Problem 82 - Calculus Single Variable

Question

Describe and sketch the curve that has the given parametric equations. \(x=3 \log _{10}(t), y=5 \log _{10}(t)\)

Step-by-Step Solution

Answer

The curve is a straight line with slope \( \frac{5}{3} \) through the origin.

Step 1: Identify the Parametric Equations

The problem provides the parametric equations: \( x = 3 \log_{10}(t) \) and \( y = 5 \log_{10}(t) \). These equations use the same parameter \( t \), and \( \log_{10}(t) \) signifies the dependency of \( x \) and \( y \) on the logarithm base 10 of \( t \).

Step 2: Understand the Domain

The domain for \( t \) in a logarithm function \( \log_{10}(t) \) is \( t > 0 \). This is because logarithms are defined only for positive values of \( t \). Thus, the parametric equations are evaluated for \( t > 0 \).

Step 3: Express y in Terms of x

To find a relation between \( x \) and \( y \), we use the equations: \( x = 3 \log_{10}(t) \) and \( y = 5 \log_{10}(t) \). Solve the first equation for \( \log_{10}(t) \): \( \log_{10}(t) = \frac{x}{3} \). Substitute into the second equation: \( y = 5 \cdot \frac{x}{3} \).

Step 4: Simplify the Relation

The equation \( y = \frac{5}{3}x \) describes the linear relation between \( y \) and \( x \). This shows that the curve is a straight line with slope \( \frac{5}{3} \). This line passes through the origin because if \( x = 0 \), then \( y = 0 \).

Step 5: Sketch the Curve

The curve represented by the parametric equations is a straight line. It can be sketched on a coordinate plane with the line passing through the origin (0,0) and having a slope of \( \frac{5}{3} \). The line extends indefinitely in the first quadrant due to the condition \( t > 0 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical expressions used to find the exponent to which a base number is raised to produce a given number. The common logarithm has a base of 10, expressed as \( \log_{10}(t) \). In other words, if \( 10^x = t \), then \( x = \log_{10}(t) \). Logarithms are useful for solving problems that involve exponential relationships by transforming them into linear forms.

Logarithmic functions are involved in many real-world applications, like measuring pH levels in chemistry or calculating sound intensity in decibels. These functions have unique properties:

• Addition: \( \log_{10}(a) + \log_{10}(b) = \log_{10}(ab) \)
• Subtraction: \( \log_{10}(a) - \log_{10}(b) = \log_{10}(\frac{a}{b}) \)
• Power: \( \log_{10}(a^b) = b\log_{10}(a) \)

Understanding logarithms helps simplify complex exponential relationships, making them easier to analyze and solve.

Domain of a Function

The domain of a function refers to all possible input values (usually referred to as \( x \)) for which the function is defined. For a logarithmic function \( \log_{10}(t) \), the domain is specifically restricted to values where \( t > 0 \). If \( t \) is zero or negative, the logarithmic function is undefined because it is not possible to raise 10 to any power to achieve a negative number or zero.

Domains ensure the function values remain real and valid. They are crucial for correctly defining a function and understanding the set of values that can be used as inputs:

• Pay special attention when functions involve logarithms or square roots, as their domains often exclude non-positive values.
• Properly identifying the domain is essential for any function manipulation, ensuring the function remains valid under all operations.

In the context of parametric equations using logarithmic functions, always check the domain to ensure logical and mathematical consistency.

Linear Equations

A linear equation is a mathematical expression that represents a straight line in a coordinate system. These equations take the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. Linear equations can describe many relationships between two variables, including our parametric example where \( y = \frac{5}{3}x \). This indicates a line passing through the origin with a slope of \( \frac{5}{3} \).

Key features of linear equations include:

• The slope \( m \) describes the steepness and direction of the line. A positive slope means the line rises, while a negative slope indicates it falls as \( x \) increases.
• The y-intercept \( b \) is the value of \( y \) when \( x \) is zero, positioning the line vertically on the graph.
• In our exercise, since \( x = 0, y = 0 \), the line intersects the origin.

Understanding linear equations is critical for analyzing and predicting relationships and behavior in mathematical and real-world contexts.

Curve Sketching

Curve sketching is an essential skill in mathematics involving drawing the graph of a function based on its equation or parametric description. A primary goal is to identify key features such as intercepts, slope, and domain to accurately represent the function on a graph.

For parametric equations \( x = 3 \log_{10}(t) \) and \( y = 5 \log_{10}(t) \), we first simplify to \( y = \frac{5}{3}x \). This indicates that the curve forms a straight line. Key pointers in sketching include:

• Determine the domain—in this case, t must be greater than zero.
• Recognize intercepts: for this straight line, it starts at the origin (0,0).
• Understand the general shape: with our linear relation, the graph remains a line with slope \( \frac{5}{3} \).

Remember, sketching curves effectively requires understanding both algebraic and geometric properties, ensuring a clear and accurate visual representation.