4.Consider the surface given by x2z+ 2yz² = 3xy2.(a) Find and at the point (1,1,1).(b) Find the tangent plane to the surface at the point (1,1,1).

Surface given, x2z+2yz2=3xy2 ....(i) a)we have to find ∂z∂x and ∂z∂y Take equation (i) and…

Answered: Consider the surface given by x²z+ 2yz2…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section: Chapter Questions

Problem 18T

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Question

4.
Consider the surface given by x2z+ 2yz² = 3xy2.
(a) Find and at the point (1,1,1).
(b) Find the tangent plane to the surface at the point (1,1,1).

Expert Solution

Step 1

Surface given,

$x^{2} z + 2 y z^{2} = 3 x y^{2} . . . . (i)$

a)we have to find $\frac{\partial z}{\partial x} and \frac{\partial z}{\partial y}$

Take equation (i) and partially differentiate it with respect to 'x' and 'y'

$x^{2} z + 2 y z^{2} = 3 x y^{2} Partially differentiate with respect to' x' we get \frac{\partial z}{\partial x} as x^{2} \frac{\partial z}{\partial x} + 2 x z + 4 y z \frac{\partial z}{\partial x} = 3 y^{2} \frac{\partial z}{\partial x} = \frac{3 y^{2} - 2 x z}{x^{2} + 4 z y} And \frac{\partial z}{\partial x} at point (1, 1, 1) {(\frac{\partial z}{\partial x})}_{(1, 1, 1)} = \frac{3 {(1)}^{2} - 2 \times 1 \times 1}{{(1)}^{2} + 4 \times 1 \times 1} = \frac{1}{5}$

$x^{2} z + 2 y z^{2} = 3 x y^{2} Partially differentiate with respect to' y' we get \frac{\partial z}{\partial y} as x^{2} \frac{\partial z}{\partial y} + 4 y z \frac{\partial z}{\partial y} + 2 z^{2} = 6 x y \frac{\partial z}{\partial y} = \frac{- 2 z^{2} + 6 x y}{x^{2} + 4 z y} And \frac{\partial z}{\partial y} at point (1, 1, 1) {(\frac{\partial z}{\partial y})}_{(1, 1, 1)} = \frac{- 2 {(1)}^{2} + 6 \times 1 \times 1}{{(1)}^{2} + 4 \times 1 \times 1} = \frac{4}{5}$

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