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

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368 Approximate Calculations [Ch. 10 Henceforth, we shall assume that all digits in the initial data are correct (if not otherwise stated) in the narrow sense. The results of intermediate calculations may contain one or two reserve digits. We note that the examples of this section are, as a rule, the results of final calculations, and for this reason the answers to them are given as approximate numbers with only correct decimals. 4. Addition and subtraction of approximate numbers. The limiting absolute error ot an algebraic sum of several numbers is equal to the sum of the limiting absolute errors of these numbers. Therefore, in order to have, in the sum of a small number of approximate numbers (all decimal places of which are correct), only correct digits (at least in the broad sense), all summands should be put into the form of that summand which has the smallest number of decimal places, and in each summand a reserve digit should be retained. Then add the resulting numbers as exact numbers, and round off the sum by one decimal place If we have to add approximate numbers that have not been rounded off, they should be rounded off and one or two reserve digits should be retained. Then be guided by the foregoing rule of addition while retaining the appropriate extra digits in the sum up to the end of the calculations. Example 1. 215.21 -f- 14.182 -f 21 .4-215.2(1) + 14.1(8)4-21 4-= 250.8. The relative error of a sum of positive terms lies between the least and greatest relative errors of these terms. The relative error of a difference is not amenable to simple counting. Particularly unfavourable in this sense is the difference of two close numbers. Example 2. In subtracting the approximate numbers 6 135 and 6.131 to four correct decimal places, we get the difference 004. The limiting relative 0.001 + 1 0/XM .J. error is 6= .. . =- , t j = 0.25. Hence, not one of the decimals 0.004 4 of the difference is correct. Therefore, it is always advisable to avoid subtracting close approximate numbers and to transform the given expression, if need be, so that this undesirable operation is omitted. 5. Multiplication and division of approximate numbers. The limiting relative error of a product and a quotient of approximate numbers is equal lo the sum of the limiting relative errors of these numbers Proceeding from Ihis and applying the rule for the number of correct decimals (3), we retain in the answer only a definite number of decimals Example 3. The product of the approximate numbers 25.3-4.12=104.236. Assuming that all decimals of the factors are correct, we find that the limiting relative error of the product is 6 = ^0.01 +^0.01=^0.003. Whence the number of correct decimals of the product is three and the result, if it is final, should be written as follows: 25.3-4 12=104, or more correctly, 25 3-4.12= 104 2 0.3. 6. Powers and roots of approximate numbers. The limiting relative error of the mth power of an approximate number a is equal to the m-fold limiting relative error of this number The limiting relative error of the mth root of an approximate number a Is the th part of the limiting relative error of the number a. 7. Calculating the error of the result of various operations on approximate numbers. If Aa lf ... , Aa,, are the limiting absolute errors of the appro-
Sec. 1] Operations on Approximate Numbers 369 xitnate numbers a t , . . . , a n , then the limiting absolute error AS of the resuU may be evaluated approximately S = /(a t, ..., a n ) from the formula The limiting relative error S is then equal to ".++ ,**- *Ji/r Example 4. Evaluate S = ln (10.3+ V^4. 4 ); the approximate numbers 10.3 and 4.4 are correct to one decimal place. Solution. Let us first compute the limiting absolute error AS in the general form: S = ln (a+ V r 1 ~b), AS = A& \ Aa We have ^TT; Aa=A6=i=2Q; 1^4.4 = 2.0976...; we leave 2.1, since the relative error of the approximate number y is equal to ^ =- -r= ^ ; is then equal to ^SOTT^JQ 20 "2" the absolute error ' we can be sure f tne first decimal place. Hence, Thus, two decimal places will be correct. Now let us do the calculations with one reserve decimal: 0005 U - UUi> ' log (10.3+ |/4 4) =5= log 12 4-1.093, In (10 3+ V 4.4)=^ 1.093-2.303 = 2.517. And we pet the answer: 2 52 8. Establishing admissible errors of approximate numbers for a given error in the result of operations on them. Applying the formulas of 7 for the quantities AS or 6S given us and considering all particular differentials U-M Ac* or the quantities \-~- equal, we calculate the admissible \dak \ * M \dak \f\ absolute errors Aa lt ... , Aa^, ... o the approximate numbers a t , . .. ,an , ... that enter into the operations (the principle of equal effects). It should be pointed out that sometimes when calculating the admissible errors of the arguments of a function it is not advantageous to use the principle of equal effects, since the practically unfulfilable In these cases latter may make it is advisable to demands that are make a reasonable redistribution of errors (if this is possible) so that the overall total error does not exceed a specified quantity. Thus, strictly speaking, the problem thus posed is indeterminate. Example 5. The volume of a "cylindrical segment", that is, *a solid cut off a circular cylinder by a plane passing through the diameter of the base (equal to 2R) at an angle a to the base, is computed from the formula 2 V =~ R* tan a. To what degree of accuracy should we measure the radius
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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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Page 325 and 326: Sec. 4] Fourier Series 319 If a fun
Page 327 and 328: Sec. 4\__Fourier Series _321 2690.
Page 329 and 330: Sec 1 1 Verifying Solutions 323 It
Page 331 and 332: Sec. 21 First-Order Differential Eq
Page 333 and 334: Sec. 3] Differential Equations with
Page 335 and 336: Sec. 3\ Differential Equations with
Page 337 and 338: Sec. 4]_First-Order Homogeneous Dif
Page 339 and 340: Sec. 5] Bernoulli's Equation 333 Co
Page 341 and 342: Sec. 6] Exact Differential Equation
Page 343 and 344: Sec. 7] First-Order Differential Eq
Page 345 and 346: Sec 8]_The Lagrange and Clairaut Eq
Page 347 and 348: Sec. 9] Miscellaneous Exercises on
Page 349 and 350: Sec. 9] Miscellaneous Exercises on
Page 351 and 352: Sec. 10] Higher-Order Differential
Page 353 and 354: Sec. 10]__Higher-Order Differential
Page 355 and 356: Sec. 11]_Linear Differential Equati
Page 357 and 358: Sec. 12] Linear Differential Equati
Page 363 and 364: Sec. 14]_Euler's Equations__357 Fin
Page 365 and 366: Sec. 15]_Systems of Differential Eq
Page 367 and 368: .Sec. 16} Integration of Differenti
Page 369 and 370: Sec. 17] Problems on Fourier's Meth
Page 371 and 372: Sec. 17]_Problems on Fourier's Meth
Page 373: Chapter X APPROXIMATE CALCULATIONS
Page 377 and 378: Sec. /]_Operations on Approximate N
Page 379 and 380: Sec, 2] Interpolation of Functions
Page 381 and 382: Sec. 2] Interpolation of Functions
Page 383 and 384: Sec. 3]_Computing the Real Roots of
Page 389 and 390: Sec. 4] Numerical Integration of Fu
Page 391 and 392: Sec. 5] Numerical Integration of Or
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
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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
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
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
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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
Page 486 and 487:
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
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492 Index Infinite discontinuities
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494 Index Operator Hamiltonian 288
Page 502:
496 Index Taylor's series Term 311,
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