The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

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  • Copyright: 3/5/2007
  • Publisher: Princeton Univ Pr
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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

Welcomep. xviii
How to Use This Book to Study for an Examp. xix
Two all-purpose study tipsp. xx
Key sections for exam review (by topic)p. xx
Acknowledgmentsp. xxiii
Functions, Graphs, and Linesp. 1
Functionsp. 1
Interval notationp. 3
Finding the domainp. 4
Finding the range using the graphp. 5
The vertical line testp. 6
Inverse Functionsp. 7
The horizontal line testp. 8
Finding the inversep. 9
Restricting the domainp. 9
Inverses of inverse functionsp. 11
Composition of Functionsp. 11
Odd and Even Functionsp. 14
Graphs of Linear Functionsp. 17
Common Functions and Graphsp. 19
Review of Trigonometryp. 25
The Basicsp. 25
Extending the Domain of Trig Functionsp. 28
The ASTC methodp. 31
Trig functions outside [0,2[pi]]p. 33
The Graphs of Trig Functionsp. 35
Trig Identitiesp. 39
Introduction to Limitsp. 41
Limits: The Basic Ideap. 41
Left-Hand and Right-Hand Limitsp. 43
When the Limit Does Not Existp. 45
Limits at [infinity] and [infinity]p. 47
Large numbers and small numbersp. 48
Two Common Misconceptions about Asymptotesp. 50
The Sandwich Principlep. 51
Summary of Basic Types of Limitsp. 54
How to Solve Limit Problems Involving Polynomialsp. 57
Limits Involving Rational Functions as x [RightArrow] ap. 57
Limits Involving Square Roots as x [RightArrow] ap. 61
Limits Involving Rational Functions as x [RightArrow infinity] ap. 61
Method and examplesp. 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 Valuesp. 72
Continuity and Differentiabilityp. 75
Continuityp. 75
Continuity at a pointp. 76
Continuity on an intervalp. 77
Examples of continuous functionsp. 77
The Intermediate Value Theoremp. 80
A harder IVT examplep. 82
Maxima and minima of continuous functionsp. 82
Differentiabilityp. 84
Average speedp. 84
Displacement and velocityp. 85
Instantaneous velocityp. 86
The graphical interpretation of velocityp. 87
Tangent linesp. 88
The derivative functionp. 90
The derivative as a limiting ratiop. 91
The derivative of linear functionsp. 93
Second and higher-order derivativesp. 94
When the derivative does not existp. 94
Differentiability and continuityp. 96
How to Solve Differentiation Problemsp. 99
Finding Derivatives Using the Definitionp. 99
Finding Derivatives (the Nice Way)p. 102
Constant multiples of functionsp. 103
Sums and differences of functionsp. 103
Products of functions via the product rulep. 104
Quotients of functions via the quotient rulep. 105
Composition of functions via the chain rulep. 107
A nasty examplep. 109
Justification of the product rule and the chain rulep. 111
Finding the Equation of a Tangent Linep. 114
Velocity and Accelerationp. 114
Constant negative accelerationp. 115
Limits Which Are Derivatives in Disguisep. 117
Derivatives of Piecewise-Defined Functionsp. 119
Sketching Derivative Graphs Directlyp. 123
Trig Limits and Derivativesp. 127
Limits Involving Trig Functionsp. 127
The small casep. 128
Solving problems-the small casep. 129
The large casep. 134
The "other" casep. 137
Proof of an important limitp. 137
Derivatives Involving Trig Functionsp. 141
Examples of differentiating trig functionsp. 143
Simple harmonic motionp. 145
A curious functionp. 146
Implicit Differentiation and Related Ratesp. 149
Implicit Differentiationp. 149
Techniques and examplesp. 150
Finding the second derivative implicitlyp. 154
Related Ratesp. 156
A simple examplep. 157
A slightly harder examplep. 159
A much harder examplep. 160
A really hard examplep. 162
Exponentials and Logarithmsp. 167
The Basicsp. 167
Review of exponentialsp. 167
Review of logarithmsp. 168
Logarithms, exponentials, and inversesp. 169
Log rulesp. 170
Definition of ep. 173
