A very easy math problem I can't do

[The_Liquid_Laser]

It's been a while since I've seen this material, but the notation doesn't look quite right to me. (Perhaps I am just not familiar with it.) From the context of the problem f is a function with multiple independent variables, e.g. f(x,y). Therefore f(x/y) doesn't make sense to me. Is this the same as f(x/y, 0) ?

[YourLocalJesus]

Good LORD! This is why we entj's need the backup of specialists

[ygolo]

Originally Posted by The_Liquid_Laser

It's been a while since I've seen this material, but the notation doesn't look quite right to me. (Perhaps I am just not familiar with it.) From the context of the problem f is a function with multiple independent variables, e.g. f(x,y). Therefore f(x/y) doesn't make sense to me. Is this the same as f(x/y, 0) ?

Aha! That interpretation works. (Clever, LL)

Note the "d" are actually denoting partials.

f(x,y)=x(x/y,0)
df(x,y)/dx=f(x/y,0)+x[df(x/y,0)/d(x/y)](d(x/y)/dx)=f(x/y,0)+(x/y)[df(x/y,0)/d(x/y)]
df(x,y)/dy=x[df(x/y,0)/d(x/y)](d(x/y)/dy)=-(x/y)^2[df(x/y,0)/d(x/y)]

Now note for all x,y:
(df(x,y)/dx)x+(df(x,y)/dx)y=xf(x/y,0)+(x^2/y)[df(x/y,0)/d(x/y)]-(x^2/y)[df(x/y,0)/d(x/y)]=xf(x/y,0)=f(x,y).

Note that this is exactly the condition we need.
f(x,y)=(df(x,y)/dx)x+(df(x,y)/dx)y

So, the equation for a tangent plane at point (x0, y0, f(x0,y0)) is:
z-f(x0,y0)=(df(x,y)/dx|x=x0)(x-x0)+(df(x,y)/dx)(y-y0)

The plane goes throught (0,0,0) if and only if:
0-f(x0,y0)=(df(x,y)/dx|x=x0)(0-x0)+(df(x,y)/dx)(0-y0)

Which is the same equation as:
f(x0,y0)=(df(x,y)/dx|x=x0)x0+(df(x,y)/dx)y0

And we know the above equation is simply:
f(x,y)=(df(x,y)/dx)x+(df(x,y)/dx)y with (x,y)=(x0,y0)

So all the tangent planes intercept (0,0,0).

[IlyaK1986]

Originally Posted by YourLocalJesus

Good LORD! This is why we entj's need the backup of specialists

This is why we ENTJs know enough that we can hold our own with the specialists, but our strength lies in knowing how to best utilize the resources that the specialists present us with.

[FDG]

Originally Posted by IlyaK1986

This is why we ENTJs know enough that we can hold our own with the specialists, but our strength lies in knowing how to best utilize the resources that the specialists present us with.

Yeah allright, but my problem with this problem in the end was about the notation of the function, i wasn't sure on how to take the derivative...I've already thnaked ygolo in private for having clarified it to me : )