2.3 Tangent Planes
Let \(T_1\) and \(T_2\) be tangents to curve \(C_1\) and \(C_2\). Then the tangent plane to the surface \(S\) at the point \(P\) is defined to be the plane that contains \(T_1\) and \(T_2\).
Equation of Tangent Plane
Suppose \(f\) has continuous first partial derivatives. An equation of the tangent plane to the surface \(z = f(x,y)\) at the point \(P(x_0, y_0, z_0)\) is \[z -z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)\]
Find the tangent plane to the elliptic paraboloid \(z = 2x^2 + y^2\) at the point \((1,1,3)\).
Solution.
\(z = 2x^2 + y^2 = f(x,y)\)
\(f_x = 4x\hspace {1cm} f_x(1,1,3) = 4\)
\(f_y = 2y\hspace {1cm} f_y(1,1,3) = 2\)
\[ z -z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)\]
\[\implies \hspace {1cm} z - 3 = 4(x-1) + 2(y-1)\]
\[z = 4x + 2y -3\]
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