Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=\ln t, y=3 \ln t, \text { for } t \text { in }(0, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = 3x\) for \(x \in (-\infty, \infty)\).

Step 1: Understand the Problem

We are given parametric equations: \(x = \ln t\) and \(y = 3\ln t\), with \(t\) in the interval \((0, \infty)\). We need to find a rectangular equation that relates \(x\) and \(y\) without the parameter \(t\).

Step 2: Express t in terms of x

Given \(x = \ln t\), we can solve for \(t\) in terms of \(x\). Taking the exponential of both sides, we have: \[t = e^x\]

Step 3: Substitute t into the y equation

Substitute \(t = e^x\) into the equation for \(y\): \[ y = 3\ln t = 3\ln(e^x) = 3x \]This simplification uses the property \(\ln(e^x) = x\).

Step 4: Identify the Appropriate Interval

Since \(t > 0\), from \(x = \ln t\), it follows that \(t = e^x\) implies \(e^x > 0\). This means that \(x\) can take any real number, so the interval for \(x\) is \((-\infty, \infty)\).

Step 5: Write the Rectangular Equation and Interval

The rectangular equation relating \(x\) and \(y\) is \(y = 3x\), and since \(x\) can be any real number, the interval is \((-\infty, \infty)\).

Key Concepts

Parametric Equations

Parametric equations are a useful way of representing curves by specifying the coordinates of the points on the curve as functions of a parameter, usually denoted by \(t\). These equations often make complex relationships easier to handle.
For instance, in our exercise, the parametric equations are \(x = \ln t\) and \(y = 3 \ln t\). Here, \(t\) is the parameter. As \(t\) varies within a certain interval, it generates a set of points \((x, y)\) that lie on a curve.
By using parametric equations, we can express more complex curves and motions, such as ellipses, circles, and cycloids, which might be cumbersome to describe with a single equation in rectangular form. Converting parametric equations into rectangular form often involves eliminating the parameter \(t\) to relate \(x\) and \(y\) directly.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It is especially useful in describing the domains and ranges for functions.
Notation typically involves brackets:

• \([a, b]\) denotes all values \(x\) where \(a \le x \le b\), including \(a\) and \(b\).
• \((a, b)\) signifies \(a < x < b\), excluding \(a\) and \(b\).
• An interval with infinity, such as \((-\infty, \infty)\), means all real numbers with no bounds.

In our exercise, the parameter \(t\) is defined within the interval \((0, \infty)\), indicating that \(t\) can be any positive number. This interval is crucial because it helps affirm the valid values for \(x\) and consequently \(y\), ensuring the function's behavior is consistently interpreted across its domain.

Exponential Function

The exponential function, expressed as \(e^x\), is a fundamental mathematical function with a constant base \(e\) (approximately 2.718). This function is incredibly important due to its unique properties, such as its constant rate of growth and applications across various scientific fields.
In the exercise context, we used the exponential function to express \(t\) in terms of \(x\) by rewriting \(x = \ln t\) as \(t = e^x\). This conversion is possible because exponentiation reverses the effect of a logarithm, so these functions are inverses.
Understanding the exponential function helps us move between different forms of expressions and solve equations that involve growth and decay processes, making it critical in both theoretical and applied mathematics.

Natural Logarithm

A natural logarithm, denoted by \(\ln\), is the logarithm to the base \(e\) (a mathematical constant approximately equal to 2.718). It's the inverse operation of exponentiation, similar to how subtraction undoes addition.
In the given example, \(\ln t\) helped us effectively transition from a parameter \(t\) to a function involving \(x\). For \(x = \ln t\), the natural logarithm allows us to express the parameter \(t\) in terms of \(x\) easily as \(t = e^x\). This relationship is crucial in simplifying the parametric equations to their rectangular form.
The properties of logarithms, like \(\ln(e^x) = x\), i.e., they "cancel out", make logarithms a powerful tool for transforming and solving equations involving exponential functions. Hence, they are widely used in calculus, scientific calculations, and statistical models.