Problem 33 - A Graphical Approach to College Algebra

Question

For each plane curve, find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t^{2}, y=2 \ln t, \text { for } t \text { in }(0, \infty)$$

Step-by-Step Solution

Answer

The rectangular equation is \(y = \ln(x)\) for \(x > 0\).

Step 1: Solve for Parameter

We start with the equation for \(x\), which is \(x = t^2\). We need to express \(t\) in terms of \(x\). Given \(x = t^2\), the inverse operation is taking the square root. Therefore, \(t = \sqrt{x}\). Since \(x = t^2\) and \(t > 0\) (as \(t\) is in \((0, \infty)\)), we take only the positive square root: \(t = \sqrt{x}\).

Step 2: Substitute and Simplify

Substitute \(t = \sqrt{x}\) into the \(y\) equation: \(y = 2 \ln t\). This gives us \(y = 2 \ln(\sqrt{x})\). Using the property of logarithms \(\ln(\sqrt{x}) = \frac{1}{2} \ln(x)\), we get \(y = 2 \times \frac{1}{2} \ln(x)\). Simplifying, we have \(y = \ln(x)\).

Step 3: Determine the Interval for \(x\)

Since \(t\) is in \((0, \infty)\), \(x = t^2\) results in \(x > 0\) (as squaring any positive \(t\) gives a positive \(x\)). Hence, the appropriate interval for \(x\) is \((0, \infty)\).

Key Concepts

Parametric Equations

Understanding parametric equations is a core skill in mathematics. Parametric equations use a parameter, commonly denoted as \( t \), to express both \( x \) and \( y \) coordinates of a curve. Instead of a traditional \( y = f(x) \) setup, each coordinate is a function of a third variable, \( t \). For instance, consider \( x = t^2 \) and \( y = 2 \ln t \). The parameter \( t \) smoothly guides the curve through its path in the plane. Key benefits include:

• Modeling complex shapes like circles or ellipses easily.
• Making calculus applications such as derivatives and integrals more flexible.

Understanding parametric equations gives you insight into a whole new dimension of analyzing and constructing mathematical curves.

Logarithmic Functions

Logarithmic functions are expressions that involve the logarithm, such as \( y = 2 \ln t \). The \( \ln \) function, also known as the natural logarithm, uses \( e \) (approximately 2.718) as its base.Let's distill essential properties of logarithmic functions:

• Inverse of Exponentials: If \( b^y = x \), then \( y = \log_b x \).
• Logarithmic Identity: \( \ln (a^b) = b \ln a \).

In the context of solving parametric equations, you apply logarithmic properties to simplify expressions. For example, \( \ln(\sqrt{x}) = \frac{1}{2} \ln(x) \) allows the function \( y = 2 \ln(\sqrt{x}) \) to simplify to \( y = \ln(x) \). Recognizing these properties helps you manipulate and understand how the logarithmic function behaves in equations.

Interval Notation

Interval Notation simplifies expressing sets of numbers on a number line. It uses brackets and parentheses to denote inclusive or exclusive values.

• Parentheses \((a, b)\): Indicates all numbers between \( a \) and \( b \) but not \( a \) and \( b \) themselves. It denotes an open interval.
• Brackets \([a, b]\): Signifies the interval includes the endpoints \( a \) and \( b \), known as a closed interval.

In parametric equations, where \( t \) spans from 0 to infinity, \( (0, \infty) \) indicates all positive real numbers. This is crucial as it influences the range of \( x \) and \( y \), both derived from \( t \). In our specific example, because \( x = t^2 \), \( x \) is positive, hence, its interval is also expressed as \( (0, \infty) \). Understanding interval notation provides clarity in defining the domain of functions and making the results more interpretable.