Problem 29 - 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. $$\begin{aligned} &x=e^{t}\\\ &y=e^{2 t} \end{aligned}$$

Step-by-Step Solution

Answer

After analyzing and visualizing the parametric equations, we see that they represent a curve in the first quadrant that extends from near the origin towards the positive direction of both axes. The orientation of the curve is positive. When the parameter 't' is eliminated, the resulting rectangular equation is \(y = x^2\), for x > 0.

Step 1: Understand the given equations

First, observe that the parameter 't' denotes the time in which the values of 'x' and 'y' evolve. The equations \(x = e^t\) and \(y = e^{2t}\) express 'x' and 'y' as exponential functions of time. When 't' increases, both x and y increase. The rates of increase are determined by the exponentials 'e^t' and 'e^{2t}', respectively. Thus, the curve will be oriented from lower left to upper right, indicating a positive orientation.

Step 2: Draw the sketch of the curve

Due to the properties of the exponential function, we know, for all \(t \in R\) (all real numbers), \(x = e^t\) implies x > 0 and \(y=e^{2t}\) implies y > 0. So, the curve of the function exists in the first quadrant. Also, the function's value increases as 't' increases. Therefore, the curve starts from very near to the origin (but doesn't touch it) and extends towards positive direction of both axes.

Step 3: Eliminate the parameter 't'

Note that \(y = e^{2t}\) can be rewritten as \(y = (e^t)^2\). Thus, \(y = x^2\), where \(x = e^t\). This is the equation of a parabola, but constrained to x > 0, since \(x = e^t > 0\).

Key Concepts

Exponential Functions

Exponential functions are mathematical functions where a constant base is raised to a variable exponent. In our problem, we have equations of the form:

• \(x = e^t\)
• \(y = e^{2t}\)

where \(e\) is Euler's number, approximately 2.718, which is a constant base often used in exponential growth and decay problems.
In these equations, 't' is a parameter generally representing time or another independent variable. As 't' increases, both 'x' and 'y' increase exponentially. But because the exponents are different—one is \(t\) and the other is \(2t\)—their rates of growth differ. Specifically, \(y\) increases more quickly than \(x\) because it involves a greater exponent. This difference is crucial for understanding the behavior and orientation of the curve they define. Exponential functions have specific characteristics such as never being zero and they always produce positive values when the base is positive. This means for any real value of 't', both 'x' and 'y' are always positive.

Rectangular Equation

A rectangular equation is an equation that involves only two variables, typically 'x' and 'y', without any parameters. In the given exercise, we need to express the relationship between 'x' and 'y' by eliminating the parameter 't'.Initially, we have two parametric equations:

• \(x = e^t\)
• \(y = e^{2t}\) or \(y = (e^t)^2\)

To find the rectangular equation, substitute \(e^t\) from the first equation into the second one:
\(y = (x)^2\).
This equation \(y = x^2\) describes a parabola. However, this parabola is only true for \(x > 0\) because of the nature of the exponential function \(x = e^t\), which is always positive.

Parameter Elimination

Eliminating a parameter translates a set of parametric equations into a single equation that describes a relationship between the two dependent variables. In the exercise, the parameter 't' needs to be eliminated to convert the parametric equations into the rectangular equation.We begin with these parametric equations:

• \(x = e^t\)
• \(y = e^{2t}\)

To eliminate 't', express 'y' in terms of 'x'. Start by noting that \(y = e^{2t}\) can be rewritten as \(y = (e^t)^2\). Then replace \(e^t\) using \(x = e^t\):
\(y = x^2\).
This step effectively removes 't', providing a direct relationship between 'x' and 'y'. By eliminating the parameter, we simplify our understanding of the curve's shape without tracking 't'. It's important to also adjust considerations like the domain and range to ensure they fit the physical or practical limits we begin with. Given \(x = e^t > 0\), the rectangular equation domain is limited to \(x > 0\).

Orientation of Curves

The orientation of a curve describes the direction in which the curve progresses as the parameter 't' increases. Understanding curve orientation is crucial in visualizing how it evolves in space over time. For the parametric equations provided:

• \(x = e^t\)
• \(y = e^{2t}\)

As 't' increases, both 'x' and 'y' exponentially grow. This causes the curve to move from the lower left to the upper right in the coordinate plane.
This specific direction defines the curve's positive orientation.
Since both \(x\) and \(y\) start at very low values (close to the origin) and increase rapidly for arithmetic increases in 't', the curve extends infinitely towards the positive direction of both axes without ever touching them at any finite 't'. This orientation is a direct visual result of the properties of exponential functions, indicating an infinite approach without ever reaching the axes, a hallmark of such exponential curves.