The Calculus Lifesaver

TABLE OF CONTENTS: Welcome xviii How to Use This Book to Study for an Exam xix Two all-purpose study tips xx Key sections for exam review (by topic) xx Acknowledgments xxiii Chapter 1: Functions, Graphs, and Lines 1 1.1 Functions 1 1.1.1 Interval notation 3 1.1.2 Finding the domain 4 1.1.3 Finding the range using the graph 5 1.1.4 The vertical line test 6 1.2 Inverse Functions 7 1.2.1 The horizontal line test 8 1.2.2 Finding the inverse 9 1.2.3 Restricting the domain 9 1.2.4 Inverses of inverse functions 11 1.3 Composition of Functions 11 1.4 Odd and Even Functions 14 1.5 Graphs of Linear Functions 17 1.6 Common Functions and Graphs 19 Chapter 2: Review of Trigonometry 25 2.1 The Basics 25 2.2 Extending the Domain of Trig Functions 28 2.2.1 The ASTC method 31 2.2.2 Trig functions outside [0; 2π] 33 2.3 The Graphs of Trig Functions 35 2.4 Trig Identities 39 Chapter 3: Introduction to Limits 41 3.1 Limits: The Basic Idea 41 3.2 Left-Hand and Right-Hand Limits 43 3.3 When the Limit Does Not Exist 45 3.4 Limits at 1 and —∞ 47 3.4.1 Large numbers and small numbers 48 3.5 Two Common Misconceptions about Asymptotes 50 3.6 The Sandwich Principle 51 3.7 Summary of Basic Types of Limits 54 Chapter 4: How to Solve Limit Problems Involving Polynomials 57 4.1 Limits Involving Rational Functions as χ → αa 57 4.2 Limits Involving Square Roots as χ → α 61 4.3 Limits Involving Rational Functions as χ → ∞ 61 4.3.1 Method and examples 64 4.4 Limits Involving Poly-type Functions as χ → ∞ 66 4.5 Limits Involving Rational Functions as χ → -∞ 70 4.6 Limits Involving Absolute Values 72 Chapter 5: Continuity and Differentiability 75 5.1 Continuity 75 5.1.1 Continuity at a point 76 5.1.2 Continuity on an interval 77 5.1.3 Examples of continuous functions 77 5.1.4 The Intermediate Value Theorem 80 5.1.5 A harder IVT example 82 5.1.6 Maxima and minima of continuous functions 82 5.2 Differentiability 84 5.2.1 Average speed 84 5.2.2 Displacement and velocity 85 5.2.3 Instantaneous velocity 86 5.2.4 The graphical interpretation of velocity 87 5.2.5 Tangent lines 88 5.2.6 The derivative function 90 5.2.7 The derivative as a limiting ratio 91 5.2.8 The derivative of linear functions 93 5.2.9 Second and higher-order derivatives 94 5.2.10 When the derivative does not exist 94 5.2.11 Differentiability and continuity 96 Chapter 6: How to Solve Differentiation Problems 99 6.1 Finding Derivatives Using the Definition 99 6.2 Finding Derivatives (the Nice Way) 102 6.2.1 Constant multiples of functions 103 6.2.2 Sums and Differences of functions 103 6.2.3 Products of functions via the product rule 104 6.2.4 Quotients of functions via the quotient rule 105 6.2.5 Composition of functions via the chain rule 107 6.2.6 A nasty example 109 6.2.7 Justification of the product rule and the chain rule 111 6.3 Finding the Equation of a Tangent Line 114 6.4 Velocity and Acceleration 114 6.4.1 Constant negative acceleration 115 6.5 Limits Which Are Derivatives in Disguise 117 6.6 Derivatives of Piecewise-Defined Functions 119 6.7 Sketching Derivative Graphs Directly 123 Chapter 7: Trig Limits and Derivatives 127 7.1 Limits Involving Trig Functions 127 7.1.1 The small case 128 7.1.2 Solving problems|the small case 129 7.1.3 The large case 134 7.1.4 The "other" case 137 7.1.5 Proof of an important limit 137 7.2 Derivatives Involving Trig Functions 141 7.2.1 Examples of Differentiating trig functions 143 7.2.2 Simple harmonic motion 145 7.2.3 