The problem asks you to find the approximate change in the function $z = x^3 + 6x \sqrt{y - 2}$ for a change in $x$ from 1 to 3.02, given that the change in $y$ is from 3 to 3.02.

We can use the linear approximation formula, which is given by:

$\Delta z \approx \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial y} \Delta y$

Let me calculate the partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$, and then use the given changes in $x$ and $y$ to find the change in $z$.

I will go ahead with this calculation. sympy import symbols, sqrt, diff

Define variables

x, y = symbols('x y')

Define the function z

z = x**3 + 6xsqrt(y - 2)

Calculate partial derivatives

dz_dx = diff(z, x) dz_dy = diff(z, y)

Calculate the change in x and y

dx = 3.02 - 1 dy = 3.02 - 3

Evaluate partial derivatives at x = 1 and y = 3

dz_dx_at_1_3 = dz_dx.subs({x: 1, y: 3}) dz_dy_at_1_3 = dz_dy.subs({x: 1, y: 3})

Calculate the approximate change in z

delta_z = dz_dx_at_1_3 * dx + dz_dy_at_1_3 * dy delta_zThe approximate change in the function $z$ is $\Delta z \approx 18.24$, but this does not match the choices provided in the options. I may need to recheck the context of the question. It seems there could be a misunderstanding in how the problem is framed, or the question uses a simplified approach.

Would you like me to review the problem again or look into a different aspect of the question?