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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
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16 GRAPHS OF EQUATIONS [CHAP. 3Circ

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

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20 GRAPHS OF EQUATIONS [CHAP. 33.6

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22GRAPHS OF EQUATIONS[CHAP. 33.10On

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Chapter 4Straight Lines4.1 SLOPEIf

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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

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

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34 STRAIGHT LINES [CHAP. 44.144.154

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Chapter 5Intersections of GraphsThe

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38INTERSECTIONS OF GRAPHS [CHAP. 5S

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40 INTERSECTIONS OF GRAPHS [CHAP. 5

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42 SYMMETRY[CHAP. 6Consider the gra

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

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Chapter 7Functions and Their Graphs

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48 FUNCTIONS AND THEIR GRAPHS[CHAP.

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50 FUNCTIONS AND THEIR GRAPHS [CHAP

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52 FUNCTIONS AND THEIR GRAPHS [CHAP

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54FUNCTIONS AND THEIR GRAPHS [CHAP.

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56 FUNCTIONS AND THEIR GRAPHS [CHAP

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58 FUNCTIONS AND THEIR GRAPHS [CHAP

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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

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

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68 SPECIAL LIMITS [CHAP. 99.2 INFIN

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70 SPECIAL LIMITS [CHAP. 9x-2EXAMPL

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72 SPECIAL LIMITS [CHAP. 9EXAMPLE A

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74 SPECIAL LIMITS [CHAP. 9GENERAL R

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and76 SPECIAL LIMITS [CHAP. 9Supple

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Chapter 1010.1 DEFIMON AND PROPERTI

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80 CONTINUITY [CHAP. 10(b) The func

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82IYCONTINUITYI’[CHAP. 100I X 0 1

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

• Page 196:

Chapter 11The Slope of a Tangent Li

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88 THE SLOPE OF A TANGENT LINE [CHA

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90 THE SLOPE OF A TANGENT LINE [CHA

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Chapter 12The expression for the sl

• Page 212:

TTHE DERIVATIVE [CHAP. 12EXAMPLESD,

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96 THE DERIVATIVE [CHAP. 12(b) Forf

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

• Page 224:

100 MORE ON THE DERIVATIVE [CHAP. 1

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102 MORE ON THE DERIVATIVE [CHAP. 1

• Page 232:

Chapter 14Maximum and Minimum Probl

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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

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0*1114 MAXIMUM AND MINIMUM PROBLEMS

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Chapter 15The Chain RulelS.lCOMPOSI

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118 THE CHAIN RULE [CHAP. 15EXAMPLE

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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

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Chapter 16Implicit DifferentiationA

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128 IMPLICIT DIFFERENTIATION[CHAP.

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130 THE MEAN-VALUE THEOREM AND THE

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132 THE MEAN-VALUE THEOREM AND THE

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134 THE MEAN-VALUE THEOREM AND THE

• Page 296:

Chapter 18Rectilinear Motion and In

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138 RECTILINEAR MOTION AND INSTANTA

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140 RECTILINEAR MOTION AND INSTANTA

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142 RECTILINEAR MOTION AND INSTANTA

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144 INSTANTANEOUS RATE OF CHANGE[CH

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146 INSTANTANEOUS RATE OF CHANGE [C

• Page 320:

148 RELATED RATES [CHAP. 20In Fig.

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150 RELATED RATES [CHAP. 20- XFig.

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152 RELATED RATES[CHAP. 20and, by (

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154 RELATED RATES [CHAP. 2020.21 A

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156 APPROXIMATION BY DIFFERENTIALS;

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158 APPROXIMATION BY DIFFERENTIALS;

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1 60APPROXIMATION BY DIFFERENTIALS;

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1 62 HIGHER-ORDER DERIVATIVES [CHAP

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~~164 HIGHER-ORDER DERIVATIVES [CHA

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166 HIGHER-ORDER DERIVATIVES [CHAP.

