Question

Let f(x, y, z) = xeyz – cos(x2 – y2 + 22) a) Find the directional derivative of f at the point (0,0,0) toward the point (1,2,0). b) Find the maximum rate of change of f at point (0,0,0). In which direction does the max rate of change at (0,0,0) does occur? (two questions here!)

Answer 1

Given f(x,y, 2) = xe ² y 2 -cos( x²y2 + 2%) ,(3,9%) : (x + - as/ x + 1) (a)

=(x* re*) yz - f-sis (22) 24 22)) (2x) fe(0,90) = ([deº +eo) bhl0) + Sir (0-0 +o) (210)) 0= (رو رہا ہے fif 2,9, 2) = a (xe Xyz -cos (x2-y4+2 =)) - **2 + sin(x2-y2722) (-by) fylo,o, o) =(0) €° (0) + sü lo-040) (-260)) - 0 toro fj (9,90) 20 fz(x, y, z) = 8 m ( xe Mya - cos (x292 +2²)) sexy & Sis 1x² - y2 +22) (22)

fal0,0,0) = (Cleº (o) + si l 0-0+o) 210) Foto=0 a fz (00,0) =0 Te Gradient Of 10,0,0) = 1 (fx Cop,o), fy(0,0,0); fz 10,0,0)) .. e <o, o, os If looo) = 40,0,07. Giwer point Plo,o,o) taward the point Let @= (1,20) . vector = P2 = (1-0, 2-0, 0-0) -21,2, 0) Vs <1,2,0) with weater uenit vector i = (1,2,07 Tol - V12+28+6012 22,07 21,2,07 15 yty

Directional derecative of flasy, zs at the point (oso,o) is the dire dion of ł is D. 110,0,0) = If (0,0,0) ou .:200,0>.< to zro)

50 (45) + 0(3) + (0) (0) - ototozo OO . | Directional derivative is of (6) Marinun rate of Change - 1 of 10,ool | IV (0)2to2 Hol2 = Voto .: Maximum rate of change =0

the direction it occurs is <0,0,07 Maxen un rate of charge =0. Direction is <0, 0,07