7. Consider the ellipse= 1, 0 < a < b.(a) Write down a set of parametric equations that describe the entire ellipse. Be sure tostate the domain of your parametric equations. (Hint: Your parametric equations shouldinvolve trigonometric functions.)(b) The curvature k of a curve C at a given point is a measure of how quickly the curvechanges direction at that point. For example, a straight line has curvature k = 0 at everypoint. For a curve C with parametric equations r = r(t), y = y(t), then at the point(r(t), y(t)) the curvature can be calculated by|(t)y"(t) – y'(t)r" (t)||(r'(t))² + (y(t))?)³/2K=Use your parametric equations from (a) to calculate the curvature of the ellipse at(2(t), y(t)).(c) Use your answer to (b) to find the point(s) on the curve at which the curvature is maxi-mized. Justify your answer with calculations.(d) Write 1-2 sentences explaining whether your answer to (c) makes sense based on theillustration of the ellipse aboe.

Answered: Image /qna-images/answer/be2ba815-56f3-434d-817b-e48100b4275d.jpg

7. Consider the ellipse = 1, 0 < a < b. (a) Write down a set of parametric equations that describe the entire ellipse. Be sure to state the domain of your parametric equations. (Hint: Your parametric equations should involve trigonometric functions.) (b) The curvaturek 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. For a curve C with parametric equations z = r(t), y = y(t), then at the point (r(t), y(t)) the curvature can be calculated by F(t)y"(t) – '(t)="(t)| (7'(t))ª + (/(t))*)®/²´ Use your parametric equations from (a) to calculate the curvature of the ellipse at (r(t), y(t)). (c) Use your answer to (b) to find the point(s) on the curve at which the curvature is maxi- mized. Justify your answer with calculations. +

7. Consider the ellipse = 1, 0 < a < b. (a) Write down a set of parametric equations that describe the entire ellipse. Be sure to state the domain of your parametric equations. (Hint: Your parametric equations should involve trigonometric functions.) (b) The curvaturek 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. For a curve C with parametric equations z = r(t), y = y(t), then at the point (r(t), y(t)) the curvature can be calculated by F(t)y"(t) – '(t)="(t)| (7'(t))ª + (/(t))*)®/²´ Use your parametric equations from (a) to calculate the curvature of the ellipse at (r(t), y(t)). (c) Use your answer to (b) to find the point(s) on the curve at which the curvature is maxi- mized. Justify your answer with calculations. +

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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