Given function $ z = f(x, y) = x^2 y^2 - xy + 2x - 2y $(a) Find the differential $ dz $.(b) Find the equation of the tangent plane to the surface at the point where $ x = -1, y = 1 $.(c) Find the unit vector in the direction of the largest increase of function $ f $ at the point $(-1, 1)$.(d) Find the Taylor polynomial of degree 2 at the point $(-1, 1)$.

Answered: 3. Given function z = f(x,y) = x²y² –…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Given function $ z = f(x, y) = x^2 y^2 - xy + 2x - 2y $

(a) Find the differential $ dz $.

(b) Find the equation of the tangent plane to the surface at the point where $ x = -1, y = 1 $.

(c) Find the unit vector in the direction of the largest increase of function $ f $ at the point $(-1, 1)$.

(d) Find the Taylor polynomial of degree 2 at the point $(-1, 1)$.

Expert Solution

Step 1

The differential can be obtained using the formula $d z = f_{x} d x + f_{y} d y$ . Here, $f_{x}, f_{y}$ are derivative with respect to x and y.

The maximum value of the directional derivative $D_{\vec{u}} f (x, y)$ at the point $(x, y)$ is $||D_{\vec{u}} f (x, y)||$ . The direction of the maximum change is in the direction of the derivative.

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