Final answer:
To estimate f(100.2, 200.2), the linear approximation method is used, which results in an estimated value of approximately 21.
Step-by-step explanation:
The student is asked to estimate the value of the function f at the point (100.2, 200.2) using the given information of the function's value and partial derivatives at the point (100, 200). This is a problem involving partial derivatives and can be solved using the linear approximation or tangent plane approximation of a multivariable function.
To estimate f(100.2, 200.2), we can use the following formula based on the tangent plane approximation:
f(a + Δx, b + Δy) ≈ f(a, b) + fx(a, b)Δx + fy(a, b)Δy,
where Δx = x - a and Δy = y - b.
In this problem we have:
- Δx = 100.2 - 100 = 0.2
- Δy = 200.2 - 200 = 0.2
- f(100, 200) = 15
- fx(100, 200) = 10
- fy(100, 200) = 20
Plugging these values into our formula, we get:
f(100.2, 200.2) ≈ 15 + (10)(0.2) + (20)(0.2) = 15 + 2 + 4 = 21.
The estimated value of f(100.2, 200.2) is therefore approximately 21.