Question

2. Define g(r, y,z) = 22 sin(y - rz) 2y. Answer the following questions (a) Compute the gradient of g at the point P (1,1,1) (b) The point P defined above is on the level surface g = C. What is the value of C? (c) Find an equation of the tangent plane to the level surface g C at the point P (d) Suppose we want to travel from the point P to the level surface g = C + 1. Estimate how far we have to go if we go in the direction that gets us there the fastest?

Answer 1

g (uit,2) - z sin(x-x2) +24 09 . -2° cos (4-72) ! 291. corcy - x2) +2 ) ! - 1+2=? (1 ) ay Ichimb aga 223 2z sin(y=x2) - x2 corcy-xz) (1,1,1) az lou) so Grad 9 (-113, -1) 6 g(1) oth So Pis on leval curt surface g(x,y,z) = ? at Pir Equation of tangent plane on 3 (224, 2) = 2 9x(11)(x-1) + 9y (1,11) (4-1) + 9. (110) (2-1) 7 – (R-1) +3(4-1) - 1 (2-1) 2.0 - X +34-2+3=0 x-3y +223 is tansent plans,

2 we have t o Fastest direction at point is given by eradient so (-1,3,-1) But we required unit vector of direction with fastest speed in line passing through p¢ in direction of ris 2 -) 15 x 21 cm x=1-7 , yz 1+30 , 221-2. we have to find Abox point at which above line intersect on g(x17, 2) = 2+1 = 3. ie z sincy- H2) +2y = 3. ie (12) sin((1+33) - (1-2)) + 2(1+32) = 3 (1-2 sin (1+30 - (1-29+2)) + 2 +62=3. (1. sin (sad) +62 = 1

Solution of a givers point of cure glaring - We hav to just find distance between (1.1) & and that point. a z 0.102 This point (estimate) is on g(47,2) = 3 d(0!!), Co.ege, 1306, 0.25€) 10-0.598)+(1-1.306) + (1-0.999 - JeCo.102) + (0.300) - Vo.1144 A 0.338295 go we have to approx 0.338295 unit distanu