A curious function 146 Chapter 8: Implicit Differentiation and Related Rates 149 8.1 Implicit Differentiation 149 8.1.1 Techniques and examples 150 8.1.2 Finding the second derivative implicitly 154 8.2 Related Rates 156 8.2.1 A simple example 157 8.2.2 A slightly harder example 159 8.2.3 A much harder example 160 8.2.4 A really hard example 162 Chapter 9: Exponentials and Logarithms 167 9.1 The Basics 167 9.1.1 Review of exponentials 167 9.1.2 Review of logarithms 168 9.1.3 Logarithms, exponentials, and inverses 169 9.1.4 Log rules 170 9.2 Definition of e 173 9.2.1 A question about compound interest 173 9.2.2 The answer to our question 173 9.2.3 More about e and logs 175 9.3 Differentiation of Logs and Exponentials 177 9.3.1 Examples of Differentiating exponentials and logs 179 9.4 How to Solve Limit Problems Involving Exponentials or Logs 180 9.4.1 Limits involving the definition of e 181 9.4.2 Behavior of exponentials near 0 182 9.4.3 Behavior of logarithms near 1 183 9.4.4 Behavior of exponentials near ∞ or -∞1 184 9.4.5 Behavior of logs near ∞ 187 9.4.6 Behavior of logs near 0 188 9.5 Logarithmic Differentiation 189 9.5.1 The derivative of χa 192 9.6 Exponential Growth and Decay 193 9.6.1 Exponential growth 194 9.6.2 Exponential decay 195 9.7 Hyperbolic Functions 198 Chapter 10: Inverse Functions and Inverse Trig Functions 201 10.1 The Derivative and Inverse Functions 201 10.1.1 Using the derivative to show that an inverse exists 201 10.1.2 Derivatives and inverse functions: what can go wrong 203 10.1.3 Finding the derivative of an inverse function 204 10.1.4 A big example 206 10.2 Inverse Trig Functions 208 10.2.1 Inverse sine 208 10.2.2 Inverse cosine 211 10.2.3 Inverse tangent 213 10.2.4 Inverse secant 216 10.2.5 Inverse cosecant and inverse cotangent 217 10.2.6 Computing inverse trig functions 218 10.3 Inverse Hyperbolic Functions 220 10.3.1 The rest of the inverse hyperbolic functions 222 Chapter 11: The Derivative and Graphs 225 11.1 Extrema of Functions 225 11.1.1 Global and local extrema 225 11.1.2 The Extreme Value Theorem 227 11.1.3 How to find global maxima and minima 228 11.2 Rolle's Theorem 230 11.3 The Mean Value Theorem 233 11.3.1 Consequences of the Mean Value Theorem 235 11.4 The Second Derivative and Graphs 237 11.4.1 More about points of inection 238 11.5 Classifying Points Where the Derivative Vanishes 239 11.5.1 Using the first derivative 240 11.5.2 Using the second derivative 242 Chapter 12: Sketching Graphs 245 12.1 How to Construct a Table of Signs 245 12.1.1 Making a table of signs for the derivative 247 12.1.2 Making a table of signs for the second derivative 248 12.2 The Big Method 250 12.3 Examples 252 12.3.1 An example without using derivatives 252 12.3.2 The full method: example 1 254 12.3.3 The full method: example 2 256 12.3.4 The full method: example 3 259 12.3.5 The full method: example 4 262 Chapter 13: Optimization and Linearization 267 13.1 Optimization 267 13.1.1 An easy optimization example 267 13.1.2 Optimization problems: the general method 269 13.1.3 An optimization example 269 13.1.4 Another optimization example 271 13.1.5 Using implicit Differentiation in optimization 274 13.1.6 A difficult optimization example 275 13.2 Linearization 278 13.2.1 Linearization in general 279 13.2.2 The Differential 281 13.2.3 Linearization summary and examples 283 13.2.4 The error in our approximation 285 13.3 Newton's Method 287 Chapter 14: L'Hôpital's Rule and Overview of