limits|proof......Page 701
A.3 Other Varieties of Limits......Page 703
A.3.1 Infinite limits......Page 704
A.3.3 Limits at 1 and infty......Page 705
A.3.4 Two examples involving trig......Page 707
A.4.1 Composition of continuous functions......Page 709
A.4.2 Proof of the Intermediate Value Theorem......Page 711
A.4.3 Proof of the Max-Min Theorem......Page 712
A.5 Exponentials and Logarithms Revisited......Page 714
A.6.2 Sums and differences of functions......Page 716
A.6.3 Proof of the product rule......Page 717
A.6.5 Proof of the chain rule......Page 718
A.6.6 Proof of the Extreme Value Theorem......Page 719
A.6.8 Proof of the Mean Value Theorem......Page 720
A.6.9 The error in linearization......Page 721
A.6.10 Derivatives of piecewise-defined functions......Page 722
A.6.11 Proof of l'H^opital's Rule......Page 723
A.7 Proof of the Taylor Approximation Theorem......Page 725
B.1 Estimating Integrals Using Strips......Page 728
B.1.1 Evenly spaced partitions......Page 730
B.2 The Trapezoidal Rule......Page 731
B.3 Simpson's Rule......Page 734
B.3.1 Proof of Simpson's rule......Page 735
B.4 The Error in Our Approximations......Page 736
B.4.1 Examples of estimating the error......Page 737
B.4.2 Proof of an error term inequality......Page 739
List of Symbols......Page 742
Index ......Page 744