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Schaum's Outline of Theory and Problems of Beginning Calculus

Schaum's Outline of Theory and Problems of Beginning Calculus

PrefaceThis

PrefaceThis Outline is limited to the essentials of calculus. It carefully develops, giving all steps, the principlesof differentiation and integration on which the whole of calculus is built. The book is suitable forreviewing the subject, or as a self-contained text for an elementary calculus course.The author has found that many of the difficulties students encounter in calculus are due to weakness inalgebra and arithmetical computation, emphasis has been placed on reviewing algebraic andarithmetical techniques whenever they are used. Every effort has been made—especially in regard tothe composition of the solved problems—to ease the beginner's entry into calculus. There are also some1500 supplementary problems (with a complete set of answers at the end of the book).High school courses in calculus can readily use this Outline. Many of the problems are adopted fromquestions that have appeared in the Advanced Placement Examination in Calculus, so that students willautomatically receive preparation for that test.The Second Edition has been improved by the following changes:1. A large number of problems have been added to take advantage of the availability of graphingcalculators. Such problems are preceded by the notation . Solution of these problems is notnecessary for comprehension of the text, so that students not having a graphing calculator will notsuffer seriously from that lack (except insofar as the use of a graphing calculator enhances theirunderstanding of the subject).2. Treatment of several topics have been expanded:(a) Newton's Method is now the subject of a separate section. The availability of calculators makesit much easier to work out concrete problems by this method.(b) More attention and more problems are devoted to approximation techniques for integration, suchas the trapezoidal rule, Simpson's rule, and the midpoint rule.(c) The chain rule now has a complete proof outlined in an exercise.3. The exposition has been streamlined in many places and a substantial number of new problems havebeen added.The author wishes to thank again the editor of the First Edition, David Beckwith, as well as the editor ofthe Second Edition, Arthur Biderman, and the editing supervisor, Maureen Walker.ELLIOTT MENDELSON

  • Page 4: To the memory of my father, Joseph,
  • Page 10: ContentsChapter 1Coordinate Systems
  • Page 14: Chapter 11The Slope of a Tangent Li
  • Page 18: Chapter 26Sine and Cosine Functions
  • Page 22: Chapter 37Inverse Trigonometric Fun
  • Page 26: Chapter 1Coordinate Systems on a Li
  • Page 30: CHAP. 13COORDINATE SYSTEMS ON A LIN
  • Page 36: 6COORDINATE SYSTEMS ON A LINE[CHAP.
  • Page 40: Chapter 2Coordinate Systems in a Pl
  • Page 44: 10 COORDINATE SYSTEMS IN A PLANE [C
  • Page 48: 12 COORDINATE SYSTEMS IN A PLANE[CH
  • Page 52: Chapter 3Graphs of EquationsConside
  • Page 56:

    16 GRAPHS OF EQUATIONS [CHAP. 3Circ

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    18GRAPHS OF EQUATIONS3 -----4--,,-2

  • Page 64:

    20 GRAPHS OF EQUATIONS [CHAP. 33.6

  • Page 68:

    22GRAPHS OF EQUATIONS[CHAP. 33.10On

  • Page 72:

    Chapter 4Straight Lines4.1 SLOPEIf

  • Page 76:

    26 STRAIGHT LINES [CHAP. 4f’Fig.

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    28 STRAIGHT LINES [CHAP. 4Therefore

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    30 STRAIGHT LINES [CHAP. 4Solve the

  • Page 88:

    32 STRAIGHT LINES [CHAP. 4Fig. 4-12

  • Page 92:

    34 STRAIGHT LINES [CHAP. 44.144.154

  • Page 96:

    Chapter 5Intersections of GraphsThe

  • Page 100:

    38INTERSECTIONS OF GRAPHS [CHAP. 5S

  • Page 104:

    40 INTERSECTIONS OF GRAPHS [CHAP. 5

  • Page 108:

    42 SYMMETRY[CHAP. 6Consider the gra

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    44 SYMMETRY [CHAP. 6(c) The line is

  • Page 116:

    Chapter 7Functions and Their Graphs

  • Page 120:

    48 FUNCTIONS AND THEIR GRAPHS[CHAP.

  • Page 124:

    50 FUNCTIONS AND THEIR GRAPHS [CHAP

  • Page 128:

    52 FUNCTIONS AND THEIR GRAPHS [CHAP

  • Page 132:

    54FUNCTIONS AND THEIR GRAPHS [CHAP.

