find the approximate change in z when the point (x,y) changes from (1,1)to (1.01, 0.90)
f(x,y) = xe^(-y) + ye^(-x)
just use the total differential
dz = Zx dx + Zy dy
= (e^-y - ye^-x)dx + (-xe^-y + e^-x)dy
= (e^-1 - 1e^-1)(.01) + (-1e^-1 + e^-1)(-.1)
= 0
seems odd, but I don't see where I might have gone astray.
To find the approximate change in z, we can use the partial derivatives of the function f(x, y) = xe^(-y) + ye^(-x).
The partial derivative with respect to x, denoted as ∂f/∂x, is found by taking the derivative of the function with respect to x while keeping y constant:
∂f/∂x = e^(-y) - ye^(-x)
Similarly, the partial derivative with respect to y, denoted as ∂f/∂y, is found by taking the derivative of the function with respect to y while keeping x constant:
∂f/∂y = -xe^(-y) + e^(-x)
To find the approximate change in z, Δz, we can use the formula:
Δz ≈ ∂f/∂x * Δx + ∂f/∂y * Δy
Here, Δx represents the change in x, and Δy represents the change in y.
Given that the point (x, y) changes from (1, 1) to (1.01, 0.90), we can calculate Δx and Δy as follows:
Δx = 1.01 - 1 = 0.01
Δy = 0.90 - 1 = -0.10
Substituting the values into the formula, we have:
Δz ≈ (∂f/∂x * Δx) + (∂f/∂y * Δy)
≈ (e^(-1) - e^(-1) * 1) * 0.01 + (-1 * e^(-1) + e^(-1)) * (-0.10)
Calculating the above expression will give the approximate change in z when the point (x, y) changes from (1, 1) to (1.01, 0.90).
To find the approximate change in z when the point (x, y) changes from (1, 1) to (1.01, 0.90), we need to calculate the partial derivatives of f(x, y) with respect to x and y and then use these derivatives to approximate the change in z.
Step 1: Calculate the partial derivative of f(x, y) with respect to x:
To find the partial derivative of f(x, y) with respect to x, we treat y as a constant and differentiate f(x, y) with respect to x:
∂f/∂x = ∂(xe^(-y))/∂x + ∂(ye^(-x))/∂x
Differentiating each term:
= e^(-y) + (-1) * ye^(-x)
Simplifying:
∂f/∂x = e^(-y) - ye^(-x)
Step 2: Calculate the partial derivative of f(x, y) with respect to y:
To find the partial derivative of f(x, y) with respect to y, we treat x as a constant and differentiate f(x, y) with respect to y:
∂f/∂y = ∂(xe^(-y))/∂y + ∂(ye^(-x))/∂y
Differentiating each term:
= (-1) * xe^(-y) + e^(-x)
Simplifying:
∂f/∂y = e^(-x) - xe^(-y)
Step 3: Approximate the change in z:
Using the partial derivatives calculated in steps 1 and 2, we can approximate the change in z by using the formulas:
Δz ≈ (∂f/∂x)Δx + (∂f/∂y)Δy
where Δx = x2 - x1 and Δy = y2 - y1 are the changes in x and y respectively, and (x1, y1) = (1, 1) and (x2, y2) = (1.01, 0.90) are the initial and final points.
Plugging in the values:
Δz ≈ (∂f/∂x)(1.01 - 1) + (∂f/∂y)(0.90 - 1)
Now you can substitute the values of (∂f/∂x) and (∂f/∂y) from steps 1 and 2 respectively, and calculate the approximate change in z.