Limits 293 14.1 L'Hôpital's Rule 293 14.1.1 Type A: 0/0 case 294 14.1.2 Type A: ±∞ / ±∞ case 296 14.1.3 Type B1 (∞ - ∞) 298 14.1.4 Type B2 (0 x ±∞) 299 14.1.5 Type C (1±∞, 00, or ∞0) 301 14.1.6 Summary of L'Hôpital's Rule types 302 14.2 Overview of Limits 303 Chapter 15: Introduction to Integration 307 15.1 Sigma Notation 307 15.1.1 A nice sum 310 15.1.2 Telescoping series 311 15.2 Displacement and Area 314 15.2.1 Three simple cases 314 15.2.2 A more general journey 317 15.2.3 Signed area 319 15.2.4 Continuous velocity 320 15.2.5 Two special approximations 323 Chapter 16: Definite Integrals 325 16.1 The Basic Idea 325 16.1.1 Some easy examples 327 16.2 Definition of the Definite Integral 330 16.2.1 An example of using the definition 331 16.3 Properties of Definite Integrals 334 16.4 Finding Areas 339 16.4.1 Finding the unsigned area 339 16.4.2 Finding the area between two curves 342 16.4.3 Finding the area between a curve and the y-axis 344 16.5 Estimating Integrals 346 16.5.1 A simple type of estimation 347 16.6 Averages and the Mean Value Theorem for Integrals 350 16.6.1 The Mean Value Theorem for integrals 351 16.7 A Nonintegrable Function 353 Chapter 17: The Fundamental Theorems of Calculus 355 17.1 Functions Based on Integrals of Other Functions 355 17.2 The First Fundamental Theorem 358 17.2.1 Introduction to antiderivatives 361 17.3 The Second Fundamental Theorem 362 17.4 Indefinite Integrals 364 17.5 How to Solve Problems: The First Fundamental Theorem 366 17.5.1 Variation 1: variable left-hand limit of integration 367 17.5.2 Variation 2: one tricky limit of integration 367 17.5.3 Variation 3: two tricky limits of integration 369 17.5.4 Variation 4: limit is a derivative in disguise 370 17.6 How to Solve Problems: The Second Fundamental Theorem 371 17.6.1 Finding indefinite integrals 371 17.6.2 Finding definite integrals 374 17.6.3 Unsigned areas and absolute values 376 17.7 A Technical Point 380 17.8 Proof of the First Fundamental Theorem 381 Chapter 18: Techniques of Integration, Part One 383 18.1 Substitution 383 18.1.1 Substitution and definite integrals 386 18.1.2 How to decide what to substitute 389 18.1.3 Theoretical justification of the substitution method 392 18.2 Integration by Parts 393 18.2.1 Some variations 394 18.3 Partial Fractions 397 18.3.1 The algebra of partial fractions 398 18.3.2 Integrating the pieces 401 18.3.3 The method and a big example 404 Chapter 19: Techniques of Integration, Part Two 409 19.1 Integrals Involving Trig Identities 409 19.2 Integrals Involving Powers of Trig Functions 413 19.2.1 Powers of sin and/or cos 413 19.2.2 Powers of tan 415 19.2.3 Powers of sec 416 19.2.4 Powers of cot 418 19.2.5 Powers of csc 418 19.2.6 Reduction formulas 419 19.3 Integrals Involving Trig Substitutions 421 19.3.1 Type 1: 421 19.3.2 Type 2: 423 19.3.3 Type 3: 424 19.3.4 Completing the square and trig substitutions 426 19.3.5 Summary of trig substitutions 426 19.3.6 Technicalities of square roots and trig substitutions 427 19.4 Overview of Techniques of Integration 429 Chapter 20: Improper Integrals: Basic Concepts 431 20.1 Convergence and Divergence 431 20.1.1 Some examples of improper integrals 433 20.1.2 Other blow-up points 435 20.2 Integrals over Unbounded Regions 437 20.3 The Comparison Test (Theory) 439 20.4 The Limit Comparison Test (Theory) 441 20.4.1 Functions asymptotic to each other 441 20.4.2 The statement of the test 443 20.5 The p-test (Theory) 444 20.6 The Absolute