The Calculus Lifesaver

by Banner, Adrian

• ISBN13: 9780691130880
• ISBN10: 0691130884
• Format: Paperback
• Copyright: 2007-03-05
• Publisher: Princeton Univ Pr

Summary

Finally, a calculus book you can pick up and actually read! Developed especially for students who are motivated to earn an A but only score average grades on exams, The Calculus Lifesaver has all the essentials you need to master calculus. Companion to any single-variable calculus textbook, Forty-eight hours of accompanying video available at www.calclifesaver.com, More than 475 examples (ranging from easy to hard) provide step-by-step reasoning, Informal, entertaining, and not intimidating, Tried and tested by hundreds of students taking freshman calculus-proven to get results, Theorems and methods justified and connections made to actual practice, Difficult topics such as improper integrals and infinite series covered in detail, Emphasis on building problem-solving skills. Book jacket.

Author Biography

Adrian Banner is Director of Research at INTECH and Lecturer in Mathematics at Princeton University

Table of Contents

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