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Find the curvature of the curve y = x at the point (1, 1).

Find the curvature of the curve y = x at the point (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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t. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n = 6 to estimate the length of the arc of the curve with equations x = t², y = t³, z = t^, 0 ≤t≤3. 8. Find the length of the curve r(t) = (2t3/2, cos 2t, sin 2t), 0 ≤t≤ 1. 9. The helix r₁(t) = cos ti+ sin tj + tk intersects the curve r₂(t) = (1 + t)i + t²j+t³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. 11. For the curve given by r(t) = (sin³t, cos³t, sin²t), 0 ≤ t ≤ π/2, find (a) the unit tangent vector, (b) the unit normal vector, tors (c) the unit binormal vector, and (d) the curvature. Kindo 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4). 13. Find the curvature of the curve y = x* at the point (1, 1). 4 14. Find an equation of the osculating circle of the curve y = x - x² at the origin. Graph both the curve and its osculating circle. 15. Find an equation of the osculating plane of the curve x = sin 2t, yt, z = cos 2t at the point (0, 7, 1). 17. A r ac 18. F in th ac 19. A i F 20. A at th (a (t (c 21. A fr ar p m 22. Fi ti 23. A di th fi:
Transcribed Image Text:t. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n = 6 to estimate the length of the arc of the curve with equations x = t², y = t³, z = t^, 0 ≤t≤3. 8. Find the length of the curve r(t) = (2t3/2, cos 2t, sin 2t), 0 ≤t≤ 1. 9. The helix r₁(t) = cos ti+ sin tj + tk intersects the curve r₂(t) = (1 + t)i + t²j+t³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. 11. For the curve given by r(t) = (sin³t, cos³t, sin²t), 0 ≤ t ≤ π/2, find (a) the unit tangent vector, (b) the unit normal vector, tors (c) the unit binormal vector, and (d) the curvature. Kindo 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4). 13. Find the curvature of the curve y = x* at the point (1, 1). 4 14. Find an equation of the osculating circle of the curve y = x - x² at the origin. Graph both the curve and its osculating circle. 15. Find an equation of the osculating plane of the curve x = sin 2t, yt, z = cos 2t at the point (0, 7, 1). 17. A r ac 18. F in th ac 19. A i F 20. A at th (a (t (c 21. A fr ar p m 22. Fi ti 23. A di th fi:
t. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n = 6 to estimate the length of the arc of the curve with equations x = t², y = t³, z = t^, 0 ≤t≤3. 8. Find the length of the curve r(t) = (2t3/2, cos 2t, sin 2t), 0 ≤t≤ 1. 9. The helix r₁(t) = cos ti+ sin tj + tk intersects the curve r₂(t) = (1 + t)i + t²j+t³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. 11. For the curve given by r(t) = (sin³t, cos³t, sin²t), 0 ≤ t ≤ π/2, find (a) the unit tangent vector, (b) the unit normal vector, tors (c) the unit binormal vector, and (d) the curvature. Kindo 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4). 13. Find the curvature of the curve y = x* at the point (1, 1). 4 14. Find an equation of the osculating circle of the curve y = x - x² at the origin. Graph both the curve and its osculating circle. 15. Find an equation of the osculating plane of the curve x = sin 2t, yt, z = cos 2t at the point (0, 7, 1). 17. A r ac 18. F in th ac 19. A i F 20. A at th (a (t (c 21. A fr ar p m 22. Fi ti 23. A di th fi:
Transcribed Image Text:t. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n = 6 to estimate the length of the arc of the curve with equations x = t², y = t³, z = t^, 0 ≤t≤3. 8. Find the length of the curve r(t) = (2t3/2, cos 2t, sin 2t), 0 ≤t≤ 1. 9. The helix r₁(t) = cos ti+ sin tj + tk intersects the curve r₂(t) = (1 + t)i + t²j+t³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. 11. For the curve given by r(t) = (sin³t, cos³t, sin²t), 0 ≤ t ≤ π/2, find (a) the unit tangent vector, (b) the unit normal vector, tors (c) the unit binormal vector, and (d) the curvature. Kindo 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sin t at the points (3, 0) and (0, 4). 13. Find the curvature of the curve y = x* at the point (1, 1). 4 14. Find an equation of the osculating circle of the curve y = x - x² at the origin. Graph both the curve and its osculating circle. 15. Find an equation of the osculating plane of the curve x = sin 2t, yt, z = cos 2t at the point (0, 7, 1). 17. A r ac 18. F in th ac 19. A i F 20. A at th (a (t (c 21. A fr ar p m 22. Fi ti 23. A di th fi:
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