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 12.1.41
Capitolo 12, Problema 12.1.41

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The line segment starting at P(0, 0) and ending at Q(2, 8)

Guida verificata passo dopo passo
1
Identify the points P(0, 0) and Q(2, 8) as the start and end points of the line segment. We want parametric equations that describe all points on the line segment between these two points.
Recall that a parametric equation for a line segment from point P(x_0, y_0) to Q(x_1, y_1) can be written as: \(x = x_0 + t(x_1 - x_0)\) \(y = y_0 + t(y_1 - y_0)\), where the parameter \(t\) varies over an interval.
Substitute the coordinates of P and Q into the parametric form: \(x = 0 + t(2 - 0) = 2t\) \(y = 0 + t(8 - 0) = 8t\).
Determine the interval for the parameter \(t\). Since the line segment starts at P when \(t=0\) and ends at Q when \(t=1\), the interval for \(t\) is \(0 \leq t \leq 1\).
Thus, the parametric equations for the line segment are \(x = 2t\), \(y = 8t\), with \(t\) in the interval \([0, 1]\).

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:
2m

Concetti chiave

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y depend on t, allowing representation of curves and line segments in a flexible way.
Video consigliato:
08:02
Parameterizing Equations

Line Segment Parameterization

To parameterize a line segment between two points P and Q, use a parameter t that varies over an interval, typically [0,1]. The coordinates are given by linear interpolation: x(t) = x_P + t(x_Q - x_P), y(t) = y_P + t(y_Q - y_P), tracing the segment as t moves from 0 to 1.
Video consigliato:
08:02
Parameterizing Equations

Parameter Interval

The parameter interval defines the range of t values for which the parametric equations describe the curve. For a line segment, choosing t in [0,1] ensures the curve starts at P when t=0 and ends at Q when t=1, covering all points in between.
Video consigliato:
05:59
Eliminating the Parameter
Pratica correlata
Domanda del libro di testo

Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.

55
views
Domanda del libro di testo

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 2 + √t, y = 2 - 4t; slope = -8

51
views
Domanda del libro di testo

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


25y² - 4x² = 100

22
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.

109
views
Domanda del libro di testo

What is the slope of the line θ=π/3?

101
views
Domanda del libro di testo

53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.


An ellipse with vertices (0, ±9) and eccentricity ¼ 

31
views