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Let z = f(x, y). Estimate f(100.2, 200.2) when f(100, 200) =15, fx(100, 200) = 10, and fy(100, 200) = 20.

User Endy
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1 Answer

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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.

User Suiwenfeng
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6.8k points
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