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The ABC's of Calculus

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The ABC's of Calculus Contents iii v Preface to the e-text edition vii 1 Functions and Their Properties 1 1.1TheMeaningofaFunction..................... 1 1.2 Function Values and the Box Method . 5 1.3TheAbsoluteValueofaFunction.................. 12 1.4 A Quick Review of Inequalities . 21 1.4.1 The triangle inequalities . 23 1.5ChapterExercises.......................... 30 1.6 Using Computer Algebra Systems (CAS), . 31 2 Limits and Continuity 33 2.1One-SidedLimitsofFunctions................... 35 2.2Two-SidedLimitsandContinuity.................. 40 2.3 Important Theorems About Continuous Functions . 59 2.4 Evaluating Limits at Infinity . 63 2.5HowtoGuessaLimit........................ 66 2.6ChapterExercises.......................... 76 3 The Derivative of a Function 79 3.1Motivation.............................. 80 3.2WorkingwithDerivatives...................... 89 3.3TheChainRule............................ 95 3.4 Implicit Functions and Their Derivatives . 108 3.5 Derivatives of TrigonometricSAMPLE Functions . 113 3.6ImportantResultsAboutDerivatives................121 3.7InverseFunctions...........................128 3.8 Inverse Trigonometric Functions . 136 3.9 Derivatives of Inverse Trigonometric Functions . 140 3.10RelatingRatesofChange......................143 ix www.math.carleton.ca/~amingare/calculus/cal104.html Protected by Copyright, - DO NOT COPY x CONTENTS 3.11 Newton’s Method for Calculating Roots . 151 3.12L’Hospital’sRule...........................161 3.13ChapterExercises..........................173 3.14ChallengeQuestions.........................174 3.15 Using Computer Algebra Systems . 175 4 Exponentials and Logarithms 177 4.1 Exponential Functions and Their Logarithms . 178 4.2 Euler’s Number, e = 2.718281828 ... 184 4.3 Euler’s Exponential Function and the Natural Logarithm . 189 4.4 Derivative of the Natural Logarithm . 193 4.5 Differentiation Formulae for General Exponential Functions . 196 4.6 Differentiation Formulae for General Logarithmic Functions . 201 4.7Applications..............................204 4.8ChapterExercises..........................210 4.9UsingComputerAlgebraSystems..................211 5 Curve Sketching 213 5.1 Solving Polynomial Inequalities . 213 5.2 Solving Rational Function Inequalities . 225 5.3GraphingTechniques.........................232 5.4 Application of Derivatives to Business and Economics . 256 5.5 Single variable optimization problems . 258 5.6ChapterExercises..........................259 6 Integration 261 6.1 Antiderivatives and the Indefinite Integral . 262 6.2DefiniteIntegrals...........................277 6.3TheSummationConvention.....................286 6.4 Area and the Riemann Integral . 291 6.5ChapterExercises..........................304 6.6UsingComputerAlgebraSystems..................306 7 Techniques of Integration 309 7.1TrigonometricIdentities.......................309 7.2 The Substitution Rule . 311 7.3IntegrationbyParts.........................323 7.3.1 The Product of a Polynomial and a Sine or Cosine . 328 7.3.2 The Product of a Polynomial and an Exponential . 331 7.3.3 The Product of a Polynomial and a Logarithm . 334 SAMPLE7.3.4 The Product of an Exponential and a Sine or Cosine . 337 7.4PartialFractions...........................346 7.4.1 Review of Long Division of Polynomials . 347 7.4.2 The Integration of Partial Fractions . 350 7.5ProductsofTrigonometricFunctions................365 7.5.1 ProductsofSinesandCosines................365 7.5.2 FourierCoefficients......................374 7.5.3 ProductsofSecantsandTangents.............378 7.6 Trigonometric Substitutions . 386 7.6.1 Completing the Square in a Quadratic (Review) . 386 7.6.2 Trigonometric Substitutions . 390 7.7 Numerical Integration . 400 7.7.1 TheTrapezoidalRule....................401 7.7.2 Simpson’s Rule for n Even.................408 7.8ImproperIntegrals..........................414 www.math.carleton.ca/~amingare/calculus/cal104.html Protected by Copyright, - DO NOT COPY CONTENTS xi 7.9 Rationalizing Substitutions . 429 7.9.1 Integrating rational functions of trigonometric expressions 432 7.10ChapterExercises..........................437 7.11UsingComputerAlgebraSystems..................443 8 Applications of the Integral 445 8.1Motivation..............................445 8.2FindingtheAreaBetweenTwoCurves...............448 8.3 The Volume of a Solid of Revolution . 464 8.4Measuringthelengthofacurve...................477 8.5MomentsandCentersofMass....................489 8.6ChapterExercises..........................502 8.7UsingComputerAlgebraSystems..................502 9 Simple Differential Equations 505 9.1WhyStudyDifferentialEquations?.................505 9.2 First-order Separable Equations . 