Question 12.8.3: Find the tangent plane at P = (x0, y0, z0) = (1, 1, 5) to th......

Essential Mathematics for Economic Analysis [1185377]

Find the tangent plane at $P = (x_{0}, y_{0}, z_{0}) = (1, 1, 5)$ to the graph of

f (x, y) = x² + 2xy + 2y²

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Because f (1, 1) = 5, P lies on the graph of f .We find that $f^{\prime}_{1}(x, y) = 2x + 2y$ and $f^{\prime}_{2}(x, y) = 2x + 4y.$ Hence, $f^{\prime}_{1}(1, 1) = 4$ and $f^{\prime}_{2}(1, 1) = 6.$ Thus, Eq. (12.8.3) yields

$z-z_{0}=f_{1}^{\prime}(x_{0},y_{0})(x-x_{0})+f_{2}^{\prime}(x_{0},y_{0})(y-y_{0})$ (12.8.3)
z − 5 = 4(x − 1) + 6(y − 1)
or, equivalently, z = 4x + 6y − 5.

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