Convergence Test 447 Chapter 21: Improper Integrals: How to Solve Problems 451 21.1 How to Get Started 451 21.1.1 Splitting up the integral 452 21.1.2 How to deal with negative function values 453 21.2 Summary of Integral Tests 454 21.3 Behavior of Common Functions near ∞ and -∞ 456 21.3.1 Polynomials and poly-type functions near ∞ and -∞ 456 21.3.2 Trig functions near ∞ and -∞ 459 21.3.3 Exponentials near ∞ and -∞ 461 21.3.4 Logarithms near ∞ 465 21.4 Behavior of Common Functions near 0 469 21.4.1 Polynomials and poly-type functions near 0 469 21.4.2 Trig functions near 0 470 21.4.3 Exponentials near 0 472 21.4.4 Logarithms near 0 473 21.4.5 The behavior of more general functions near 0 474 21.5 How to Deal with Problem Spots Not at 0 or ∞ 475 Chapter 22: Sequences and Series: Basic Concepts 477 22.1 Convergence and Divergence of Sequences 477 22.1.1 The connection between sequences and functions 478 22.1.2 Two important sequences 480 22.2 Convergence and Divergence of Series 481 22.2.1 Geometric series (theory) 484 22.3 The nth Term Test (Theory) 486 22.4 Properties of Both Infinite Series and Improper Integrals 487 22.4.1 The comparison test (theory) 487 22.4.2 The limit comparison test (theory) 488 22.4.3 The p-test (theory) 489 22.4.4 The absolute convergence test 490 22.5 New Tests for Series 491 22.5.1 The ratio test (theory) 492 22.5.2 The root test (theory) 493 22.5.3 The integral test (theory) 494 22.5.4 The alternating series test (theory) 497 Chapter 23: How to Solve Series Problems 501 23.1 How to Evaluate Geometric Series 502 23.2 How to Use the nth Term Test 503 23.3 How to Use the Ratio Test 504 23.4 How to Use the Root Test 508 23.5 How to Use the Integral Test 509 23.6 Comparison Test, Limit Comparison Test, and p-test 510 23.7 How to Deal with Series with Negative Terms 515 Chapter 24: Taylor Polynomials, Taylor Series, and Power Series 519 24.1 Approximations and Taylor Polynomials 519 24.1.1 Linearization revisited 520 24.1.2 Quadratic approximations 521 24.1.3 Higher-degree approximations 522 24.1.4 Taylor's Theorem 523 24.2 Power Series and Taylor Series 526 24.2.1 Power series in general 527 24.2.2 Taylor series and Maclaurin series 529 24.2.3 Convergence of Taylor series 530 24.3 A Useful Limit 534 Chapter 25: How to Solve Estimation Problems 535 25.1 Summary of Taylor Polynomials and Series 535 25.2 Finding Taylor Polynomials and Series 537 25.3 Estimation Problems Using the Error Term 540 25.3.1 First example 541 25.3.2 Second example 543 25.3.3 Third example 544 25.3.4 Fourth example 546 25.3.5 Fifth example 547 25.3.6 General techniques for estimating the error term 548 25.4 Another Technique for Estimating the Error 548 Chapter 26: Taylor and Power Series: How to Solve Problems 551 26.1 Convergence of Power Series 551 26.1.1 Radius of convergence 551 26.1.2 How to find the radius and region of convergence 554 26.2 Getting New Taylor Series from Old Ones 558 26.2.1 Substitution and Taylor series 560 26.2.2 Differentiating Taylor series 562 26.2.3 Integrating Taylor series 563 26.2.4 Adding and subtracting Taylor series 565 26.2.5 Multiplying Taylor series 566 26.2.6 Dividing Taylor series 567 26.3 Using Power and Taylor Series to Find Derivatives 568 26.4 Using Maclaurin Series to Find Limits 570 Chapter 27: Parametric Equations and Polar Coordinates 575 27.1 Parametric Equations 575 27.1.1 Derivatives of parametric equations 578 27.2 Polar Coordinates 581 27.2.1 Converting to and