  • Page 136:

    56 FUNCTIONS AND THEIR GRAPHS [CHAP

  • Page 140:

    58 FUNCTIONS AND THEIR GRAPHS [CHAP

  • Page 144:

    60 LIMITS [CHAP. 8PROPERTY 111.EXAM

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    h,62 LIMITS [CHAP. 8Notice that the

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    64 LIMITS[CHAP. 8tYII-t------IIII-I

  • Page 156:

    66LIMITS [CHAP. 88.10 (a)(b)x4 - 1F

  • Page 160:

    68 SPECIAL LIMITS [CHAP. 99.2 INFIN

  • Page 164:

    70 SPECIAL LIMITS [CHAP. 9x-2EXAMPL

  • Page 168:

    72 SPECIAL LIMITS [CHAP. 9EXAMPLE A

  • Page 172:

    74 SPECIAL LIMITS [CHAP. 9GENERAL R

  • Page 176:

    and76 SPECIAL LIMITS [CHAP. 9Supple

  • Page 180:

    Chapter 1010.1 DEFIMON AND PROPERTI

  • Page 184:

    80 CONTINUITY [CHAP. 10(b) The func

  • Page 188:

    82IYCONTINUITYI’[CHAP. 100I X 0 1

  • Page 192:

    84CONTINUITY[CHAP. 10YY0T'-3 -2 -1

  • Page 196:

    Chapter 11The Slope of a Tangent Li

  • Page 200:

    88 THE SLOPE OF A TANGENT LINE [CHA

  • Page 204:

    90 THE SLOPE OF A TANGENT LINE [CHA

  • Page 208:

    Chapter 12The expression for the sl

  • Page 212:

    TTHE DERIVATIVE [CHAP. 12EXAMPLESD,

  • Page 216:

    96 THE DERIVATIVE [CHAP. 12(b) Forf

  • Page 220:

    h,h,98 THE DERIVATIVE [CHAP. 12(b)

  • Page 224:

    100 MORE ON THE DERIVATIVE [CHAP. 1

  • Page 228:

    102 MORE ON THE DERIVATIVE [CHAP. 1

  • Page 232:

    Chapter 14Maximum and Minimum Probl

  • Page 236:

    106 MAXIMUM AND MINIMUM PROBLEMS [C

  • Page 240:

    108 MAXIMUM AND MINIMUM PROBLEMS [C

  • Page 244:

    110 MAXIMUM AND MINIMUM PROBLEMS [C

  • Page 248:

    112 MAXIMUM AND MINIMUM PROBLEMS [C

  • Page 252:

    0*1114 MAXIMUM AND MINIMUM PROBLEMS

  • Page 256:

    Chapter 15The Chain RulelS.lCOMPOSI

  • Page 260:

    118 THE CHAIN RULE [CHAP. 15EXAMPLE

  • Page 264:

    4)1)120 THE CHAIN RULE [CHAP. 15The

  • Page 268:

    122 THE CHAIN RULE [CHAP. 15The onl

  • Page 272:

    124 THE CHAIN RULE [CHAP. 1515.15 F

  • Page 276:

    Chapter 16Implicit DifferentiationA

  • Page 280:

    128 IMPLICIT DIFFERENTIATION[CHAP.

  • Page 284:

    130 THE MEAN-VALUE THEOREM AND THE

  • Page 288:

    132 THE MEAN-VALUE THEOREM AND THE

  • Page 292:

    134 THE MEAN-VALUE THEOREM AND THE

  • Page 296:

    Chapter 18Rectilinear Motion and In

  • Page 300:

    138 RECTILINEAR MOTION AND INSTANTA

  • Page 304:

    140 RECTILINEAR MOTION AND INSTANTA

  • Page 308:

    142 RECTILINEAR MOTION AND INSTANTA

  • Page 312:

    144 INSTANTANEOUS RATE OF CHANGE[CH

  • Page 316:

    146 INSTANTANEOUS RATE OF CHANGE [C

  • Page 320:

    148 RELATED RATES [CHAP. 20In Fig.

  • Page 324:

    150 RELATED RATES [CHAP. 20- XFig.

  • Page 328:

    152 RELATED RATES[CHAP. 20and, by (

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    154 RELATED RATES [CHAP. 2020.21 A

  • Page 336:

