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7-14 Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases. 7. r(t) = (sin t, t) 9. r(t)= (1, 21, 2t) 11. r(t)=(3, 1,2 - 1²) 12. r(t) = 2 cos ti + 2 sin tj + k 13. r(t) = t²i + tªj + tºk 8. r(t) = (1²-1, t) 10. r(t) = (sin t, t, cos mt) Tor

7-14 Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases. 7. r(t) = (sin t, t) 9. r(t)= (1, 21, 2t) 11. r(t)=(3, 1,2 - 1²) 12. r(t) = 2 cos ti + 2 sin tj + k 13. r(t) = t²i + tªj + tºk 8. r(t) = (1²-1, t) 10. r(t) = (sin t, t, cos mt) Tor

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I V 854 5. lim 118 III X- 6. lim te 1-8 X- 7-14 Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. 7. r(t) = (sin t, t) 9. r(t) = (t, 2 t, 2t) 11. r(t) =(3, t, 2 - 1²) 12. r(t) = 2 cos ti + 2 sin tj + k poo 13. r(t) = t²i + tªj + tºk 14. r(t) = cos ti-cos tj + sin t k (1+, tar 17. P(2, 0, 0), 19. P(0, -1, 1), CHAPTER 13 Vector Functions P.ET MOITD02 t ¹27³ - t (10%, 2 (27) +1,1 sin -—-) 15-16 Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve. 15. r(t) = (t, sin t, 2 cos t) 16. r(t) = (t, t, t²) X 17-20 Find a vector equation and parametric equations for the line segment that joins P to Q. ZA 1- e tan ¹t, t 21-26 Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices. II ZA ZA -21 Q(6, 2, -2) (12, 13, 14) 8. r(t) = (t²-1, t) 10. r(t) = (sin πt, t, cos mt) in 18. P(-1, 2, -2), Q(-3, 5, 1) 20. P(a, b, c), Q(u, v, w) foogle+ XK notamine ben bavlo TERSE VINIVASA VI ZA X svo songe gnieoroim ni svino sobre to SOB to signe le boireito (anoge ZA y 9 ∙y Sunwollt ZA mil sdb ball 9- 21. x = t cos t, 22. x = cos t, 23. x = 1, y = 1/(1+1²), z = 1² 24. x = cos t, y=t, z=tsint, t≥0 y = sint, z = 1/(1 + f²) 25. x = cos 8t, y = sin 8t, z = 26. x = cos²t, y = sin²t, z = t 27. Show that the curve with parametric equations x = t cost, y = t sin t, z = t lies on the cone z² = x² + y², and use this fact to help sketch the curve. y = sin t, z = cos 2t 0.81 e0.8r, t≥ 0 y = cos t, z 28. Show that the curve with parametric equations x = sint, sin²t is the curve of intersection of the surfaces 2 2 x² and x² + y² = 1. Use this fact to help sketch the curve. Z = = 29. Find three different surfaces that contain the curve r(t) = 2ti + e'j + e²¹ k. ROLLY SYS 30. Find three different surfaces that contain the curve r(t) = t²i+ In tj+ (1/t) k. 21.0 31. At what points does the curve r(t) = ti+ (2t - t²) k inter- sect the paraboloid z = x² + y²? 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x² + y² + z²: = 5? 33-37 Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. 33. r(t) = (cos t sin 2t, sin t sin 2t, cos 2t) 34. r(t) = (te', e¯', t) 35. r(t) = (sin 3t cos t, t, sin 3t sin sin t) 36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cos t) 37. r(t) = (cos 2t, cos 3t, cos 4t) 38. Graph the curve with parametric equations x = sin t, y = sin 2t, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes. 39. Graph the curve with parametric equations x = (1 + cos 16t) cos t y = (1 + cos 16t) sin t z = 1 + cos 16t 21DA3X3 1.81 Explain the appearance of the graph by showing that it lies on a cone. to ob Sot 40. Graph the curve with parametric equations x = √1 -0.25 cos² 10t cos t y=√1-0.25 cos² 10t sin t z = 0.5 cos 10t 23-11-5
Transcribed Image Text:I V 854 5. lim 118 III X- 6. lim te 1-8 X- 7-14 Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. 7. r(t) = (sin t, t) 9. r(t) = (t, 2 t, 2t) 11. r(t) =(3, t, 2 - 1²) 12. r(t) = 2 cos ti + 2 sin tj + k poo 13. r(t) = t²i + tªj + tºk 14. r(t) = cos ti-cos tj + sin t k (1+, tar 17. P(2, 0, 0), 19. P(0, -1, 1), CHAPTER 13 Vector Functions P.ET MOITD02 t ¹27³ - t (10%, 2 (27) +1,1 sin -—-) 15-16 Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve. 15. r(t) = (t, sin t, 2 cos t) 16. r(t) = (t, t, t²) X 17-20 Find a vector equation and parametric equations for the line segment that joins P to Q. ZA 1- e tan ¹t, t 21-26 Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices. II ZA ZA -21 Q(6, 2, -2) (12, 13, 14) 8. r(t) = (t²-1, t) 10. r(t) = (sin πt, t, cos mt) in 18. P(-1, 2, -2), Q(-3, 5, 1) 20. P(a, b, c), Q(u, v, w) foogle+ XK notamine ben bavlo TERSE VINIVASA VI ZA X svo songe gnieoroim ni svino sobre to SOB to signe le boireito (anoge ZA y 9 ∙y Sunwollt ZA mil sdb ball 9- 21. x = t cos t, 22. x = cos t, 23. x = 1, y = 1/(1+1²), z = 1² 24. x = cos t, y=t, z=tsint, t≥0 y = sint, z = 1/(1 + f²) 25. x = cos 8t, y = sin 8t, z = 26. x = cos²t, y = sin²t, z = t 27. Show that the curve with parametric equations x = t cost, y = t sin t, z = t lies on the cone z² = x² + y², and use this fact to help sketch the curve. y = sin t, z = cos 2t 0.81 e0.8r, t≥ 0 y = cos t, z 28. Show that the curve with parametric equations x = sint, sin²t is the curve of intersection of the surfaces 2 2 x² and x² + y² = 1. Use this fact to help sketch the curve. Z = = 29. Find three different surfaces that contain the curve r(t) = 2ti + e'j + e²¹ k. ROLLY SYS 30. Find three different surfaces that contain the curve r(t) = t²i+ In tj+ (1/t) k. 21.0 31. At what points does the curve r(t) = ti+ (2t - t²) k inter- sect the paraboloid z = x² + y²? 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x² + y² + z²: = 5? 33-37 Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. 33. r(t) = (cos t sin 2t, sin t sin 2t, cos 2t) 34. r(t) = (te', e¯', t) 35. r(t) = (sin 3t cos t, t, sin 3t sin sin t) 36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cos t) 37. r(t) = (cos 2t, cos 3t, cos 4t) 38. Graph the curve with parametric equations x = sin t, y = sin 2t, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes. 39. Graph the curve with parametric equations x = (1 + cos 16t) cos t y = (1 + cos 16t) sin t z = 1 + cos 16t 21DA3X3 1.81 Explain the appearance of the graph by showing that it lies on a cone. to ob Sot 40. Graph the curve with parametric equations x = √1 -0.25 cos² 10t cos t y=√1-0.25 cos² 10t sin t z = 0.5 cos 10t 23-11-5
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