Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am asked to find the equations of the tangent lines at three different points for the following function:

$$y^{5}-y-x^{2}=-1$$

I am provided with a graph/sketch of the function and asked to find the tangent lines to the three different points at $x=1$. I found the following with implicit differentiation:

$$\frac{dy}{dx}=\frac{2x}{5y^{4}-1}$$

My problem is finding the points where the curve intersects $x=1$. It is pretty clear from the graph (though not explicitly marked) that the points will be $y=-1$,$y=0$, and $y=1$. Wolfram Alpha confirms this http://www.wolframalpha.com/input/?i=y%5E5-y-x%5E2%3D-1+and+x%3D1. I just don't know if seeing them visually is enough or if I have to derive the intersection points mathematically.

Thank you very much.

share|improve this question
add comment

1 Answer 1

up vote 1 down vote accepted

Well, have you tried solving? When $x=1$, you have $y^5 - y -(1)^2 = -1$, which means $y^5 = y$, hence either $y=0$ or $y^4 = 1$, which means $y=\pm 1$. Satisfied?

Hope that helps =)

share|improve this answer
    
Just so you know that in general there are also the complex solutions $\pm i$ but I don't think they were appropriate in the context. –  Patrick Da Silva Nov 3 '11 at 3:36
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.