Q. 9 The function聽g(x,y)=x2-y2聽is g... [FREE SOLUTION]

91影视

Chapter 12: Q. 9 (page 943)

The function $g (x, y) = x^{2} - y^{2}$ is graphed in the figure as follows. use the definition of the partial derivatives to show that $f_{x} (0, 0) = f_{y} (0, 0) = 0$ what are the equations of the lines tangent to the surface on the x and y directions at (0,0,0). what is the equation of the plane containing these two lines.

Short Answer

Expert verified

The value of the partial derivative is $f_{x} (0, 0) = 0 f_{y} (0, 0) = 0$

The equation of the lines tangent to the surface in the x and y direction at (0,0,0) is 0.

Equation of the plane containing these two line is z=0.

Step by step solution

Given information

We are given an equation $g (x, y) = x^{2} - y^{2}$ .

Use the definition of partial derivatives to compute the partial derivatives

Using the definition we have

$\lim_{h 鈫� 0} \frac{g (x + h, y) - g (x, y)}{h} \lim_{h 鈫� 0} \frac{{(x + h)}^{2} - y^{2} - x^{2} + y^{2}}{h} \lim_{h 鈫� 0} \frac{x^{2} + 2 x h + h^{2} - y^{2} - x^{2} + y^{2}}{h} \lim_{h 鈫� 0} 2 x + h \lim_{h 鈫� 0} 2 x A t p o i n t (0, 0) w e h a v e t h i s v a l u e e q u a l t o z e r o$

Similarly we have,

$\lim_{k 鈫� 0} \frac{f (x, y + k) - f (x, y)}{k} \lim_{k 鈫� 0} \frac{x^{2} - {(y + k)}^{2} - x^{2} + y^{2}}{k} \lim_{k 鈫� 0} - 2 y + k = - 2 k A t p o i n t (0, 0) w e h a v e t h i s v a l u e e q u a l t o z e r o$

Compute equations of the tangent line

The equation of tangent line in x direction is given as

$z = f (0, 0) + f_{x} (0, 0) z = 0$

The equation of tangent line in y direction is given as

$z = f (0, 0) + f_{y} (0, 0) t z = 0$

Now we find the equation of the plane containing these lines to do this we find the cross product of

$N = (1, 0, f_{y} (0, 0)) 脳 (0, 1, f x (0, 0)) N = (0, 0, - 1)$

And now we take dot product of N and vector (x,y,z)

We get,

$(0, 0, - 1) (x, y, z) - z = 0$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Given information

Use the definition of partial derivatives to compute the partial derivatives

Compute equations of the tangent line

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Pure Maths

Geometry

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

The functiong(x,y)=x2-y2is graphed in the figure as follows. use the definition of the partial derivatives to show thatfx(0,0)=fy(0,0)=0 what are the equations of the lines tangent to the surface on the x and y directions at (0,0,0). what is the equation of the plane containing these two lines.

Short Answer

Step by step solution

Given information

Use the definition of partial derivatives to compute the partial derivatives

Compute equations of the tangent line

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Pure Maths

Geometry

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.