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

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402 Answers b) /'(!)= lim 1^15^= lim^_ ==00; c) /'- Ax = lim = lim l sinAxl r , 2fe+l 2 \\ * + A *JL ~ I =1. 368. 5x 4 '= lim LîJLJ=-l; sin Ax | ,. r , , /2*-fl 12x 2 + 2. 369. l + 2x 2x 8 . 370. A*->+o Ax 3 1 5x2 6ax5 ji 371. . 372. mat m - l + b(m-\-n)t rn + n - 1 . 373. 374. T . 375. -JL ' 2x 5x 8 1 i JL 2 3x- 4 . 376. -* x 9 . Hint. y = x zx* = x 8 . 377. wy X , , 2 3x l/x . U*U. (c + dx) . o*7A 379. OIS7. 9y2 Â fW-4-9ft VJA -7- z-o ! . on 380. OOU. 1 * 4*- ** 2 (x 2 5x + 5) 2 x 2 (2x I) 2 1. l . r^ 7= 2. 382. 5 cos x 3 sinx. 383. - . 384. 2 2 Y 2 (1 V 2) sin 2x (sinx cos x) 385. / 2 sin/. 386. = y' 0. 387. cot x . ^- sin 2 x ' 388. arcsinx-f- * j/~i X 2 " y O *>y* v 5 389. x arc tan x. 390. xV(x-f-7). 391. xex . 392. e x ^f-. 393. x . 394. e* (cos x sin x). 395. x 2 e x . 396. e x 1 /ârc sin x+ ^ . 397. * \ K 1 (21n *~ 1) x2/ In . 2 x 398. 3x 2 lnx. 399. 1+1H/-A.400. x x ^1 1 . 401. sinh x + x coshx. 2 x 2 x In 10 x 402 toco.h^dnh* cosh 403 _ tanh2x 4()4 -3(xln* + sinh*cosM 2 x xln 2 xsmh 2 x 2x 2 1 1 arc; 405. . 406. sinh x arc sin -f- y. x y x*\ arc coshx 1 -|-2x arc tanhx 3a fax + b \ 2 407. . . .... , x 2 Vx2 l (1x 2 2 c ) fax-}-b\ 2 I I . \ c / 411. 12fl6+18^ 2 //. 412. 16x(3 + 2x 2 8 . ) 413. . 414. . (2x I) 8 /I x 2 41B . fex2 - 416. -.1/j/l;-!. 418. 1-tan 2 x + tan 4 >X x 8 2 ) f X ^ COS 2 X 419. "~ * 420. 2 15 cos 2 x sinx. 421.^^^--- .Hint. x = sin- 2 2 sin 1 + 2 x V cot x sin 8 2/ 422. sin * (1 3cosx) 3 423> sin> ^ 424> 3 cos x + 2 sinx cos*x 2 V^lSsinx lOcosx 1 425. _2cos^ Ssin^^ ^ cos 4 x 2 1^1 x2 ' >^1-|- arc sinx 427. L _3jari!*)l > ^ 2(l+Jt ! ' ' )Vârctanjc /"l x* (1 + x 2 2 ) (arc tan x)
429. . 430. X cos(x 5 3 Answers 403 - 2 2* -}- 1) , x 432. (2x-5) X 5x+l) -. 433. asin(ax + p). 434. sin(2/+q>). X 2 COS 2 435. 2 ?21i . 436. sin x 2 X ""* s . n2 ^ a . 437. x cos 2x 2 sin 3x 2 . 438. Solution. "~ 2 l,- (2x)' = - . 439. . 440. T 1 . V^l (2x) 2 lA 4x 2 x/V 1 2 J^jc x 2 442. -p^jf - 443. 10xe~* 2 441. "~ 1 1 + x 2 . 444. 2*5~* f In 5. 445. 2xi0 2* (1 + x In 10). pX 2 446. sin2 t + 2 t / cos 2 f In 2. 447. 448. . 449. cot x log e. 450. 1 ^ . 451. ^^ . x 452. (- 2 . 453. x xlnx L + J x (e -}-5sinx ] 4 arc sin x) VI 454. (1 -j- In x) x (1 -j-x ) arc tanx 2x \ In x-j- 1 x / x \ ' x 3 455. Solution, y' (sin 3 sin 2 5x cos 5x5 cos 2 - x f x\ \ x2 H- sin 3 5x2 cos f sin -^ -- 5x)' cos2 - + sin 3 5x I cos 2 -=- ) = ) 6 \ 6 J 3 = 15 sin 2 5xcos5xcos 2 + xx ~sin 8 5xcos -r- sin -- . 4x + 3 x 2 -f4x 6 x' x 1 456. 457. . 458. . 459. x . 463. 465. 4x 3 (a 2x 3 ) (a 5x 3 ). 474. sin 3 x cos 2 x. 476. 10tan5xsec 2 5x. 477. xcosx 2 . 478. 3/ 1 sin 2/. sin 4 x cos 4 x 479. 3cos*cos2*. 480. tan'*. 481. -f - . 482 . = sin 4 x 2 K a sin 2 x + p cos 2 ---- x . . 1 arc sin x (2 arc cos x arc sin . x) OB 2 AQC -. 484. r 485. - - . 486. 2 ]/"l x2 ' 2 x 1 V^x 1+x 2 487. V 491. , 492. arc sin , ... 1^2x x a V 1 25x 2 arc sin 5x . '
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TEXT FLY WITHIN THE BOOK ONLY
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OUP-43 30-1.71 5,000 OSMANIA UNIVER