A question about compound interestp. 173
The answer to our questionp. 173
More about e and logsp. 175
Differentiation of Logs and Exponentialsp. 177
Examples of differentiating exponentials and logsp. 179
How to Solve Limit Problems Involving Exponentials or Logsp. 180
Limits involving the definition of ep. 181
Behavior of exponentials near 0p. 182
Behavior of logarithms near 1p. 183
Behavior of exponentials near [infinity] or -[infinity]p. 184
Behavior of logs near [infinity]p. 187
Behavior of logs near 0p. 188
Logarithmic Differentiationp. 189
The derivative of x[superscript a]p. 192
Exponential Growth and Decayp. 193
Exponential growthp. 194
Exponential decayp. 195
Hyperbolic Functionsp. 198
Inverse Functions and Inverse Trig Functionsp. 201
The Derivative and Inverse Functionsp. 201
Using the derivative to show that an inverse existsp. 201
Derivatives and inverse functions: what can go wrongp. 203
Finding the derivative of an inverse functionp. 204
A big examplep. 206
Inverse Trig Functionsp. 208
Inverse sinep. 208
Inverse cosinep. 211
Inverse tangentp. 213
Inverse secantp. 216
Inverse cosecant and inverse cotangentp. 217
Computing inverse trig functionsp. 218
Inverse Hyperbolic Functionsp. 220
The rest of the inverse hyperbolic functionsp. 222
The Derivative and Graphsp. 225
Extrema of Functionsp. 225
Global and local extremap. 225
The Extreme Value Theoremp. 227
How to find global maxima and minimap. 228
Rolle's Theoremp. 230
The Mean Value Theoremp. 233
Consequences of the Mean Value Theoremp. 235
The Second Derivative and Graphsp. 237
More about points of inflectionp. 238
Classifying Points Where the Derivative Vanishesp. 239
Using the first derivativep. 240
Using the second derivativep. 242
Sketching Graphsp. 245
How to Construct a Table of Signsp. 245
Making a table of signs for the derivativep. 247
Making a table of signs for the second derivativep. 248
The Big Methodp. 250
Examplesp. 252
An example without using derivativesp. 252
The full method: example 1p. 254
The full method: example 2p. 256
The full method: example 3p. 259
The full method: example 4p. 262
Optimization and Linearizationp. 267
Optimizationp. 267
An easy optimization examplep. 267
Optimization problems: the general methodp. 269
An optimization examplep. 269
Another optimization examplep. 271
Using implicit differentiation in optimizationp. 274
A difficult optimization examplep. 275
Linearizationp. 278
Linearization in generalp. 279
The differentialp. 281
Linearization summary and examplesp. 283
The error in our approximationp. 285
Newton's Methodp. 287
L'Hopital's Rule and Overview of Limitsp. 293
L'Hopital's Rulep. 293
Type A: 0/0 casep. 294
Type A: [PlusMinus infinity] / [PlusMinus infinity] casep. 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 typesp. 302
Overview of Limitsp. 303
Introduction to Integrationp. 307
Sigma Notationp. 307
A nice sump. 310
Telescoping seriesp. 311
Displacement and Areap. 314
Three simple casesp. 314
A more general journeyp. 317
Signed areap. 319
Continuous velocityp. 320
Two special approximationsp. 323
Definite Integralsp. 325
The Basic Ideap. 325
Some easy examplesp. 327
Definition of the Definite Integralp. 330
An example of using the definitionp. 331
Properties of Definite Integralsp. 334
Finding Areasp. 339
Finding the unsigned areap. 339
Finding the area between two curvesp. 342
Finding the area between a curve and the y-axisp. 344
Estimating Integralsp. 346
A simple type of estimationp. 347
Averages and the Mean Value Theorem for Integralsp. 350
The Mean Value Theorem for integralsp. 351
A Nonintegrable Functionp. 353
The Fundamental Theorems of Calculusp. 355
Functions Based on Integrals of Other Functionsp. 355
The First Fundamental Theoremp. 358
Introduction to antiderivativesp. 361
The Second Fundamental Theoremp. 362
Indefinite Integralsp. 364
How to Solve Problems: The First Fundamental Theoremp. 366
