Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.2

Chapter 12, Problem 12.R.2

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 guidance

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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Key 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.

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.

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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Textbook Question

53–57. Conic sections

b. Use analytical methods to determine the location of the foci, vertices, and directrices.

4x² + 8y² = 16

Textbook Question

Cartesian conversion Write the equation x=y ² in polar coordinates and state values of θ that produce the entire graph of the parabola.

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Textbook Question

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).

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Textbook Question

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

Textbook Question

22–23. Arc length Find the length of the following curves.

x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. A set of parametric equations for a given curve is always unique.