Final answer:
To compare ∆z and dz, calculate the actual change in z for the specified changes in x and y, and use partial derivatives to estimate the change for dz. Perform calculations for both given functions and compare the numerical values, rounded to four decimal places.
Step-by-step explanation:
To compare the values of ∆z and dz, we first need to understand that ∆z represents the actual change in z when the variables x and y change, while dz represents the differential, which is an approximation of the change in z using the derivative.
- The first function given is z = 8x² + y². To find ∆z, we calculate the value of z at the new point and subtract the value of z at the original point. For dz, we take the partial derivatives with respect to x and y, evaluate them at the original point, and multiply by the respective changes in x and dx, and y and dy.
∆z for the first function: z(0.95, 1.9) - z(1, 2) - dz for the first function: (dz/dx)(dx) + (dz/dy)(dy)
The second function is z = x² - xy + 5y². Similarly, to find ∆z, compute the difference in the function values at the new and original point, and for dz, use the partial derivatives like before.
- ∆z for the second function: z(1.03, -1.05) - z(1, -1)
- dz for the second function: (dz/dx)(dx) + (dz/dy)(dy)
Calculations will provide the numerical comparison between ∆z and dz for each case after rounding to four decimal places.