7. Consider the ellipse + = 5 = 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 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. For a curve C with parametric equations z = x(t), y = y(t), then at the point (r(t), y(t)) the curvature can be calculated by F(t)y"(t) – y (t)z"(t)| (z(t))? + (3'(t))?)³/2 Use your parametric equations from (a) to calculate the curvature of the ellipse at (x(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 + = 5 = 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 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. For a curve C with parametric equations z = x(t), y = y(t), then at the point (r(t), y(t)) the curvature can be calculated by F(t)y"(t) – y (t)z"(t)| (z(t))? + (3'(t))?)³/2 Use your parametric equations from (a) to calculate the curvature of the ellipse at (x(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:7. Consider the ellipse + =
5 = 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 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. For a curve C with parametric equations z = x(t), y = y(t), then at the point
(r(t), y(t)) the curvature can be calculated by
F(t)y"(t) – y (t)z"(t)|
(z(t))? + (3'(t))?)³/2
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.
(d) Write 1-2 sentences explaining whether your answer to (c) makes sense based on the
illustration of the ellipse above.
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