Question

Find grad  if at (1,1,1).​

Answer 1

Answer:

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Step-by-step explanation:

et f(x,y)=x2y. (a) Find ∇f(3,2). (b) Find the derivative of f in the direction of (1,2) at the point (3,2).

Solution: (a) The gradient is just the vector of partial derivatives. The partial derivatives of f at the point (x,y)=(3,2) are:

∂f∂x(x,y)∂f∂x(3,2)=2xy=12∂f∂y(x,y)∂f∂y(3,2)=x2=9

Therefore, the gradient is

∇f(3,2)=12i+9j=(12,9).

(b) Let u=u1i+u2j be a unit vector. The directional derivative at (3,2) in the direction of u is

Duf(3,2)=∇f(3,2)⋅u=(12i+9j)⋅(u1i+u2j)=12u1+9u2.(1)

To find the directional derivative in the direction of the vector (1,2), we need to find a unit vector in the direction of the vector (1,2). We simply divide by the magnitude of (1,2).

u=(1,2)∥(1,2)∥=(1,2)12+22−−−−−−√=(1,2)5√=(1/5√,2/5√).

Plugging this expression for u=(u1,u2) into equation (1) for the directional derivative, and we find that the directional derivative at the point (3,2) in the direction of (1,2) is