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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 12.1.43
12장, 문제 12.1.43

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 segment of the parabola y=2x ²−4, where −1≤x≤5

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Identify the given curve equation: \(y = 2x^{2} - 4\) and the domain for \(x\) is \(-1 \leq x \leq 5\).
Choose a parameter to represent \(x\). A natural choice is to let the parameter \(t\) equal \(x\), so set \(x = t\).
Express \(y\) in terms of the parameter \(t\) by substituting \(x = t\) into the equation: \(y = 2t^{2} - 4\).
Write the parametric equations as \(x = t\) and \(y = 2t^{2} - 4\) with the parameter interval \(-1 \leq t \leq 5\).
Verify that the parametric equations describe the segment of the parabola by checking that as \(t\) varies over the interval, the points \((x,y)\) trace the curve segment.

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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 third variable, called the parameter. Instead of y as a function of x, both x and y are defined in terms of a parameter t, allowing more flexible representations of curves, including segments and complex shapes.
추천 영상:
08:02
Parameterizing Equations

Interval of the Parameter

The interval of the parameter defines the range of values that the parameter t can take, which corresponds to the portion of the curve being described. Choosing the correct interval ensures the parametric equations represent only the desired segment of the curve, such as restricting x between -1 and 5.
추천 영상:
05:59
Eliminating the Parameter

Converting Cartesian to Parametric Form

To convert a Cartesian equation like y = 2x² - 4 into parametric form, assign the parameter t to x (e.g., x = t), then express y in terms of t using the original equation (y = 2t² - 4). This method directly translates the curve into parametric equations, with t varying over the specified x-interval.
추천 영상:
05:59
Eliminating the Parameter
관련 실천
교과서 질문

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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교과서 질문

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

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교과서 질문

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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교과서 질문

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

x² = -6y; (-6, -6)

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교과서 질문

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.


An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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교과서 질문

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 horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

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