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tangent plane[′tan·jənt ′plān]
The tangent plane to a surface S at a point M is the plane that passes through the point M and that is characterized by the property that the distance from this plane to the variable point M’ on the surface S is infinitesimal in comparison with the distance MM’ as M ’ approaches M. If a surface S has the equation z = f(x, y), then the equation of the tangent plane at the point (x0, y0, z0), where z0 = f(x0, y0), has the form
z − z0 = A(x - x0) + B(y - y0)
if and only if the function f(x, y) has a total differential at thepoint (x0, y0). In this case, A and B are the values of the partialderivatives ∂f/∂x and ∂f/∂x at the point (x0y0) (seeDIFFER-ENTIAL CALCULUS).