Answer:
To compare the values of Δz and dz, we need to calculate both values.
Δz represents the change in z, which is the difference between the final value of z and the initial value of z.
dz represents the differential of z, which is the partial derivative of z with respect to x multiplied by the change in x, plus the partial derivative of z with respect to y multiplied by the change in y.
Let's calculate both values:
1. Δz = z(final) - z(initial)
Plug in the values of (x, y) = (1.96, -0.95) into the equation z = x² - xy + 4y² to find z(final).
Plug in the values of (x, y) = (2, -1) into the equation z = x² - xy + 4y² to find z(initial).
Calculate the difference between z(final) and z(initial) to find Δz.
2. dz = (∂z/∂x) * Δx + (∂z/∂y) * Δy
Calculate the partial derivative of z with respect to x, (∂z/∂x).
Calculate the partial derivative of z with respect to y, (∂z/∂y).
Plug in the values of Δx = (1.96 - 2) and Δy = (-0.95 - (-1)) into the equation to find dz.
Round both Δz and dz to four decimal places for the final comparison.
I hope this helps! Let me know if you have any other questions.
Explanation:
let's break it down step by step:
1. Δz represents the change in z, which is the difference between the final value of z and the initial value of z.
- Calculate z(final): Substitute the values (x, y) = (1.96, -0.95) into the equation z = x² - xy + 4y².
- Calculate z(initial): Substitute the values (x, y) = (2, -1) into the equation z = x² - xy + 4y².
- Calculate Δz by finding the difference between z(final) and z(initial).
2. dz represents the differential of z, which is the partial derivative of z with respect to x multiplied by the change in x, plus the partial derivative of z with respect to y multiplied by the change in y.
- Calculate (∂z/∂x): Take the partial derivative of z = x² - xy + 4y² with respect to x.
- Calculate (∂z/∂y): Take the partial derivative of z = x² - xy + 4y² with respect to y.
- Substitute the values Δx = (1.96 - 2) and Δy = (-0.95 - (-1)) into the equation dz = (∂z/∂x) * Δx + (∂z/∂y) * Δy.
Round both Δz and dz to four decimal places for the final comparison.
I hope this clarifies the step-by-step process! Let me know if you have any other questions.