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Chapt_09_1 - 9 Vector Calculus EXERCISES 9.1 Vector...

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9 9 Vector Calculus EXERCISES 9.1 Vector Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. Note: the scale is distorted in this graph. For t = 0, the graph starts at (1 , 0 , 1). The upper loop shown intersects the xz -plane at about (286751 , 0 , 286751). 438
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9.1 Vector Functions 10. 11. x = t , y = t , z = t 2 + t 2 = 2 t 2 ; r ( t ) = t i + t j + 2 t 2 k 12. x = t , y = 2 t , z = ± t 2 + 4 t 2 + 1 = ± 5 t 2 1; r ( t ) = t i + 2 t j ± 5 t 2 1 k 13. x = 3cos t , z = 9 9cos 2 t = 9sin 2 t , y = 3sin t ; r ( t ) = 3cos t i + 3sin t j + 9sin 2 t k 14. x = sin t , z = 1, y = cos t ; r ( t ) = sin t i + cos t j + k 439
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9.1 Vector Functions 15. r ( t ) = sin2 t t i + ( t 2) 5 j + ln t 1 /t k . Using L’Hˆ opital’s Rule, lim t 0 + r ( t ) = 2cos2 t 1 i + ( t 2) 5 j + 1 /t 1 /t 2 k = 2 i 32 j . 16. (a) lim t α [ 4 r 1 ( t ) + 3 r 2 ( t )] = 4( i 2 j + k ) + 3(2 i + 5 j + 7 k ) = 2 i + 23 j + 17 k (b) lim t α r 1 ( t ) · r 2 ( t ) = ( i 2 j + k ) · (2 i + 5 j + 7 k ) = 1 17. r ( t ) = 1 t i 1 t 2 j ; r ( t ) = 1 t 2 i + 2 t 3 j 18. r ( t ) = t sin t, 1 sin t ; r ( t ) = t cos t sin t, cos t 19. r ( t ) = 2 te 2 t + e 2 t , 3 t 2 , 8 t 1 ; r ( t ) = 4 te 2 t + 4 e 2 t , 6 t, 8 20. r ( t ) = 2 t i + 3 t 2 j + 1 1 + t 2 k ; r ( t ) = 2 i + 6 t j 2 t (1 + t 2 ) 2 k 21. r ( t ) = 2sin t i + 6cos t j r ( π/ 6) = i + 3 3 j 22. r ( t ) = 3 t 2 i + 2 t j r ( 1) = 3 i 2 j 23. r ( t ) = j 8 t (1 + t 2 ) 2 k r (1) = j 2 k 24. r ( t ) = 3sin t i + 3cos t j + 2 k r ( π/ 4) = 3 2 2 i + 3 2 2 j + 2 k 25. r ( t ) = t i + 1 2 t 2 j + 1 3 t 3 k ; r (2) = 2 i + 2 j + 8 3 k ; r ( t ) = i + t j + t 2 k ; r (2) = i + 2 j + 4 k Using the point (2 , 2 , 8 / 3) and the direction vector r (2), we have x = 2 + t , y = 2 + 2 t , z = 8 / 3 + 4 t . 26. r ( t ) = ( t 3 t ) i + 6 t t + 1 j +(2 t +1) 2 k ; r (1) = 3 j +9 k ; r ( t ) = (3 t 2 1) i + 6 ( t + 1) 2 j +(8 t +4) k ; r (1) = 2 i + 3 2 j +12 k . Using the point (0 , 3 , 9) and the direction vector r (1), we have x = 2 t , y = 3 + 3 2 t , z = 9 + 12 t . 27. d dt [ r ( t ) × r ( t )] = r ( t ) × r ( t ) + r ( t ) × r ( t ) = r ( t ) × r ( t ) 28. d dt [ r ( t ) · ( t r ( t ))] = r ( t ) · d dt ( t r ( t )) + r ( t ) · ( t r ( t )) = r ( t ) · ( t r ( t ) + r ( t )) + r ( t ) · ( t r ( t )) = r ( t ) · ( t r ( t )) + r ( t ) · r ( t ) + r ( t ) · ( t r ( t )) = 2 t ( r ( t ) · r ( t )) + r ( t ) · r ( t ) 440
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9.1 Vector Functions 29. d dt [ r ( t ) · ( r ( t ) × r ( t ))] = r ( t ) · d dt ( r ( t ) × r ( t )) + r ( t ) · ( r ( t ) × r ( t )) = r ( t ) · ( r ( t ) × r ( t ) + r ( t ) × r ( t )) + r ( t ) · ( r ( t ) × r ( t )) = r ( t ) · ( r ( t ) × r ( t )) 30. d dt [ r 1 ( t ) × ( r 2 ( t ) × r 3 ( t ))] = r 1 ( t ) × d dt ( r 2 ( t ) × r 3 ( t )) + r 1 ( t ) × ( r 2 ( t ) × r 3 ( t )) = r 1 ( t ) × ( r 2 ( t ) × r 3 ( t ) + r 2 ( t ) × r 3 ( t )) + r 1 ( t ) × ( r 2 ( t ) × r 3 ( t )) = r 1 ( t ) × ( r 2 ( t ) × r 3 ( t )) + r 1 ( t ) × ( r 2 ( t ) × r 3 ( t )) + r 1 ( t ) × ( r 2 ( t ) × r 3 ( t )) 31. d dt r 1 (2 t ) + r 2 1 t = 2 r 1 (2 t ) 1 t 2 r 2 1 t 32. d dt [ t 3 r ( t 2 )] = t 3 (2 t ) r ( t 2 ) + 3 t 2 r ( t 2 ) = 2 t 4 r ( t 2 ) + 3 t 2 r ( t 2 ) 33. 2 1 r ( t ) dt = 2 1 t dt i + 2 1 3 t 2 dt j + 2 1 4 t 3 dt k = 1 2 t 2 2 1 i + t 3 2 1 j + t 4 2 1 k = 3 2 i + 9 j + 15 k 34. 4 0 r ( t ) dt = 4 0 2 t + 1 dt i + 4 0 t dt j + 4 0 sin πt dt k = 1 3 (2 t + 1) 3 / 2 4 0 i 2 3 t 3 / 2 4 0 j 1 π cos πt 4 0 k = 26 3 i 16 3 j 35. r ( t ) dt = te t dt i + e 2 t dt j + te t 2 dt k = [ te t e t + c 1 ] i + 1 2 e 2 t + c 2 j + 1 2 e t 2 + c 3 k = e t ( t 1) i + 1 2 e 2 t j + 1 2 e t 2 k + c , where c = c 1 i + c 2 j + c 3 k . 36. r ( t ) dt = 1 1 + t 2 dt i + t 1 + t 2 dt j + t 2 1 + t 2 dt k = [tan 1 t + c 1 ] i + 1 2 ln(1 + t 2 ) + c 2 j + 1 1 1 + t 2 dt k = [tan 1 t + c 1 ] i + 1 2 ln(1 + t 2 ) + c 2 j + [ t tan 1 t + c 3 ] k = tan 1 t i + 1 2 ln(1 + t 2 ) j + ( t tan 1 t ) k + c , where c = c 1 i + c 2 j + c 3 k .
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