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

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428_Answers_ the function /(x) = -- on the interval [0,1] where the division points have -- the form xk =l+ n (& = !, 2..... n). Therefore, lim sn = f ^2- = In 2. n -> oo J 1 H-* t520. j-7 o . 1521. ~ . 1522. 152 p -|- 1 o = - oo334- 1523 - "T 1524 - ^r o 1525 - - -4o 1526. -jln-| 4 . 1527. In . j 1528. 35~-321n3. 1529. arc tan 3 arc tan 2 = = arc tan . 1530. ln-| . 1531. ^. 1532. 1 -1= . V 5 O o 4 o 1535. ~ In 1 1533. ~. 1534. ~. 1536. %-+\-. 1537. |- 1538. In 2. 1539. 1 cosh 1540. 0. 1541. ^7= + ir. 1542. arc tang ~. 1543. sinh 1 = -i- (* } . 4 2 9^36 \ e J 1544. tanh(ln3) tanh (In 2) = -^- . 1545. ^- + -|-sinh 2ji. 1546. 2. 1547. Di- verges. 1548. ; , if pl; diverges, if p0. 1575. Hint. ? (p)=\f (x) dx+( f (x) dx, where 1 -xp ~ l e~*. The first integral converges when p>0, the second when p is arbitrary. 1576. No. 1577. 2 /~~2 (" /7rff. 1578 00 IL 2 2 In 3 1580. C^f r f !?"**#. 1581. x = (6 a)/+a. 1582.4 21n3. 1583.8 ^JT. J 1+' 1 21^3 1584. 2~ , 1585. -^ 1586. -^ . 1587. 1~~. 1588. /""S- 2 1/5 2 1^1 4-^ 4 3 1589. 4 n. 1590. ~ 1" H2. 1591. In . ^-|^ 1592. ~ + T- 1593 - nr o y ^ 4 o
Answer^ 429 1594. -- . 1599. -- 1 I 600 - 1- 1601. 1 6(>2. ^rr^ 4-(^ + l^. 1603. 1. 1604. z 2 Z o 2 . 2 . 1605. z\ b z- I 606 - Solution. r(p+l)=C xPe~* dx. Applying the formula of integration by parts, we put *P U, e~ x dx = dv. Whence and du = px. p ~ l dx, v= e~* If p is a natural number, then, applying formula (*) p times and taking into account that we get: t608 . ^7 1>35 " (2fe 2k 246 2-4-6 2 1-3-5 ... (2k , ii n = 2k-\- 1 is an odd number _ 63jt / _ 128 . / 9 " 3l5 ' 10 ~" 512 ' 00 is an even number; V W ... - 1609. ^B ^, '-Hr^ Hint. Put 1610. a) Plus; b) minus; c) plus Hint. Sketch the graph of the integrand for values of the argument on the interval of integration 1611. a) First; b) second; c) first. 1612. ~ 1613.a. 1614. ~. 1615. -| 1617. 2
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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
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Sec. 12] Linear Differential Equati
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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
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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
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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 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 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
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478 Appendix IV. Trigonometric Func
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480 Appendix VI. Some Curves (for R
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482 Appendix 11. Graphs //-=secx an
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486 Appendix 24. Strophoid, * u s^-
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Absolute error Absolute value 367 o
Page 496 and 497:
490 Index Differential equations of
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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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