Variation 1: variable left-hand limit of integrationp. 367
Variation 2: one tricky limit of integrationp. 367
Variation 3: two tricky limits of integrationp. 369
Variation 4: limit is a derivative in disguisep. 370
How to Solve Problems: The Second Fundamental Theoremp. 371
Finding indefinite integralsp. 371
Finding definite integralsp. 374
Unsigned areas and absolute valuesp. 376
A Technical Pointp. 380
Proof of the First Fundamental Theoremp. 381
Techniques of Integration, Part Onep. 383
Substitutionp. 383
Substitution and definite integralsp. 386
How to decide what to substitutep. 389
Theoretical justification of the substitution methodp. 392
Integration by Partsp. 393
Some variationsp. 394
Partial Fractionsp. 397
The algebra of partial fractionsp. 398
Integrating the piecesp. 401
The method and a big examplep. 404
Techniques of Integration, Part Twop. 409
Integrals Involving Trig Identitiesp. 409
Integrals Involving Powers of Trig Functionsp. 413
Powers of sin and/or cosp. 413
Powers of tanp. 415
Powers of secp. 416
Powers of cotp. 418
Powers of cscp. 418
Reduction formulasp. 419
Integrals Involving Trig Substitutionsp. 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 substitutionsp. 426
Summary of trig substitutionsp. 426
Technicalities of square roots and trig substitutionsp. 427
Overview of Techniques of Integrationp. 429
Improper Integrals: Basic Conceptsp. 431
Convergence and Divergencep. 431
Some examples of improper integralsp. 433
Other blow-up pointsp. 435
Integrals over Unbounded Regionsp. 437
The Comparison Test (Theory)p. 439
The Limit Comparison Test (Theory)p. 441
Functions asymptotic to each otherp. 441
The statement of the testp. 443
The p-test (Theory)p. 444
The Absolute Convergence Testp. 447
Improper Integrals: How to Solve Problemsp. 451
How to Get Startedp. 451
Splitting up the integralp. 452
How to deal with negative function valuesp. 453
Summary of Integral Testsp. 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 0p. 469
Polynomials and poly-type functions near 0p. 469
Trig functions near 0p. 470
Exponentials near 0p. 472
Logarithms near 0p. 473
The behavior of more general functions near 0p. 474
How to Deal with Problem Spots Not at 0 or [infinity]p. 475
Sequences and Series: Basic Conceptsp. 477
Convergence and Divergence of Sequencesp. 477
The connection between sequences and functionsp. 478
Two important sequencesp. 480
Convergence and Divergence of Seriesp. 481
Geometric series (theory)p. 484
The nth Terra Test (Theory)p. 486
Properties of Both Infinite Series and Improper Integralsp. 487
The comparison test (theory)p. 487
The limit comparison test (theory)p. 488
The p-test (theory)p. 489
The absolute convergence testp. 490
New Tests for Seriesp. 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 Problemsp. 501
How to Evaluate Geometric Seriesp. 502
How to Use the nth Term Testp. 503
How to Use the Ratio Testp. 504
How to Use the Root Testp. 508
How to Use the Integral Testp. 509
Comparison Test, Limit Comparison Test, and p-testp. 510
How to Deal with Series with Negative Termsp. 515
Taylor Polynomials, Taylor Series, and Power Seriesp. 519
Approximations and Taylor Polynomialsp. 519
Linearization revisitedp. 520
Quadratic approximationsp. 521
Higher-degree approximationsp. 522
Taylor's Theoremp. 523
Power Series and Taylor Seriesp. 526
Power series in generalp. 527
Taylor series and Maclaurin seriesp. 529
Convergence of Taylor seriesp. 530
A Useful Limitp. 534
How to Solve Estimation Problemsp. 535
Summary of Taylor Polynomials and Seriesp. 535
Finding Taylor Polynomials and Seriesp. 537
Estimation Problems Using the Error Termp. 540
First examplep. 541
Second examplep. 543
Third examplep. 544
Fourth examplep. 546
Fifth examplep. 547
General techniques for estimating the error termp. 548
Another Technique for Estimating the Errorp. 548
