yuki 5 years ago 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.

1. anonymous

Do you mean algebraically?

2. Yuki

exactly

3. Yuki

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

4. anonymous

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

5. anonymous

I assume you're plotting y vs x?

6. Yuki

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.

7. Yuki

woops, I meant -10

8. anonymous

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.

9. Yuki

I appreciate it

10. anonymous

$y=\frac{x \pm \sqrt{x^2-4(10-x^3)}}{2}$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.

11. anonymous

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

12. Yuki

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.

13. anonymous

14. anonymous

Do you have plotting software?

15. Yuki

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

16. anonymous
17. anonymous

It's free ^^

18. Yuki

Thanks.