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.R.7a
Capitolo 12, Problema 12.R.7a

7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

Guida verificata passo dopo passo
1
Identify the given parametric equations: \(x = 8\cos t + 1\) and \(y = 8\sin t + 2\) with the parameter \(t\) in the interval \(0 \leq t \leq 2\pi\).
Recall the Pythagorean identity: \(\cos^2 t + \sin^2 t = 1\). This will help us eliminate the parameter \(t\) by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\).
Isolate \(\cos t\) and \(\sin t\) from the parametric equations: \(\cos t = \frac{x - 1}{8}\) and \(\sin t = \frac{y - 2}{8}\).
Substitute these expressions into the Pythagorean identity to get an equation involving only \(x\) and \(y\): \(\left(\frac{x - 1}{8}\right)^2 + \left(\frac{y - 2}{8}\right)^2 = 1\).
Simplify the equation by multiplying both sides by \(64\) (since \(8^2 = 64\)) to obtain the Cartesian equation of the curve without the parameter \(t\).

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, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
Video consigliato:
08:02
Parameterizing Equations

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
Video consigliato:
05:59
Eliminating the Parameter

Tangent Lines to Parametric Curves

The tangent line to a parametric curve at a given parameter value t is found by computing derivatives dx/dt and dy/dt, then using dy/dx = (dy/dt)/(dx/dt). This slope helps write the equation of the tangent line at the specified point on the curve.
Video consigliato:
Percorso guidato
06:49
Differentiation of Parametric Curves
Pratica correlata
Domanda del libro di testo

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

69
views
Domanda del libro di testo

27–32. Polar curves Graph the following equations.


r = 3 sin 4θ

85
views
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

43
views
Domanda del libro di testo

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x²/4 + y²/25 = 1

53
views
Domanda del libro di testo

Polar conversion Consider the equation r=4/(sinθ+cosθ). 


a. Convert the equation to Cartesian coordinates and identify the curve it describes.  

27
views
Domanda del libro di testo

Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?

b. r=(½)+sinθ

31
views