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x+y+z= 2. 13-14 (a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then dysg by reparametrize the curve with respect to arc length starting from P, and (b) find the point 4 units along the curve (in the direction of increasing t) from P. 13. r(t) = (5 t)i + (4t-3)j + 3t k, P(4, 1, 3) 14. r(t) = e' sin tit 4 and the plane

x+y+z= 2. 13-14 (a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then dysg by reparametrize the curve with respect to arc length starting from P, and (b) find the point 4 units along the curve (in the direction of increasing t) from P. 13. r(t) = (5 t)i + (4t-3)j + 3t k, P(4, 1, 3) 14. r(t) = e' sin tit 4 and the plane

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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868 13.3 EXERCISES 1-6 Find the length of the curve. 1. r(t) = (t, 3 cos t, 3 sin t), -5 ≤ 1≤5 2. r(t) = (21, 1², 1³), 0≤ 1 ≤ 1 CHAPTER 13 Vector Functions MOBA 3. r(t) = √√√2ti + e'j+e¹k, 0≤ t ≤1 4. r(t) = cos ti+ sin tj + In cos tk, 5. r(t) = i + t²j+t³k, 0≤ t ≤<l 6. r(t) = t² i + 9tj + 4t3/2 k, 1 < t < 4 7-9 Find the length of the curve correct to four decimal places. (Use a calculator to approximate the integral.) 102 7. r(t) = (t², 1³, 14), 0≤ t ≤2 8. r(t) = (t, e, te ¹), 1<t <3 9. r(t) = (cos πt, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0) 10. Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length of this curve correct to four decimal places. 11. Let C be the curve of intersection of the parabolic cylinder x² = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36). 0 < t < π/4 12. Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x² + y2² = 4 and the plane x + y + z = 2. - 13-14 (a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from. P, and (b) find the point 4 units along the curve (in the direction of increasing t) from P. 13. r(t) = (5 t)i + (4t-3)j + 3t k, P(4, 1, 3) 14. r(t) = e' sin ti + e' cos tj + √Ze'k, P(0, 1, √√2) r(t) = 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = tive direction. Where are you now? 3 sin t, y = 4t, z = 3 cost in the posi- 16. Reparametrize the curve 2 t² + 1 -₁)₁ it p 2t t² + 1 j 24. Find the curvature of r(t) = (t², In t, t In t) at the pitnoo 8 Is snpoint (1, 0, 0). 100077 3 with respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametriza- tion in its simplest form. What can you conclude about the curve? 0 blam limmer 501 51 17-20 (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the curvature. 17. r(t) = (1, 3 cos t, 3 sin t) 18. r(t) = (t², sin t t cos t, cost+t sin t), t>0 cups 021-23 Use Theorem 10 to find the curvature. 21. r(t) = t³j+t² k 22. r(t) = ti + t²j+e'k 23. r(t) = √√√6t² i + 2t j + 2t³ k 19. r(t) =(√21, e', e-¹) 20. r(t) = (1, 1², 1²) - 26. Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0). UMSOR 27-29 Use Formula 11 to find the curvature. 0 27. y = x² 28. y = tan x BN 25. Find the curvature of r(t) = (t, t², ³) at the point (1, 1, 1). FUSE 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 30. y = ln x 31. y = e* slonta ISUOS 32. Find an equation of a parabola that has curvature 4 at the origin. YA 33. (a) Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. (b) Estimate the curvature at P and at Q by sketching the osculating circles at those points. aloda 0 29. y = xe P C 20 Qatchslug20 X (
Transcribed Image Text:868 13.3 EXERCISES 1-6 Find the length of the curve. 1. r(t) = (t, 3 cos t, 3 sin t), -5 ≤ 1≤5 2. r(t) = (21, 1², 1³), 0≤ 1 ≤ 1 CHAPTER 13 Vector Functions MOBA 3. r(t) = √√√2ti + e'j+e¹k, 0≤ t ≤1 4. r(t) = cos ti+ sin tj + In cos tk, 5. r(t) = i + t²j+t³k, 0≤ t ≤<l 6. r(t) = t² i + 9tj + 4t3/2 k, 1 < t < 4 7-9 Find the length of the curve correct to four decimal places. (Use a calculator to approximate the integral.) 102 7. r(t) = (t², 1³, 14), 0≤ t ≤2 8. r(t) = (t, e, te ¹), 1<t <3 9. r(t) = (cos πt, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0) 10. Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length of this curve correct to four decimal places. 11. Let C be the curve of intersection of the parabolic cylinder x² = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36). 0 < t < π/4 12. Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x² + y2² = 4 and the plane x + y + z = 2. - 13-14 (a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from. P, and (b) find the point 4 units along the curve (in the direction of increasing t) from P. 13. r(t) = (5 t)i + (4t-3)j + 3t k, P(4, 1, 3) 14. r(t) = e' sin ti + e' cos tj + √Ze'k, P(0, 1, √√2) r(t) = 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = tive direction. Where are you now? 3 sin t, y = 4t, z = 3 cost in the posi- 16. Reparametrize the curve 2 t² + 1 -₁)₁ it p 2t t² + 1 j 24. Find the curvature of r(t) = (t², In t, t In t) at the pitnoo 8 Is snpoint (1, 0, 0). 100077 3 with respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametriza- tion in its simplest form. What can you conclude about the curve? 0 blam limmer 501 51 17-20 (a) Find the unit tangent and unit normal vectors T(t) and N(t). (b) Use Formula 9 to find the curvature. 17. r(t) = (1, 3 cos t, 3 sin t) 18. r(t) = (t², sin t t cos t, cost+t sin t), t>0 cups 021-23 Use Theorem 10 to find the curvature. 21. r(t) = t³j+t² k 22. r(t) = ti + t²j+e'k 23. r(t) = √√√6t² i + 2t j + 2t³ k 19. r(t) =(√21, e', e-¹) 20. r(t) = (1, 1², 1²) - 26. Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0). UMSOR 27-29 Use Formula 11 to find the curvature. 0 27. y = x² 28. y = tan x BN 25. Find the curvature of r(t) = (t, t², ³) at the point (1, 1, 1). FUSE 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 30. y = ln x 31. y = e* slonta ISUOS 32. Find an equation of a parabola that has curvature 4 at the origin. YA 33. (a) Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. (b) Estimate the curvature at P and at Q by sketching the osculating circles at those points. aloda 0 29. y = xe P C 20 Qatchslug20 X (
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