Final answer:
To find Δw and dw for the function w=xy^2z^3 with a change in variables from (3,2,1) to (2.8,2.1,1.05), calculate the changes in variables (Δx, Δy, Δz), evaluate the partial derivatives at the initial point, and use these to calculate the total differential dw. Then compute Δw by substituting the initial and final values into the original function w.
Step-by-step explanation:
To compute Δw and the total differential dw for w=xy^2z^3, you first identify the changes in the variables x, y, and z. The difference in values for each variable when moving from point (3,2,1) to (2.8,2.1,1.05) is: Δx = 2.8 - 3, Δy = 2.1 - 2, and Δz = 1.05 - 1. The total differential dw is given by dw = ∂w/∂x * dx + ∂w/∂y * dy + ∂w/∂z * dz, where dx, dy, dz are the changes in x, y, z respectively, and ∂ denotes the partial derivative. To calculate Δw you substitute the initial and final values into the function w to get Δw = w(x_2,y_2,z_2) - w(x_1,y_1,z_1).
Evaluating the partial derivatives at the initial point (3,2,1), we can then plug in the values of dx, dy, and dz to find dw. Lastly, we calculate the actual change in w by substituting the initial and final values of x, y, and z into the formula for w to find Δw.