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When we talk about horizontal tangency on a curve, we're essentially looking for points where the tangent to the curve is perfectly horizontal. This occurs where the slope of the curve, or the derivative of the y-component with respect to the parameter \( \theta \), is zero. Mathematically, this is when \( \frac{dy}{d\theta} = 0 \). For the given curve, the y-component is \( y = \cos \theta \), thus its derivative is \( \frac{dy}{d\theta} = -\sin \theta \).

• A horizontal tangent happens when \( -\sin \theta = 0 \), which simplifies to \( \sin \theta = 0 \).
• Since \( \sin \theta \) equals zero at integer multiples of \( \pi \), such as \( \theta = 0, \pi, 2\pi, \ldots \), substitute these values into the parametric equations to find the actual points.

Substitute these into the original equations to find:

• If \( \theta = 0 \), then \( x = \cos^2(0) = 1 \) and \( y = \cos(0) = 1 \).
• If \( \theta = \pi \), then \( x = \cos^2(\pi) = 1 \) and \( y = \cos(\pi) = -1 \).

Points of vertical tangency occur where the tangent line is vertical. Such points occur when the derivative of the x-component with respect to \( \theta \) is zero, which implies that \( \frac{dx}{d\theta} = 0 \). The x-component given is \( x = \cos^2 \theta \), and by taking the derivative, we have \( \frac{dx}{d\theta} = -2\cos\theta\sin\theta \).

• Setting \( -2\cos\theta\sin\theta = 0 \), we solve for \( \cos \theta = 0 \) or \( \sin \theta = 0 \).
• A vertical tangent happens when \( \cos \theta = 0 \), occurring at odd multiples of \( \frac{\pi}{2} \) such as \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \).

Substitute these values into the parametric equations:

• If \( \theta = \frac{\pi}{2} \), then \( x = \cos^2(\frac{\pi}{2}) = 0 \) and \( y = \cos(\frac{\pi}{2}) = 0 \).
• If \( \theta = \frac{3\pi}{2} \), then \( x = \cos^2(\frac{3\pi}{2}) = 0 \) and \( y = \cos(\frac{3\pi}{2}) = 0 \).

Thus, the curve has points of vertical tangency when it crosses the y-axis at these points.

Parametric derivatives allow us to study curves that are defined by separate equations for x and y in terms of a third parameter, often \( \theta \). By using parametric derivatives, we can find the slope of the tangent to the curve by comparing the rates of change for x and y with respect to the parameter. This involves:

• Finding \( \frac{dx}{d\theta} \) using the derivative of the x-function. In our case, \( \frac{dx}{d\theta} = -2\cos\theta\sin\theta \).
• Finding \( \frac{dy}{d\theta} \) using the derivative of the y-function, where \( \frac{dy}{d\theta} = -\sin\theta \).

The slope of the tangent to the curve at any point is then found using the formula \( \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \). Simplifying this gives us the ratio of changes and tells us how steeply the curve rises or falls at a given point. This process helps in identifying points of horizontal or vertical tangency by setting \( \frac{dy}{dx} \) to zero or undefined respectively.

A graphing utility is a helpful tool that allows visualization of mathematical equations and concepts. When studying parametric curves like the one given, it's beneficial to use a graphing utility because it helps:

• Visualize the curve and confirm points of horizontal or vertical tangency.
• Ensures our analytic solutions correspond to the actual shape of the curve.

With the graphing utility, you can input the parametric equations \( x = \cos^2\theta \) and \( y = \cos\theta \) to see the curve. Compare the points of tangency found analytically by checking if they match visually with the flat sections (horizontal tangents) or the places where the curve changes direction sharply (vertical tangents). The graphing utility reinforces our understanding by providing a visual representation of abstract mathematical concepts.