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Tangent Planes and Differentials

Tangent Planes and Differentials - 12.6 Tangent Planes and...

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12.6 Tangent Planes and Differentials In R 2 : Tangent Line For x near a , The Linearization of f(x) at x = a is Conclusion: In R 3 : Tangent Plane 3 conditions for the tangent plane of z = f ( x, y ) at ( a, b ): 1. 2. 3. 1
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The equation of the Tangent Plane to z = f ( x, y ) at ( a, b ) Example 1 Find the equation of the plane tangent to f ( x, y ) = xy 2 + y cos( x - 1) at P (1 , 2). Defn. Linearization of f ( x, y ) at ( a, b ) The linearization of a function f ( x, y ) at a point ( a, b ) is Note Back to Example l The linearization of f ( x, y ) = xy 2 + y cos( x - 1) at P (1 , 2) is 2
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The equation of the Tangent Plane to f ( x, y, z ) = k at P ( a, b, c ) Note The Line Normal to f ( x, y, z ) = k at P ( a, b, c ) Example 2 Find the plane tangent to x 2 + 2 xy - y 2 + z 2 = 7 at P (1 , - 1 , 3). 3
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Defn. Linearization of f ( x, y, z ) at ( a, b, c ) The linearization of a function f ( x, y, z ) at a point ( a, b, c ) is Back to Example 2 Find the linearization of f ( x, y, z ) = x 2 + 2 xy - y 2 + z 2 at P (1 , - 1 , 3). Estimating Directional Change in f The change in f as we move a small distance ds from a point P in the direction
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