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

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228_Functions of Several Variables_[Ch. 6 Consequently, And so at f-g-, ~) the function reaches a maximum. Since f(x, */)=0 on the contour of the triangle, this maximum will be the greatest value, which is to say that the product will be greatest, if x = y = a x // = --, and the a 3 greatest value is equal to -^=- . Note The ploblem can also be solved by the methods of a conditional extremum, by seeking the maximum of the function u = xyz on the condition that x + y + z = a. 2034. From among all rectangular parallelepipeds with a given volume V, find the one whose total surface is the least. of 2035. For what dimensions does an open rectangular a given capacity V have the smallest surface? 2036. Of all triangles of a given perimeter 2p, find bathtub the one that has the greatest area. 2037. Find a rectangular parallelepiped of a given surface S with greatest volume. 2038. Represent a positive number a in the form of a product of four positive factors which have the least possible sum. 2039. Find a point M (x, y), on an x^- plane, the sum of the squares of the distances of which from three straight lines (x = Q, f/ = 0, x y+l=0) is the least 2040. Find a possible. triangle of a given perimeter 2p, which, upon being revolved about one of its sides, generates a solid of greatest volume. 2041. Given in a plane are three material points P, (x lf # t ), P*( x z' #2)* ^i(*> #3) with masses m m m what lf 2 , position of the point P(x, y) will the quadratic moment (the moment of inertia) of the given system of points relative to the point P (i.e., the sum m.P.P 2 +m P P 2 2 2 +m P P 3 3 2 ) be the least? 2042. Draw a plane through the point M (a, b t c) to form a tetrahedron of least volume with the planes of the coordinates. 2043. Inscribe in an ellipsoid a rectangular parallelepiped of greatest volume. 2044. Determine the outer dimensions of an open box with a given wall thickness 8 and capacity (internal) V so that the smallest quantity of material is used to make it. 3 . For
Sec. 14] Finding the Greatest and Smallest Values of Functions 229 2045. At what point of the ellipse r2 w 2 -T- + TT-1 ' a 2 does the tangent line to it form with the coordinate axes a triangle of smallest area? 2046*. Find the axes of the ellipse b z 5x 2 -f- 8xy + 5tf = 9. 2047. Inscribe in a given sphere a cylinder having the greatest total surface. 2048. The beds of two rivers (in a certain region) approximately represent a parabola y = x* and a straight line x y 2 = 0. It is required to connect these rivers by a straight canal of least length. Through what points will it pass? 2049. Find the shortest distance from the point M(l, 2, 3) to the straight line x __ y 2 ~~ _ 1 3 ~~ 2 ' 2050*. The points A and B are situated in different optical media separated by a straight line (Fig. 72). The velocity of *1 Fii
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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
Page 183 and 184: Sec. 12] Applying Definite Integral
Page 185 and 186: 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 and 198: 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: 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 and 250: Sec. 20] Curvature and Torsion of a
Page 251 and 252: 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
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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
Page 468 and 469:
460 Answers %-l n=o (n)=^=%. 2873.
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462 Answers Chapter IX 2704. Yes. 2
Page 472 and 473:
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
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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^-
Page 494 and 495:
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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