Textbook Question
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
A hyperbola with vertices (±4, 0) and foci (±6, 0)
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.
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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.
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05:53Finding Differentials
Related Practice
Textbook Question
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
r = 3
Textbook Question
85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)
A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope of length 0≤a≤2 (see figure). What is the area of the region (inside the corral) that the goat can graze? Check your answer with the special cases a=0 and a=2.
Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
(x - 1)² + y² = 1
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Textbook Question
Spiral arc length Consider the spiral r=4θ, for θ≥0.
a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.