    156 APPROXIMATION BY DIFFERENTIALS;

  • Page 340:

    158 APPROXIMATION BY DIFFERENTIALS;

  • Page 344:

    1 60APPROXIMATION BY DIFFERENTIALS;

  • Page 348:

    1 62 HIGHER-ORDER DERIVATIVES [CHAP

  • Page 352:

    ~~164 HIGHER-ORDER DERIVATIVES [CHA

  • Page 356:

    166 HIGHER-ORDER DERIVATIVES [CHAP.

  • Page 360:

    168 THE SECOND DERIVATIVE AND GRAPH

  • Page 364:

    170 THE SECOND DERIVATIVE AND GRAPH

  • Page 368:

    172 THE SECOND DERIVATIVE AND GRAPH

  • Page 372:

    174 THE SECOND DERIVATIVE AND GRAPH

  • Page 376:

    23.7 If, for all x,f’(x) > 0 andf

  • Page 380:

    178 THE SECOND DERIVATIVE AND GRAPH

  • Page 384:

    180 MORE MAXIMUM AND MINIMUM PROBLE

  • Page 388:

    182 MORE MAXIMUM AND MINIMUM PROBLE

  • Page 392:

    184 MORE MAXIMUM AND MINIMUM PROBLE

  • Page 396:

    186 ANGLE MEASURE [CHAP. 25and so o

  • Page 400:

    188 ANGLE MEASURE [CHAP. 25Solved P

  • Page 404:

    Chapter 2626.1 GENERAL DEFINITIONSi

  • Page 408:

    192 SINE AND COSINE FUNCTIONS [CHAP

  • Page 412:

    194 SINE AND COSINE FUNCTIONS [CHAP

  • Page 416:

    ~~~ ~196 SINE AND COSINE FUNCTIONS

  • Page 420:

    198 SINE AND COSINE FUNCTIONS [CHAP

  • Page 424:

    200 SINE AND COSINE FUNCTIONS [CHAP

  • Page 428:

    Chapter 27Graphs and Derivatives of

  • Page 432:

    204 GRAPHS AND DERIVATIVES OF SINE

  • Page 436:

    206 GRAPHS AND DERIVATIVES OF SINE

  • Page 440:

    208 GRAPHS AND DERIVATIVES OF SINE

  • Page 444:

    210 GRAPHS AND DERIVATIVES OF SINE

  • Page 448:

    212GRAPHS AND DERIVATIVES OF SINE A

  • Page 452:

    Chapter 28The Tangent andOther Trig

  • Page 456:

    216 THE TANGENT AND OTHER TRIGONOME

  • Page 460:

    218 THE TANGENT AND OTHER TRIGONOME

  • Page 464:

    220 THE TANGENT AND OTHER TRIGONOME

  • Page 468:

    222 ANTIDERIVATIVES [CHAP. 29EXAMPL

  • Page 472:

    224 ANTIDERIVATIVES [CHAP. 29(ii) F

  • Page 476:

    226 ANTIDERIVATIVES [CHAP. 2929.5 A

  • Page 480:

    228 ANTIDERIVATIVES [CHAP. 2929.13

  • Page 484:

    230 THE DEFINITE INTEGRAL [CHAP. 30

  • Page 488:

    232 THE DEFINITE INTEGRAL [CHAP. 30

  • Page 492:

    234 THE DEFINlTE INTEGRAL (CHAP.[I)

  • Page 496:

    ’)236 THE DEFINITE INTEGRAL [CHAP

  • Page 500:

    Chapter 31The Fundamental Theorem o

  • Page 504:

    240 THE FUNDAMENTAL THEOREM OF CALC

  • Page 508:

    242 THE FUNDAMENTAL THEOREM OF CALC

  • Page 512:

    244 THE FUNDAMENTAL THEOREM OF CALC

  • Page 516:

    246 THE FUNDAMENTAL THEOREM OF CALC

  • Page 520:

    248 THE FUNDAMENTAL THEOREM OF CALC

  • Page 524:

    250 APPLICATlONS OF INTEGRATION I:

  • Page 528:

    252 APPLICATIONS OF INTEGRATION I:

  • Page 532:

    254 APPLICATIONS OF INTEGRATION I:

  • Page 536:

    256 APPLICATIONS OF INTEGRATION I:

  • Page 540:

    258 APPLlCATlONS OF INTEGRATION 11:

  • Page 544:

