Find d y / d x and d^2 y / d x^2, and find the slope and...

Find $d y / d x$ and $d^{2} y / d x^{2}$, and find the slope and concavity (if possible) at the given value of the parameter.
$$
\begin{array}{ll}
\underline{\text { Parametric Equations }} & \underline{\text { Point }} \\
x=2 t, y=3 t-1 &\quad t=3
\end{array}
$$

Key Concepts

Parametric Equations

A parametric equation represents a curve by expressing the coordinates of the points on the curve as functions of an independent parameter. This concept allows for the description of curves that may not be functions in the traditional y = f(x) form, and it is fundamental for analyzing motion and curves in a different coordinate system.

First Derivative of Parametric Equations

The first derivative dy/dx in parametric form is found by taking the derivative of y with respect to the parameter and dividing it by the derivative of x with respect to the same parameter, i.e., dy/dx = (dy/dt) / (dx/dt). This application of the chain rule is crucial for determining the slope of the tangent to the curve at a given point.

Second Derivative of Parametric Equations

The second derivative, d²y/dx², measures the curvature or concavity of the curve. It is computed by differentiating dy/dx with respect to the parameter and then dividing by dx/dt. This process requires careful application of the chain rule to correctly capture how the slope changes as the parameter varies.

Slope and Concavity Evaluation

Determining the slope and concavity at a specific parameter value involves evaluating the first and second derivatives at that value. The slope gives the angle of the tangent line to the curve, while the concavity, indicated by the sign of the second derivative, reveals whether the curve bends upward or downward. These concepts are essential for understanding the geometry and behavior of the curve at a given point.

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