m251S04test1 - MATH 251 Test I Friday Name 1 2 3 4 5.Total...

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MATH 251 Test I Friday 2/20/2004 1. 2. 3. 4. 5.- Total Name There are a total of 5 problems. No calculators are allowed. At the top of each page, write down your initials. 1. Use the differential to find an approximate value for [8(10.06)2 + (4.96)3 + 3(5.02)2j1/3, with 4-digit accuracy. 1()(, ~, ~) := (8''£ L + 'jlt3}2 J y~ At ~ ctf - d +- Ai. + ~ 6.li +- '01 A<--- - ~ g~ I ~ d (15%) AX= ID.Ob-lo = o.ob A 1 = 4.1b.-!) =- - O.o't A ~ = 5'.0.2 - ':; =. O. ();;t { X :: I D ~iL '~:!i }= S- 2t:: -L ( 'L .)- - 'L J -; d- I I 6 ) ~ X 3 tx t -J +1' ' /6 X ("''1,1s)::(lo)!:",t) 3" ~l! .1 0 nL ~ ~ ~"-3 (lIx'-+ ~!tb'T'.~f = t I~ (j)(S-) ~= f(~~'-+1Lt¥'-r;'bt 11~(6)t5) .- ~_.-..'" ",' '- <?.J.V l\ L- t ( 10 Db 4- 0 I 5 () 1 ,) ,- : f It 0 !> S- ) ~ ~\ - ,} 'lbJ' : ) ~ ,.' . - - . . . - --. ~ -bo Obo) (0.06) -I- 3~ (15) (-b.O It) +~ (~D) (OIO~) ::: 0.03;2..-6,01 +o.uo2..:::. 0.024- Tk f CI 0 . Ob, 4-.q 6) S:" 2.) ~ [ c) to. c> 1- t :::-10 . b 2..'t 1
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Your initials 2. Let n be a positive integer. Define f(XI' X2,..., xn) = (xi + x~+ ... + x~)(2-n)/S, for (Xl,X2,..., xn) =f- (0,0,...,0). (a) Evaluate () (); f(Xt,X2"" ,xn), where 1:S i,j :S n. Xi Xj (b) Show that f satisfies the partial differential equation (10%) ,iuw.. {)2f {)2f ()2f ~ 2 + ~ 2 + .. . + ~ 2 = O. vXI vX2 uXn ( ) ",1 ~ 2-)t 4. .4 - 2.-)1 "I- 2. ~ ~ -1 0 . 2. 2- 'oX' - ~ ~ (X1+X;z. +--. +X~) . iX, ('Xc'",..+X;) J \It J = (2-Jt) ('~t-t ~-' +X~)-Z, X'j 4 . i'1 0 11 -- = 'C> ~ 2--:1.. , - 'Z- :l 2. x' aXi~Xj ("->1.) l ~X;~x, + - -.-I-'X.) ~ - X1 H"d --..h.) ;xJ ~ '" "1'\. 1. 2--T-l _!\.
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