Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of Problems 1-20, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples 1-4). $$ x=t^{3}-4 t, y=t^{2}-4 ;-3 \leq t \leq 3 $$

Step-by-Step Solution

Answer

The curve is not closed and not simple. The Cartesian equation is \( x = (y+4)^{3/2} - 4(y+4)^{1/2} \) and \( x = -(y+4)^{3/2} + 4(y+4)^{1/2} \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = t^3 - 4t \) and \( y = t^2 - 4 \). The parameter \( t \) is ranging from \(-3\) to \(3\).

Step 2: Determine the Range of Values

Calculate the values of \(x\) and \(y\) at the endpoints and determine how the curve behaves within the given range of \(t\). By substituting the values of \(t\) from \(-3\) to \(3\), generate several points to understand the path of the curve.

Step 3: Graph the Curve

Using the points obtained by substituting the values of \(t\) into the parametric equations, plot these points on the Cartesian plane. Analyze the changes in the direction of the curve as \(t\) varies to sketch the path accurately.

Step 4: Check for Closure and Simplicity

Determine whether the curve comes back to its starting point (closure) and whether it crosses itself (simplicity) by analyzing the graph. A curve is closed if it starts and ends at the same point, and it is simple if it doesn't intersect itself.

Step 5: Eliminate the Parameter to Find the Cartesian Equation

To eliminate the parameter \(t\), express \(t\) in terms of \(y\) from the equation \( y = t^2 - 4 \), giving \( t = \pm \sqrt{y+4} \). Substitute this expression for \(t\) into the equation for \(x\): \[ x = (\pm \sqrt{y+4})^3 - 4(\pm \sqrt{y+4}). \]

Step 6: Simplify the Cartesian Equation

From \( t = \pm \sqrt{y+4} \), the expression becomes:\( x = (y+4)^{3/2} - 4(y+4)^{1/2} \) and \( x = -(y+4)^{3/2} + 4(y+4)^{1/2} \). This gives the Cartesian equation of the curve in the implicit form.

Key Concepts

Curve Sketching

Curve sketching is a fundamental technique in mathematics to visualize the behavior of a given parametric equation. When dealing with parametric equations like \( x = t^3 - 4t \) and \( y = t^2 - 4 \), it's important to assess how the curve progresses as the parameter \( t \) changes.

To sketch the curve:

• Start by substituting various values of \( t \) into both equations to generate points that lie on the curve.
• Use these points to plot the curve on a Cartesian plane.
• Observe how the curve moves as \( t \) transitions from \(-3\) to \(3\) to understand its full shape.

By plotting these points systematically, you get a clear picture of the trajectory and bending of the curve. This allows you to see patterns and infer critical properties such as curve closure and intersection points.

Cartesian Equation

A Cartesian equation represents a relationship between x and y in a standard form, free from any parameters. To transform parametric equations into Cartesian form, the process of eliminating the parameter \( t \) is used.

To eliminate the parameter:

• Find \( t \) in terms of a single variable, typically using one of the parametric equations.
• For our example, starting with \( y = t^2 - 4 \), you solve for \( t \): \( t = \pm \sqrt{y + 4} \).
• Substitute this expression for \( t \) back into the equation for \( x \), forming \( x = (\pm \sqrt{y+4})^3 - 4(\pm \sqrt{y+4}) \).

What you get is a Cartesian equation that captures the relation between \( x \) and \( y \) directly, aiding in a uniform understanding of the curve in question.

Curve Closure

The concept of curve closure revolves around whether or not a curve will return to its starting point as it is traversed once. To verify closure in parametric equations:

• Check if the starting and ending values of the parameter result in identical points on the curve.
• For instance, when \( t = -3 \), calculate \( x \) and \( y \); then do the same for \( t = 3 \).
• Compare the points: if they are the same, the curve is closed; if not, the curve remains open.

In our example, these calculations reveal that the curve does not loop back to its initial point, indicating an open path without closure.

Curve Simplicity

Curve simplicity examines if a curve intersects itself at any point. A simple curve differs from others by not intersecting itself. When evaluating simplicity:

• Observe the plotted curve for any crossings or overlaps.
• Simple curves will not revisit a point already plotted, except at their endpoints.
• For the parametric equations in question, visualize the curve using plotted points to identify any self-intersections.

If no self-intersections are visible from the curve's path, it is considered simple. In our given range and behavior, this particular curve does not cross itself, affirming its simplicity.