91Ó°ÊÓ

A tangent line to a curve is horizontal when its slope is zero. In the context of a parametric curve, defined by equations x = f(t) and y = g(t), the slope of the tangent line at any point is given by \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). To find horizontal tangents, we set \( \frac{dy}{dx} = 0 \). This implies that the numerator in the expression \( \frac{dy}{dt} \) needs to be zero while the denominator \( \frac{dx}{dt} \) is non-zero.

For the given problem, we have \( y = t^3 - 3t \). The derivative \( \frac{dy}{dt} = 3t^2 - 3 \). To find when the tangent is horizontal, solve \( 3t^2 - 3 = 0 \), which simplifies to \( t^2 = 1 \). Hence, \( t = 1 \) or \( t = -1 \).

These values of \( t \) correspond to the points \( (1, -2) \) and \( (1, 2) \) on the curve. At these points, the tangent lines are horizontal, meaning they run parallel to the x-axis.

The slope of a tangent line indicates how steep the line is at a particular point on a curve. In the parametric equations \( x = t^2 \) and \( y = t^3 - 3t \), the slope of the tangent line \( \frac{dy}{dx} \) is determined using the chain rule: \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \).

The derivatives \( \frac{dy}{dt} = 3t^2 - 3 \) and \( \frac{dx}{dt} = 2t \) lead to:

• \( \frac{dy}{dx} = \frac{3t^2 - 3}{2t} \)

To interpret the slope properly, remember:

• A positive slope means the curve is increasing at that point.
• A negative slope means the curve is decreasing.
• A zero slope (horizontal tangent) signifies a local minimum or maximum in the curve.
• An undefined slope indicates a vertical tangent, often associated with cusps or sharp turns on the curve.

This understanding of slope aids in visualizing the behavior and tendency of the curve at specific points.

The intersection of a curve with the x-axis occurs where the y-coordinate is zero. For parametrically defined curves, such as \( x = f(t) \) and \( y = g(t) \), find values of \( t \) such that \( g(t) = 0 \).

In this example, we have \( y = t^3 - 3t \). Setting \( y = 0 \) gives \( t^3 - 3t = 0 \). Factoring yields \( t(t^2 - 3) = 0 \), leading to solutions \( t = 0, \sqrt{3}, -\sqrt{3} \). These correspond to points on the x-axis:

• \( t = 0 \) gives point \( (0, 0) \)
• \( t = \sqrt{3} \) gives point \( (3, 0) \)
• \( t = -\sqrt{3} \) gives point \( (3, 0) \)

At these intersections, we examine the slope to understand the curve's behavior:

• For \( t = 0 \), the slope is undefined (vertical tangent).
• For \( t = \sqrt{3} \), the slope is \( \sqrt{3} \).
• For \( t = -\sqrt{3} \), the slope is \( -\sqrt{3} \).

The intersections provide insight into points of tangency and orientation changes, crucial for understanding curve movement and shape.