Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

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30_Introduction to Analysis_[Ch. ! When solving the problems that follow, it is useful to know that if the limit lim/(x) exists and is positive, then Example tO. Prove that Solution. We have lim ln X-*0 X X-+Q lim [In /(*)] = In [Hm f (x)]. x-+a X-+Q Formula (*) is frequently used in the solution of problems. 253. lim [In (2*+!) *- 254. li . - X 255. limfjlnl/J-i^). 260*. llmn(^/a \) (a>0). ,_* \ " lX/ n^ V) pCLX ptX 256. lim *[ln(jt+l) Inx]. 261. lim-- . * 0). b) lim x * (see Problems 103 and 104). Find the* following limits that occur on one side: 264. a) lira *_^ . b)Jirn* p===. 265. a/lLutanh*; *-*-* fa Hm *" + 267 - a ) lim *-- b) limtanh*, b) Hm *->+ *-*+ where tanh^ = ^^~. 268. a) lim 266. a) lira V ; b) |im i 1+ ' T
Sec. 31 Limits 31 269. a) 270. lim-^4i; a) ' Hm-^-; x~ * Construct the graphs of the following functions: 271**. y = \im (cos 2 "*). n->oo 272*. y=lim n-*c * * i x 273. y = \im J/V-t-a 2 a->o 274. t/ = li;n| 275. t/ = li -*< n (x^O). . 276. Transform the following mixed periodic fraction into a common fraction: a = 0.13555... Regard it as the limit of the corresponding finite fraction. 277. What will happen to the roots of the quadratic equation if the coefficient a approaches zero while the coefficients b and c are constant, and fc^=0? 278. Find the limit of the interior angle of a regular n-gon as n > oo. 279. Find the limit of the perimeters of regular n-gons inscribed in a circle of radius R and circumscribed about it as n - oo. 20. Find the limit of the sum of the lengths of the ordinates of the curve y = e~*cos nx, drawn at the points x = 0, 1, 2, ..., n, as n *oo. 281. Find the limit of the sum of the areas of the squares constructed on the ordinates of the curve as on bases, where x=^l, 2, 3, ..., n, provided that n *oo. 282. Find the limit of the perimeter of a broken line M^.. .Mn inscribed in a logarithmic spiral
Page 2 and 3: TEXT FLY WITHIN THE BOOK ONLY
Page 5: OUP-43 30-1.71 5,000 OSMANIA UNIVER
Page 8 and 9: T. C. BapaneHKoe. C. M. Koean, r Jl
Page 10 and 11: TO THE READER MIR Publishers would
Page 12 and 13: Contents Sec. 6. Integrating Certai
Page 14 and 15: 8 Chapter Contents Sec. 13. Linear
Page 17 and 18: Sec. 1. Functions Chapter I INTRODU
Page 19 and 20: Sec 1] Functions 13 8. Find the rat
Page 21 and 22: Sec. 1]_Functions_15 34. Prove that
Page 23 and 24: Sec. 2] Graphs of Elementary functi
Page 25 and 26: Sec. 2] Graphs of Elementary functi
Page 27 and 28: Sec. 2] Graphs of Elementary Functi
Page 29 and 30: Sec. 3] Limits 23 For the existence
Page 31 and 32: Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
Page 33 and 34: Sec. #] Limits 27 209. lim 210. lim
Page 35: ^Sec 3] Limits Therefore, Example 9
Page 39 and 40: Sec. 4] Infinitely Small and Large
Page 41 and 42: Sec. 4] Infinitely Small and Large
Page 43 and 44: Sec. 5] Continuity of Functions 37
Page 45 and 46: Sec. 5]_Continuity of Functions_39
Page 47 and 48: Sec. 5] Continuity of Functions 334
Page 49 and 50: Sec. 1] Calculating Derivatives Dir
Page 51 and 52: Sec. 1] Calculating Derivatives Dir
Page 53 and 54: Sec. 2]__Tabular Differentiation_47
Page 55 and 56: Sec. 2]_Tabular Differentiation_49
Page 57 and 58: Sec. 2] Tabular Differentiation_ 51
Page 59 and 60: Sec. 2] Tabular Differentiation 504
Page 61 and 62: Sec. 2] Tabular Differentiation 55
Page 63 and 64: Sec. 3] The Derivatives of Function
Page 65 and 66: Sec. 3] The Derivatives of Function
Page 67 and 68: Sec 4] Geometrical and Mechanical A
Page 69 and 70: Sec. 4\ Geometrical and Mechanical
Page 71 and 72: Sec. 4] Geometrical and Mechanical
Page 73 and 74: Sec. 5]_Derivatives of Higher Order
Page 75 and 76: Sec. 5]_Derivatives of Higher Order
Page 77 and 78: Sec. 6] Differentials of First and
Page 79 and 80: Sec. 6}_Differentials of First and
Page 81 and 82: Sec. 7]_Mean-Value Theorems_75 745.
