Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

\(x=e^{t}, \quad y=e^{-2 t}, \quad t\) in \(\mathbb{R}\)

Step-by-Step Solution

Answer

The relationship between \(x\) and \(y\) is \(y = x^{-2}\).

Step 1: Understanding the Exercise

We need to convert the parametric equations into a single equation that relates the variables \(x\) and \(y\). The given equations are \(x = e^t\) and \(y = e^{-2t}\).

Step 1: Express \(t\) in terms of \(x\)

From the first equation \(x = e^t\), we take the natural logarithm of both sides to express \(t\) in terms of \(x\). This gives us \(t = \ln(x)\).

Step 2: Substitute \(t\) into \(y\)'s Expression

Now, substitute \(t = \ln(x)\) into the equation for \(y\), which is \(y = e^{-2t}\). This gives us \(y = e^{-2 \ln(x)}\).

Step 3: Simplify the Expression for \(y\)

Use the power rule \(e^{a \cdot b} = (e^a)^b\) to simplify \(y = e^{-2 \ln(x)}\). This is equivalent to \(y = (e^{\ln(x)})^{-2}\). Since \(e^{\ln(x)} = x\), we have \(y = x^{-2}\).

Step 5: Conclusion

The parametric equations can be represented in the simplified form \(y = x^{-2}\), which is an equation relating \(x\) and \(y\) directly.

Key Concepts

Convert Parametric to Cartesian

Parametric equations are used to express the coordinates of the points making up a geometric object, such as a curve, by using one or more parameters. To convert a set of parametric equations to a Cartesian equation, which relates the variables directly, you follow a straightforward process of substitution.
Given the parametric form:

• \(x = e^t\)
• \(y = e^{-2t}\)

First, express the parameter, here \(t\), in terms of one of the variables. From the equation \(x = e^t\), solve for \(t\) to get \(t = \ln(x)\). Now, substitute \(t\) into the equation for \(y\), yielding \(y = e^{-2t} = e^{-2 \ln(x)}\). Simplifying this equation using exponent rules transforms the given parametric equations into the Cartesian form \(y = x^{-2}\). This directly relates \(x\) and \(y\) without involving the parameter \(t\).
Understanding and implementing these steps not only aids in converting parametric forms but also strengthens your grasp of solving problems that involve multiple variables and parameters.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a useful mathematical function to understand due to its many applications in various fields. It is the inverse function of the exponential function. For a positive number \(x\), the natural logarithm \(\ln(x)\) is defined as the power to which the base \(e\) (approximately 2.718) must be raised to produce that number \(x\).
This function is particularly helpful in converting expressions involving exponential terms when dealing with equations. For instance, when given \(x = e^t\), applying the natural logarithm to both sides allows us to isolate \(t\), resulting in \(t = \ln(x)\). This transformation is indispensable when shifting between exponential and logarithmic forms, streamlining the process of mixing and rearranging exponential equations.
Natural logarithms not only simplify otherwise complex expressions but also enable easy manipulation and solution of equations in calculus and algebra.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent, typically represented as \(f(x) = a^x\). The base \(a\) is a positive constant, the most common being the natural base \(e\). These functions model a variety of real-life scenarios from population growth to radioactive decay and are characterized by their rapid increase or decrease.
In the context of the given exercise, the equations \(x = e^t\) and \(y = e^{-2t}\) involve exponential functions. These equations demonstrate the properties of exponential functions, such as how changing the exponent affects growth or decay, a concept especially visible in \(y = e^{-2t}\), which simplifies to \(y = x^{-2}\) in its Cartesian form.
A fundamental property used in converting these parametric equations is \(e^{a \cdot b} = (e^a)^b\), which allows simplification of complex expressions. Exponential functions and their manipulation are crucial for many advanced concepts in mathematics, engineering, and sciences, showing how the initial growth or decay behavior of these functions can be reinterpreted in context-specific simplified forms.