Final Answer:
The values of Δz and dz, when (x, y) changes from (1, 2) to (0.95, 2.1), are approximately Δz = -0.7216 and dz = -0.7216.
Step-by-step explanation:
The change in the function z = 7x²y², denoted as Δz, is calculated by evaluating the function at the new point (0.95, 2.1) and subtracting the value at the initial point (1, 2): Δz = 7(0.95)²(2.1)² - 7(1)²(2)² ≈ -0.7216. This represents the difference in the function value due to the change in the input coordinates.
The differential of z, dz, represents the total derivative of z with respect to its variables x and y. For this function, dz is obtained by taking the partial derivative with respect to x and y and then multiplying by the corresponding changes in x and y. Evaluating this at the given points yields dz = (14xy²Δx) + (14x²yΔy), where Δx = 0.95 - 1 and Δy = 2.1 - 2. Calculating this expression gives dz ≈ -0.7216. The close agreement between Δz and dz reflects the linear approximation provided by the total differential, reinforcing the concept that dz approximates the change in the function due to small changes in the input variables.
Understanding the relationship between Δz and dz is essential in calculus, particularly in the context of multivariable functions. The comparison between these values illustrates the concept of differentials and their role in estimating changes in a function as the input variables undergo small variations.