Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

a parametric representation of a curve is given. $$ x=4-t, y=\sqrt{t} ; 0 \leq t \leq 4 $$

Step-by-Step Solution

Answer

The parametric curve represents \( y = \sqrt{4 - x} \) for \( 0 \leq x \leq 4 \).

Step 1: Understanding the Parameters

First, recognize that the parametric equations given are \( x = 4 - t \) and \( y = \sqrt{t} \) where \( t \) is the parameter ranging from 0 to 4. Each value of \( t \) gives a corresponding \( (x, y) \) point on the curve.

Step 2: Substitute Parameters to Find Ends

Start by substituting the boundary values of \( t \) into the expressions for \( x \) and \( y \) to find the points where the curve begins and ends. For \( t = 0 \), \( x = 4 \) and \( y = 0 \). For \( t = 4 \), \( x = 0 \) and \( y = 2 \). So, the curve begins at (4, 0) and ends at (0, 2).

Step 3: Eliminate the Parameter

To express \( y \) in terms of \( x \), eliminate \( t \). From \( x = 4-t \), solve for \( t \): \( t = 4 - x \). Substitute into \( y = \sqrt{t} \) to get \( y = \sqrt{4 - x} \).

Step 4: Determine the Curve's Range

Find the range of \( x \). Since \( t \) varies from 0 to 4, \( x = 4 - t \), and hence \( x \) varies from 4 to 0. Therefore, the curve's range is from \( (4, 0) \) to \( (0, 2) \)

Key Concepts

Curve Sketching

Curve sketching refers to the process of drawing a curve using its parametric equations. When dealing with parametric equations, we express the coordinates of points on a curve as functions of a parameter, often denoted as \( t \). In this exercise, the parametric equations are \( x = 4 - t \) and \( y = \sqrt{t} \). These equations enable us to understand how both \( x \) and \( y \) change as \( t \) changes. By selecting various values of \( t \) within the given range \( 0 \leq t \leq 4 \), we can compute the corresponding values of \( x \) and \( y \), and plot these points to visualize the curve.
Also, by determining the end points—\( (4, 0) \) when \( t = 0 \) and \( (0, 2) \) when \( t = 4 \)—we establish the starting and ending point of the curve. These insights help both in sketching the curve and comprehending its behavior across the specified parameter range.

Eliminating Parameters

Eliminating parameters involves rewriting the parametric equations to express one variable in terms of the other, thereby removing the parameter \( t \) from the equations. This process allows us to more easily see the relationship between \( x \) and \( y \). From the given equations, start with the expression for \( x \), which is \( x = 4 - t \). We solve for \( t \) by rearranging the equation to get \( t = 4 - x \).
Next, we substitute this expression into the second equation \( y = \sqrt{t} \). This substitution yields \( y = \sqrt{4 - x} \), which is a non-parametric equation that describes the relationship between \( x \) and \( y \). This equation is essential for understanding the curve in terms of traditional coordinate geometry, offering a new perspective that can be analyzed without the parameter \( t \).

Range of a Curve

Determining the range of a curve involves understanding the possible set of values for \( x \) and \( y \) that the curve can attain. For the given parametric equations, the parameter \( t \) runs from 0 to 4. As a result, \( x = 4 - t \) varies from 4 to 0 as \( t \) increases from 0 to 4. Similarly, by considering \( y = \sqrt{t} \), we conclude that \( y \) spans from 0 to 2.

• The range for \( x \) is concluded by substituting the boundary values of \( t \): maximum \( x \) when \( t = 0 \) is 4, and the minimum \( x \) when \( t = 4 \) is 0.
• The range for \( y \) spans from the minimum calculated at \( t = 0 \) to the maximum when \( t = 4 \), which are 0 and 2, respectively.

Understanding the range helps in plotting the entire curve correctly and comprehensively, as it indicates the full extent of the curve on the coordinate plane. It also ensures that any graphing or analysis of the curve takes into account all possible values that the curve can exhibit.