Skip to main content
Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh.12 - Parametric and Polar Curves문제 12.1.11a
12장, 문제 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 

검증된 단계별 안내
1
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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
가이드 코스
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.
추천 영상:
가이드 코스
06:49
Differentiation of Parametric Curves
관련 실천
교과서 질문

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?

26
views
교과서 질문

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
교과서 질문

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).  

40
views
교과서 질문

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

32
views
교과서 질문

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

67
views
교과서 질문

(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.

54
views