Question

Let h(x,y,z)=eˣ sin(2y+z). Calculate ∇h.

Answer 1

To calculate the gradient (∇h) of the function h(x,y,z)=eˣ sin(2y+z), we need to find the partial derivatives of h with respect to x, y, and z. The partial derivatives are eˣ sin(2y+z) for x, 2eˣ cos(2y+z) for y, and eˣ cos(2y+z) for z. Therefore, the gradient ∇h is (eˣ sin(2y+z), 2eˣ cos(2y+z), eˣ cos(2y+z)).

Step-by-step explanation:

To calculate the gradient (∇h) of the function h(x,y,z)=eˣ sin(2y+z), we need to find the partial derivatives of h with respect to x, y, and z.

So, ∂h/∂x = eˣ sin(2y+z), ∂h/∂y = 2eˣ cos(2y+z), and ∂h/∂z = eˣ cos(2y+z).

Therefore, ∇h = (∂h/∂x, ∂h/∂y, ∂h/∂z) = (eˣ sin(2y+z), 2eˣ cos(2y+z), eˣ cos(2y+z)).

Answer 2

To calculate ∇h for the function h(x,y,z)=eⁿ sin(2y+z), the partial derivatives with respect to x, y, and z are computed resulting in ∇h = .

Step-by-step explanation:

To calculate the gradient of the function h(x,y,z)=ex sin(2y+z), denoted as ∇h, we need to take the partial derivatives of h with respect to each variable, x, y, and z.

The partial derivative with respect to x is:

∂h/∂x = ex sin(2y+z)

The partial derivative with respect to y is:

∂h/∂y = 2ex cos(2y+z)

And the partial derivative with respect to z is:

∂h/∂z = ex cos(2y+z)

So, the gradient is the vector:

∇h = <∂h/∂x, ∂h/∂y, ∂h/∂z> = x sin(2y+z), 2ex cos(2y+z), ex cos(2y+z)>

This results in the gradient being a vector field that points in the direction of the greatest rate of increase of the function h.