14.4%20Ex%2035-37

14.4%20Ex%2035-37 - 516 C H A P T E R 14 C A L C U L U S O...

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516 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS (ET CHAPTER 13) x y (2 2, 2 2) 4 p 4 N () 35. Find the unit normal vector N ( t ) to r ( t ) = h 4 , sin 2 t , cos 2 t i . SOLUTION We frst fnd the unit tangent vector: T ( t ) = r 0 ( t ) k r 0 ( t ) k (1) We have r 0 ( t ) = d dt h 4 , sin 2 t , cos 2 t i = h 0 , 2cos2 t , 2sin2 t i = 2 h 0 , cos 2 t , sin 2 t i k r 0 ( t ) k= 2 q 0 2 + cos 2 2 t + ( sin 2 t ) 2 = 2 0 + 1 = 2 Substituting in (1) gives: T ( t ) = 2 h 0 , cos 2 t , sin 2 t i 2 = h 0 , cos 2 t , sin 2 t i The normal vector is the ±ollowing vector: N ( t ) = T 0 ( t ) k T 0 ( t ) k (2) We compute the derivative o± the unit tangent vector and its length: T 0 ( t ) = d h 0 , cos 2 t , sin 2 t i = h 0 , t , t i=− 2 h 0 , sin 2 t , cos 2 t i k T 0 ( t ) 2 q 0 2 + sin 2 2 t + cos 2 2 t = 2 0 + 1 = 2 Substituting in (2) we obtain: N ( t ) = 2 h 0 , sin 2 t , cos 2 t i 2 = h 0 , sin 2 t , cos 2 t i 36. Sketch the graph o± r ( t ) = - t , t 3 ® .S ince r 0 ( t ) = - 1 , 3 t 2 ® , the unit normal N ( t ) points in one o± the two directions ± - 3 t 2 , 1 ® . Which sign is correct at t = 1? Which is correct at t =− 1?
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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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14.4%20Ex%2035-37 - 516 C H A P T E R 14 C A L C U L U S O...

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