Suppose that f(x,y,z)=x2yz−xyz3
is a function of three variables.
1. Find the gradient of f(x,y,z).
Answer: ∇f(x,y,z)=
2. Evaluate the gradient at the point P(−1,2,2).
Answer: ∇f(−1,2,2)=
3. Find the rate of change of f(x,y,z)
at P
in the direction of the vector u=⟨0,45,−35⟩.
Answer: Duf(−1,2,2)=
1. To find the gradient of f(x,y,z), we need to find the partial derivatives of f with respect to each variable:
∂f/∂x = 2xyz - yz^3
∂f/∂y = x^2z - xz^3
∂f/∂z = x^2y - 3xyz^2
Therefore, the gradient of f(x,y,z) is ∇f(x,y,z) = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩ = ⟨2xyz - yz^3, x^2z - xz^3, x^2y - 3xyz^2⟩.
2. To evaluate the gradient at the point P(-1,2,2), substitute these values into the expression for the gradient:
∇f(-1,2,2) = ⟨2(-1)(2)(2) - (2)(2)^3, (-1)^2(2)(2) - (-1)(2)^3, (-1)^2(2) - 3(-1)(2)^2⟩
= ⟨-8 - 16, 4 + 8, 2 + 12⟩
= ⟨-24, 12, 14⟩.
Therefore, ∇f(-1,2,2) = ⟨-24, 12, 14⟩.
3. To find the rate of change of f at point P in the direction of the vector u = ⟨0,45,-35⟩, we need to take the dot product of the gradient ∇f(-1,2,2) and the unit vector in the direction of u:
Duf(-1,2,2) = ∇f(-1,2,2) · (u/||u||),
where ||u|| is the magnitude of u.
First, find the magnitude of u:
||u|| = √(0^2 + 45^2 + (-35)^2) = √(2025 + 1225) = √3250 ≈ 57.008.
Next, find the unit vector in the direction of u:
u/||u|| = ⟨0/57.008, 45/57.008, -35/57.008⟩ ≈ ⟨0, 0.789, -0.614⟩.
Finally, substitute these values into the dot product formula:
Duf(-1,2,2) = ⟨-24, 12, 14⟩ · ⟨0, 0.789, -0.614⟩
= (-24)(0) + (12)(0.789) + (14)(-0.614)
= 9.468 - 8.996
≈ 0.472.
Therefore, Duf(-1,2,2) ≈ 0.472.
To find the gradient of f(x,y,z), we need to find the partial derivatives with respect to each variable:
1. ∂f/∂x = 2xyz - yz^3
2. ∂f/∂y = x^2z - xz^3
3. ∂f/∂z = x^2y - 3xyz^2
Therefore, the gradient of f(x,y,z) is ∇f(x,y,z) = (2xyz - yz^3) i + (x^2z - xz^3) j + (x^2y - 3xyz^2) k.
Now, let's evaluate the gradient at the point P(-1,2,2):
1. ∇f(-1,2,2) = (2(-1)(2)(2) - (2)(2)^3) i + ((-1)^2(2)(2) - (-1)(2)^3) j + ((-1)^2(2) - 3(-1)(2)^2) k
= (-8 - 16) i + (4 - 16) j + (2 + 12) k
= -24 i - 12 j + 14 k
So, the gradient at the point P(-1,2,2) is ∇f(-1,2,2) = -24 i - 12 j + 14 k.
Finally, let's find the rate of change of f(x,y,z) at P in the direction of the vector u = (0, 45, -35):
1. Compute the dot product of the gradient at P and the vector u:
Duf(-1,2,2) = (-24)(0) + (-12)(45) + (14)(-35)
= 0 - 540 - 490
= -1030
Therefore, the rate of change of f(x,y,z) at P in the direction of the vector u is Duf(-1,2,2) = -1030.