Answer
$\frac{dy}{dx} = \frac{3}{4}$
$\frac{d^2y}{dx^2} = 0$
In t = 3:
The slope is equal to $3/4$, and there is no concavity.
Work Step by Step
1. Find dx/dt and dy/dt
$x = 4t$
$dx = (4)dt$
$dx/dt = 4$
$y = 3t -2$
$dy = (3 + 0)dt$
$dy/dt = 3$
2. Calculate dy/dx:
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3}{4}$
3. Find $d^2y/dx^2$
$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}[dy/dx]}{dx/dt} = \frac{\frac{d}{dt}[3/4]}{4} = \frac{0}{4} = 0$
4. Find the slope and the concavity:
The slope is dy/dx in t = 3, which is: $\frac{3}{4}$
Since the second derivative is equal to 0, this equation doesn't have any concavity (upward or downward).
