High School

For which values of t is the parametric curve x=913, y=t+t², concave up? (Enter your answer using interval notation i.e., (a,b), [a,b), (a,b] or [a,b])

Answer :

Final answer:

The parametric curve x=913, y=t+t² is concave up for all values of t.

Explanation:

To determine the concavity of the parametric curve x=913, y=t+t², we need to find the second derivative of y with respect to t. The second derivative will tell us whether the curve is concave up or concave down. Let's differentiate y twice and find the interval of t-values where the second derivative is positive.

First, differentiate y: y' = 1 + 2t

Then differentiate y' with respect to t: y'' = 2

The second derivative, y'', is a constant 2. Since it is positive for all values of t, the parametric curve is concave up for all values of t.

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