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 12.1.35
Capitolo 12, Problema 12.1.35

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=tan t, y=sec ² t−1 

Guida verificata passo dopo passo
1
Start with the given parametric equations: \(x = \tan t\) and \(y = \sec^{2} t - 1\).
Recall the Pythagorean identity involving tangent and secant: \(\sec^{2} t = 1 + \tan^{2} t\).
Substitute \(\sec^{2} t\) in the expression for \(y\) using the identity: \(y = (1 + \tan^{2} t) - 1\).
Simplify the expression for \(y\) to get \(y = \tan^{2} t\).
Since \(x = \tan t\), replace \(\tan t\) with \(x\) in the expression for \(y\) to eliminate the parameter \(t\), resulting in \(y = x^{2}\).

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:
1m

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, often denoted as t. Instead of y being directly related to x, both x and y depend on t, allowing the description of more complex curves.
Video consigliato:
08:02
Parameterizing Equations

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. For example, the identity sec² t - tan² t = 1 is essential for relating sec² t and tan t, which helps eliminate the parameter t.
Video consigliato:
7:17
Verifying Trig Equations as Identities

Eliminating the Parameter

Eliminating the parameter involves rewriting parametric equations to form a single equation in x and y by removing the parameter t. This often requires expressing one variable in terms of t and substituting into the other, using identities or algebraic manipulation.
Video consigliato:
05:59
Eliminating the Parameter
Pratica correlata
Domanda del libro di testo

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 lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

58
views
Domanda del libro di testo

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

72
views
Domanda del libro di testo

81–88. Arc length Find the arc length of the following curves on the given interval.


x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

37
views
Domanda del libro di testo

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

y² - x²/64 = 1; (6, -5/4)

71
views
Domanda del libro di testo

15–22. Sets in polar coordinates Sketch the following sets of points.


1 < r < 2 and π/6 ≤ θ ≤ π/3

31
views
Domanda del libro di testo

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

141
views