Schaum's Outline of Theory and Problems of Beginning Calculus

PrefaceThis **Outline** is limited to the essentials **of** calculus. It carefully develops, giving all steps, the principles**of** 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 **and**arithmetical 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 ins**of**ar as the use **of** a graphing calculator enhances theirunderst**and**ing **of** the subject).2. Treatment **of** several topics have been exp**and**ed:(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 pro**of** 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 **of**the 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

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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

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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)

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100 MORE ON THE DERIVATIVE [CHAP. 1

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102 MORE ON THE DERIVATIVE [CHAP. 1

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Chapter 14Maximum and Minimum Probl

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106 MAXIMUM AND MINIMUM PROBLEMS [C

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108 MAXIMUM AND MINIMUM PROBLEMS [C

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110 MAXIMUM AND MINIMUM PROBLEMS [C

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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

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122 THE CHAIN RULE [CHAP. 15The onl

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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

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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

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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.

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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

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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

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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

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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

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212GRAPHS AND DERIVATIVES OF SINE A

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Chapter 28The Tangent andOther Trig

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216 THE TANGENT AND OTHER TRIGONOME

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218 THE TANGENT AND OTHER TRIGONOME

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220 THE TANGENT AND OTHER TRIGONOME

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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

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230 THE DEFINITE INTEGRAL [CHAP. 30

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232 THE DEFINITE INTEGRAL [CHAP. 30

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234 THE DEFINlTE INTEGRAL (CHAP.[I)

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’)236 THE DEFINITE INTEGRAL [CHAP

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Chapter 31The Fundamental Theorem o

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240 THE FUNDAMENTAL THEOREM OF CALC

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242 THE FUNDAMENTAL THEOREM OF CALC

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244 THE FUNDAMENTAL THEOREM OF CALC

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246 THE FUNDAMENTAL THEOREM OF CALC

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248 THE FUNDAMENTAL THEOREM OF CALC

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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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Chapter 3434.1 DEFlMTlONWe already

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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

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Chapter 36L’HGpital’s Rule ; Ex

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286 L'HOPITAL'S RULE; EXPONENTIAL G

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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

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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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336 ANSWERS. TO SUPPLEMENTARY PROBL

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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

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350 ANSWERS TO SUPPLEMENTARY PROBLE

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352 ANSWERS TO SUPPLEMENTARY PROBLE

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354 ANSWERS TO SUPPLEMENTARY PROBLE

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356 ANSWERS TO SUPPLEMENTARY PROBLE

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358 ANSWERS TO SUPPLEMENTARY PROBLE

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360 ANSWERS TO SUPPLEMENTARY PROBLE

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R-v QdA 1d 1

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364+I)ANSWERS TO SUPPLEM ENTARY PRO

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366 ANSWERS TO SUPPLEMENTARY PROBLE

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368 ANSWERS TO SUPPLEMENTARY PROBLE

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370 ANSWERS TO SUPPLEMENTARY PROBLE

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Collinear points, 30Common logarith

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Infinite limits, 68Inflection point

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RRadian, 185Radicals, 118Range, 47R