Textbook Question
57–62. Polar equations for conic sections Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work.
r = 1/(2 - 2 sin θ)
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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 12.3.14
11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.
r = 4 + sin θ; (4, 0) and (3, 3π/2)
Verified step by step guidance1
Recall that for a polar curve given by \(r = f(\theta)\), the slope of the tangent line \(\frac{dy}{dx}\) can be found using the formulas for \(x\) and \(y\) in terms of \(r\) and \(\theta\): \(x = r \cos \theta\) and \(y = r \sin \theta\).
Express \(x\) and \(y\) as functions of \(\theta\): \(x(\theta) = (4 + \sin \theta) \cos \theta\) and \(y(\theta) = (4 + \sin \theta) \sin \theta\).
Find the derivatives \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) by applying the product rule to each:
\(\frac{dx}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \cos \theta]\) and
\(\frac{dy}{d\theta} = \frac{d}{d\theta}[(4 + \sin \theta) \sin \theta]\).
Use the chain rule and product rule to compute these derivatives explicitly, remembering that \(\frac{d}{d\theta}(\sin \theta) = \cos \theta\) and \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\).
Calculate the slope of the tangent line at each given point by evaluating \(\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) at the corresponding \(\theta\) values: \(\theta = 0\) for the point \((4,0)\) and \(\theta = \frac{3\pi}{2}\) for the point \((3, \frac{3\pi}{2})\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Curves
Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar curves are equations expressed as r = f(θ), describing how the radius changes with the angle. Understanding this system is essential to analyze points and curves given in polar form.
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Intro to Polar Coordinates
Slope of Tangent Lines in Polar Coordinates
The slope of a tangent line to a polar curve at a point is found by converting the polar equation to Cartesian coordinates or using the formula dy/dx = (dr/dθ sin θ + r cos θ) / (dr/dθ cos θ - r sin θ). This formula relates derivatives of r with respect to θ to the slope in the xy-plane.
Recommended video:
05:13Slopes of Tangent Lines
Differentiation with Respect to θ
To find the slope of the tangent line, you must differentiate r = 4 + sin θ with respect to θ. This involves applying basic differentiation rules to trigonometric functions, which is crucial for computing dr/dθ and subsequently the slope dy/dx.
Recommended video:
05:53Finding Differentials
Related Practice
Textbook Question
Tangent lines for a hyperbola Find an equation of the line tangent to the hyperbola x²/a² + y²/b² = 1 at the point (x₀, y₀)
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Textbook Question
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola with focus at (3, 0)
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Textbook Question
9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.
(-4, 3π/2)
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Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r cos θ = -4
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Textbook Question
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
The left half of the parabola y=x ² +1, originating at (0, 1)
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