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Q. 28

Found in: Page 880


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

For each of the vector-valued functions, find the unit tangent vector.



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Step by Step Solution

Step1. Given Information

Consider r(t)=sin2t,cos2t,t The objective is to find the unit tangent vector to r(t) The unit tangent vector The unit tangent vector of r(t) denoted by T(t) is defined as T(t)=r(t)r(t) Consider r(t)=sin2t,cos2t,t First we compute r(t) : r(t)=ddtsin2t,cos2t,t=ddt(sin2t),ddt(cos2t),ddt(t)=2cos2t,2sin2t,1

Step2. Continue

r(t)=2cos2t,2sin2t,1=(2cos2t)2+(2sin2t)2+(1)2=4cos22t+4sin22t+1=4cos22t+sin22t+1=5 since sin22t+cos22t=1 The unit tangent vector to r(t) is T(t)=r(t)r′′(t)=2cos2t,2sin2t,15 Thus the unit tangent vector to the given vector - valued function is 152cos2t,2sin2t,1

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