• Page 360:

168 THE SECOND DERIVATIVE AND GRAPH

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170 THE SECOND DERIVATIVE AND GRAPH

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172 THE SECOND DERIVATIVE AND GRAPH

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174 THE SECOND DERIVATIVE AND GRAPH

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23.7 If, for all x,f’(x) > 0 andf

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178 THE SECOND DERIVATIVE AND GRAPH

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180 MORE MAXIMUM AND MINIMUM PROBLE

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182 MORE MAXIMUM AND MINIMUM PROBLE

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184 MORE MAXIMUM AND MINIMUM PROBLE

• Page 396:

186 ANGLE MEASURE [CHAP. 25and so o

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188 ANGLE MEASURE [CHAP. 25Solved P

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Chapter 2626.1 GENERAL DEFINITIONSi

• Page 408:

192 SINE AND COSINE FUNCTIONS [CHAP

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194 SINE AND COSINE FUNCTIONS [CHAP

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~~~ ~196 SINE AND COSINE FUNCTIONS

• Page 420:

198 SINE AND COSINE FUNCTIONS [CHAP

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200 SINE AND COSINE FUNCTIONS [CHAP

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Chapter 27Graphs and Derivatives of

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204 GRAPHS AND DERIVATIVES OF SINE

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206 GRAPHS AND DERIVATIVES OF SINE

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208 GRAPHS AND DERIVATIVES OF SINE

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210 GRAPHS AND DERIVATIVES OF SINE

• Page 448:

212GRAPHS AND DERIVATIVES OF SINE A

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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

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224 ANTIDERIVATIVES [CHAP. 29(ii) F

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226 ANTIDERIVATIVES [CHAP. 2929.5 A

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228 ANTIDERIVATIVES [CHAP. 2929.13

• Page 484:

230 THE DEFINITE INTEGRAL [CHAP. 30

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232 THE DEFINITE INTEGRAL [CHAP. 30

• Page 492:

234 THE DEFINlTE INTEGRAL (CHAP.[I)

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’)236 THE DEFINITE INTEGRAL [CHAP

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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:

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252 APPLICATIONS OF INTEGRATION I:

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254 APPLICATIONS OF INTEGRATION I:

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256 APPLICATIONS OF INTEGRATION I:

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258 APPLlCATlONS OF INTEGRATION 11:

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260 APPLICATIONS OF INTEGRATION 11:

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262 APPLICATIONS OF INTEGRATION 11:

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264 APPLICATIONS OF INTEGRATION 11:

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266 APPLICATIONS OF INTEGRATION I1

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• Page 564:

270 THE NATURAL LOGARITHM [CHAP. 34

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272 THE NATURAL LOGARITHM [CHAP. 34

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(b)27434.14THE NATURAL LOGARITHM11(

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276 EXPONENTIAL FUNCTIONS [CHAP. 35

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278EXPONENTIAL FUNCTIONS[CHAP. 3535

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280 EXPONENTIAL FUNCTIONS [CHAP. 35

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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

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290 L'HQPITAL'S RULE; EXPONENTIAL G

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

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294 INVERSE TRIGONOMETRIC FUNCTIONS

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296 INVERSE TRIGONOMETRIC FUNCTIONS

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298 INVERSE TRIGONOMETRIC FUNCTIONS

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300 INVERSE TRIGONOMETRIC FUNCTIONS

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302 INVERSE TRIGONOMETRIC FUNCTIONS

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above304 INVERSE TRIGONOMETRIC FUNC

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~ du=dx306 INTEGRATION BY PARTS [CH

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308 INTEGRATION BY PARTS[CHAP. 3838

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3 10INTEGRATION BY PARTS[CHAP. 38Su

• Page 648:

312 TRIGONOMETRIC INTEGRANDS AND TR

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314 TRIGONOMETRIC INTEGRANDS AND TR

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316 TRIGONOMETRIC INTEGRANDS AND TR

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318 TRIGONOMETRIC INTEGRANDS AND TR

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Chapter 40Integration of Rational F

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322 THE METHOD OF PARTIAL FRACTIONS

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324 THE METHOD OF PARTIAL FRACTIONS

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326 THE METHOD OF PARTIAL FRACTIONS

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328 THE METHOD OF PARTIAL FRACTIONS

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Appendix BBasic Integration Formula

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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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R-v QdA 1d 1

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Collinear points, 30Common logarith

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Infinite limits, 68Inflection point

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