How Do You Determine the Concavity of a Parametric Equation?

[tnutty]

I found a second derivative to the parametric equation :

x = t - e^t

y = t +e^-t


d^2y / dx^2 = e^-t+e^t-2 / (1-e^t)^3

I tried to do some tricks to it, but could not figure out its concavity, any help?

 

[Hogger]

I think you found the wrong second derivative. What did you do to get that?

 

[tnutty]

Its correct, because the online H.W said it is. Now it asks for its concavity. By the way this is Calc2, and not the concavity lesson on Calc1, if that helps any.

 

[Mark44]

I agree with Hogger that your second derivative is wrong. I don't much care what the online HW said. The exponent on the expression in the denominator should be 2, not 3.
Also, you need parentheses surrounding the three terms in the numerator.

Without getting too elaborate with the LaTeX coding, your equation would be improved by looking like this:
d²y/dx² = (e^t - 2 + e^-t)/(1 - e^t)²

As it turns out, and fortunately for you, e^t - 2 + e^-t can be factored. It's a perfect square trinomial. Having it factored makes it much easier to determine when d²y/dx² is positive and when it's negative, which you'll need to determine concavity.