1873_solutions

In the event that m e 0 it is clear that e has

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Unformatted text preview: t  b. To show that  is continuous at the number t we shall show that for every number  0 we have lim  u  . lim  u u t Suppose that ut  0. Using the fact that the function  is continuous at t, we choose a number   0 such that a  t   t  t  b and such that the inequality d  t , u  4 holds whenever t   u  t  . Now we choose a partition u 0 , u 1 , u 2 , , u n of the interval a, b such that n b d  uu 1 ,  uj  j1 4 . Since the inclusion of extra points in this partition makes the sum n d  uj 1 ,  uj j1 larger, we may assume, without loss of generality that t is a point in the partition and that if t  u k then both of the points u k 1 and u k1 lie in the interval t , t   . t  uk t uk 1 Now given any m  1, 2, , n we clearly have 403 t u k 1 m d  uj 1 ,  uj  um j1  u m is the length of the restriction of the curve  to the interval u m , b , we also have and, since  b n d  uj ,  uj 1 b  um . jm1 Therefore if m  1, 2, , n then since n b m d  uu 1 ,  uj  n  um d  uj j1 ,  uj 1  b  um j1 d  uj 1 ,  uj jm1 we conclude that m d  uj  um 1 ,  uj  j1 4 . We conclude that k 1  u k 1  d  uj 1 ,  uj  d  uj 1 ,  uj  d  uk j1 4 k1  ,  uk 1  d  u k ,  u k 1 j1   uk 1  4  4   4 4 Thus lim  u u t  u k 1   u k 1  3  lim  u  . ut 4 6. Suppose that  is a rectifiable curve with domain a, b in a metric space X. Prove that for every number there exists a number   0 such that whenever P is a partition of a, b and P   we have |L P,  L  |  . 0 Hint: Examine the method of proof of Darboux’s theorem. 7. Suppose that  1 and  2 are curves in R k with domain a, b . The sum  1   2 and dot product  1   2 of  1 and  2 are defined by the equations 1  2 t  1 t  2 t and 1  2 t  1 t  2 t for each t a, b . If f is a real valued function defined on the interval a, b then we define f 1 by the equation f 1 t  f t  1 t for every t a, b . Prove that if, for a given number t a, b the derivatives  1 t and  2 t and f t exist then 1  2 t  1 t  2 t 1  2 t  1 t  2 t  1 t  2 t f 1 t  f t  1 t  f t  1 t All of these facts follow directly from the definitions. 8. Prove that if  is a differentiable curve in R k and if the function  is constant then    is the constant function zero. Interpret this fact geometrically. If we define f t   t 2 then, for each t, 404 0  f t   t   t   t   t  2 t   t . A geometric interpretation of this fact is that if the path  runs around in a sphere with center O then the velocity  , being perpendicular to the line that runs from O to , is tangential to the sphere. 9. If a curve  in R k is differentiable at a number t and if  t t is defined by the equation t Tt  t O then the unit tangent T t of  at the number . This is standard stuff and can be found in any elementary calculus text. a. Prove that if  t exists then the derivative T t of T exists at the number t and we have...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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