Textbook Question
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = t² + 4, y = -t, for -2 < t < 0; (5, 1)
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16. Parametric Equations & Polar Coordinates
Parametric Equations
1:55 minutesProblem 12.R.2
Textbook Question
Eliminate the parameter in the parametric equations x=1+sin t, y=3+2 sin t, for 0≤t≤π/2, and describe the curve, indicating its positive orientation. How does this curve differ from the curve x=1+sin t, y=3+2 sin t, for π/2≤t≤π?
Verified step by step guidance1
Start by writing down the given parametric equations: \(x = 1 + \sin t\) and \(y = 3 + 2 \sin t\), with the parameter \(t\) in the interval \(0 \leq t \leq \frac{\pi}{2}\).
Express \(\sin t\) from the first equation: \(\sin t = x - 1\).
Substitute \(\sin t = x - 1\) into the second equation to eliminate the parameter \(t\): \(y = 3 + 2(x - 1)\).
Simplify the resulting equation to get the Cartesian form of the curve: \(y = 3 + 2x - 2\) which simplifies to \(y = 2x + 1\).
To describe the curve's positive orientation, consider the interval \(0 \leq t \leq \frac{\pi}{2}\) where \(\sin t\) increases from 0 to 1, so \(x\) increases from 1 to 2 and \(y\) increases from 3 to 5. For the interval \(\frac{\pi}{2} \leq t \leq \pi\), \(\sin t\) decreases from 1 back to 0, so the curve is traced in the opposite direction, moving from \((2,5)\) back to \((1,3)\).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Eliminating the Parameter in Parametric Equations
Eliminating the parameter involves expressing one variable directly in terms of the other, removing the parameter (t). This is done by solving one parametric equation for the parameter and substituting into the other, resulting in a Cartesian equation that describes the curve.
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Eliminating the Parameter
Understanding the Domain and Orientation of Parametric Curves
The domain of the parameter t determines which portion of the curve is traced. Orientation refers to the direction in which the curve is drawn as t increases. Different intervals of t can produce different segments or directions on the same curve.
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Analyzing Curve Differences Based on Parameter Intervals
Changing the interval of t can alter the part of the curve traced and its orientation. Comparing curves for different t-intervals helps identify how the shape, position, and direction of traversal differ, even if the parametric formulas are similar.
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02:59Finding Area Between Curves that Cross on the Interval
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