This problem is asking for the second derivative of the parametric curve given by $x = t^2$ and $y = t^3 - 3t$. The goal is to compute the second derivative $\frac{d^2 y}{dx^2}$ and determine the concavity of the curve (whether it is concave up or down) at different points.

Let's break it down:

Step 1: First derivative $\frac{dy}{dx}$

We know that: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

Given:

• $x = t^2$, so $\frac{dx}{dt} = 2t$
• $y = t^3 - 3t$, so $\frac{dy}{dt} = 3t^2 - 3$

Now, we can compute the first derivative: $\frac{dy}{dx} = \frac{3t^2 - 3}{2t}$

Step 2: Second derivative $\frac{d^2 y}{dx^2}$

Using the relation for the second derivative: $\frac{d^2 y}{dx^2} = \frac{d}{dt} \left( \frac{dy}{dx} \right) \div \frac{dx}{dt}$

First, we need to compute $\frac{d}{dt} \left( \frac{dy}{dx} \right)$: $\frac{d}{dt} \left( \frac{dy}{dx} \right) = \frac{d}{dt} \left( \frac{3t^2 - 3}{2t} \right)$

Using the quotient rule for differentiation: $\frac{d}{dt} \left( \frac{3t^2 - 3}{2t} \right) = \frac{(2t)(6t) - (3t^2 - 3)(2)}{(2t)^2}$

Simplifying the numerator: $= \frac{12t^2 - 2(3t^2 - 3)}{4t^2} = \frac{12t^2 - 6t^2 + 6}{4t^2} = \frac{6t^2 + 6}{4t^2} = \frac{6(t^2 + 1)}{4t^2}$

Now we can calculate $\frac{d^2 y}{dx^2}$: $\frac{d^2 y}{dx^2} = \frac{6(t^2 + 1)}{4t^2} \div 2t = \frac{6(t^2 + 1)}{8t^3}$

This is the second derivative.

Step 3: Concavity

The sign of $\frac{d^2 y}{dx^2}$ determines the concavity of the curve. If $\frac{d^2 y}{dx^2} > 0$, the curve is concave up, and if $\frac{d^2 y}{dx^2} < 0$, the curve is concave down.

• When $t > 0$, $\frac{d^2 y}{dx^2} > 0$ (concave up).
• When $t < 0$, $\frac{d^2 y}{dx^2} < 0$ (concave down).

Would you like more details or have any questions about the steps? Here are 5 related questions for you:

1. Can you verify the calculation of the first derivative $\frac{dy}{dx}$?
2. What is the geometric interpretation of the first and second derivatives for a curve?
3. How would the concavity change if the parametric equations were slightly different?
4. How do the values of $t$ relate to inflection points on the curve?
5. How would you interpret concavity for this curve in the context of motion or physics?

Tip: When working with parametric curves, it's helpful to always differentiate with respect to the parameter $t$, as it simplifies handling derivatives and analyzing curve properties.