91Ó°ÊÓ

When studying tangent lines, a horizontal tangent is where the slope of the line is zero. This happens when the change in y with respect to x, or the derivative \( \frac{dy}{dx} \), is zero. To find a horizontal tangent for a curve defined by parametric equations, we calculate the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \), and then find \( \frac{dy}{dx} \) using the formula \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). The challenge is to determine \( t \) where \( \frac{dy}{dx} = 0 \). This involves solving the equation \( 6t - 9t^4 = 0 \), leading to specific solutions for \( t \) that correspond to points where the tangent is horizontal, such as \( (0,0) \) and another point found through substitution.

A vertical tangent occurs when the curve's rate of change in x terms is zero but not in y terms. Mathematically, this means \( \frac{dx}{dt} = 0 \) but \( \frac{dy}{dt} eq 0 \). In our exercise, we use the expression for \( \frac{dx}{dt} \) to find such a point. We set the equation \( 3 - 6t^3 = 0 \) and solve for \( t \). This value of \( t \) is used to substitute back into the parametric equations of x and y to find the exact coordinates where the vertical tangent line occurs. This provides a clear understanding of how the vertical tangent is defined independently of the horizontal.

Derivatives are fundamental when working with tangent lines. They help us understand how a curve behaves and changes. For parametric equations, you must calculate \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) separately. Using the derivative rules, especially the quotient rule, is necessary when the equations are fractions. For instance, differentiating \( x = \frac{3t}{1+t^3} \) and \( y = \frac{3t^2}{1+t^3} \) requires careful application of these rules to obtain \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \). From these, \( \frac{dy}{dx} \) is found by dividing \( \frac{dy}{dt} \) by \( \frac{dx}{dt} \), crucial for determining the slope of tangent lines.

Parametric equations express a set of related quantities as functions of a parameter, usually \( t \). They are incredibly useful for describing complex curves that can't be represented with a single function. In this context, \( x = \frac{3t}{1+t^3} \) and \( y = \frac{3t^2}{1+t^3} \) are parametric equations with \( t \) as an independent parameter. Understanding these allows us to trace out the curve as \( t \) changes. Each value of \( t \) provides coordinates \( (x,y) \) on the plane. This understanding is key when we need to analyze features like tangent lines, because it reveals how changes in \( t \) affect the entire curve.