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

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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 right answer with least computation. Week 4 Homework: 13.3 curvature (2nd Wed) Problem 13.3.16 Use formula (9) to find the curvature of r ( t ) = < t 2 , 2 t, ln t > .
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week04 - Math 23 Sections 110-113 B Dodson Week 4 Homework...

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