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. $$ \begin{array}{ll} \underline{\text { Parametric Equations }} & \underline{\text { Point }} \\ x=\cos ^{3} \theta, y=\sin ^{3} \theta &\quad \theta=\frac{\pi}{4} \end{array} $$

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.
$$
\begin{array}{ll}
\underline{\text { Parametric Equations }} & \underline{\text { Point }} \\
x=\cos ^{3} \theta, y=\sin ^{3} \theta &\quad \theta=\frac{\pi}{4}
\end{array}
$$

Key Concepts

Concavity

Concavity describes the manner in which a curve bends. It is determined by the sign of the second derivative: if d²y/dx² is positive, the curve is concave upward (cup-shaped), and if it is negative, the curve is concave downward (cap-shaped). This concept is crucial in understanding the local geometric behavior of the curve and indicates the nature of any possible inflection points.

Parametric Equations

Parametric equations express the coordinates of the points that make up a geometric object as functions of a parameter. This approach is particularly useful for describing curves where standard Cartesian equations may be difficult or impossible to define, allowing for the independent variation of each coordinate with respect to the parameter.

First Derivative of a Parametrically Defined Function

The first derivative of a curve defined by parametric equations is obtained by differentiating both x and y with respect to the parameter and then taking the ratio of these derivatives. This derivative, dy/dx = (dy/d?) / (dx/d?), represents the slope of the tangent line to the curve at any given value of the parameter, provided that the derivative dx/d? is non-zero.

Second Derivative for Parametric Equations

For parametric curves, the second derivative provides information on the curvature or concavity of the graph. It is calculated by differentiating the first derivative with respect to the parameter and then dividing by dx/d?, i.e., d²y/dx² = (d/d? (dy/dx)) / (dx/d?), as long as dx/d? does not vanish. This derivative is essential for determining how the slope changes along the curve.

Slope

The slope of a curve at a particular point is defined as the value of the first derivative at that point. In the context of parametric curves, the slope indicates the rate at which the y-coordinate changes with respect to the x-coordinate at the specified parameter value, providing insight into the direction and steepness of the curve.

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