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30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 31. y=ealthis 017, MOTUJO2 30. y = ln x

30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 31. y=ealthis 017, MOTUJO2 30. y = ln x

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 CHAPTER 13 Vector Functions 13.3 EXERCISES 1-6 Find the length of the curve. 1. r(t) = (1, 3 cos 1, 3 sin t), -5=1=5 2. r(t) = (21, 1², 1³), 0≤1≤1 3. r(t) = √√2ri + e'j+e'k, 0≤1≤1 4. r(t) = cos ti+ sin tj + In cos tk, 0≤ t ≤ π/4 5. r(t) = i + 1²j+ 1³k, 0≤ t ≤ 1 6. r(t) = 1² 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.) 7. r(t) = (1², 13, 14), 0≤ t ≤ 2 8. r(t) = (1, e, te), 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). 12. Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x² + y² = 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. sill 18 13. r(t) = (5 t)i + (4t − 3)j + 3t k, 14. r(t) = e' sin ti + e' cos tj + √√2e'k, 2 t² + 1 137-60 r(t) 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x 3 sin t, y = 4t, z = 3 cost in the posi- tive direction. Where are you now? = 16. Reparametrize the curve = -¹) ₁. P(4, 1, 3) P(0, 1, √2) + 2t 1² + 1 noo is ons point (1, 0, 0). alemax 1 MO1.1.1. 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) 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? 18. r(t) = (1², sin f t cost, cost + f sin t), t>0 19. r(t) =(√√2t, e', e-¹) 20. r(t) = (1, 11², 1²) 21-23 Use Theorem 10 to find the curvature. 21. r(t) = t³j+ t² k 22. r(t) = ti + t²j+e'k 23. r(t) = √√√61² i + 2t j + 2t³ k 24. Find the curvature of r(t) = (t², In t, t In t ) at the 25. Find the curvature of r(t) = (t, t²2, 1³) at the point (1, 1, 1). cos t, 26. Graph the curve with parametric equations x = y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0). tamon a 27-29 Use Formula 11 to find the curvature. 0-27. y = x4 28. y = tan x planes mor 30. y = ln x 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x →→ ∞o? 31. y = e YA 32. Find an equation of a parabola that has curvature 4 at the origin. 1+ 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. 20 0 83AUDA 23. y= xe* 1 P 2010 au 90 Q C 37 mate X
Transcribed Image Text:868 CHAPTER 13 Vector Functions 13.3 EXERCISES 1-6 Find the length of the curve. 1. r(t) = (1, 3 cos 1, 3 sin t), -5=1=5 2. r(t) = (21, 1², 1³), 0≤1≤1 3. r(t) = √√2ri + e'j+e'k, 0≤1≤1 4. r(t) = cos ti+ sin tj + In cos tk, 0≤ t ≤ π/4 5. r(t) = i + 1²j+ 1³k, 0≤ t ≤ 1 6. r(t) = 1² 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.) 7. r(t) = (1², 13, 14), 0≤ t ≤ 2 8. r(t) = (1, e, te), 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). 12. Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x² + y² = 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. sill 18 13. r(t) = (5 t)i + (4t − 3)j + 3t k, 14. r(t) = e' sin ti + e' cos tj + √√2e'k, 2 t² + 1 137-60 r(t) 15. Suppose you start at the point (0, 0, 3) and move 5 units along the curve x 3 sin t, y = 4t, z = 3 cost in the posi- tive direction. Where are you now? = 16. Reparametrize the curve = -¹) ₁. P(4, 1, 3) P(0, 1, √2) + 2t 1² + 1 noo is ons point (1, 0, 0). alemax 1 MO1.1.1. 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) 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? 18. r(t) = (1², sin f t cost, cost + f sin t), t>0 19. r(t) =(√√2t, e', e-¹) 20. r(t) = (1, 11², 1²) 21-23 Use Theorem 10 to find the curvature. 21. r(t) = t³j+ t² k 22. r(t) = ti + t²j+e'k 23. r(t) = √√√61² i + 2t j + 2t³ k 24. Find the curvature of r(t) = (t², In t, t In t ) at the 25. Find the curvature of r(t) = (t, t²2, 1³) at the point (1, 1, 1). cos t, 26. Graph the curve with parametric equations x = y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0). tamon a 27-29 Use Formula 11 to find the curvature. 0-27. y = x4 28. y = tan x planes mor 30. y = ln x 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x →→ ∞o? 31. y = e YA 32. Find an equation of a parabola that has curvature 4 at the origin. 1+ 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. 20 0 83AUDA 23. y= xe* 1 P 2010 au 90 Q C 37 mate X
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