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T. C. BapaneHKoe. C. M. Koean, r Jl
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TO THE READER MIR Publishers would
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Contents Sec. 6. Integrating Certai
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8 Chapter Contents Sec. 13. Linear
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Sec. 1. Functions Chapter I INTRODU
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Sec 1] Functions 13 8. Find the rat
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Sec. 1]_Functions_15 34. Prove that
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary functi
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Sec. 2] Graphs of Elementary Functi
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Sec. 3] Limits 23 For the existence
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Sec. 3] 179. Hm (Vn + 1 \f~n). n -+
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Sec. #] Limits 27 209. lim 210. lim
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^Sec 3] Limits Therefore, Example 9
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Sec. 31 Limits 31 269. a) 270. lim-
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Sec. 4] Infinitely Small and Large
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Sec. 4] Infinitely Small and Large
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Sec. 5] Continuity of Functions 37
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Sec. 5]_Continuity of Functions_39
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Sec. 5] Continuity of Functions 334
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Sec. 1] Calculating Derivatives Dir
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Sec. 1] Calculating Derivatives Dir
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Sec. 2]__Tabular Differentiation_47
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Sec. 2]_Tabular Differentiation_49
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Sec. 2] Tabular Differentiation_ 51
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Sec. 2] Tabular Differentiation 504
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Sec. 2] Tabular Differentiation 55
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Sec. 3] The Derivatives of Function
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Sec. 3] The Derivatives of Function
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Sec 4] Geometrical and Mechanical A
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Sec. 4\ Geometrical and Mechanical
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Sec. 4] Geometrical and Mechanical
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Sec. 5]_Derivatives of Higher Order
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Sec. 5]_Derivatives of Higher Order
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Sec. 6] Differentials of First and
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Sec. 6}_Differentials of First and
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Sec. 7]_Mean-Value Theorems_75 745.
