0 Gradient implies?

[kenhung123]

For the function , it has gradient of 0 at x=0
Doesn't that mean the gradient is almost horizontal?
I thought that gradient=0 means the change in y must be close to 0 as change in x cannot be 0 otherwise undefined
But that function in fact has a tangent that is vertical at x=0...

[moekamo]

dy/dx is undefined at x=0  for this function, since it will have a negative power...

[kenhung123]

Is it 0?
Anyway how did they get a vertical tangent?

[/0]

When , and , so there is a vertical tangent there.

[kenhung123]

Hmm how do you know to use dx/dy :S

[vidang]

^
when it talks about the gradient

[shinny]

As moekamo said, doing dy/dx will just get you an undefined gradient at this point, hence you'll know it is vertical. I don't know how you got 0, try your working out again. I don't think you need to use dx/dy, although /0 might have had reasons for it (whether illustrative or for mathematical correctness).

[kenhung123]

Oh yea...it is undef :S I was assuming 0 the power of anything=0 but not in the case of negative powers!
Thanks for clearing that up!
So undef=vertical tangent
0=horizontal tangent?

[shinny]

Oh yea...it is undef :S I was assuming 0 the power of anything=0 but not in the case of negative powers!
Thanks for clearing that up!
So undef=vertical tangent
0=horizontal tangent?

Yep.

[stonecold]

uhh, don't you mean that it is undefined because the denominator of the derivative will = 0 ?

[shinny]

uhh, don't you mean that it is undefined because the denominator of the derivative will = 0 ?

Same thing really; negative power just makes the x become the denominator, hence making the denominator 0.

[/0]

In some cases, an undefined could also indicate a cusp, as in . Here, would also be undefined, which tells you a vertical gradient shouldn't exist. Also, in , is undefined at , and yet a vertical tangent does not exist.

is really only useful for finding vertical tangents. If you imagine flipping the x and y axes, all the vertical tangents become horizontal, so they should be where .

[kenhung123]

Ah, I see what you mean. So dx/dy is like a test to see if a vertical tangent exists IF dy/dx does not exist.

[/0]

Yep

[kenhung123]

Thanks all!