week04 - Math 23 Sections 110-113 B. Dodson Week 4...

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Math 23 Sections 110-113 B. Dodson Week 4 Homework: 13.1, 13.2 vector functions, derivatives 13.3 arc length, curvature 13.4 velocity, acceleration Problem 13.2.9: Find the derivative of the vector function ± r ( t ) = < t 2 , 1 - t, t > . Solution: We just take the derivative of the components, ± r ± ( t ) = < ( t 2 ) ± , (1 - t ) ± , ( t ) ± > = < 2 t, - 1 , 1 2 t >, where ( t 1 2 ) ± = 1 2 t - 1 2 . Week 4 Homework: 13.3 arc length, curvature (1st Wed) 13.4 velocity, acceleration Problem 13.2.17: If ± r ( t ) = < 6 t 5 , 4 t 3 , 2 t >, find the unit tangent vector ± T when t = 1 .
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2 . Solution: First, ± r ± ( t ) = < 30 t 4 , 12 t 2 , 2 > = 2 < 15 t 3 , 6 t 2 , 1 >, and ± r ± (1) = 2 < 15 , 6 , 1 >, so the length | ± r ± (1) | = 2 225 + 36 + 1 = 2 252 , so ± T (1) = 1 252 < 15 , 6 , 1 > . Note that finding ± T ( t ) first, before setting t = 1 , makes the problem harder; while setting t = 1 in ± r ( t ) before differentiating changes the derivative to ± 0 : the above steps are in exactly the correct order to get the
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This note was uploaded on 02/29/2008 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .

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week04 - Math 23 Sections 110-113 B. Dodson Week 4...

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