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

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54 Differentiation of Functions_[C/t. 2 633. t/= 1Vsin* 534. y = 1 In Ji- + 1 In J^| + 1 arc tan x. 535. /(*) = 1 In (1 + *)- In (*'-*+ 1) + arc tan 536. f(*) = 537. y = s'mh*2x. 542. y=r. arc cosh In*. 538. y = e a* cosh px. 543. = arc tanh (tan x). 539. y = ianh*2x. 544. # = arc coth (sec*). 540. y = Insinh2jc. 545. y = arc tanh y^ . 541. t/ = arcsinh~. 546. t/ = -2-( A:a~ 547. r/= -^+-- 548. Find y', if: a) y=\x\\ b) = *|*|. Construct the graphs of the functions y and y'. 549. Find #' if 550. Find f (x) if 551. Calculate /' (0) if 1 x for x
Sec. 2] Tabular Differentiation 55 555. Find / (0) -f xf (0) of the function f(x)=e-*. 556. Find /(3) + (x 3)/' (3) of the function f(x) = Y\ + x. 557. Given the functions f(x) = tar\x and = (p(*) ln(l x)> find n0) tind q/(or 558. Given the functions /(x)=l x and = l cp(jc) sin^ r find nna 2^ . ff (1) 559. Prove that the derivative of an even function is an odd function, tion. and the derivative of an odd function is an even func- 560. Prove that the derivative of a periodic function is also a periodic function. 561. Show that the function y = xy' xe~* satisfies the equation = d-x)y- 562. Show that the function y = xe~? satisfies the e
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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
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 and 36: ^Sec 3] Limits Therefore, Example 9
Page 37 and 38: Sec. 31 Limits 31 269. a) 270. lim-
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: Sec. 2] Tabular Differentiation 504
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
Page 97 and 98: Sec. 2]_The Direction of Concavity.
Page 99 and 100: Sec. 3]_Asymptotes_93 Example 2. Fi
Page 101 and 102: Sec. 3]_Asymptotes_95 Solution. Sin
Page 103 and 104: Sec 4] Graphing Functions by Charac
Page 105 and 106: 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
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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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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
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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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Page 257 and 258:
Page 259 and 260:
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
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Sec. 3]_Computing the Real Roots of
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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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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
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398 Answers Form a table of values:
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400 Answer* 271. Solution. If x & k
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402 Answers b) /'(!)= lim 1^15^= li
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404_Answers 494. , . 495. - - . 1 2
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-406 Answers 640. Hint. The equatio
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408_Answers_ 755. e xcosa sin(*sina
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410_Answers_4 where R is the radius
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412 Answers 1/~JT f Q \ J/max = TT=
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414 Answers 976. j/mi n =1.32 when
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416 Answers 1037. j/ 1040. 3x* 13 J
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418 1 1137. 4- 6 3a . In I ocot3x|
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420 Answers 4^8 Problem 1218*. 1221
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422 Answers X arctan(* + 2). 1294.
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424 Answers sin 5* , 1374. -!- 1377
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426 Answers 3 1442. In l-x2 ' '" 2
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428_Answers_ the function /(x) = --
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430 Answers 1651. 3na 2 . 1652. n(b
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432 Answers inertia /=Y 2jttf 1 r *
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434 Answers along the large lower s
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436 Answers cos^. ,810. * ox x* x d
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438_Answers 1905. 1914. u(*. !/) =
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440 Answers 1983. 3* + 4j/-H2z 169
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442 Answers is ~ |/2 + -~=, the alt
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444 Answers (binormal). 2103. bx *=
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444_Answers_x = \ ' > (principal no
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446 Answers 2 a a a 2139. ( dy(f(x,
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448_Answers_ r 4 **/* [ rj-cos*(p--
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450 Answers 2243. 2244. 1 i VT=T* ^
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452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
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456__Answers_ and when x | ^ =- , t
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458 Answers
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460 Answers %-l n=o (n)=^=%. 2873.
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462 Answers Chapter IX 2704. Yes. 2
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464 Answers 2822. g-C*+; y=2*. 2824
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466 Answers x + C z . 2914. #=
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468_Answers_ A $007. 1) x = C cosco
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470 Answers 3081. x = C l e 3082. x
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472 Answers ,*. f (2"+ 1)JtJf dx. H
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474 Answers 3131. a) 0.211; 0.389;
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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
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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