Finding Tangent Plane to z = x²+xy+y² at (1,2,7)

Calculus 3 · Axiom Academy

EXAMPLE Finding the Tangent Plane to z = x^2 + xy + y^2 at (1, 2, 7) Build a tangent plane from partial derivatives: the surface's best linear approximation at a point. Find the equation of the tangent plane to the surface z = x^2 + xy + y^2 at the point (1, 2, 7) . The tangent plane is the flat sheet that just grazes the surface there — the best linear approximation to the surface near that point. Nice work — you built the tangent plane to z = x^2 + xy + y^2 at (1, 2, 7) from the ground up. Tangent plane formula: for a surface z = f(x,y) , the plane at (x_0, y_0, z_0) is z - z_0 = f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) . Partial derivatives: compute f_x by treating y as constant and f_y by treating x as constant — here f_x = 2x + y and f_y = x + 2y . Evaluate at the point: plug in (1, 2) to get the slopes f_x(1,2) = 4 and f_y(1,2) = 5 . Result: the tangent plane is z = 4x + 5y - 7 , the best linear approximation to the surface near (1, 2, 7) . Check: the point of tangency must lie on the plane — 4(1) + 5(2) - 7 = 7 , so z = 7 ︎ The tangent plane extends the idea of a tangent line from curves to surfaces — the foundation for differentiability, linear approximation, and optimization in higher dimensions.

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