Second Derivative of Parametric Equations

Question

For each parametric curve above, compute the second derivative $$\frac{d^2y}{dx^2}$$ at the indicated evaluation point and rank the curves in order of increasing concavity (i.e., increasing $$\frac{d^2y}{dx^2}$$). The approximate values are: Curve 1: 2, Curve 2: 6, Curve 3: -0.25, Curve 4: -1. Thus, order from least to greatest is required.
 
| Curve      | Parametric Equations               | Evaluation (for $\frac{d^2y}{dx^2}$): use $t=1$ |
|------------|------------------------------------|-------------------------------------------------------|
| Curve 1    | $$x=t, \quad y=t^2$$                |                                                       |
| Curve 2    | $$x=t, \quad y=t^3$$                |                                                       |
| Curve 3    | $$x=t, \quad y=\sqrt{t}$$           |                                                       |
| Curve 4    | $$x=t, \quad y=\ln(t)$$             |                                                       |

Accepted Answer

Curve 4, Curve 3, Curve 1, Curve 2

Answer

Curve 3, Curve 4, Curve 1, Curve 2

Answer

Curve 4, Curve 1, Curve 3, Curve 2

Answer

Curve 1, Curve 3, Curve 4, Curve 2

Curve	Parametric Equations	Evaluation (for \(\frac{d^2y}{dx^2}\)): use \(t=1\)
Curve 1	$$x=t, \quad y=t^2$$
Curve 2	$$x=t, \quad y=t^3$$
Curve 3	$$x=t, \quad y=\sqrt{t}$$
Curve 4	$$x=t, \quad y=\ln(t)$$

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