    260 APPLICATIONS OF INTEGRATION 11:

  • Page 548:

    262 APPLICATIONS OF INTEGRATION 11:

  • Page 552:

    264 APPLICATIONS OF INTEGRATION 11:

  • Page 556:

    266 APPLICATIONS OF INTEGRATION I1

  • Page 560:

    Chapter 3434.1 DEFlMTlONWe already

  • Page 564:

    270 THE NATURAL LOGARITHM [CHAP. 34

  • Page 568:

    272 THE NATURAL LOGARITHM [CHAP. 34

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    (b)27434.14THE NATURAL LOGARITHM11(

  • Page 576:

    276 EXPONENTIAL FUNCTIONS [CHAP. 35

  • Page 580:

    278EXPONENTIAL FUNCTIONS[CHAP. 3535

  • Page 584:

    280 EXPONENTIAL FUNCTIONS [CHAP. 35

  • Page 588:

    282 EXPONENTIAL FUNCTIONS [CHAP. 35

  • Page 592:

    Chapter 36L’HGpital’s Rule ; Ex

  • Page 596:

    286 L'HOPITAL'S RULE; EXPONENTIAL G

  • Page 600:

    288 L'H~PITAL'S RULE; EXPONENTIAL G

  • Page 604:

    290 L'HQPITAL'S RULE; EXPONENTIAL G

  • Page 608:

    ~~~~~~~~ ~ ~~~ ~ ~ ~ ~ ~ ~ ~Chapter

  • Page 612:

    294 INVERSE TRIGONOMETRIC FUNCTIONS

  • Page 616:

    296 INVERSE TRIGONOMETRIC FUNCTIONS

  • Page 620:

    298 INVERSE TRIGONOMETRIC FUNCTIONS

  • Page 624:

    300 INVERSE TRIGONOMETRIC FUNCTIONS

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    302 INVERSE TRIGONOMETRIC FUNCTIONS

  • Page 632:

    above304 INVERSE TRIGONOMETRIC FUNC

  • Page 636:

    ~ du=dx306 INTEGRATION BY PARTS [CH

  • Page 640:

    308 INTEGRATION BY PARTS[CHAP. 3838

  • Page 644:

    3 10INTEGRATION BY PARTS[CHAP. 38Su

  • Page 648:

    312 TRIGONOMETRIC INTEGRANDS AND TR

  • Page 652:

    314 TRIGONOMETRIC INTEGRANDS AND TR

  • Page 656:

    316 TRIGONOMETRIC INTEGRANDS AND TR

  • Page 660:

    318 TRIGONOMETRIC INTEGRANDS AND TR

  • Page 664:

    Chapter 40Integration of Rational F

  • Page 668:

    322 THE METHOD OF PARTIAL FRACTIONS

  • Page 672:

    324 THE METHOD OF PARTIAL FRACTIONS

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    326 THE METHOD OF PARTIAL FRACTIONS

  • Page 680:

    328 THE METHOD OF PARTIAL FRACTIONS

  • Page 684:

    Appendix BBasic Integration Formula

  • Page 688:

    0"1"2"3"4"5"6"7"8"9"10"11"12"13"14"

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    Appendix F-X0.000.050.100.150.200.2

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    336 ANSWERS. TO SUPPLEMENTARY PROBL

  • Page 700:

    338ANSWERS TO SUPPLEMENTARY PROBLEM

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    340 ANSWERS TO SUPPLEMENTARY PROBLE

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    342 ANSWERS TO SUPPLEMENTARY PROBLE

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    344 ANSWERS TO SUPPLEMENTARY PROBLE

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    346 ANSWERS TO SUPPLEMENTARY PROBLE

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    348 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 724:

    350 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 728:

    352 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 732:

    354 ANSWERS TO SUPPLEMENTARY PROBLE

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    356 ANSWERS TO SUPPLEMENTARY PROBLE

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    358 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 744:

    360 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 748:

    R-v QdA 1d 1

  • Page 752:

    364+I)ANSWERS TO SUPPLEM ENTARY PRO

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    366 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 760:

    368 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 764:

    370 ANSWERS TO SUPPLEMENTARY PROBLE

  • Page 768:

    Collinear points, 30Common logarith

  • Page 772:

    Infinite limits, 68Inflection point

  • Page 776:

    RRadian, 185Radicals, 118Range, 47R

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