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

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434 Answers along the large lower side of the rectangle, and the y-axis, perpendicular to it in the middle. 1778. Solution. S=\ dv, on the other hand, 7^=a, u a J v t a at whence dt = dv, and consequently, the acceleration time is /=l- = S. 1780. M*=: \ (xt)ktdt + Ax=j(l'x*). 1781. Q =0.12 TRI* 9 cal. Hint. Use the Joule-Lenz law. 2 1782. V = (i/ Chapter VI x 2 )*. 1783. S = - (x + t/) /"4z 2 -}-3 (xy) 2 -f. _... 4 Oje// /? l/ j^S. 1786. /
_Answers__435 parabola y = ~- x*(x* + y > 0); ,k) the entire j/-plane; 1) the entire *t/-plane, with the exception of the coordinate origin; m) that part of the plane located above the parabola y* = x and to the right of the (/-axis, including the points of the t/-axis and excluding the points of the parabola (*:^0, y > V x)\ n) the entire place except points of the straight lines *=1 and t/ = 0; o) the family of concentric circles 2nk < x2 + y 2 < n (2k + 1 ) ( = 0, 1, 2, ...). 1793. a) First octant (including boundary); b) First, Third, Sixth and Eighth octants (excluding the boundary); c) a cube bounded by the planes x= 1, 1 y~ and z 1, including its faces; d) a sphere of radius 1 with centre at the origin, including its surface 1794. a) a plane; the level lines are straight lines parallel to the straight line *-f = */ 0; b) a paraboloid of revolution; the level lines are concentric circles with centre at the origin; c) a hyperbolic paraboloid; the level lines are equilateral hyperbolas; d) second-order cone; the level lines are equilateral hyperbolas; e) a parabolic cylinder, the generatrices of which are parallel to the straight line x + t/rf- 1 0; the level lines are parallel lines; f) the lateral surface of a quadrangular pyramid; the level lines are the outlines of squares; g)_level lines are parabolas y^-Cx z \ h) the level lines are parabolas y C ]fx ; i) the level Ifnes 2 2 are the circles C (* + y ) = 2*. 1795. a) Parabolas y^Cx* (C > 0); b) hyperbolas xy^C(\ C |< 1); c) circles jt 2 -f*/ 2 = C 2 ; d) straight lines y = ax-{-C; c) straight lines y-=Cx(x^Q). 1796. a) Planes parallel to the plane x-\-y-\-z^=Q\ b) concentric spheres with centre at origin; c) for u > 0, one-sheet hyperboloids of revolution about the z-axis; for u < 0, two-sheet hyperboloids of revolution about the same axis; both families of surfaces are divided by the cone *2 4-r/ 2 z a = (u = 0). 1797. a) 0; b) 0;c) 2; d) ek \ e) limit does not exist; f) limit does not exist. Hint. In Item(b) pass to polar coordinates In Items (e) and (f), consider the variation of x and y along the straight lines y kx and show that the given expression may tend to different limits, depending an the choice of k. 1798. Continuous. 1799. a) Discontinuity at je = 0, y 0; b) all points of the straight line x = y (line of discontinuity); c) line of discontinuity is the circle 2 = l; d) the tines of discontinuity are the coordinate axes. 1800 Hint. Putting y = =^ y l const, we get the function = (?,(*) , which is continuous everywhere, since for y l ^ the denominator * 2 2 -|-f/ ^0, and when f/ 1 ^0, q^M^O. Similarly, when jt = = *, const, the function
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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
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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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Page 389 and 390: Sec. 4] Numerical Integration of Fu
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Page 398 and 399: Table 3. Basic Table for Calculatin
Page 400 and 401: 394 Approximate Calculations \Ch 10
Page 402 and 403: ANSWERS Chapter I 1. Solution. Sinc
Page 404 and 405: 398 Answers Form a table of values:
Page 406 and 407: 400 Answer* 271. Solution. If x & k
Page 408 and 409: 402 Answers b) /'(!)= lim 1^15^= li
Page 410 and 411: 404_Answers 494. , . 495. - - . 1 2
Page 412 and 413: -406 Answers 640. Hint. The equatio
Page 414 and 415: 408_Answers_ 755. e xcosa sin(*sina
Page 416 and 417: 410_Answers_4 where R is the radius
Page 418 and 419: 412 Answers 1/~JT f Q \ J/max = TT=
Page 420 and 421: 414 Answers 976. j/mi n =1.32 when
Page 422 and 423: 416 Answers 1037. j/ 1040. 3x* 13 J
Page 424 and 425: 418 1 1137. 4- 6 3a . In I ocot3x|
Page 426 and 427: 420 Answers 4^8 Problem 1218*. 1221
Page 428 and 429: 422 Answers X arctan(* + 2). 1294.
Page 430 and 431: 424 Answers sin 5* , 1374. -!- 1377
Page 432 and 433: 426 Answers 3 1442. In l-x2 ' '" 2
Page 434 and 435: 428_Answers_ the function /(x) = --
Page 436 and 437: 430 Answers 1651. 3na 2 . 1652. n(b
Page 438 and 439: 432 Answers inertia /=Y 2jttf 1 r *
Page 442 and 443: 436 Answers cos^. ,810. * ox x* x d
Page 444 and 445: 438_Answers 1905. 1914. u(*. !/) =
Page 446 and 447: 440 Answers 1983. 3* + 4j/-H2z 169
Page 448 and 449: 442 Answers is ~ |/2 + -~=, the alt
Page 450 and 451: 444 Answers (binormal). 2103. bx *=
Page 452 and 453: 444_Answers_x = \ ' > (principal no
Page 454 and 455: 446 Answers 2 a a a 2139. ( dy(f(x,
Page 456 and 457: 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
Page 462 and 463: 454_Answers__ TtD the flux, use the
Page 464 and 465: 456__Answers_ and when x | ^ =- , t
Page 466 and 467: 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
Page 486 and 487: 478 Appendix IV. Trigonometric Func
Page 488 and 489: 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
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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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