Second Derivatives of Parametric Equations

Concave up $$\left[0,\ \infty\right)$$

Concave down $$\left[0,\infty\right)$$

Concave up $$\left[0,\ 4\right]$$ Concave down $$\left[4,\ \infty\right)$$

Concave down $$\left[0,\ 4\right]$$ .

Concave up $$\left[4,\ \infty\right)$$

$$\frac{d^2y}{dx^2}=\frac{144t^3}{3t^2+1}$$

$$\frac{d^2y}{dx^2}=\frac{3t^2+1}{144t^3}$$

$$\frac{d^2y}{dx^2}=\frac{-36t^2-12}{\left(3t^2-1\right)^3}$$

$$\frac{d^2y}{dx^2}=\frac{36t^2+12}{\left(3t^2-1\right)^3}$$

Slope: $$2$$

Slope $$-\frac{1}{2}$$

Concave Up

Concave Down

$$2\csc^2\Theta$$

$$-2\csc^3\ \Theta$$

$$\frac{-2\sin\Theta}{\cos^2\Theta}$$

$$-2\sec^3\Theta$$

$$\frac{\sin\left(3t\right)}{4}$$

$$-\frac{9\sin\left(3t\right)}{64}$$

$$\frac{3\cos\left(3t\right)}{16}$$

Undefined