from polar coordinates 582 27.2.2 Sketching curves in polar coordinates 585 27.2.3 Finding tangents to polar curves 590 27.2.4 Finding areas enclosed by polar curves 591 Chapter 28: Complex Numbers 595 28.1 The Basics 595 28.1.1 Complex exponentials 598 28.2 The Complex Plane 599 28.2.1 Converting to and from polar form 601 28.3 Taking Large Powers of Complex Numbers 603 28.4 Solving zn = w 604 28.4.1 Some variations 608 28.5 Solving ez = w 610 28.6 Some Trigonometric Series 612 28.7 Euler's Identity and Power Series 615 Chapter 29: Volumes, Arc Lengths, and Surface Areas 617 29.1 Volumes of Solids of Revolution 617 29.1.1 The disc method 619 29.1.2 The shell method 620 29.1.3 Summary . . . and variations 622 29.1.4 Variation 1: regions between a curve and the y-axis 623 29.1.5 Variation 2: regions between two curves 625 29.1.6 Variation 3: axes parallel to the coordinate axes 628 29.2 Volumes of General Solids 631 29.3 Arc Lengths 637 29.3.1 Parametrization and speed 639 29.4 Surface Areas of Solids of Revolution 640 Chapter 30: Differential Equations 645 30.1 Introduction to Differential Equations 645 30.2 Separable First-order Differential Equations 646 30.3 First-order Linear Equations 648 30.3.1 Why the integrating factor works 652 30.4 Constant-coefficient Differential Equations 653 30.4.1 Solving first-order homogeneous equations 654 30.4.2 Solving second-order homogeneous equations 654 30.4.3 Why the characteristic quadratic method works 655 30.4.4 Nonhomogeneous equations and particular solutions 656 30.4.5 Finding a particular solution 658 30.4.6 Examples of finding particular solutions 660 30.4.7 Resolving conicts between yP and yH 662 30.4.8 Initial value problems (constant-coefficient linear) 663 30.5 Modeling Using Differential Equations 665 Appendix A Limits and Proofs 669 A.1 Formal Definition of a Limit 669 A.1.1 A little game 670 A.1.2 The actual definition 672 A.1.3 Examples of using the definition 672 A.2 Making New Limits from Old Ones 674 A.2.1 Sums and Differences of limits|proofs 674 A.2.2 Products of limits|proof 675 A.2.3 Quotients of limits|proof 676 A.2.4 The sandwich principle|proof 678 A.3 Other Varieties of Limits 678 A.3.1 Inffinite limits 679 A.3.2 Left-hand and right-hand limits 680 A.3.3 Limits at ∞ and -∞ 680 A.3.4 Two examples involving trig 682 A.4 Continuity and Limits 684 A.4.1 Composition of continuous functions 684 A.4.2 Proof of the Intermediate Value Theorem 686 A.4.3 Proof of the Max-Min Theorem 687 A.5 Exponentials and Logarithms Revisited 689 A.6 Differentiation and Limits 691 A.6.1 Constant multiples of functions 691 A.6.2 Sums and Differences of functions 691 A.6.3 Proof of the product rule 692 A.6.4 Proof of the quotient rule 693 A.6.5 Proof of the chain rule 693 A.6.6 Proof of the Extreme Value Theorem 694 A.6.7 Proof of Rolle's Theorem 695 A.6.8 Proof of the Mean Value Theorem 695 A.6.9 The error in linearization 696 A.6.10 Derivatives of piecewise-defined functions 697 A.6.11 Proof of L'Hôpital's Rule 698 A.7 Proof of the Taylor Approximation Theorem 700 Appendix B Estimating Integrals 703 B.1 Estimating Integrals Using Strips 703 B.1.1 Evenly spaced partitions 705 B.2 The Trapezoidal Rule 706 B.3 Simpson's Rule 709 B.3.1 Proof of Simpson's rule 710 B.4 The Error in Our Approximations 711 B.4.1 Examples of estimating the error 712 B.4.2 Proof of an error term inequality 714 List of Symbols 717 Index 719

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