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

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192_Functions of Several Variables_[Ch. then 1859. Find ^ if I860. Find if r, where x = z = u v , where u = sin x, v = cos #, 1861. Find g and ~ if 1862. Find g and ~ if 2 = arc tan and */ = #*. = x y t 1863. Find ~ and j if where y = (x). z = f(u, v), where a = A: 1 1864. Find ~ and ~ if ow oy 1 t/ , v^e 2 = arc tan , where x = usmv, // = a 1865. Find ~ and | if 1866. Show that if z = /(u), where w = x^/ + ^. - 3*), where x = R cos 9 cos \|>> cos cp sin ip, = and *_a. 1867. Find if = /(*. y> *). where y = ( 1868. Show that if where / is a differentiable function, then dz dz e = R sin 9,
Sec. 6]_Derivative in a Given Direction_19S 1869. Show that the function w = f(u, v), where u = x + at, v = y + bt satisfy the equation dw dw , dT= a dt 1870. Show that the function satisfies the equation + = J 1871. Show that the function satisfies the equation x-^ + y -^ = j 1872. Show that the function 2 satisfies the equation (A- , dw y 2 ) ^- + xy-^ = 1873. The side of a rectangle x -^20 m increases at the rate of 5 m/sec, the other side f/ = 30 m decreases at 4 m/sec. What is the rate of change of the perimeter and the area of the rectangle? 1874. The equations of motion of a material point are What is the rate of recession of this point from the coordinate origin? 1875. Two boats start out from A at one time; one moves northwards, the other in a northeasterly direction. Their velocities are respectively 20 km/hr and 40 km/hr. At what rate does the distance between them increase? Sec. 6. Derivative in a Given Direction and the Gradient of a Function 1. The derivative of a function in a given direction. The derivative of a function z = /(#, y) in a given direction / = PP, is 7- 1900
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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
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.
Page 173 and 174: Sec. 10] The Area of a Surface of R
Page 175 and 176: Sec. II] Moments. Centres of Gravit
Page 177 and 178: Sec 11] Moments. Centres of Gravity
Page 179 and 180: &?c. 12] Applying Definite Integral
Page 181 and 182: Sec. 12] Applying Definite Integral
Page 187 and 188: Sec. 1) Basic Notions [SI Putting *
Page 189 and 190: Sec. 1]_Basic Notions_183 1790*. Le
Page 191 and 192: Sec. 3] Partial Derivatives 185 180
Page 193 and 194: Sec. 4]_Total Differential of a Fun
Page 195 and 196: Sec. 4}_Total Differential of a Fun
Page 197: Sec. 5]_Differentiation of Composit
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
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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
Page 438 and 439:
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--
Page 458 and 459:
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
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
Page 488 and 489:
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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