Question

6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).

Answer 1

classmate Date Page Page No - 1 Ans: Let Fl@, y, z)=40+42+72 6002, 1. The gradient vector of flavo), 2) is denoted by 'De and is given by VF ( 2 , 7, 2 (fa fy, Fe) i cili :DF (2139:32) = ( 22-607, 24-687g 22-62y), (2) We know, tangent plane to the surface given by FLY, 2) = k at (x, t) has the equation, fi (70) Jo, ze) (1-60) + Fy (20,90,%) 9-40) F Fe (20,9,KO) (270)29 ir tangent plane of the levelsusface Play 2 - ² ty e f 2² 627 2 = 4 at (0, 0, 2) is given by Ft (0,032) (96-0) + fy (0,0,2)(y-0) + Fz (0,0,2)(2-2) 20 رسميا for = 22-642 314601092) = 0 similaely, fy = 0 and Rabbit 10 .: eqn () becomes Ol1-0) +0 [Y-0) + 402-28 13 - 80 2.3

classmate fase Mas 2 Date Page is 7-2. :252 is equation equation or tangent plane,

The level sueface flu,9, 1) =4 defines a function. 2=fuy) Now It 2 = f(ay) then f12y)-2=0 Now We define a new function Fligy,3) = f(0,y)-x. NOω oz FR - 2 (f (199)-2) = fq -LT) ox fy Off(4,3)-x) = fy Od fes & ff(194) -2) :-) Llili). intf(X,Y,Z) Efte fyy)) eqn of tangent egs (2) But from ean 'U), lii), (ii) egn (1) becomes fall to youd ftatott fg (404) utsfato fa (0o, yo) (x-2) + fyllo, yo) (Y-yo) = 12-20)=0 plane i given by H.2= fllo, yo) + f(0, Y.) (X-X) + fy (20 %) (9-yo) at pt ll, lg22 2 = f(0,0) + fx (0,0) (0-0) + tycoo) (4-0 put a = -0.002, y = 0.003 z = f(0,0) ffy (000) (-0.002)+ fy (0,0) 60.003.)