Question

1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a.

Answer 1

Condide the function tx,5, 2) x Som 322 P3,0,1) ( nit vedtor- v 2 For 9radient ff at P grodient poiut ,y,2) i the Veeto Y the funotion cut Ne pnow a Vflp) (p) 212+j^-f 322+a (2,9, 2) 22 vfl2) 2xo, 6x17 K4x3 V fP) 12, 06 Ang the direction of The adient at p has maximum tncreale f th funetion at p. The in hsdireetion Unit Vector K/2, 0,6> Je20t6

VP) K106 K121016 44+36 610,3 3 J5 2, > the ate change f need to pend We at P Punction in the direotion of it gradient grad rent atp given by the magnitude ofthe t 144t36 Vf) Directional derivative of f at p ih the dration at P in the dlireuhbh af a vnit veeto r vis given bthe veetoy produt vf(p) U -12 0- -9 x2 J2