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3. Consider the curve C with parametric equations r(t) = cos(2t), y(t) = sin(t), where -≤t≤ a) Find a Cartesian equation for C. Then make a rough sketch of the curve. b) The curvature к of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature K = 0 at every point. At any point, the curvature can be calculated by k(t) = (¹ + (2)*)* Show that the curvature of the curve C is: k(t) = 4 (1 +16 sin² t) **

3. Consider the curve C with parametric equations r(t) = cos(2t), y(t) = sin(t), where -≤t≤ a) Find a Cartesian equation for C. Then make a rough sketch of the curve. b) The curvature к of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature K = 0 at every point. At any point, the curvature can be calculated by k(t) = (¹ + (2)*)* Show that the curvature of the curve C is: k(t) = 4 (1 +16 sin² t) **

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 53E
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3. Consider the curve C with parametric equations a(t) = cos(2t), y(t) = sin(t), where ≤t≤ 1. a) Find a Cartesian equation for C. Then make a rough sketch of the curve, b) The curvature K of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature x = 0 at every point. At any point, the curvature can be calculated by k(t) Show that the curvature of the curve C is: k(t) = dy dx² (¹ + (2) ²) ³ 4 (1 + 16 sin² t) // - K
Transcribed Image Text:3. Consider the curve C with parametric equations a(t) = cos(2t), y(t) = sin(t), where ≤t≤ 1. a) Find a Cartesian equation for C. Then make a rough sketch of the curve, b) The curvature K of a curve C at a given point is a measure of how quickly the curve changes direction at that point. For example, a straight line has curvature x = 0 at every point. At any point, the curvature can be calculated by k(t) Show that the curvature of the curve C is: k(t) = dy dx² (¹ + (2) ²) ³ 4 (1 + 16 sin² t) // - K
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