(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 k = 0 at every point. For a curve C with parametric equations x = (x(t), y(t)) the curvature can be calculated by x(t), y = y(t), then at the point |a (t)y" (t) – 3' (t)a" (t)| (7'(t))? + (y'(t))?)³/2 K = Use your parametric equations from (a) to calculate the curvature of the ellipse at (x(t), y(t)).
(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 k = 0 at every point. For a curve C with parametric equations x = (x(t), y(t)) the curvature can be calculated by x(t), y = y(t), then at the point |a (t)y" (t) – 3' (t)a" (t)| (7'(t))? + (y'(t))?)³/2 K = Use your parametric equations from (a) to calculate the curvature of the ellipse at (x(t), y(t)).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please explain step b
Transcribed Image Text:7. Consider the ellipse
a2
y?
= 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 k = 0 at every
point. For a curve C with parametric equations x =
(x(t), y(t)) the curvature can be calculated by
x(t), y = y(t), then at the point
|a'(t)g" (t) – y'(t)æ"(t)|
(r'(t))² + (y'(t))²)³/2
K =
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.
(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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