test4_sol_Winter_08 - MATH 2008/09B NAME(PLEASE PRINT...

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MATH 2008/09B Test 4 l\IIarch 18, 2008 NAME(PLEASE PRINT): s:ilJ:D"'~ STUDENT NO.: . Instructions: 1. This test has 4 questions and a total of 40 marks. 2. Calculators are NOT allowed. (Ql) Let the function f(x, y) = e3xy - y2 and the point P(O, -1) be given; (a) [6 Marks] Compute the directional derivative of f at the point P in the direction from the point (2,3) to the point (3,1); -'> (), :::: (3... \"')- (d J :'» c:: ( \ )- .).. "') ~ &( _ (\)_2) _ (.J- -3-\ V :::: - - ;;-:-;: -1/ - ) r;:: ) liC/ ilf-if ~ v5" -) 3Jd 3~Y) 'J"(~ (f:;>:) fJ):: (3~ e. ~ 3~ e-;).'d ~ (0) - \) ::::(- 3) .;2) ---') -> :: "'J f (OJ-I) 11 - f '\1) J(~-J).::;- v;=- . -,-';) :) . 'lJ ------ -3 !L--l --- ..-- - 'IS' l;- ~ 1:t f( OJ-I) IS~ C~M:e) ! (b) [4 Marks] Find the maximum rate of change of direction in which it occurs . f at the point P and the . ~ \ f (OJ-I) = \ 'V f (Or\) = '(-3.1~) \ ~ ;q;;; ::: \!i3 NDte -- d / I L (), 1'1 :.--- I luLl i))'\f 11.t-"\' ~ 7 0 n oY YY'CLA/l ie. I'TW-- ~ ~. ) LA. :::: '7F IS aJS6 ole;
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(Q2) [10 Marks] Find the local maximum and local minimum values, and saddle points of the function f(x, y) = 4xy - x3 - 2y2, A =0 ~> 1-::;')(:::0 =~ P: (dIO) <:i ~ % =>]",'- % )p, (11/ W f .:::: - b':>l f _4-L =-4 ;J. 'X J :1.-:1 - ) If! L j) (x, 'J) = f;~ f; 'I - ({"1) = +;;L 't:;( - I 6 P6InJs) -h" ~ D \ -rtrf-f f<»~ ---' , ~o w-t JaafJ /J1-CL)( Loc"p-1Y'''X -v~ :-0 r (It/S/~) "It( %)(73) -c 4;3)~;}{ ~o/ "' i-{ %1 -&3l-4 %) L 2. - -(':6) (4- %-;;.) Z (- ;;2 === (413) 3) _~ :e- ~r
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(Q3) [10 Marks] Use the method of Lagrange Multipliers
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