Question

Estimate f(2.05, 3.95) assuming that f(2, 4) = 5, fx(2,4) = 0.3, fy(2,4) = -0.2 (Use decimal notation. Give your answer to three decimal places.) f(2.05, 3.95) = ...

Answer 1

To estimate f(2.05, 3.95) using the given information, we can use the formula f(2.05, 3.95) ≈ f(2, 4) + fx(2,4) * Δx + fy(2,4) * Δy. Substituting the given values, the estimate is approximately 5.01.

Step-by-step explanation:

To estimate f(2.05, 3.95), we can use the given information: f(2, 4) = 5, fx(2,4) = 0.3, and fy(2,4) = -0.2.

We can approximate the change in x and y values as follows:

Δx = 2.05 - 2 = 0.05

Δy = 3.95 - 4 = -0.05

Now, use the formula:

f(2.05, 3.95) ≈ f(2, 4) + fx(2,4) * Δx + fy(2,4) * Δy

Substituting the given values:

f(2.05, 3.95) ≈ 5 + 0.3 * 0.05 - 0.2 * 0.05

Simplifying:

f(2.05, 3.95) ≈ 5.01

Answer 2

To estimate f(2.05, 3.95), we use the function's value and its partial derivatives at the nearby point (2, 4) to compute the linear approximation. The estimated value turns out to be approximately 5.025.

Step-by-step explanation:

We can estimate the value of the function f(2.05, 3.95) using a linear approximation, which involves the function's value and its partial derivatives at a nearby point. Given that f(2, 4) = 5, fx(2,4) = 0.3, and fy(2,4) = -0.2, we can approximate the change in f as follows:

Δf ≈ fx(2,4) Δx + fy(2,4) Δy

Where Δx = 2.05 - 2 = 0.05 and Δy = 3.95 - 4 = -0.05.

So,

Δf ≈ (0.3)(0.05) + (-0.2)(-0.05)

Δf ≈ 0.015 + 0.01

Δf ≈ 0.025

Therefore, we estimate that:

f(2.05, 3.95) ≈ f(2, 4) + Δf ≈ 5 + 0.025 ≈ 5.025