Functions of two variables-doubt!

[Nupur]

This question is a from Hammond! My answers do not match with that given in the book and neither do I get the way it's done in the book! Can anyone please give these questions a try??

Q. Find the domain of the following two functions:
https://www.evernote.com/shard/s242/sh/6fe4a0f9-79cd-47ba-a6eb-0f925f5bc36e/bf8c5e7fa791286aaaca70e0c9ff92f5

[Noel]

Q.1 the domain is the intersection of x>=y and x^2+y^2>=1 i.e. the area of the graph under the line y=x and outside of the unit circle centered at origin

Q.2 domain is the common area of intersection of the parabolas y=x^2 and y^2=x

[Homer Simpson]

thank you Noel!

[Nupur]

Yeah! My answers were close to yours! Your answers make perfect sense Noel! Then the answers given in Hammond must be for some other questions and wrongly printed as answers to these questions! Have a look at what the book says the solutions should be! It does not make sense!?

[Noel]

Hmm..the answers seem to be incorrect because if the answer for the first question is x^2+y^2<=2 then this is the region inside the circle centered at origin with radius under-root 2..hence the point (0,0) must be in the domain..but substituting these values in the expression under-root(x^2+y^2-1) we get root(-1) which is not defined in the real number system

[Noel]

Similarly for question 2..the solution provided by hammond is the region within the concentric circles x^2+y^2=1 and x^2+y^2=4 which again seems incorrect

[Nupur]

Exactly! Thanks.. Noel!
Can you answer this one also from the same topic (Functions a of Several Variables-Tangent Planes):
Prove that all tangent planes to z=xf(y/x) pass through the origin.

[The Villain]

Tangent plane at (a,b) will be given by
z-z1=del z/del x (x-a) + del z/del y(y-b)
We have z1=af(b/a)
now find del z / delx and del z/del y at (a,b)
we get eqn as
z-af(b/a)=f(b/a)-(b/a)f'(b/a)(x-a)+f'(b/a)(y-b)
the tangent plane will pass thtough origin if x=y=0
substitute the vlues x=0 and y=0
 we will get z=0
hence proved.. :-)

[Noel]

[Nupur]

Got it! Thanks Ron and Noel! :)