Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=\frac{1}{s}, y=s ; 1 \leq s<10 $$

Step-by-Step Solution

Answer

The curve is a segment of the hyperbola \( y = \frac{1}{x} \) for \( 1 \leq y < 10 \).

Step 1: Determine the Parametric Domain

First, we will consider the given domain of the parameter \( s \). According to the problem, the parameter \( s \) ranges from \( 1 \) to \( 10 \) (excluding \( 10 \)). This domain is: \( 1 \leq s < 10 \).

Step 2: Express x in terms of s

The parametric equations are given as \( x = \frac{1}{s} \). To understand how \( x \) varies, substitute values of \( s \) within its domain. As \( s \) increases from 1 to just under 10, \( x \) will decrease from \( \frac{1}{1} = 1 \) to \( \frac{1}{10} = 0.1 \).

Step 3: Express y in terms of s

From the parametric equation, \( y = s \), therefore \( y \) varies directly as \( s \) does. As \( s \) ranges from 1 to just under 10, \( y \) will also range from 1 to just under 10.

Step 4: Analyze the x-y Relationship

Since \( y = s \) and \( x = \frac{1}{s} \), we can eliminate \( s \) to find a relationship between \( x \) and \( y \). By substituting \( s = y \) into the expression for \( x \), we get \( x = \frac{1}{y} \). This indicates the curve is the segment of the hyperbola defined by \( y = \frac{1}{x} \) for \( 1 \leq y < 10 \).

Step 5: Describe the Curve

The parametric representation describes a section of a hyperbolic curve. As \( s \) increases, \( x \) decreases and \( y \) increases, resulting in the curve moving from the point \((1,1)\) to a point close to \((0.1,10)\), staying on \( y = \frac{1}{x} \).

Key Concepts

Curve Analysis

Curve analysis involves understanding the behavior and characteristics of a curve represented by a set of parametric equations. In this context, we are examining how the curve is traced by the motion of a point. The provided parametric equations describe a hyperbola, specifically the branch where both coordinates change in opposite directions with respect to the parameter \( s \). As the parameter \( s \) increases from 1 to just under 10, the following occurs:

• \( x \) decreases from 1 to 0.1.
• \( y \) increases from 1 to a value just below 10.

These observations allow us to deduce that the curve represents the upper right branch of the hyperbola \( y = \frac{1}{x} \), as long as the values of \( y \) are between 1 and just under 10. Curve analysis also helps identify how the points move along this path, thereby defining the nature of the curve segment.

Parametric Representation

Parametric representation allows us to express a curve using parameters instead of just \( x \)-values or \( y \)-values. Here, the parameter \( s \) is used to define the curve with equations \( x = \frac{1}{s} \) and \( y = s \).
This approach provides flexibility, enabling us to describe complex curves that cannot be represented as a single function \( y = f(x) \) or \( x = f(y) \). By using a parameter:

• It simplifies the representation of geometric shapes.
• Allows the expression of curves with restricted domains and ranges easily.
• Facilitates the analysis of the curve's characteristics as the parameter systematically generates coordinate pairs.

In our specific problem, the parametric representation introduces the idea that each \( s \) value generates a unique \( (x, y) \) pair, portraying the real-time evolution of the curve. Hence, it is essential in visualizing the movement of a point in space as the parameter changes.

Domain of Parameter

The domain of a parameter is crucial in defining which segments of a curve are traced by the parametric equations. For the equations \( x = \frac{1}{s} \) and \( y = s \), the domain of \( s \) is given as \( 1 \leq s < 10 \).
Understanding the domain involves:

• Specifying where the parameter \( s \) begins and ends.
• Ensuring the values lead to all valid points on the curve within the specified interval without reaching undefined points.

In this problem, the domain ensures that the curve begins at point \( (1,1) \) and extends up to but never reaching \( (0.1, 10) \). Restricting \( s \) this way prevents reaching undefined values like negative or zero denominators in the equation for \( x \). This careful parameter management ensures the curve is drawn accurately and completely within the desired span.