Question

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e-x2 - 5y2 - 722 where Tis measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P12,-1, 3) in the direction towards the point (5, -3, 6). °C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

Answer 1

Vector from P (2,2,3) to (5,-3, 6) is TX,Y,E) = 100eth 54²-72² @ find sale of change of temp at the point P (2,-1,3) in the direction towards the point ( 5,-3,6). Rate of change of temp. at p in the clir. of (5,-3,6) is Du T(2,-1,3), where u is a unit rector from P(2,-213) to (51-3,6) 5-2, -3+1, 6-37 (3,-2,3 13) :. Us (3,-2,3) 3 Jotutg Vinter 122 Jཉན Now Ta (49,2) -200xé 1² 5y? 722 -72 100 Esy? 722 (-29) Tv (2,-1, 3) = -200(2)24-5-63 -400 e -x? 59?-124 Ty (249,2)= 100 (loy) = -loooye 20-113) = -1000(-Ve -x²59²722 Ty -72 Ž 1000872

-p?sy72% (142) Tz (*19, 2) = love -1400(2), +² 54² 722 Tz (2,-1,3) = -4/200 72 .3 Du T(2,-1,3) = V T(2,-1,3) · (Pavel Forum J22 Jaz (-400072 1000 e -4200272 ) -72 3 Jaa 1 2 - 12000-72 -72 2000 € Jag 12600 72 २२ -72 -15800 e २२ Next Max. sale at which the temp. increases at the point p to be (it's max. ullon two vectors in some Du T(2,-1,3) -400072, 1000672,-4200€ dir.) -72 943/14168410) 5|200€ (-2,5;21)., (ou isomis verder 11200 272) [423°419°47212 So lu121. 2000 725 470 cm.