Taylor and Power Series: How to Solve Problemsp. 551
Convergence of Power Seriesp. 551
Radius of convergencep. 551
How to find the radius and region of convergencep. 554
Getting New Taylor Series from Old Onesp. 558
Substitution and Taylor seriesp. 560
Differentiating Taylor seriesp. 562
Integrating Taylor seriesp. 563
Adding and subtracting Taylor seriesp. 565
Multiplying Taylor seriesp. 566
Dividing Taylor seriesp. 567
Using Power and Taylor Series to Find Derivativesp. 568
Using Maclaurin Series to Find Limitsp. 570
Parametric Equations and Polar Coordinatesp. 575
Parametric Equationsp. 575
Derivatives of parametric equationsp. 578
Polar Coordinatesp. 581
Converting to and from polar coordinatesp. 582
Sketching curves in polar coordinatesp. 585
Finding tangents to polar curvesp. 590
Finding areas enclosed by polar curvesp. 591
Complex Numbersp. 595
The Basicsp. 595
Complex exponentialsp. 598
The Complex Planep. 599
Converting to and from polar formp. 601
Taking Large Powers of Complex Numbersp. 603
Solving z[superscript n] = wp. 604
Some variationsp. 608
Solving e[superscript z] = wp. 610
Some Trigonometric Seriesp. 612
Euler's Identity and Power Seriesp. 615
Volumes, Arc Lengths, and Surface Areasp. 617
Volumes of Solids of Revolutionp. 617
The disc methodp. 619
The shell methodp. 620
Summary...and variationsp. 622
Variation 1: regions between a curve and the y-axisp. 623
Variation 2: regions between two curvesp. 625
Variation 3: axes parallel to the coordinate axesp. 628
Volumes of General Solidsp. 631
Arc Lengthsp. 637
Parametrization and speedp. 639
Surface Areas of Solids of Revolutionp. 640
Differential Equationsp. 645
Introduction to Differential Equationsp. 645
Separable First-order Differential Equationsp. 646
First-order Linear Equationsp. 648
Why the integrating factor worksp. 652
Constant-coefficient Differential Equationsp. 653
Solving first-order homogeneous equationsp. 654
Solving second-order homogeneous equationsp. 654
Why the characteristic quadratic method worksp. 655
Nonhomogeneous equations and particular solutionsp. 656
Finding a particular solutionp. 658
Examples of finding particular solutionsp. 660
Resolving conflicts between y[subscript P] and y[subscript H]p. 662
Initial value problems (constant-coefficient linear)p. 663
Modeling Using Differential Equationsp. 665
Limits and Proofsp. 669
Formal Definition of a Limitp. 669
A little gamep. 670
The actual definitionp. 672
Examples of using the definitionp. 672
Making New Limits from Old Onesp. 674
Sum and differences of limits-proofsp. 674
Products of limits-proofp. 675
Quotients of limits-proofp. 676
The sandwich principle-proofp. 678
Other Varieties of Limitsp. 678
Infinite limitsp. 679
Left-hand and right-hand limitsp. 680
Limits at [infinity] and -[infinity]p. 680
Two examples involving trigp. 682
Continuity and Limitsp. 684
Composition of continuous functionsp. 684
Proof of the Intermediate Value Theoremp. 686
Proof of the Max-Min Theoremp. 687
Exponentials and Logarithms Revisitedp. 689
Differentiation and Limitsp. 691
Constant multiples of functionsp. 691
Sums and differences of functionsp. 691
Proof of the product rulep. 692
Proof of the quotient rulep. 693
Proof of the chain rulep. 693
Proof of the Extreme Value Theoremp. 694
Proof of Rolle's Theoremp. 695
Proof of the Mean Value Theoremp. 695
The error in linearizationp. 696
Derivatives of piecewise-defined functionsp. 697
Proof of l'Hopital's Rulep. 698
Proof of the Taylor Approximation Theoremp. 700
Estimating Integralsp. 703
Estimating Integrals Using Stripsp. 703
Evenly spaced partitionsp. 705
The Trapezoidal Rulep. 706
Simpson's Rulep. 709
Proof of Simpson's rulep. 710
The Error in Our Approximationsp. 711
Examples of estimating the errorp. 712
Proof of an error term inequalityp. 714
List of Symbolsp. 717
Indexp. 719
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