Find the second derivative of the parametric equations , .Round your answer to two decimal places,if necessary.

Question

Find the second derivative  of the parametric equations  ,  .Round your answer to two decimal places,if necessary. ​

Quizplus · Accepted Answer

The second derivative of a parametric equation is given by:
$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)=\dfrac{d}{dt}\left(\dfrac{dy/dt}{dx/dt}\right)$
Now, $x(t)=\dfrac{t^2}{2}$ and $y(t)=t^3-3t$
Hence, $dx/dt=t$ and $dy/dt=3t^2-3$
Therefore, $\dfrac{d^2y}{dx^2}=\dfrac{d}{dt}\left(\dfrac{3t^2-3}{t}\right)=\dfrac{-3t^2+6}{t^2}=-3+\dfrac{6}{t^2}$
Now, we don't have a single value of $t$ to compute the second derivative. Hence, we can't find the exact value of the second derivative.

However, we can see that as $t$ approaches infinity, the second derivative approaches $0$. Therefore, the answer is A.

Find the Second Derivative of the Parametric Equations