# A5 - ~ B t and ~ B(1(f ﬁnd the equation of the osculating plane at t = 1(g Regarding ~ r t as giving the motion of a particle in space(with

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MATH 222, Fall 2011 Calculus 3 Assignment 3, due Friday, October 21, 2011 1. For the helix deﬁned parametrically by ~ r ( t ) = h 3cos t, 3sin t, 4 t i , (a) ﬁnd the arc length s ( t ) travelled from 0 to t > 0, and speciﬁcally from 0 to 6 π ; (b) ﬁnd the parametrization of the curve with respect to arc length from 0 to t > 0; (c) ﬁnd the unit tangent ~ T ( t ) and ﬁnd ~ T (6 π ); (d) ﬁnd the principle unit normal ~ N ( t ) and ~ N (6 π ); (e) ﬁnd the binormal vector ~ B ( t ) and ~ B (6 π ); (f) ﬁnd the equation of the osculating plane at t = 6 π . (g) Regarding ~ r ( t ) as giving the motion of a particle in space (with re- spect to time t ) ﬁnd the acceleration ~a ( t ) and the tangent and normal component of acceleration. evaluate these at t = 6 π . 2. For the curve deﬁned parametrically by ~ r ( t ) = h 1 2 t 5 , 1 3 t 5 , 1 6 t 5 i , (a) ﬁnd the arc length s ( t ) travelled from 0 to t > 0, and speciﬁcally from 0 to 1; (b) ﬁnd the parametrization of the curve with respect to arc length from 0 to t > 0; (c) ﬁnd the unit tangent ~ T ( t ) and ﬁnd ~ T (1); (d) ﬁnd the principle unit normal ~ N ( t ) and ~ N (1); (e) ﬁnd the binormal vector

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Unformatted text preview: ~ B ( t ) and ~ B (1); (f) ﬁnd the equation of the osculating plane at t = 1. (g) Regarding ~ r ( t ) as giving the motion of a particle in space (with re-spect to time t ) ﬁnd the acceleration ~a ( t ) and the tangent and normal component of acceleration. Evaluate these at t = 1. 3. Suppose that ~ r ( t ) deﬁnes a curve parametrically for a ≤ t ≤ b , and that ~ r ( t ) 6 = ~ 0 on [ a,b ]. Show that d dt | ~ r ( t ) | = ~ r ( t ) · ~ r 00 ( t ) | ~ r ( t ) | . [Hint: What is | ~ r ( t ) | 2 and what is its derivative?] 1 4. Suppose that ~ r ( s ) deﬁnes a curve parametrically with respect to arc length and that ~ r ( s ) is nonzero on the curve. Show that d ~ B ds is orthogonal to both ~ B ( s ) and ~ T ( s ). Conclude that there is a scalar function τ ( s ) such that d ~ B ds =-τ ( s ) ~ N . (This function τ is known as the torsion of the curve.) 2...
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## This note was uploaded on 01/11/2012 for the course MATH 100 taught by Professor Loveys during the Fall '11 term at McGill.

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A5 - ~ B t and ~ B(1(f ﬁnd the equation of the osculating plane at t = 1(g Regarding ~ r t as giving the motion of a particle in space(with

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