Question

Problem 4 a5 pts) La f(x,y,z) = 3x2+2+2 (a) Draw a few of the level surfaces (4.3.2) = c for admissible values of cand classify the type of surface these are (b) Compute the directional derivative of fat (1.2.3) in the direction of the vector û= 2.2.1). (c) Find the value and direction of the maximum rate of change off at the point(1.2.3).

Answer 1

f(x, y, z) = 3x² + 2y² + z²zc Here ao are the all admissible values of c. sre The loved level surfaces are the curves are the ellipse ellipsoid whose in the xy plane. (b) f(0,8, Z) = 32?+ 2y?+Z?. directional desivative of f at (1,2,3) in direction of the vector û = 1/3 (2, 2, 1) as. the lim of (6 (1,2,3)+X ů) - 8 (1,2,3) xto = lim f( (1,2,3) + (21 27 2 ) ) – 8 (1,2,37 A>o z lim f (1+21 1 , 2 + 21, 3+27) _ f (1,2,3) aso hin 3(1+233*+ 2 (2+2)+(3+3)° -20 217? +102X+180 - 20 z lim a o a

= lim 2 ² + 102x+ 20-20 ayo 102 102 change Direction of f at of the maximum rate of the pt (1,2,3) is 75 (1,2,3) f(x,y,z) = 3x²+24² + Z² . as (x, y, z) = 6xê + 4y 9 tazê .:88 (1,2,3) = 6î +8 ft bê = (6,8,6) the value is, I V8 (1,2,3)| and = √6²+8² +62 = √136 22√34 .