Question

Make (1, 2, 3)

A. Choose a specific point Po = (a, b, fa,b) on the surface z=f(x,y), other than (0,0,0). 1. Calculate the gradient of fat (a,b). Show your work. 2. Calculate the unit vector in which fincreases most rapidly at (a,b). Show your work. 3. Calculate the directional derivative of fat (a,b) in the direction of greatest increase. Show your work. 4. Determine the two unit vectors in which the directional derivative of fat (a,b) is equal to zero. Show your work.

Answer 1

A choose a specific point Po= (a, b, f(a, b)) the surface z=f(x, y) other than on » The gradient of fat (a,b) is we know that grad f = 7 f = (18 th of Now grad f = 7 f aut (a, b) is of laats op len we need to find the one't rector in which to increases mostrapidly 2x19 at cab).

WoW Now 7 f = (of +3 3 ) Now Vf at (a, b) is poleti f increases Now the one't vedar in which most rapidly at (a, b) is 6 40, 1, 1,9 Vete, oltClou) pe 17411 = vete]2+ 100) so Tho So - there timer 0,9 + 3 34 ,ir the unit rector in the direction foncorases most rapidly at Point 12, 6).

3 we need to calculate the directional desivative of f at a,b in the direction of greatest increase ine along B = Pelan + 16, vet (200)? Celt, len ? The directional derivative along unit rector û is Du f(a,b) = sqflan î = ( 24 12.9+9941209 12465 - habia e laj? ve 10,5) = 4 19.05 )? - ret lan) f ( 1 )