Calculus

Calculus - Chapter 11 Vector-Valued Functions Chapter 11...

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Chapter 11 Vector-Valued Functions 352 Chapter 11 Section 11.1 5. –5 0 5 –5 –2.5 0 2.5 5 –5 –2.5 0 2.5 5 x y z (0, 0, 1) (3, 1, 1) (6, 4, 3) 7. –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z (0, 2, 1) for t = (1, 2, 1) for t = 0 π 2 π 2 , 11. 0 y x 2 2 –2 13. 0 y x 10 –10 –10 15. –4 –2 0 2 4 –4 –2 0 2 4 –4 –2 2 4 x y z 0 17. –4 –2 0 2 4 –4 –2 0 2 4 –4 –2 0 2 4 x y z 19. –4 0 4 –4 –2 0 2 4 –4 –2 0 2 4 x y z 21. –5 0 5 –5 –2.5 0 2.5 5 –5 –2.5 0 2.5 5 x y z 23. –5 0 5 –5 0 5 –5 0 5 x y z 25. –10 0 10 –10 0 10 –10 0 10 x y z 9. 0 y x 2 2 –2

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Chapter 11 Vector-Valued Functions 353 27. –10 0 10 –10 0 10 –10 0 10 x y z 29. –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z 31. –2 0 2 –2 0 2 –2 0 2 x y z 33. -2 0 2 –2 0 2 –2 0 2 x y z 35. a. F b. C c. E d. A e. B f. D 37. –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z 2 22 2 0 2 2 0 () s in,cos, 2s in2 (s i n) ( c o s) (2 s i n 2 ) 14 s i n2 10.54 by numerical integration tt t t st t t d t td t π π =− =−+ + =+ r 39. –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z 2 2 0 2 0 ( ) sin , cos , 16sin16 [ sin ] [ cos ] [ 16sin16 ] 256sin 16 21.56 by numerical integration t t t t d t t π π π π − π π + π π + + H 41. –5 0 5 –5 0 5 –5 0 5 x y z 2 2 2 2 0 2 42 0 1 ,2,3 1( 2 )( 3 ) 941 9.57 by numerical integration t t d t d t = + + r
Chapter 11 Vector-Valued Functions 354 43. 22 cos 2 cos sin tt t =− –1 1 –1 0 1 –1 0 1 0 45. The two curves are identical, with the same endpoints. They are just parameterized using different t -values. 47. g ( t ) and h ( t ) are portions of () , , , . t t t < < H g ( t ) = r ( t ) with 11 t −≤ ≤ , and h ( t ) = r ( t ) with 0 t . 49. –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z –2 –1 0 1 2 –2 –1 0 1 2 –2 –1 0 1 2 x y z The graph is an ellipse in 3-space, and it is periodic with period 2 π . However, for T much larger than 2 π , the points plotted become too few and "jump" around the ellipse, causing the jagged lines.

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Chapter 11 Vector-Valued Functions 355 Section 11.2 5. 22 0 00 0 lim 1, , sin l im( 1 ) ,l im ims in 1, 1, 0 t t t tt t te t tet →→ =− 7. 0 000 sin 1 lim , cos , 1 sin 1 lim , lim cos , lim 1 1, 1, 1 t ttt t t →→→ + + = 9. 2 0 2 0 lim ln , 1, 3 lim ln , lim 1, lim( 3) t t t t +− =+ which does not exist, because of the undefined limit of the x -component. 11. 1 t , because t = 1 is an excluded value for the x - component. 13. 2 n t π ( n odd), because the x -component is undefined for 33 ..., , , , , ... 2 2 t ππ π π 15. 0 t , because the y -component is undefined for t < 0. 17. () 4 2 3 3 3 , 1 , 16 4, , 21 dd d d dt dt dt dt t t t t    + r 19. 2 2 (sin ), (sin ), (cos ) cos , 2 cos , sin d d t dt dt dt dt tt t t = r 21. 2 2 2 ( ), ( ), (sec2 ) 2 , 2 , 2sec2 tan 2 t t d d et t dt dt dt dt te t t t = = r 23. s in,cos (0) 1, 0 ; (0) 0, 1 0, 1 ; 1, 0 1 ,0; 0 , 1 t == π=− π= r rr 0 y x 2 2 –2 25. s in,1 ,cos (0) 1 ,0 ,0 ; (0) 0 ,1 0, , 1 ; 2 1 , ,0; 0 t π π=− π r –2 –4 0 2 4 –4 –2 0 2 4 –4 –2 0 2 4 x y z 27.
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Calculus - Chapter 11 Vector-Valued Functions Chapter 11...

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