Problem 25 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=t^{3}\) \(y=3\ \textrb{ln}\ t \)

Step-by-Step Solution

Answer

The graph of the curve can be obtained by plotting points using different values of t. The resulting rectangular equation after eliminating t is \(y = 3 \textrm{ ln }(x^{1/3})\) and its domain is (0, Inf).

Step 1: Sketch the curve

You can sketch the curve by plotting different values of t into the parametric equations. Such points will give you a good sense of the shape of the curve.

Step 2: Eliminate the parameter

To eliminate the parameter t, you can solve one of the equations for t and then substitute the result into the other equation. Here you can solve the equation \(x = t^{3}\) for t to get \(t = x^{1/3}\). Substituting this into the second equation \(y = 3 \textrm{ ln } t\) will give \(y = 3 \textrm{ ln }(x^{1/3})\).

Step 3: Adjust the domain of the resulting rectangular equation

From \(y = 3 \textrm{ ln }(x^{1/3})\), you can see that the domain of the function is all real numbers x > 0. This means the function is not defined for x ≤ 0. Hence, the domain should be adjusted as such ;(0, Inf).

Key Concepts

Sketching Curves

When it comes to visualizing parametric equations, sketching curves is a fundamental skill. To begin, plot a series of points by substituting sample values of the parameter, often 't', into your equations for 'x' and 'y'. Assign these values carefully and consider using a table to maintain organization. As an example, if we have parameters like

\(x=t^{3}\) and \(y=3\,\textrm{ln}\,t\), start with values of 't' that make calculations simpler, such as 0, 1, or -1, bearing in mind the natural logarithm's domain. Connect these plotted points to visualize the curve.

Mark the orientation of the curve by indicating the direction in which 't' increases. This can often reveal the behavior of the curve over time or another independent parameter. A well-drawn curve provides insight not only into its shape but also into its directionality and often, its limits and possible points of inflection.

Eliminating the Parameter

Eliminating the parameter from parametric equations is essential for converting them into a single rectangular (also known as Cartesian) equation. This process involves isolating the variable 't' in one of the equations and substituting this expression into the other. For instance, from the system \(x=t^{3}\) and \(y=3\,\textrm{ln}\,t\), solving for 't' gives \(t = x^{1/3}\). By substituting into the second equation, you can derive \(y = 3\,\textrm{ln}\,(x^{1/3})\), which no longer explicitly involves the parameter 't'.

Remember that this process requires a one-to-one relationship between 't' and the other variables to ensure a valid substitution. If this condition is not met, the resulting equation may not properly represent the initial parametric form.

Rectangular Equations

Rectangular equations are expressions in the standard 'y = f(x)' format. They are preferred for most functions because they highlight the input-output relationship unique to standard Cartesian coordinates. Upon eliminating the parameter, one receives a rectangular equation corresponding to the original parametric equations. For example, the parametric equations \(x=t^{3}\) and \(y=3\,\textrm{ln}\,t\) convert to the rectangular equation \(y = 3\,\textrm{ln}\,(x^{1/3})\).

This form makes it evident that 'y' is a function of 'x', which is suitable for classical algebraic manipulation, graphing, calculus, and other mathematical analysis. Students must ensure that their final equation is properly simplified and represents the original relationship as accurately as possible.

Domain Adjustment

Parametric equations often come with inherent domain restrictions that must be accounted for when rewriting them as rectangular equations. The domain indicates the set of all possible input values 'x' for which the function is defined. After eliminating the parameter, revise the domain to reflect any limitations imposed by the operations used. For our example, where the final equation is \(y = 3\,\textrm{ln}\,(x^{1/3})\), realize that the domain must be restricted to \(x>0\) because logarithms are undefined for nonpositive numbers.

Therefore, the domain of the rectangular equation is adjusted to be '(0, ∞)', indicating that the function only accepts positive 'x' values. Students should always state the domain after simplifying the rectangular equation to avoid misrepresenting the function's true scope and to prevent invalid operations, such as taking the logarithm of a negative number.