Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.102
Capitolo 12, Problema 12.2.102

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

Guida verificata passo dopo passo
1
Understand the given spiral equation: the Spiral of Archimedes is defined by the polar equation \(r = a\theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle in radians.
Set the parameter \(a\) to the given values, first \(a = 1\) and then \(a = -1\), to see how the spiral changes with positive and negative values of \(a\).
For each value of \(a\), plot points by choosing several values of \(\theta > 0\) (for example, \(\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, 2\pi\), etc.), then calculate the corresponding \(r\) using \(r = a\theta\).
Convert each polar coordinate \((r, \theta)\) to Cartesian coordinates using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to help with graphing on the Cartesian plane.
Determine the direction of the spiral as \(\theta\) increases: since \(r\) increases linearly with \(\theta\) for \(a=1\), the spiral moves outward counterclockwise; for \(a=-1\), \(r\) becomes negative, which reflects the point across the origin, causing the spiral to move outward in the opposite direction. Indicate these directions on your graph.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how to plot points in this system is essential for graphing spirals like the Spiral of Archimedes.
Video consigliato:
05:32
Intro to Polar Coordinates

Spiral of Archimedes

The Spiral of Archimedes is defined by the equation r = aθ, where r increases linearly with θ. This spiral moves outward at a constant rate as θ increases, and the sign of a determines the direction in which the spiral expands.

Direction of Spiral Generation

As θ increases, the spiral is traced out in a specific direction, typically counterclockwise for positive θ. The sign of the parameter a affects whether the spiral expands outward or inward and influences the orientation of the curve.
Video consigliato:
05:21
Finding Limits by Direct Substitution
Pratica correlata
Domanda del libro di testo

Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by

x = 80t, y = −4.9t² + 3000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

30
views
Domanda del libro di testo

31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

52
views
Domanda del libro di testo

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 3 sin θ and r = 3 cos θ

54
views
Domanda del libro di testo

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. 

56
views
Domanda del libro di testo

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.


r² - 8r cos(θ - π/2) = 9

46
views
Domanda del libro di testo

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = x²

72
views