Tangent Planes and Differentials

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

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Unformatted text preview: 12.6 Tangent Planes and Differentials In R2 : Tangent Line For x near a, The Linearization of f(x) at x = a is Conclusion: In R3 : Tangent Plane 3 conditions for the tangent plane of z = f (x, y ) at (a, b): 1. 2. 3. 1 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 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 x2 + 2xy − y 2 + z 2 = 7 at P (1, −1, 3). 3 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 ) = x2 + 2xy − 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 u is Example By how much will g (x, y, z ) = x + x cos z − y sin z + y change if the point P (x, y, z ) moves from P0 (2, −1, 0) a distance of ds = 0.2 units toward the point P1 (0, 1, 2)? 4 Defn. Total differential of f If we move from (a, b) to a point (a + dx, b + dy ) nearby, the resulting change in the linearization of f , called the total differential of f , is Example 1 Suppose T (x, y ) = y cos x gives the temperature of a heat source at a given position (x, y ). Suppose your location is measured to be (0, 1) with potential maximum error |dx| = 0.1 and |dy | = 0.2. Estimate the maximum possible error in the computed temperature. Example 2 Around the point (1, 0) is f (x, y ) = x2 (y + 1) more sensitive to changes in x or y ? 5 ...
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Tangent Planes and Differentials - 12.6 Tangent Planes and...

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