Page 83 and 84: Sec. 8] Taylor's Formula 77 765. a)
Page 85 and 86: Sec. 9]_ V Hospital-Bernoulli Rule
Page 87 and 88:
Sec. 9] L' Hospital-Bernoulli Rule
Page 89 and 90:
Chapter III THE EXTREMA OF A FUNCTI
Page 91 and 92:
Sec. 1] The Extrema of a Function o
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Page 95 and 96:
Page 97 and 98:
Sec. 2]_The Direction of Concavity.
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Sec. 3]_Asymptotes_93 Example 2. Fi
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Sec. 3]_Asymptotes_95 Solution. Sin
Page 103 and 104:
Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
Page 107 and 108:
Sec. />] Differential of an Arc. Cu
Page 109 and 110:
Sec 5] Differential of an Arc. Curv
Page 111 and 112:
Sec. 5]_Differential of an Arc. Cur
Page 113 and 114:
Sec. 1. Direct Integration 1. Basic
Page 115:
Sec 1] Direct Integration_109 1048*
Page 119 and 120:
Sec. 2] Integration by Substitution
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Sec. 2] Integration by Substitution
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Sec. 3\_Integration by Parts_H7 Exa
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Sec 4] Standard Integrals Containin
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Sec. 5] Integration of Rational Fun
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Sec. 5] In t egration of Rational F
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Sec. 6] Integrating Certain Irratio
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Sec. 6] Integrating Certain Irratio
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Sec. 7]_Integrating Trigonometric F
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Sec. 7]_Integrating Trigonometric F
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Sec. 8] Integration of Hyperbolic F
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Sec. 11] Using Reduction Formulas 1
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Sec. 12] Miscellaneous Examples on
Page 145 and 146:
Sec. 1] The Definite Integral as th
Page 147 and 148:
Sec. 2]_Evaluating Definite Integra
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Sec. 3]_Improper Integrals_143 Sec.
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Sec. 3] Improper Integrals 145 Solu
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Sec. 4] Change of Variable in a Def
Page 155 and 156:
Sec. 5] Integration by Parts 149 Bu
Page 157 and 158:
Sec. 6] Mean-Value Theorem 151 In p
Page 159 and 160:
Sec. 7] The Areas of Plane Figures
Page 161 and 162:
Sec. 7] The Areas of Plane Figures
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Sec. 7\_The Areas of Plane Figures_
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Sec. 8] The Arc Length of a Curve 1
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Sec, 9] Volumes of Solids 161 1673.
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Sec. 9] Volumes of Solids 163' Solu
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Sec. 9]_Volumes of Solids_165 1693.
