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

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440 Answers 1983. 3* + 4j/-H2z 169 = 0. 1985. x + 4y + 6z = 21 1986. = / a 2 -f & 2 2 -f c 1987 Ai the points (1, 1. 0), the tangent planes are parallel to the xz-plane; at the points (0, 0, 0) and (2, 0, 0), to the j/z-plane. There are no points on the surface at which the tangent plane is parallel to l _ _ n the x#-plane. 1991. -^ . 1994. Projection on the xi/-plane: 1. * = ~ ._w_ )=0 ( Projection on the /z-plane: < 3y* 2 Projection on the xz-plane: I^Q = | \ 3x 2 , A Hint. The line of tangency of the surface with the cylin- 4-z 2 1=0. | der projecting this surface on some plane is a locus at which the tangent plane to the given surface is perpendicular to the plane of the projection 1996. /(* + /t, y + k) = ax z + 2bxy + cy*-{- 2(ax + by)h + 2(b* + cy) k i-ah 2 + -f 26/ifc-K/z 2 1997. /(*, /)=! (xH-2) 2 + 2(jc-f2)(t/~l)-f3((/ 1)2, 1998. A/(x, i/) = 2/i + fc-M 2 + 2/i/e-f/i 2 *. 1999. / (x, y, z) = 2 2 (x I) 4- (// I) + + (2 -_1)H2(*-1) (*/-!)--( 2005 . a) Un?^ 2003. b) + (3 nz_4M)p2j 2006. a) 1.0081; b)J).902. Hint. Apply Taylor's formula for r the functions: a) f(x,y)=\ x yy in the neighbourhood of the point (1,1); b) f (x, y) = y* in the neighbourhood of the point (2,1). 2007. z= 1 -}-2(x~-l) -(t/-l)~-8(x~l) 2 + 10 ( X-l)(y-l)-3 (y- !)'+.. 2008. zmm = when x=l, i/=0 2009. NocxVcmum. 2010. z -= min 1 whenx=l, 0=0. 2011. zmax=108 = 3,t/ = 2.20l2. zmin = 8 when x = Y^ y = 1^2 and when x = There is no extremum for x=j/ = 0. 2013. 2 max ^ at 3 V 3 . . a b , a b ab the points X^-TTL, t/---^ and*= at the points ^ = y== a " d * == T^ ""Ff '~'Vf' y== F?' 2 14 ' ;?Tnax " =1 when jc = = i/ 0. 2015. zm i n = wne" x=0 = 0; nonrigorous maximum = 1"\ at points of the circle * 2 2017. min = 3- + y 2 = 1. 2016. zmax = 1^3 whenx= 1, y = 1. 4 21 when x = -j , / = , z=l. 2018. w min = 4 when JC = Y J/1' z== ^ 2019. The equation defines two functions, of which one has a maximum (zmax 8) when *=1, f/=2; the other has a minimum (Zimn = 2jwhenx 1,0 = 2, at points of the circle (x 1) 2 2 + (t/ -f 2 -^ ; 25, eacn of these functions has a boundary ext-emum = (z 3). Hint. The functions mentioned in the answer are explicitly defined by the equalities
_Answers_441 e=3 1^25 (x\)* (f/-f2) 2 and consequently exist only inside and on the boundary of thecircle (x I) 2 -|- (l/ + 2> 2 = 25, at the points of which both functions assume the value 2 = 3. This value is the least for the first function and is the greatest for the second. 2020. One of the functions defined by the equation has a maximum (*max"-~~~ 2) for x 1, r/ = 2, the other has a minimum (2min = 1) for x = 1, y 2, both functions have a boundary extremum at the points of the curve 4jt s 2 4i/ 12* + 16r/ 33=0. 2021. 2 max =-j for *=#=-. 2022. 2max = 5 for x=l, = f/ 2; 2min ---- 5 for x = 1, #--2 2023. imta for * = , ^~. 2024. ~. -- . max j/ = 5 + *:i, 2025. = 9 for x = 1, = j/ 2, 2= 2, "max^ 9 f r *=1, = r/ 2, 2 = 2. 2026. MMIX -=fl f r *= fl = I/ 2 = 0; Wmin^C for x = t/=0 2^C. 2027. w max = 2.4 2 .6Mor x = 2, y=4, 2=6. 2028. u max -4 4 / 27 at the points (f f I)' (4- f T)= (f f T)'--- 2, 1) (2, 1, 2) (1, 2, 2). 2030. a) Greatest value 2 = 3 for x = 0, y=l; b) smallest value 2 = 2 for x== 1, j/ = 0. 2031. a) Greatest value 2= , for /^ /"T 2 /""2" TFT for x== ^ V j "3 ^^ T 3 ; smallest value z== "~ y I/ -IT; b) greatest value 2 = 1 for *= 1, y-^0; smallest value z = \ for x = 0, t/= 1. 2032. Greatest value 2 = ^ for x = t/ = Y (internal maximum); smallest value 2 = for x y-~Q (boundary minimum). 2033. Greatest value 2 = 13 for x = 2, y = 1 (boundary maximum); smallest value ? = 2 for x y = \ (internal minimum) and for x = 0, y = 1 (boundary minimum). 2034. Cube. 2035. J/2V, j/2V, -- ^2V* 2036. Isosceles triangle. 2037. Cube. 2038. a = \/a \/a > 2 2040. Sides of the triangle are ~p,~/>, and . fe 44 \/a >/a. 2041. Jgaa 2039. Ai f v ll 1a , i> m, -i- m 2 -f m, 2Q42 + a + b s=s3i c 2Q43. The dimensions of the parallelepiped *2a 2b 2c are , -rp 7^-* "T^' where a, b, and c are the semi- Y 3 V 3 V 3 axes of the ellipsoid. 2044. * = j/ = 26+2V, * = -- 2045. x= - , y ~T 2046 - Major axis, 2a = 6, minor axis, 26 = 2. Hint. The square of the distance of the point (x,y) of the ellipse from its centre (coordinate origin) 2 2 is equal to x . -f-r/ The problem reduces to finding the extremum of the function x*-\-y* provided 5*2 2 + Bxy + = 5t/ 9. 2047. The radius of the base of the cylinder ^ , -|- J .
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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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Sec. 4] Numerical Integration of Fu
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Sec. 5] Numerical Integration of Or
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Sec. 5] Numerical Integration of Or
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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 440 and 441: 434 Answers along the large lower s
Page 442 and 443: 436 Answers cos^. ,810. * ox x* x d
Page 444 and 445: 438_Answers 1905. 1914. u(*. !/) =
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
Page 490 and 491: 482 Appendix 11. Graphs //-=secx an
Page 492 and 493: 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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