Final answer:
To compare ∆z and dz, the student must calculate both the actual change in z when x and y are slightly altered, and the differential of z using partial derivatives and the changes in x and y.
Step-by-step explanation:
The question involves finding the change in the function z when the variables x and y are slightly changed. ∆z represents the actual change in z, while dz refers to the differential of z - an infinitesimal change that approximates ∆z when the changes in x and y are very small. To compare ∆z and dz, one can use the original function z = x² - xy - 7y² and its partial derivatives with respect to x (which is 2x - y) and y (which is -x - 14y).
The student is asked to evaluate the changes when x changes from 2 to 2.03 and y changes from -1 to -0.95. We can calculate dz using the partial derivatives and the small changes in x and y (dx and dy) and then compare it to the actual change ∆z after substituting the new values of x and y into the given function.