Textbook Question
Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
a. For what value of p is P tangent to H?
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem R.12.14
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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Identify the given curve equation: \(x = y^3 + y + 1\).
Note the segment starts at the point \((1, 0)\) and ends at \((11, 2)\), so the parameter will be based on \(y\) ranging from 0 to 2.
Choose the parameter \(t\) to represent \(y\), so set \(y = t\) with \(t\) in the interval \([0, 2]\).
Express \(x\) in terms of \(t\) using the given curve equation: \(x = t^3 + t + 1\).
Write the parametric equations as \(x(t) = t^3 + t + 1\) and \(y(t) = t\) for \(t\) in \([0, 2]\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y are given in terms of t, allowing more flexible descriptions of curves, including those that are not functions in the traditional sense.
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Parameterizing Equations
Curve Segment and Parameter Interval
When describing a specific segment of a curve parametrically, the parameter t is restricted to an interval that corresponds to the start and end points. Identifying the correct parameter values for these points ensures the parametric equations trace the desired segment.
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Non-uniqueness of Parametric Representations
Parametric equations for a curve are not unique; different parameterizations can describe the same curve segment. This flexibility allows choosing convenient parameter functions or intervals to simplify calculations or meet specific conditions.
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Related Practice
Textbook Question
53–57. Conic sections
b. Use analytical methods to determine the location of the foci, vertices, and directrices.
4x² + 8y² = 16
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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
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?
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Textbook Question
67–72. Derivatives Consider the following parametric curves.
a. Determine dy/dx in terms of t and evaluate it at the given value of t.
x = cos t, y = 8 sin t; t = π/2
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Textbook Question
(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933
a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.
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