Problem 33 - Precalculus

Question

(A) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary. $$\begin{aligned} &x=t^{3}\\\ &y=3 \ln t \end{aligned}$$

Step-by-Step Solution

Answer

The rectangular equation representing the curve is \(y = 3 \ln (\sqrt[3]{x})\) and it covers the whole real number domain.

Step 1: Sketch the Curve

Plot points by assigning values to the parameter t (including negative values, zero and positive values). Plot these points on a graph and draw the curve. The orientation of the curve is determined by the direction of the curve as t increases.

Step 2: Eliminate the Parameter

You eliminate the parameter by isolating t in one of the equations and substituting it into the other equation. From the equation \(x = t^3\), you can solve for t to get \(t = \sqrt[3]{x}\). Substituting \(t\) into second equation leads to the new equation \(y = 3 \ln (\sqrt[3]{x})\).

Step 3: Adjust the Domain

The domain needs to cover the values that t can take to form the curve. In this case however, because t is cube rooted, all real numbers are possible so no adjustment is needed.

Key Concepts

Sketching Curves

Understanding the process of sketching curves from parametric equations is an essential skill in mathematics, particularly when visualizing complex functions. In the given exercise, the parametric equations are defined by \(x=t^{3}\) and \(y=3 \ln t\). To begin sketching, we assign various values to the parameter \(t\), spanning negative values, zero, and positive values.

Each value of \(t\) provides a corresponding point \((x,y)\) that can be plotted on a coordinate graph. Seeing the progression of these points as \(t\) increases helps infer the orientation or the direction of the curve. In this case, as \(t\) increases, the value of \(x\) will increase more rapidly due to the cubic relationship. Conversely, as \(t\) approaches zero and becomes negative, the logarithmic function becomes undefined, indicating that the curve does not extend into negative \(x\) values.

By connecting the plotted points smoothly, considering the nature of both the cubic function and the logarithmic function, the sketch of the curve begins to take shape. This visual representation is a powerful tool for understanding the behavior and properties of the parametric equations at a glance.

Eliminating Parameters

The technique of eliminating parameters is used to convert parametric equations into a single rectangular, or Cartesian, equation. To eliminate the parameter \(t\) from our equations, \(x=t^{3}\) and \(y=3 \ln t\), we first express \(t\) in terms of \(x\). From the first equation, we get \(t = \sqrt[3]{x}\).

Next, we substitute this expression for \(t\) into the second equation. This gives us the rectangular form of the equation: \(y = 3 \ln (\sqrt[3]{x})\). This new equation represents the same curve as the original parametric equations, but it is now in a format that is more familiar and often easier to work with in terms of analyzing functions and their properties.

By eliminating the parameter, students will find it easier to study the characteristics of the curve, such as its increasing or decreasing nature, concavity, and asymptotic behavior, using standard calculus tools.

Domain of Rectangular Equation

The domain of a rectangular equation represents the set of all possible input values (or \(x\)-values) for which the equation is defined. Determining the domain is critical when sketching curves because it informs us about the extent of the curve on the graph.

In our case, after eliminating the parameter to find the rectangular equation \(y = 3 \ln (\sqrt[3]{x})\), we must consider the domain. The cube root function, \(\sqrt[3]{x}\), is defined for all real numbers, so there are no restrictions on \(x\) due to the cube root. However, the logarithmic function is only defined for positive numbers. Thus, the domain of \(x\) must be limited to positive values, since \(t\) (and consequently \(\sqrt[3]{x}\)) must be greater than zero for the logarithm to be defined.

Typically, adjusting the domain might require excluding certain values that make the function undefined, like zero or negatives for logarithms. But in this instance, as \(t\) ranges over positive real numbers, the domain of the rectangular equation naturally consists of all positive reals, and no further adjustments are necessary. Understanding and adjusting the domain is a crucial step to ensure the accuracy of the curve representation on the graph.