512 9.3LawsofGrowthandDecay.....................518 9.4UsingComputerAlgebraSystems..................525 10 Multivariable Optimization Techniques 527 10.1FunctionsofMoreThanOneVariable...............527 10.2Continuity...............................528 10.2.1Discontinuityatapoint...................529 10.3PartialDerivatives..........................531 10.4 Higher Order Partial Derivatives . 533 10.5 The Chain Rule for Partial Derivatives . 535 10.6 Extrema of Functions of Two Variables . 540 10.6.1MaximaandMinima.....................540 10.6.2 The method of Lagrange multipliers . 544 10.7ChapterExercises..........................551 11 Advanced Topics 553 11.1 Infinite Sequences . 553 11.2 Sequences with Infinite Limits . 560 11.3LimitsfromtheRight........................563 11.4LimitsfromtheLeft.........................569 11.5Summary...............................575 11.6Continuity...............................576 11.7 Limits of Functions at Infinity . 578 11.8 Infinite Limits of FunctionsSAMPLE . 581 11.9TheEpsilon-DeltaMethodofProof.................585 12 Appendix A: Review of Exponents and Radicals 597 13 Appendix B: The Straight Line 603 14 Appendix C: A Quick Review of Trigonometry 609 14.1 The right-angled isosceles triangle (RT45) . 610 14.2TheRT30triangle..........................610 14.3 The basic trigonometric functions . 611 14.4Identities...............................613 14.4.1TheLawofSines.......................614 14.4.2TheLawofCosines.....................615 14.4.3 Identities for the sum and difference of angles . 616 www.math.carleton.ca/~amingare/calculus/cal104.html Protected by Copyright, - DO NOT COPY xii CONTENTS 15 Appendix D: The Natural Domain of a Function 623 Solutions Manual 627 1.1 .....................................627 1.2ExerciseSet1(page10).......................627 1.3ExerciseSet2(page19).......................627 1.4ExerciseSet3(page28).......................628 1.5 Chapter Exercises (page 30 ) . 629 Solutions 631 2.1ExerciseSet4(page39).......................631 2.2ExerciseSet5(page45).......................631 2.2ExerciseSet6(page49).......................631 2.2ExerciseSet7(page56).......................632 2.2ExerciseSet8(page57).......................632 2.3 .....................................633 2.4ExerciseSet9(page65).......................633 2.5 .....................................633 2.6ChapterExercises(page76).....................633 Solutions 635 3.1ExerciseSet10(page88)......................635 3.2ExerciseSet11(page93)......................635 3.3 Exercise Set 12 (page 105) . 636 3.4 Exercise Set 13 (page 112) . 637 3.5 Exercise Set 14 (page 119) . 637 3.6 Exercise Set 15 (page 127) . 637 3.7 Exercise Set 16 (page 134) . 639 3.8 Exercise Set 17 (page 139) . 639 3.9 Exercise Set 18 (page 142) . 639 3.10 Special Exercise Set (page 149) . 640 3.11 Exercise Set 19 (page 160) . 640 3.12 Exercise Set 20 (page 172) . 641 3.13 Chapter Exercises (page 173) . 641 Solutions 643 4.1 Exercise Set 21 (page 183) . 643 4.2 Exercise Set 22 (page 188) . 643 SAMPLE4.3 .....................................644 4.4 Exercise Set 23 (page 196) . 644 4.5 Exercise Set 24 (page 199) . 644 4.6 Exercise Set 25 (page 203) . 645 4.7 .....................................646 4.8 Chapter Exercises (page 210) . 646 Solutions 649 5.1 Exercise Set 26 (page 216) . 649 5.2 Exercise Set 27 (page 223) . 649 5.3 Exercise Set 28 (page 230) . 650 5.4 .....................................652 5.5 .....................................652 5.6 Single variable optimization problems (page 258) . 653 5.7 Chapter Exercises: Use Plotter (page 259) . 654 www.math.carleton.ca/~amingare/calculus/cal104.html Protected by Copyright, - DO NOT COPY CONTENTS xiii Solutions 657 6.1 Exercise Set 29 (page 275) . 657 6.2 Exercise Set 30 (page 289) . 657 6.3 Exercise Set 31 (page 284) . 659 6.4 .....................................660 6.5 Chapter Exercises (page 304) . 660 Solutions 665 7.1 .....................................665 7.2 Exercise Set 32 (page 321) . 665 7.3 Exercise Set 33 (page 345) . 666 7.4 .....................................667 7.4.1 Exercise Set 34 (page 349) . 667 7.4.2 Exercise Set 35 (page 364) . 668 7.5 .....................................670 7.5.1 Exercise Set 36 (page 373) . 670 7.5.2 Exercise Set 37 (page 385) . 671 7.6 .....................................672 7.6.1 Exercise Set 38 (page 390) . 672 7.6.2 Exercise Set 39 (page 398) . 673 7.7 .....................................675 7.7.1 ................................675 7.7.2 Exercise Set 40 (page 411) . 675 7.8 Exercise Set 41 (page 426) . 677 7.9 Chapter Exercises (page 437) . 678 Solutions 693 8.1 .....................................693 8.2 Exercise Set 42 (page 452) . 693 8.2.1 Exercise Set 43 (page 462) . 694 8.3 Exercise Set 44 (page 476) . 695 8.4 Exercise Set 45 (page 487) . 696 8.5 Exercise Set 46 (page 500) . 697 8.6 Chapter Exercises (page 502) . 698 Solutions
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