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Chapter 12: Q. 8 (page 943)

The function f(x,y)=|xy|is graphed in the figure as follows. use the definition of the partial derivatives to show that fx(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

Expert verified

The value of the partial derivative is fx(0,0)=0fy(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

01

Given information 

We are given a functionf(x,y)=|xy|

02

Use the definition of partial derivatives to compute the partial derivatives

Using the definition we have

limh鈫�0f(x+h,y)-f(x,y)hlimh鈫�0|(x+h)y|-|xy|hlimh鈫�0hyhlimh鈫�0yatpoint(0,0)wehavethislimitequaltozero

Similarly, We have

limk鈫�0f(x,y+k)-f(x,y)klimk鈫�0|x(y+k)|-|xy|klimk鈫�0xkklimk鈫�0xAtpoint(0,0)wehavethislimitequaltozero

Hence the partial derivative is zero

03

Compute equations of the tangent line

The equation of tangent line in x direction is given as

z=f(0,0)+fx(0,0)tz=0

The equation of tangent line in y direction is given as

z=f(0,0)+fy(0,0)tz=0

Now we find the equation of the plane containing these lines

to do this we find the cross product of

N=(0,1,fy(0,0))脳(1,0,fx(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

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