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Sec. 10] The Area of a Surface of R
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Sec. II] Moments. Centres of Gravit
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Sec 11] Moments. Centres of Gravity
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&?c. 12] Applying Definite Integral
Page 181 and 182:
Sec. 12] Applying Definite Integral
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Page 185 and 186:
Page 187 and 188:
Sec. 1) Basic Notions [SI Putting *
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Sec. 1]_Basic Notions_183 1790*. Le
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Sec. 3] Partial Derivatives 185 180
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Sec. 4]_Total Differential of a Fun
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Sec. 4}_Total Differential of a Fun
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Sec. 5]_Differentiation of Composit
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Sec. 6]_Derivative in a Given Direc
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Sec. 6] Derivative in a Given Direc
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Sec. 7]_Higher-Order Derivatives an
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Sec. 7] Higher-Order Derivatives an
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Sec. 7]_Higher-Order Derivatives an
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Sec. ti\_Integration of Total Diffe
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Sec. 9]_Differentiation of Implicit
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Sec. 9] Differentiation of Implicit
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Sec. 9]_Differentiation of Implicit
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Sec. 10]_Change of Variables_2H 196
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Sec. 10]_Change of Variables_213 or
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Sec. 10] Change of Variables 215 an
Page 223 and 224:
Sec. 11]_The Tangent Plane and the
Page 225 and 226:
Sec. 11]_The Tangent Plane and the
Page 227 and 228:
Sec. 12] Taylor's Formula for a Fun
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Sec. 13]_The Extremum of a Function
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Sec. 13] The Extremutn of a Functio
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Sec. 14] Finding the Greatest and S
Page 235 and 236:
Sec. 14] Finding the Greatest and S
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Sec. 15] Singular Points of Plane C
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Sec. 16] Envelope 233 Solution. The
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Sec. 18] The Vector Function of a S
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Sec. 18]_The Vector Function of a S
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Sec. 19] The Natural Trihedron of a
Page 247 and 248:
Sec. 19] The Natural Trihedron of a
Page 249 and 250:
Sec. 20] Curvature and Torsion of a
Page 251 and 252:
Sec. 20]_Curvature and Torsion of a
Page 253 and 254:
Sec. 1] The Double Integral in Rect
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Page 257 and 258:
Page 259 and 260:
Sec. 2]_Change of Variables in a Do
Page 261 and 262:
Sec. 2]_Change of Variables in a Do
Page 263 and 264:
Sec 3]_Computing Areas_257 2177. Co
Page 265 and 266:
Sec. 5]_Computing th Area* of Surfa
Page 267 and 268:
Sec. 6]_Applications of the Double
Page 269 and 270:
Sec. 7] Triple Integrals 2o3 Evalua
Page 271 and 272:
Sec 7\ Triple Integrals 265 We ther
Page 273 and 274:
gee. 7]_Triple Integrals_267 2251.
Page 275 and 276:
Sec. 8] Improper Integrals Dependen
Page 277 and 278:
Sec. 8]_Improper Integrals Dependen
Page 279 and 280:
See. 9] Line Integrals 273 Test for
Page 281 and 282:
Sec. 9]____Line Inlegtah____275 whe
Page 283 and 284:
Sec. 9]_Line Integrals__277 express
Page 285 and 286:
Sec. 9] Lim Integrals 279 (/-axis;
Page 287 and 288:
Sec 9] Line Integrals 281 Evaluate
Page 289 and 290:
Sec. 9] Line Integrals 2S& D. Appli
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Sec. 10] Surface Integrals 285 When
Page 293 and 294:
ec. 11]_The Ostrogradsky-Gauss Form
Page 295 and 296:
Sec. 12] Fundamentals of Field Theo
Page 297 and 298:
Sec. 12] Fundamentals of Field Theo
Page 299 and 300:
Sec. 1. Number Series Chapter VIII
Page 301 and 302:
Sec. 1]_Number Series_295 c) D'Alem
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Sec /\_Number Series_297 Note. For
Page 305 and 306:
2418. + + 0419 J___1_ . 4- 4- 4- '
Page 307 and 308:
Sec. 1] Number Series 2464. cos-2-)
Page 309 and 310:
Sec. 1]_Number Series_303 - V r(2-Q
Page 311 and 312:
Sec. 2]_Functional Series_305 In th
Page 313 and 314:
Sec. 2]_Functional Series_307 [the
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Sec. 2] Functional Series 309 Deter
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Sec. 3]_Taylor's Series_311 v8 s r
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Sec. 3] Taylor's Series 313 Since a
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Sec 3} Taylor's Series 315 2615. ln
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Sec. 3] Taylor's Series 317 2651. F
Page 325 and 326:
Sec. 4] Fourier Series 319 If a fun
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Sec. 4\__Fourier Series _321 2690.
