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273s09pt1.Key - MATH 273 PRACTICE TEST 1 NAME ~r ~ ~ 1(10...

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MATH 273 PRACTICE TEST 1 .. NAME: __ ~~~r- _ ~ 1. (10 points). Suppose that z = z(x, y) is a function definedimplicitly by the equation z3 + xz - y2 = 1. ~ Use implicit differentiation to find g~ and g~. ~ 32 J a c + ,.~ -+ at. X -:::::0 -ax' ~ ax &:: -l: &: (3~:2. +-'X) ~ -"l -=i 32;.2+X 0')( ax ~-t 31;. 'J.. de +- 'X. 9t ~8 =0 -' 2 . dS ~ tr ~ (3l--/--x:) -=- ~ 1 d~ - ~~ 8j - 2J An . 8z _ z 8z _ 2y swer. ox - -3z'+x' oy -~. 2. (10 points). Find an equation of the tangent plane to the surface given by the graph of f(x, y) = 2x2 + 4y2 - 5 at the point pel, ~,O). +x ('Xt~ ') -= 'i»:\ (1,'/l4) ~ L/ 9 j (7vI~):: g~\ (,,'/~) -=- ~ -=? 1 ('X-I) -\-). (~-~'-f):::: 2-0 =;> \. 'i ('I< -\ )+1 (~_I),) - 2:: -=- 0 l Answer: 4(x - 1) + 2(y - ~) - z = o. 1
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~ ~ 3. (10points). Compute all first-order partial derivatives of f(7", 8, t) =:. (1 - 7"2 - 8 2 - t2)e- rst . ~= f '~ - e. -y-s~[~( t.st ( l-r2- 0'"- t2J] i ~t- =_ ",,-r5t[DIo t- ct: (I-I" '_.5 2 - e)] S . 1 ~ = _e- r6r [~t+ r.:>(\-Y'l.-s~-t;l I - 4. (10points). Prove that . x2y l~m(x,y)-+(O,O) x 3 + y3 does not exist. Hint: find two different direction of approach (x, y) -> (0,0) leading to different "limits". ~U- f(AI~):::' I</~. N~'. f(o}O)= ~ 1\3+\13 0~ ~ 'X-evilS: (x,tJ) _,;\0.(01 :: ~ =- 0 J X f.0 ~ f6c.:\) --10 Q6 (?C1j) ~(o,o) . X~+D:!> X ~ Y-f.L 'X-oJ)G(S. ll) ~ dtrri(j'IN- $iM- y-=:X' ("X. JC ) ~2.( -x) -e, ~ -=..L -=t 9 (?<l ) -7 1 (LS (x l (}) ~ (OJ 0) ",,:3 +'"X.:3 ;;.:x?
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