Textbook Question
45–60. Areas of regions Find the area of the following regions.
The region outside the circle r = 1/2 and inside the circle r = cos 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 12.3.2
Explain why the slope of the line θ=π/2 is undefined.
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Recall that the slope of a line in the Cartesian plane is given by the ratio of the change in y to the change in x, which is \(\text{slope} = \frac{\Delta y}{\Delta x}\).
The line \(\theta = \frac{\pi}{2}\) in polar coordinates corresponds to all points where the angle from the positive x-axis is \(90^\circ\), which means the line is vertical.
A vertical line has the same x-coordinate for all points, so the change in x, \(\Delta x\), is zero along this line.
Since the slope formula involves division by \(\Delta x\), and here \(\Delta x = 0\), the slope expression becomes \(\frac{\Delta y}{0}\), which is undefined because division by zero is not defined in mathematics.
Therefore, the slope of the line \(\theta = \frac{\pi}{2}\) is undefined because it is a vertical line with no horizontal change.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Slope of a Line
The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x (rise over run). It indicates how much y changes for a unit change in x.
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Slopes of Tangent Lines
Vertical Lines and Undefined Slope
Vertical lines have no horizontal change (Δx = 0), making the slope calculation involve division by zero, which is undefined. Therefore, vertical lines do not have a defined slope.
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05:13Slopes of Tangent Lines
Angle θ = π/2 in Polar Coordinates
The line θ = π/2 in polar coordinates corresponds to the vertical line x = 0 in Cartesian coordinates. Since it is vertical, its slope is undefined due to zero horizontal displacement.
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Related Practice
Textbook Question
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
Textbook Question
102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.
Spiral of Archimedes: r = aθ
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Textbook Question
Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment.
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Textbook Question
What is the polar equation of the vertical line x = 5?
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Textbook Question
25–28. Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.
r = 4 cos θ
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