Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$

Short answer

The corresponding rectangular equation for the given parametric equations is \(y = \frac{x^{2/3}}{2}\). The curve is oriented in the direction where both x and y are increasing.

Step-by-step solution

Plot the curve

The best way to plot this curve would be to consider few points for \(t\) and then plot the corresponding \(x\) and \(y\). For \(t = -2, -1, 0, 1, 2\), we get five points. Do this and then connect those points smoothly.

Indicate the orientation

Note that as \(t\) increases, \(x\) and \(y\) both increase. This shows that the curve is oriented towards the direction in which both \(x\) and \(y\) are increasing.

Write the corresponding rectangular equation

To write the corresponding rectangular equation, we could express \(t\) in terms of either \(x\) or \(y\) from one of the equations, and then substitute it into the other. From \(x = t^{3}\), we get \(t = \sqrt[3]{x}\). Substituting this into the second equation, we get \(y = \frac{(\sqrt[3]{x})^{2}}{2}\), which simplifies to \(y = \frac{x^{2/3}}{2}\). This is the corresponding rectangular equation.