Change of variables Consider the parameterized curvesr(t) = ⟨ƒ(t), g(t), h(t)⟩ and R(t) = ⟨ƒ(u(t)0, g(u(t)), h(u(t))⟩, where ƒ, g, h, and u are continuously differentiable functions and u has an inverse on [a, b].a. Show that the curve generated by r on the intervala ≤ t ≤ b is the same as the curve generated by R onu-1(a) ≤ t ≤ u-1(b) (or u-1(b) ≤ t ≤ u-1(a)).b. Show that the lengths of the two curves are equal.(Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.)

Change of variables Consider the parameterized curvesr(t) = ⟨ƒ(t), g(t), h(t)⟩ and R(t) = ⟨ƒ(u(t)0, g(u(t)), h(u(t))⟩, where ƒ, g, h, and u are continuously differentiable functions and u has an inverse on [a, b].a. Show that the curve generated by r on the intervala ≤ t ≤ b is the same as the curve generated by R onu-1(a) ≤ t ≤ u-1(b) (or u-1(b) ≤ t ≤ u-1(a)).b. Show that the lengths of the two curves are equal.(Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.)

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

ChapterP: Prerequisites

Section: Chapter Questions

Problem 8PS

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