Question

Find ∇f at the given point, f(x, y, z) = e^(x+y)sin(z) + (y+1)cos⁻¹(x), (0,0,π/6).

Answer 1

The gradient vector ∇f of the function f at point (0,0,π/6) is calculated to be (0.5, 1, √3/2) by finding the partial derivatives with respect to x, y, and z, then evaluating them at the specified point.

Step-by-step explanation:

The student is asking to find the gradient vector, denoted as ∇f, of the function f(x, y, z) = e^{(x+y)}sin(z) + (y+1)cos^{-1}(x) at a specific point (0,0,π/6). To find the gradient of f, we will take the partial derivatives of f with respect to each variable, x, y, and z, and then evaluate those derivatives at the given point.

• The partial derivative with respect to x, ∂f/∂x, is: e^{(x+y)}sin(z) - sin(π/6)/(1-x^2)^{1/2}.
• The partial derivative with respect to y, ∂f/∂y, is: e^{(x+y)}sin(z) + 1.
• The partial derivative with respect to z, ∂f/∂z, is: e^{(x+y)}cos(z).

Now we evaluate these derivatives at (0,0,π/6):

• ∂f/∂x at (0,0,π/6) is 0.5.
• ∂f/∂y at (0,0,π/6) is 1.
• ∂f/∂z at (0,0,π/6) is √3/2.

Therefore, the gradient vector ∇f at (0,0,π/6) is (0.5, 1, √3/2).