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Chapter 7: Problem 12

Graph each pair of parametric equations in the rectangular coordinate system. Determine the domain (the set of x-coordinates) and the range (the set of y-coordinates). $$x=\ln t, y=t+3, \text { for }-2

Short Answer

Expert verified
Domain: (-2.302, 0.693), Range: (3, 5)

Step by step solution

01

- Understand the Parametric Equations

The given parametric equations are: \[ x = \ln t, \quad y = t + 3 \]The parameter range is: \[ -2 < t < 2 \]
02

- Determine the Domain

The domain (set of x-coordinates) is governed by the equation \( x = \ln t \). The natural logarithm function is only defined for \( t > 0 \). Thus, the applicable range of \( t \) is from 0 to 2:\[ 0 < t < 2 \]Substituting these values into the equation for \( x \):When \( t = 0.1 \), \[ x = \ln(0.1) \approx -2.302 \]When \( t = 2 \), \[ x = \ln(2) \approx 0.693 \]So, the domain is:\[ \left(-2.302, 0.693 \right) \]
03

- Determine the Range

The range (set of y-coordinates) is governed by the equation \( y = t + 3 \). Using the applicable range of \( t \) (i.e., 0 to 2):When \( t = 0 \), \[ y = 0 + 3 = 3 \]When \( t = 2 \), \[ y = 2 + 3 = 5 \]So, the range is:\[ (3, 5) \]
04

- Graph the Parametric Equations

To graph the parametric equations, plot points for several values of \( t \) within the range 0 to 2 and draw a curve through these points. For instance, plot points for \( t = 0.1, 0.5, 1, 1.5, 2 \). The approximate points are:\[ (\ln(0.1), 3.1), (\ln(0.5), 3.5), (\ln(1), 4), (\ln(1.5), 4.5), (\ln(2), 5) \]A smooth curve through these points forms the graph of the parametric equations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain
In parametric equations, the domain is the set of all possible x-coordinates. Here, the x-coordinate comes from the equation \ x = \ln t.
The natural logarithm function, \ln t, is only defined for positive values of t. This means t must be greater than 0.
Given \( -2 < t < 2 \), we need to adjust t to the range \( 0 < t < 2 \).
By substituting t values within \( 0 < t < 2 \) into \( x = \ln t \), we get the following domain:
  • For \( t = 0.1 \), \[ x = \ln(0.1) \approx -2.302 \]
  • For \( t = 2 \), \[ x = \ln(2) \approx 0.693 \]
Hence, the domain is
\( (-2.302, 0.693) \).
Range
The range in parametric equations is the set of all possible y-coordinates. In this problem, y is given by the equation \ y = t + 3.
To find the y-values, we use the adjusted t range of \( 0 < t < 2 \):
  • For \( t = 0 \), \[ y = 0 + 3 = 3 \]
  • For \( t = 2 \), \[ y = 2 + 3 = 5 \]
Thus, the range for y is:
\( (3, 5) \).
Graphing Parametric Equations
Graphing parametric equations involves plotting the points generated by the equations \( x = \ln t \) and \( y = t + 3 \), and then connecting these points with a smooth curve.
First, choose several t-values within the range \( t = 0.1, 0.5, 1, 1.5, 2 \) and find corresponding x and y coordinates:
  • For \( t = 0.1 \), \( x \approx -2.302, y = 3.1 \)
  • For \( t = 0.5 \), \( x \approx -0.693, y = 3.5 \)
  • For \( t = 1 \), \( x = 0, y = 4 \)
  • For \( t = 1.5 \), \( x \approx 0.405, y = 4.5 \)
  • For \( t = 2 \), \( x \approx 0.693, y = 5 \)
Plot these points on the rectangular coordinate system and draw a curve through them.
Natural Logarithm
A natural logarithm, denoted as \( \ln \), is the logarithm to the base e, where e is approximately 2.71828.
The function \( \ln t \) is only defined for positive numbers (\

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