Question 366019
To determine the concavity we need the second derivative of y with respect to x. To find this we will find the second derivative of x with respect to t and the second derivative of y with respect to t. (Since the notation used in Calculus is not as standard as other parts of Math are and since Algebra.com's software does not make derivative notation easy, I am going to use more words than notation to explain what I'm doing.)<br>
{{{dx/dt = 1 + 1/t}}}
2nd derivative of x with respect to t = {{{(-1)/t^2}}}<br>
{{{dy/dt = 1 - 1/t}}}
2nd derivative of y with respect to t = {{{1/t^2}}}<br>
The second derivative of y with repect to x would be the ratio of the two second derivatives above:
2nd derivative of y with respect to x = (2nd derivative of y with respect to t)/(2nd derivative of x with respect to t)
or
2nd derivative of y with respect to x = {{{(1/t^2)/((-1)/t^2)}}}
which simmplifies to:
2nd derivative of y with respect to x = -1<br>
Since there is no variable in the 2nd derivative of y with respect to x, the concavity is a constant -1. In short, concavity is negative <i>everywhere</i>. This means the curve is concave downward <i>everywhere</i>.<br>
So the answer to "For which values of t is the curve ... concave upward?" is: There are no values of t where the curve is concave upward.