This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View Full
Document
 Spring '03
 MECothren
 Multivariable Calculus

Click to edit the document details