Textbook Question
A polar conic section Consider the equation r² = sec2θ
a. Convert the equation to Cartesian coordinates and identify the curve.
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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.R.40b
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r =3 − 6 cos θ
Verified step by step guidance1
Recognize that the curve is given in polar form: \(r = 3 - 6 \cos \theta\). To find the slope of the tangent line at the origin, we need to express the curve in Cartesian coordinates or use the formula for the slope of a tangent line in polar coordinates.
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). The slope of the tangent line \(\frac{dy}{dx}\) can be found using the chain rule as \(\frac{dy/d\theta}{dx/d\theta}\).
Compute \(\frac{dr}{d\theta}\) by differentiating \(r = 3 - 6 \cos \theta\) with respect to \(\theta\). This gives \(\frac{dr}{d\theta} = 6 \sin \theta\).
Use the formulas for derivatives of \(x\) and \(y\) with respect to \(\theta\):
\(\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos \theta - r \sin \theta\)
\(\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin \theta + r \cos \theta\)
Evaluate \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\) at the value(s) of \(\theta\) where the curve passes through the origin (i.e., where \(r=0\)). Then find the slope of the tangent line as \(\frac{dy/d\theta}{dx/d\theta}\) at those \(\theta\) values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Curves
Polar coordinates represent points using a radius and angle (r, θ) instead of Cartesian (x, y). Understanding how curves are defined in polar form, like r = 3 - 6 cos θ, is essential for analyzing their properties and converting between coordinate systems when needed.
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05:32
Intro to Polar Coordinates
Slope of Tangent Lines in Polar Coordinates
The slope of a tangent line to a polar curve at a given point can be found using the formula dy/dx = (dr/dθ sin θ + r cos θ) / (dr/dθ cos θ - r sin θ). This relates the derivatives of r with respect to θ to the slope in Cartesian terms, enabling the determination of tangent slopes at specific angles.
Recommended video:
05:13Slopes of Tangent Lines
Implicit Differentiation and Derivatives
Implicit differentiation is used to find derivatives when variables are interdependent, such as r and θ in polar equations. Calculating dr/dθ and applying the chain rule allows us to find the rate of change needed to determine the slope of tangent lines at points like the origin.
Recommended video:
05:14Finding The Implicit Derivative
Related Practice
Textbook Question
19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve
The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2
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Textbook Question
14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The circle x ² + y ² =9, generated clockwise
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Textbook Question
58–59. Tangent lines Find an equation of the line tangent to the following curves at the given point. Check your work with a graphing utility.
x²/16 - y²/9 = 1; (20/3, -4)
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The hyperbola y²/2 - x²/4 = 1 has no x-intercept.
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Textbook Question
53–57. Conic sections
d. Make an accurate graph of the curve.
x = 16y²
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