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

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problema 12.1.11a

Capitolo 12, Problema 12.1.11a

11–14. Working with parametric equations Consider the following parametric equations.
a. Make a brief table of values of t, x, and y.
b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).

x=2 t,y=3t−4;−10≤d≤10

Guida verificata passo dopo passo

Identify the parametric equations given: \(x = 2t\) and \(y = 3t - 4\), with the parameter \(t\) ranging from \(-10\) to \(10\).

Create a table of values by choosing several values of \(t\) within the interval \([-10, 10]\). For each chosen \(t\), calculate the corresponding \(x\) and \(y\) values using the parametric equations: \(x = 2t\) and \(y = 3t - 4\).

Plot the points \((x, y)\) from the table on the Cartesian plane. Each point corresponds to a specific value of \(t\).

Draw the curve by smoothly connecting the plotted points, representing the parametric curve defined by the equations. Make sure the curve reflects the continuous change of \(t\) from \(-10\) to \(10\).

Indicate the positive orientation on the curve by adding arrows showing the direction of increasing \(t\), which corresponds to moving from points with smaller \(t\) values to larger \(t\) values.

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:

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 more complex curves and motions.

Video consigliato:

08:02

Parameterizing Equations

Creating a Table of Values

To analyze parametric equations, select values of the parameter t within the given interval, then compute corresponding x and y values. This table helps visualize points on the curve and understand how the curve evolves as t changes.

Video consigliato:

Percorso guidato

06:37

Average Value of a Function

Plotting Parametric Curves and Orientation

Plotting involves graphing the (x, y) pairs from the table to form the curve. The positive orientation shows the direction of increasing t, indicating how the curve is traced over time, which is important for understanding motion or directionality.

Video consigliato:

Percorso guidato

06:49

Differentiation of Parametric Curves

Pratica correlata

Domanda del libro di testo

Area of roses Assume m is a positive integer.

a. Even number of leaves: What is the relationship between the total area enclosed by the 4m-leaf rose r=cos(2mθ) and m?

views

Domanda del libro di testo

Reflection property of parabolas: Consider the parabola y = x²/(4p) with its focus at F(0, p). The goal is to show that the angle of incidence (α) equals the angle of reflection (β).

a. Let P(x₀, y₀) be a point on the parabola. Show that the slope of the tangent line at P is tan θ = x₀/(2p).

296

views

Domanda del libro di testo

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

a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).

views

Domanda del libro di testo

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = cos t, y = 8 sin t; t = π/2

views

Domanda del libro di testo

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = t + 1/t, y = t − 1/t; t = 1

views

Domanda del libro di testo

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.

views