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Unformatted text preview: 1 Mathematics 1c: Solutions, Homework Set 4 Due: Monday, April 26 at 10am. 1. (10 Points) Section 4.1, Exercise 14 Show that, at a local maximum or minimum of the quantity k r ( t ) k , r ( t ) is perpendicular to r ( t ) . Solution. Notice first that at the time t where a local maximum or minimum for k r ( t ) k occurs, a local maximum or minimum for k r ( t ) k 2 = r ( t ) · r ( t ) also occurs. And at those particular t ’s, the first derivative of k r ( t ) k 2 is equal to zero. Therefore 0 = ( r ( t ) · r ( t )) = r ( t ) · r ( t ) + r ( t ) · r ( t ) = 2 r ( t ) · r ( t ) , which means that r ( t ) is perpendicular to r ( t ). 2. (10 Points) Section 4.1, Exercise 18. Let c be a path in R 3 with zero accel- eration. Prove that c is a straight line or a point. Solution. Write c ( t ) = ( x ( t ) ,y ( t ) ,z ( t )). If c 00 ( t ) = , then x 00 ( t ) = y 00 ( t ) = z 00 ( t ) ≡ 0. These equations imply that x ( t ) = c 1 ,y ( t ) = c 2 , and z ( t ) = c 3 , where c 1 ,c 2 ,c 3 are all constants. Continuing, we see that this implies thatare all constants....
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
- Spring '08