(a) Show that the curvature of a'smooth curve r(t) = f(t)i+g(t)j defined by twice-differentiable functions x = f(t) and y g(t) is given by the formula |ij – yä K = (i² + j?)

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 20T
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(a) Show that the curvature of a'smooth curve r(t) = f(t)i+g(t)j defined
by twice-differentiable functions x =
f(t) and y
g(t) is given by
the formula
K =
(i? + j?)
(b) Find 7, T, N, and B at the given value of t for
T(t) = (cos t)ê + (sin t)j – k, t =
4
Then find equations for the osculating, normal, and rectifying planes
at that value of t.
Transcribed Image Text:(a) Show that the curvature of a'smooth curve r(t) = f(t)i+g(t)j defined by twice-differentiable functions x = f(t) and y g(t) is given by the formula K = (i? + j?) (b) Find 7, T, N, and B at the given value of t for T(t) = (cos t)ê + (sin t)j – k, t = 4 Then find equations for the osculating, normal, and rectifying planes at that value of t.
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