What is included with this book?
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 |
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