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Sec 1 1 Verifying Solutions 323 It
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Sec. 21 First-Order Differential Eq
Page 333 and 334:
Sec. 3] Differential Equations with
Page 335 and 336:
Sec. 3\ Differential Equations with
Page 337 and 338:
Sec. 4]_First-Order Homogeneous Dif
Page 339 and 340:
Sec. 5] Bernoulli's Equation 333 Co
Page 341 and 342:
Sec. 6] Exact Differential Equation
Page 343 and 344:
Sec. 7] First-Order Differential Eq
Page 345 and 346:
Sec 8]_The Lagrange and Clairaut Eq
Page 347 and 348:
Sec. 9] Miscellaneous Exercises on
Page 349 and 350:
Sec. 9] Miscellaneous Exercises on
Page 351 and 352:
Sec. 10] Higher-Order Differential
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Sec. 10]__Higher-Order Differential
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Sec. 11]_Linear Differential Equati
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Sec. 12] Linear Differential Equati
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Page 361 and 362:
Page 363 and 364:
Sec. 14]_Euler's Equations__357 Fin
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Sec. 15]_Systems of Differential Eq
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.Sec. 16} Integration of Differenti
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Sec. 17] Problems on Fourier's Meth
Page 371 and 372:
Sec. 17]_Problems on Fourier's Meth
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Chapter X APPROXIMATE CALCULATIONS
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Sec. 1] Operations on Approximate N
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Sec. /]_Operations on Approximate N
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Sec, 2] Interpolation of Functions
Page 381 and 382:
Sec. 2] Interpolation of Functions
Page 383 and 384:
Sec. 3]_Computing the Real Roots of
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Page 395 and 396:
Page 398 and 399:
Table 3. Basic Table for Calculatin
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394 Approximate Calculations \Ch 10
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ANSWERS Chapter I 1. Solution. Sinc
Page 404 and 405:
398 Answers Form a table of values:
Page 406 and 407:
400 Answer* 271. Solution. If x & k
Page 408 and 409:
402 Answers b) /'(!)= lim 1^15^= li
Page 410 and 411:
404_Answers 494. , . 495. - - . 1 2
Page 412 and 413:
-406 Answers 640. Hint. The equatio
Page 414 and 415:
408_Answers_ 755. e xcosa sin(*sina
Page 416 and 417:
410_Answers_4 where R is the radius
Page 418 and 419:
412 Answers 1/~JT f Q \ J/max = TT=
Page 420 and 421:
414 Answers 976. j/mi n =1.32 when
Page 422 and 423:
416 Answers 1037. j/ 1040. 3x* 13 J
Page 424 and 425:
418 1 1137. 4- 6 3a . In I ocot3x|
Page 426 and 427:
420 Answers 4^8 Problem 1218*. 1221
Page 428 and 429:
422 Answers X arctan(* + 2). 1294.
Page 430 and 431:
424 Answers sin 5* , 1374. -!- 1377
Page 432 and 433:
426 Answers 3 1442. In l-x2 ' '" 2
Page 434 and 435:
428_Answers_ the function /(x) = --
Page 436 and 437:
430 Answers 1651. 3na 2 . 1652. n(b
Page 438 and 439:
432 Answers inertia /=Y 2jttf 1 r *
Page 440 and 441:
434 Answers along the large lower s
Page 442 and 443:
436 Answers cos^. ,810. * ox x* x d
Page 444 and 445:
438_Answers 1905. 1914. u(*. !/) =
Page 446 and 447:
440 Answers 1983. 3* + 4j/-H2z 169
Page 448 and 449:
442 Answers is ~ |/2 + -~=, the alt
Page 450 and 451:
444 Answers (binormal). 2103. bx *=
Page 452 and 453:
444_Answers_x = \ ' > (principal no
Page 454 and 455:
446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457:
448_Answers_ r 4 **/* [ rj-cos*(p--
Page 458 and 459:
450 Answers 2243. 2244. 1 i VT=T* ^
Page 460 and 461:
452 Answers 2 t 2297. ) ~ll . |-[(I
Page 462 and 463:
454_Answers__ TtD the flux, use the
Page 464 and 465:
456__Answers_ and when x | ^ =- , t
Page 466 and 467:
458 Answers
Page 468 and 469:
460 Answers %-l n=o (n)=^=%. 2873.
Page 470 and 471:
462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473:
464 Answers 2822. g-C*+; y=2*. 2824
Page 474 and 475:
466 Answers x + C z . 2914. #=
Page 476 and 477:
468_Answers_ A $007. 1) x = C cosco
Page 478 and 479:
470 Answers 3081. x = C l e 3082. x
Page 480 and 481:
472 Answers ,*. f (2"+ 1)JtJf dx. H
Page 482 and 483:
474 Answers 3131. a) 0.211; 0.389;
Page 484 and 485:
476 Appendix III. Inverse Quantitie
Page 486 and 487:
478 Appendix IV. Trigonometric Func
Page 488 and 489:
480 Appendix VI. Some Curves (for R
Page 490 and 491:
482 Appendix 11. Graphs //-=secx an
Page 492 and 493:
486 Appendix 24. Strophoid, * u s^-
Page 494 and 495:
Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
Page 498 and 499:
492 Index Infinite discontinuities
Page 500 and 501:
494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
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