Question

x^3 + xy -y^2 = 10.

is there any way to find the vertical asymptote by hand?
I can do it graphically, but it looks like a problem that is solvable by hand.

Answer 1

Do you mean algebraically?

Answer 2

exactly

Answer 3

I tried implicit differentiation but I couldn't work it out.

Answer 4

You solve for y, then check out what happens as you move through x. You check for where the function collapses.

Answer 5

I assume you're plotting y vs x?

Answer 6

by letting
\[-2y^2+xy+(x^3+10) = 0 \]

and solving for y using the quadratic formula
gave me a curve and I was able to figure out the solution,
but if there is any way to do it algebraically it would be great.

I tried to differentiate the explicit function, too
but that took a very long time and I doubt that it is the natural way.

Answer 7

woops, I meant -10

Answer 8

Well, you solve for y as per quadratic and then look at the result to see if there are points where parts of the function will be undefined. Let me do the problem.

Answer 9

I appreciate it

Answer 10

\[y=\frac{x \pm \sqrt{x^2-4(10-x^3)}}{2}\]will be a function for y for either the positive or negative square root. The only place this should fall apart is for those x's such that the radicand x^2-4(10-x^3), is less than zero.

Answer 11

For x < about 2.07, the function shouldn't exist. Apart from that, but there are no vertical asymptotes...

Answer 12

it seems like the problem looks easy but the answer is not.
Thanks for the help though, it feels good to have other people
making same conclusions as me.

Answer 13

So did that help? Was your question answered?

Answer 14

Do you have plotting software?

Answer 15

I just graphed it on my graphing calculator TI-86 and
it did all the job I wanted it to do.

Answer 16

http://www.geogebra.org/cms/

Answer 17

It's free ^^

Answer 18

Thanks.