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Question
Let z(x, y) = x2y + 3xy4, x, y ∈ R, Find the linear approximation for z at (2, –1)
Solution
z(x, y) = x2y + 3xy4 at (2, –1)
Here `(x_0, y_0)` = (2, –1)
`(delz)/(delx) = 2xy + 3y^4`
`(delz)/(dely) = x^2 + 12xy^3`
At (2, –1)
z = `(2)^2(- 1) + 3(2) (- 1)^4`
= `- 4 + 6`
= 2
`(delz)/(delx) = 2(2)(- 1) + 3(- 1)^4`
= `- 4 + 3`
= –1
`(delz)/(dely) = (2)^2 12(2)(- 1)^3`
= `4 - 24`
= – 20
Linear approximation is given by
L(x, y) = `z(x_0 + y_0) + ((delz)/(delx))_(((x_0, y_0))) (x - x_0) + ((delz)/(dely))_(((x_0, y_0))) (y - y_0)`
= `2 + (- 1)(x - 2) - 20(y + 1)`
= `2 - x + 2 - 20y - 20`
= ` x - 20y 16`
= `- (x + 20y + 16)`
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