Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The point on a parabola closest to the focus is the vertex.
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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 12.2.106d
(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933
d. Plot the curve with various values of k. How many fingers can you produce?
Verified step by step guidance1
Understand the given polar curve equation: \(r = f(\theta) = \cos(a\theta) - 1.5\), where \(a = (1 + 12\pi)^{\frac{1}{2\pi}}\). This defines the radius \(r\) as a function of the angle \(\theta\).
Recognize that the parameter \(a\) controls the frequency of the cosine function inside the polar equation, which affects the number of 'fingers' or lobes in the plot.
To explore how the number of fingers changes, vary the parameter \(k\) in the expression for \(a\), for example by replacing \(a\) with \(k\) in the function \(r = \cos(k\theta) - 1.5\) and plotting for different values of \(k\).
For each chosen value of \(k\), plot the curve in polar coordinates over a full rotation, typically \(\theta\) from \(0\) to \(2\pi\), to observe the shape and count the number of distinct lobes or fingers formed.
Analyze the plots to determine the relationship between \(k\) and the number of fingers: generally, the number of fingers corresponds to the integer part of \(k\) or related to the frequency of the cosine term, so by increasing \(k\), you can produce more fingers.
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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 an angle (r, θ), allowing curves to be defined as functions of θ. Understanding how to interpret and plot r = f(θ) is essential for visualizing shapes like finger curves, where the radius changes with the angle.
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Intro to Polar Coordinates
Parameter Influence on Curve Shape
Parameters within the function, such as 'a' in r = cos(aθ) - 1.5, control the frequency and number of oscillations in the curve. Varying these parameters changes the number of 'fingers' or lobes in the plot, so analyzing their effect helps predict and count the features formed.
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Guided course
05:59Eliminating the Parameter
Use of Technology for Plotting
Graphing software or calculators can plot complex polar functions efficiently, allowing exploration of how different parameter values affect the curve. Using technology helps visualize the number of fingers produced and understand the behavior of the function dynamically.
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04:48Finding Volume Using Disks
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The point (3,π/2) lies on the graph of r=3 cos 2θ.
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Textbook Question
Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).
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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. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.
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Textbook Question
Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).
c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.
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