For each of the following oriented curves, split it up into pieces and parameterize each piece, taking care that your parameterizations respect the orientation of the curve. (a) The curve in R2 that goes counterclockwise around the triangle with vertices (0,0), (3,1), and (2, 3). (b) The curve in R2 that goes from (-1, 1) to (2, 4) by following the parabola y = x², then returns along a straight line. (c) The curve in R³ that goes from (2,0,0) to (-2, 0, 0) along a half-circle in the xy-plane, and returns along a half-circle in the zz-plane. Both half-circles should be centered at the origin; the first should stay in the y ≥ 0 half of the xy-plane, and the second should stay in the z≥ 0 half of the xz-plane.
For each of the following oriented curves, split it up into pieces and parameterize each piece, taking care that your parameterizations respect the orientation of the curve. (a) The curve in R2 that goes counterclockwise around the triangle with vertices (0,0), (3,1), and (2, 3). (b) The curve in R2 that goes from (-1, 1) to (2, 4) by following the parabola y = x², then returns along a straight line. (c) The curve in R³ that goes from (2,0,0) to (-2, 0, 0) along a half-circle in the xy-plane, and returns along a half-circle in the zz-plane. Both half-circles should be centered at the origin; the first should stay in the y ≥ 0 half of the xy-plane, and the second should stay in the z≥ 0 half of the xz-plane.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:2. For each of the following oriented curves, split it up into pieces and parameterize each piece,
taking care that your parameterizations respect the orientation of the curve.
(a) The curve in R2 that goes counterclockwise around the triangle with vertices (0,0),
(3, 1), and (2, 3).
(b) The curve in R2 that goes from (-1, 1) to (2,4) by following the parabola y = x², then
returns along a straight line.
(c) The curve in R³ that goes from (2,0, 0) to (-2, 0, 0) along a half-circle in the xy-plane,
and returns along a half-circle in the xz-plane.
Both half-circles should be centered at the origin; the first should stay in the y ≥ 0 half
of the xy-plane, and the second should stay in the z ≥ 0 half of the xz-plane.
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