Problem 3 - Essential Calculus

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. $$ x=t^{3}, \quad y=\frac{t^{2}}{2} $$

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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.

Key Concepts

Rectangular Equation

A rectangular equation is one that relates two variables, typically x and y, without involving any parameter. When dealing with parametric equations, which describe curves using a third variable, like t, the goal is often to eliminate this parameter. This lets us express y directly as a function of x, simplifying analysis and graphing. To convert from parametric to rectangular form, we solve one of the parametric equations for the parameter and substitute it into the second. For the parametric equations \( x = t^3 \) and \( y = \frac{t^2}{2} \), we solve \( x = t^3 \) to get \( t = \sqrt[3]{x} \). After substituting \( t = \sqrt[3]{x} \) into the y equation, we obtain the rectangular form: \[ y = \frac{x^{2/3}}{2} \] This expression describes the original parametric curve by relating x and y directly. Understanding both parametric and rectangular equations is crucial for deeper insights into the shape and behavior of the curve.

Curve Sketching

Curve sketching involves drawing a curve represented by equations to visualize its shape and behavior. Starting with parametric equations requires plotting a few key points for different values of the parameter \( t \). Choose values like \( t = -2, -1, 0, 1, 2 \) to start. Plug these into the equations \( x = t^3 \) and \( y = \frac{t^2}{2} \) to get coordinates such as: - \( t = -2 \) gives \( x = -8, y = 2 \)- \( t = -1 \) gives \( x = -1, y = 0.5 \)- \( t = 0 \) gives \( x = 0, y = 0 \)- \( t = 1 \) gives \( x = 1, y = 0.5 \)- \( t = 2 \) gives \( x = 8, y = 2 \)Plot these points on a graph and connect them smoothly. Doing so illustrates how the curve behaves for different t-values. In this exercise, the graph is symmetric with respect to the y-axis, but not the x-axis, showing a cup-like shape.

Parameter Elimination

Parameter elimination is a technique used to transform parametric equations into a single rectangular equation. It's essential for simplifying the understanding of curves described parametrically. Given the parametric equations \( x = t^3 \) and \( y = \frac{t^2}{2} \), the first step is to express the parameter \( t \) in terms of x. Solve the equation \( x = t^3 \) to get \( t = \sqrt[3]{x} \). Then, substitute \( t \) back into the equation for \( y \). Replacing \( t \) in \( y = \frac{t^2}{2} \) with \( \sqrt[3]{x} \), gives us: \[ y = \frac{(\sqrt[3]{x})^2}{2} \]This calculation results in the rectangular equation \( y = \frac{x^{2/3}}{2} \). This approach simplifies parametric curves by removing the intermediary variable, making the relationship between x and y explicit.

Orientation of Curve

The orientation of a curve refers to the direction in which the curve progresses as the parameter increases. Understanding this provides insights into the behavior and movement of the curve over its domain.With the given equations \( x = t^3 \) and \( y = \frac{t^2}{2} \), as \( t \) increases, both x and y increase. This indicates that the curve moves upwards and towards the right for positive values of \( t \), and downwards and towards the left for negative \( t \).To visualize the orientation, consider plotting the points for various t-values and observing the sequence in connecting these points. Tracking the progression helps determine the curve's orientation. It's crucial to remember that orientation can affect how a curve is interpreted. Identifying it correctly ensures accurate description and communication of the curve's path and behavior.