# The Calculus Lifesaver

by: Banner, Adrian

**ISBN 13:**## 9780691130880

**ISBN 10:**## 0691130884

**Format:**Paperback**Copyright:**03/05/2007**Publisher:**Princeton Univ Pr

Note: Not guaranteed to come with supplemental materials (access cards, study guides, lab manuals, CDs, etc.)

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### 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

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

### Table of Contents

Read moreWelcome | 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 |

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