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25. Find the curvature of r(t) = (t, t², t³) at the point (1, 1, 1).

25. Find the curvature of r(t) = (t, t², t³) at the point (1, 1, 1).

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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aces. er fC ve e 7 19. r(t) 20. r(t) = (1, 21², 1²) 339MANE 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, ln t, t ln t ) at the rpoint (1, 0, 0). 25. Find the curvature of r(t) = (t, t², t³) at the point (1, 1, 1). 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). DE 27-29 Use Formula 11 to find the curvature. 27. y = x¹ 28. y = tan x 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 31. y = e* 30. y = 29. y = xe* In x 32. Find an equation of a parabola that has curvature 4 at the origin. 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 point
Transcribed Image Text:aces. er fC ve e 7 19. r(t) 20. r(t) = (1, 21², 1²) 339MANE 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, ln t, t ln t ) at the rpoint (1, 0, 0). 25. Find the curvature of r(t) = (t, t², t³) at the point (1, 1, 1). 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). DE 27-29 Use Formula 11 to find the curvature. 27. y = x¹ 28. y = tan x 30-31 At what point does the curve have maximum curvature? What happens to the curvature as x → ∞? 31. y = e* 30. y = 29. y = xe* In x 32. Find an equation of a parabola that has curvature 4 at the origin. 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 point
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