91Ó°ÊÓ

Understanding horizontal tangency in the context of parametric equations requires looking at the slope of the tangent line to the curve. In this case, a horizontal tangent occurs where the slope is zero. When we have a parametric equation, such as the one in our exercise \( x=t+4, y=t^{3}-3t \), we can find the horizontal tangents by setting the derivative \( y'(t) \) equal to zero:

For our equation, \( y'(t) = 3t^{2}-3 \), solving for \( y'(t) = 0 \) provides us with specific values of \( t \) that correspond to horizontal tangency. In our example, we find two points of horizontal tangency, occurring at \( t = -1 \) and \( t = 1 \) which correspond to the points (5, -2) and (3, 2) on the graph.

Vertical tangency is characterized by a tangent line that is vertical, or in other words, the slope of the tangent is undefined or infinite. To locate points of vertical tangency in parametric equations, we investigate where the denominator of the derivative expression, \( dx/dt \) approaches zero. With \( x'(t) = 1 \) for our provided exercise, we note that there are no values of \( t \) where \( x'(t) \) is zero since it remains constant. Consequently, this implies that no vertical tangency will occur for the curve described by the given set of parametric equations.

To find points of tangency, we must first understand how to compute the derivative of a parametric equation. The derivative \( y'(t) \) represents the slope of the tangent line and is calculated by dividing the derivative of \( y \) with respect to \( t \) by the derivative of \( x \) with respect to \( t \) i.e., \( y'(t) = \frac{dy/dt}{dx/dt} \). In this exercise, the derivatives \( \frac{dx}{dt} = 1 \) and \( \frac{dy}{dt} = 3t^{2}-3 \) are used to compute \( y'(t) \) which is then analyzed to determine the points of tangency. This step is crucial for understanding the behavior of the graph at different points.

Once we have theoretically determined the points of horizontal and vertical tangency, it's essential to visually confirm the findings with a graphing utility. By plotting the curve defined by the parametric equations \( x=t+4 \) and \( y=t^{3}-3t \), we can inspect the presence of horizontal tangency at the points found in our calculations. Using a graphing utility serves as a visual concretization of the abstract derivatives and algebraic manipulations we perform, providing an excellent tool to affirm that the computed points (5, -2) and (3, 2) align with the graph's apparent horizontal tangent lines. Notably, for this specific curve, there will be no vertical tangent lines to confirm, as illustrated when the derivative of \( x \) with respect to \( t \) never equals zero.