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Unformatted text preview: 15.4 Tangent Planes and Linear Approximations Tangent Planes and Linear Approximations Definition 15.4.1 Let z = f ( x, y ) determine the surface S . If f has continuous first order partial deriva- tives at ( x , y ) , then the tangent plane to the surface S at a point P ( x , y , z ) is the plane that contains the point P and the tangent lines to the curves of intersections with the vertical planes x = x and y = y at the point P . Is there a general formula for the tangent plane at a given point? Lets see. The point P ( x , y , z ) is the point on the plane. In the plane y = y , the tangent line to the curve of intersection has para- metric equations x = t, y = y , z = f x ( x , y )( t- x ) + f ( x , y ) , and the tangent line to the curve of intersection in the vertical plane x = x has parametric equations x = x , y = t, z = f y ( x , y )( t- y ) + f ( x , y ) . Problem: Find a normal vector to the tangent plane containing the two tangent lines given above, and then, find an equation of the tangent plane.then, find an equation of the tangent plane....
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