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

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Hence, on the basis of formulas (1) and (2), we get and 1 R ~( fl i + 6i) JDL VaT+b*^ a a /. "fl' + a 1 1 a*b = b ~ 2 a Q (a 2 + b*)~~ a 2 ' 2 + b Thus, for a screw-line, the curvature and torsion are constants. 3 Frenet formulas: dr _ v ^v ___ T iP ^P_ v 7s~"~R ' 5s ~~ ' ~~~~R~T~~Q ds~~~~o"' 2104. Prove that if the curvature at all points of a line is zero, then the line is a straight line. 2105. Prove that if the torsion at all points of a curve is zero, then the 2106. curve is a Prove that plane curve. the curve x=\+3t + 2t 2 , y = 22t + 5t\ z=lt* is a plane curve; find the plane in which it lies. 2107. Compute the curvature of the following curves: a) x = cost, y = s'mt, z = cosh / at the / point = 0; b) x* 2 // -| z 2 2 = l, y -2x + z = Q at the point (1, 1, 1). 2108. Compute the curvature and torsion at curves: any point of the a) je = e'cos/, y = e i sint, z e*\ b) x^acosht, y asiuht. z = at (hyperbolic screw-line). 2109. Find the radii of curvature and torsion at an arbitrary point (x, y, z) of the curves: a) x 2 = b) x 9 = 2110. Prove that the tangential and normal components of acceleration w are expressed by the formulas dv v 2 VOif T
Sec. 20]_Curvature and Torsion of a Space Curve_24S 2111. A point is in = la cost +ja uniform motion along a screw-line r =* sin t + btk with velocity v. Compute its acceleration w. 2112. The equation of motion is Determine, at times / and /=!: 1) the curvature of the trajectory and 2) the tangential and normal components of the acceleration.
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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
Page 199 and 200: Sec. 6]_Derivative in a Given Direc
Page 201 and 202: Sec. 6] Derivative in a Given Direc
Page 203 and 204: Sec. 7]_Higher-Order Derivatives an
Page 205 and 206: Sec. 7] Higher-Order Derivatives an
Page 207 and 208: Sec. 7]_Higher-Order Derivatives an
Page 209 and 210: Sec. ti\_Integration of Total Diffe
Page 211 and 212: Sec. 9]_Differentiation of Implicit
Page 213 and 214: Sec. 9] Differentiation of Implicit
Page 215 and 216: Sec. 9]_Differentiation of Implicit
Page 217 and 218: Sec. 10]_Change of Variables_2H 196
Page 219 and 220: Sec. 10]_Change of Variables_213 or
Page 221 and 222: 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
Page 229 and 230: Sec. 13]_The Extremum of a Function
Page 231 and 232: Sec. 13] The Extremutn of a Functio
Page 233 and 234: Sec. 14] Finding the Greatest and S
Page 235 and 236: Sec. 14] Finding the Greatest and S
Page 237 and 238: Sec. 15] Singular Points of Plane C
Page 239 and 240: Sec. 16] Envelope 233 Solution. The
Page 241 and 242: Sec. 18] The Vector Function of a S
Page 243 and 244: Sec. 18]_The Vector Function of a S
Page 245 and 246: Sec. 19] The Natural Trihedron of a
Page 247 and 248: Sec. 19] The Natural Trihedron of a
Page 249: Sec. 20] Curvature and Torsion of a
Page 253 and 254: Sec. 1] The Double Integral in Rect
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
Page 291 and 292: 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
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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. #=
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
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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