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Sec. 8] Taylor's Formula 77 765. a)
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Sec. 9]_ V Hospital-Bernoulli Rule
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Sec. 9] L' Hospital-Bernoulli Rule
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Chapter III THE EXTREMA OF A FUNCTI
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Sec. 1] The Extrema of a Function o
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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
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Sec 4] Graphing Functions by Charac
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CM CM -h CO O non ' > C " -~ a - >
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Sec. />] Differential of an Arc. Cu
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Sec 5] Differential of an Arc. Curv
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Sec. 5]_Differential of an Arc. Cur
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Sec. 1. Direct Integration 1. Basic
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Sec 1] Direct Integration_109 1048*
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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
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Sec. 1] The Definite Integral as th
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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
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Sec. 5] Integration by Parts 149 Bu
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Sec. 6] Mean-Value Theorem 151 In p
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Sec. 7] The Areas of Plane Figures
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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
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Sec. 12] Applying Definite Integral
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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
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Sec. 11]_The Tangent Plane and the
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Sec. 11]_The Tangent Plane and the
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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
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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
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Sec. 19] The Natural Trihedron of a
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Sec. 20] Curvature and Torsion of a
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Sec. 20]_Curvature and Torsion of a
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Sec. 1] The Double Integral in Rect
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Sec. 2]_Change of Variables in a Do
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Sec. 2]_Change of Variables in a Do
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Sec 3]_Computing Areas_257 2177. Co
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Sec. 5]_Computing th Area* of Surfa
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Sec. 6]_Applications of the Double
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Sec. 7] Triple Integrals 2o3 Evalua
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Sec 7\ Triple Integrals 265 We ther
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gee. 7]_Triple Integrals_267 2251.
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Sec. 8] Improper Integrals Dependen
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Sec. 8]_Improper Integrals Dependen
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See. 9] Line Integrals 273 Test for
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Sec. 9]____Line Inlegtah____275 whe
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Sec. 9]_Line Integrals__277 express
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Sec. 9] Lim Integrals 279 (/-axis;
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Sec 9] Line Integrals 281 Evaluate
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Sec. 9] Line Integrals 2S& D. Appli
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Sec. 10] Surface Integrals 285 When
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ec. 11]_The Ostrogradsky-Gauss Form
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Sec. 12] Fundamentals of Field Theo
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Sec. 12] Fundamentals of Field Theo
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Sec. 1. Number Series Chapter VIII
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Sec. 1]_Number Series_295 c) D'Alem
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Sec /\_Number Series_297 Note. For
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2418. + + 0419 J___1_ . 4- 4- 4- '
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Sec. 1] Number Series 2464. cos-2-)
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Sec. 1]_Number Series_303 - V r(2-Q
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Sec. 2]_Functional Series_305 In th
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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
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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
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Sec. 3] Differential Equations with
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Sec. 3\ Differential Equations with
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Sec. 4]_First-Order Homogeneous Dif
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Sec. 5] Bernoulli's Equation 333 Co
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Sec. 6] Exact Differential Equation
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Sec. 7] First-Order Differential Eq
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Sec 8]_The Lagrange and Clairaut Eq
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Sec. 9] Miscellaneous Exercises on
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Sec. 9] Miscellaneous Exercises on
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Sec. 10] Higher-Order Differential
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Sec. 10]__Higher-Order Differential
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Sec. 11]_Linear Differential Equati
Page 357 and 358: Sec. 12] Linear Differential Equati
Page 363 and 364: Sec. 14]_Euler's Equations__357 Fin
Page 365 and 366: Sec. 15]_Systems of Differential Eq
Page 367 and 368: .Sec. 16} Integration of Differenti
Page 369 and 370: Sec. 17] Problems on Fourier's Meth
Page 371 and 372: Sec. 17]_Problems on Fourier's Meth
Page 373 and 374: Chapter X APPROXIMATE CALCULATIONS
Page 375 and 376: Sec. 1] Operations on Approximate N
Page 377 and 378: Sec. /]_Operations on Approximate N
Page 379 and 380: 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 389 and 390: Sec. 4] Numerical Integration of Fu
Page 391 and 392: Sec. 5] Numerical Integration of Or
Page 398 and 399: Table 3. Basic Table for Calculatin
Page 400 and 401: 394 Approximate Calculations \Ch 10
Page 402 and 403: 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 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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