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Unformatted text preview: Section 15.4 Tangent Planes and Linear Approximations Generalizing the Tangent Line Recall that one of the primary results in Calculus 1 was to approximate functions with lines. Specifically, the tangent line to a function at a point. We generalize this idea to functions of more than one variable. 1. Tangent Planes Suppose a surface S has equation z = f ( x, y ) where f is continuous and differentiable. Let P ( x , y , z ) be a point on S . We want to define the tangent plane of S at P to be the plane which best approximates S at P . We do this as follows: In the plane y = y , there is a 2-d curve defined by z = f ( x, y ). At the point P , we can find the tangent line to this curve using partial derivatives - call it T 1 . In the plane x = x , there is a 2-d curve defined by z = f ( x , y ). At the point P , we can find the tangent line to this curve using partial derivatives - call it T 2 . For T 1 and T 2 , we can find the direction vectors. Specifically, we shall have vectorv 1 = vector i + f x vector k as the direction vector in the x direction and vectorv 2 = vector j + f y vector k in the y-direction (WHY?) We define the tangent plane T to be the plane which contains both vectors vectorv 1 and vectorv 2 . Note that this will always give use a plane because the two vectors determined above will never be parallel and neither will ever be vector (WHY?). Calculation of an equation for the tangent plane is fairly 1 2 straight forward using vector operations. Specifically, we first find the normal vector to the plane: vectorv 1 vectorv 2 = ( vector i + f x vector k ) ( vector j + f y vector k ) = vector i vector j + vector i f y vector k + f x vector...
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