Answered: Tangent planes Find an equation of the…

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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Tangent planes Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations).

z = 4 - 2x² - y²; (2, 2, -8) and (-1, -1, 1)

Expert Solution

Step 1 To find:

An equation of the plane tangent to the following surfaces at the given points (two planes and two equations):

$z = 4 - 2 x^{2} - y^{2}$ ; $(2, 2, - 8)$ and $(- 1, - 1, 1)$ .

Step 2 Concept used:

Equation of the tangent to surface $z = f (x, y)$ .

$f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0}) - (z - z_{0}) = 0$

Step 3

Here, $f (x, y) = 4 - 2 x^{2} - y^{2}$

Differentiate $f (x, y)$ with respect to x.

$\begin{array}{rcl} f_{x} & = & 0 - (2) (2 x) - 0 \\ = & - 4 x \end{array}$

Differentiate $f (x, y)$ with respect to y.

$\begin{array}{rcl} f_{y} & = & 0 - 0 - 2 y \\ = & - 2 y \end{array}$

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Tangent planes Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). z = 4 - 2x2 - y2; (2, 2, -8) and (-1, -1, 1)