91Ó°ÊÓ

The process of finding the slope of a curve that is described by parametric equations involves a form of differentiation that accounts for the variables being functions of a third parameter. Instead of using the standard rise over run formula for slope used in Cartesian coordinates, slope in the parametric form is represented as the rate of change of y with respect to x, denoted as \( \frac{dy}{dx} \).

In the given exercise, the parametric equations are \( x = \cos(\theta) \) and \( y = 3\sin(\theta) \). To find the slope at any point, we first find the derivatives \( \frac{dx}{d\theta} = -\sin(\theta) \) and \( \frac{dy}{d\theta} = 3\cos(\theta) \) individually. Then, the slope \( \frac{dy}{dx} \) is obtained by dividing the derivative of y by the derivative of x with respect to \(\theta\), which gives us \( \frac{dy}{dx} = -3\cot(\theta) \). For the slope at \(\theta = 0\), we plug in the value only to encounter an indeterminate result of negative infinity, suggesting a vertical tangent line at this point.

Concavity of a curve described by parametric equations is determined by the second derivative, similar to functions in the Cartesian coordinate system. However, the calculation requires a bit more work, as we first need to find the first derivative with respect to the parameter and then differentiate it again.

In our problem, after computing \( \frac{dy}{dx} \), we then differentiate it with respect to \(\theta\) to get our preliminary second derivative followed by dividing by \( \frac{dx}{d\theta} \) to find \( \frac{d^2y}{dx^2} \). The result, \( \frac{d^2y}{dx^2} = 3\csc(\theta) \), provides insight into the concavity. Nevertheless, at \(\theta = 0\), similar to the slope, the concavity is undefined since \(\csc(0)\) does not exist. It signifies a point of inflection as well as an indeterminate concavity.

When differentiating parametric equations, we compute the derivatives of each coordinate function with respect to the parameter. These derivatives are then manipulated to find the derivative of one coordinate with respect to the other, which encapsulates the behavior of the curve.

To begin with, compute \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \) for the given parametric functions. The differentiation process follows the standard rules as with functions of a single variable, ensuring to apply the chain rule where needed. In our case, differentiation gave us \( \frac{dx}{d\theta} = -\sin(\theta) \) and \( \frac{dy}{d\theta} = 3\cos(\theta) \). These individual derivatives are fundamental steps in elucidating wider curve characteristics like slope and concavity.

The second derivative test for concavity in parametric equations helps us understand the concavity of the curve at a particular value of the parameter. In Cartesian coordinates, a positive second derivative indicates concavity upwards, and a negative second derivative indicates concavity downwards. The test in parametric form operates on the same principle.

After finding \( \frac{dy}{dx} \) by dividing \( \frac{dy}{d\theta} \) by \( \frac{dx}{d\theta} \) as performed earlier, we must differentiate \( \frac{dy}{dx} \) once more with respect to \(\theta\) and divide by \( \frac{dx}{d\theta} \) to get \( \frac{d^2y}{dx^2} \). This second derivative is a crucial test for concavity. In our exercise, again, since \( \theta = 0\) yields values that make \( \frac{d^2y}{dx^2} \) undefined, we cannot use the second derivative test for concavity at this specific point on the curve, asserting the need for a further contextual analysis to determine curve behavior.