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Question
If w(x, y) = x3 – 3xy + 2y2, x, y ∈ R, find the linear approximation for w at (1, –1)
Solution
w(x, y) = x3 – 3xy + 2y2, at (1, –1)
Linear approximation is given by
L(x, y, z) = `"w"(x_0, y_0) + (delw)/(delx) "|"_(((x_0, y_0))) (x - x_0) + (delw)/(dely) "|"_(((x_0, y_0))) (y - y_0)`
w(1, –1) = 1 + 3 + 2 = 6
`(delw)/(delx) = 3x^2 - 3y`
`(delw)/(dely) = -3x + 4y`
`(delw)/(delx) |_(((1, -1)))` = 3 + 3 = 6
`(delw)/(dely) |_(((1, -1)))` =– 3 – 4 = –7
∴ L(x, y, z) = 6 + 6(x – 1) – 7(y + 1)
L(x, y, z) = 6x – 7y – 7
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