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

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U6__Indefinite Integrals \Ch. 4 1205. (f^+idx 1206*. f ff__ y 4 x2 J x J x2 1208. Evaluate the integral r dx 1207. J /*(!-*) by means of the substitution x=sin 2 /. 1209. Find by applying the hyperbolic substitution x = as\n\\t. Solution. We have: ]Aa 2 + x2 = 2 Solution. We have: ]Aa ]/~a 2 + x2 = 2 ]/~a + a2 suih 2 /=a cosh ^ and dx=a cosh /d/. Whence \ y a z -\-x 2 dx= ^ a cosh t-a cosh f ctf = Since and we finally get __ x ~ a where C 1=^C In a is a new arbitrary constant. 1210. Find putting x = a cosh/. Sec. 3. Integration by Parts C } ~~~2~ A formula for integration on by parts. If H =
Sec. 3\_Integration by Parts_H7 Example 1. Find Putting u In*, dv xdx we % \ x In xdx. dx x* have da , v ~~9 2 dx x * Whence Sometimes, to reduce a given integral to tabular form, one has to apply the fcrmula of integration by parts several times. In certain cases, integration by parts yields an equation from which the desired integral is determined. Example 2. Find We have \ e* cos x dx. V e* cos x dx = \ e x d (sin x) e* sin x \ e x sin AT dx e x sin x + Hence, whence + \ ex d (cos x) e* sin x -\- e x cos x \ e* cos x dx. \ e* cos x dx e x sm .v-j-e^cos x V e x cos x dx, cos v dx -- ~ (sin x -f- cos .v) -f C. Applying the formula of integration by parts, find the following integrals: 1211. ^ \nxdx. 1221. { x sin x cos x d\ 1212. Jarclanjcdx. a 1222* (jt +5x+6)cos $ 1213. Jarcsin A-rfjc. 1223. ^ x* \nxdx. 1214. Jjcsiiucr/.v. 1215. Jjccos3A'Jx. 1224. Jln'xd*. 1225. (^djc. 1216. Urfjc. 1226. f^d*. J J K A 1217. Jx-2-*rfx. 1227. Jjcarctanjcdjc. 1218**. JjV'd*. 1228. Jjcarcsmxdjc. 1219*. (x 2 ^2A' + 5)^*dA:. 1229. \n (x + V T~x*) dx. 1220*. '- 1230 '
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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
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
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
Page 121: Sec. 2] Integration by Substitution
Page 125 and 126: Sec 4] Standard Integrals Containin
Page 127 and 128: Sec. 5] Integration of Rational Fun
Page 129 and 130: Sec. 5] In t egration of Rational F
Page 131 and 132: Sec. 6] Integrating Certain Irratio
Page 133 and 134: Sec. 6] Integrating Certain Irratio
Page 135 and 136: Sec. 7]_Integrating Trigonometric F
Page 137 and 138: Sec. 7]_Integrating Trigonometric F
Page 139 and 140: Sec. 8] Integration of Hyperbolic F
Page 141 and 142: Sec. 11] Using Reduction Formulas 1
Page 143 and 144: 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
Page 149 and 150: Sec. 3]_Improper Integrals_143 Sec.
Page 151 and 152: Sec. 3] Improper Integrals 145 Solu
Page 153 and 154: 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
Page 163 and 164: Sec. 7\_The Areas of Plane Figures_
Page 165 and 166: Sec. 8] The Arc Length of a Curve 1
Page 167 and 168: Sec, 9] Volumes of Solids 161 1673.
Page 169 and 170: Sec. 9] Volumes of Solids 163' Solu
Page 171 and 172: 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
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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* ^
Page 460 and 461:
452 Answers 2 t 2297. ) ~ll . |-[(I
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454_Answers__ TtD the flux, use the
Page 464 and 465:
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
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
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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
Page 490 and 491:
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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