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36-37 Plot the space curve and its curvature function K(1). Comment on how the curvature reflects the shape of the curve. 36. r(t)= (t - sin t, 1 cost, 4 cos(1/2)), 018 37. r(t)= (te', e, √√21), -5=1=5

36-37 Plot the space curve and its curvature function K(1). Comment on how the curvature reflects the shape of the curve. 36. r(t)= (t - sin t, 1 cost, 4 cos(1/2)), 018 37. r(t)= (te', e, √√21), -5=1=5

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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37,43
and N(t). 34-35 Use a graphing calculator or computer to graph both the curve and its curvature function K(x) on the same screen. Is the graph of k what you would expect? 34. y=x¹2x² or (b) E 36-37 Plot the space curve and its curvature function k(t). Comment on how the curvature reflects the shape of the curve. 36. r(t)= (1 - sin t, 1 cos 1, 4 cos(1/2)), 0≤t≤ 8T (te', e, √21), 37. r(t) = 38. 38-39 Two graphs, a and b, are shown. One is a curve y = f(x) and the other is the graph of its curvature function y = x(x). Identify each curve and explain your choices. 39. a FROUNUBOO0 b 35. y=x2 -5=1=5 221 (99) 15/ sqola zuoninoo and bublin K = 40. (a) Graph the curve r(t) = (sin 3t, sin 21, sin 3t). At how many points on the curve does it appear that the curva- ture has a local or absolute maximum? (b) Use a CAS to find and graph the curvature function.i Does this graph confirm your conclusion from part (a)? a |xÿ - yx | [x² + y²13/2 y = b sin wt m 2 (0 mabing where the dots indicate derivatives with respect to t. in (1) ibolov ori ai timil at bris dignel 43-45 Use the formula in Exercise 42 to find the curvature. 43. x = t², y = t³ 44. x = a cos wt, 45. x = e' cos t, y e' sin t SECTION 13.3 Arc Length and Cur 47-48 Find the vectors T, N, and B at the gi 47. r(t) = (1²,1³,1), (1,3,1) 48. r(t) = (cos t, sin r, In cos t), (1,0,0) 46. Consider the curvature at x = 0 for each member of the family of functions f(x) = e. For which members is K (0) largest? out of Joogaon dirw sometib to saada 49-50 Find equations of the normal plane an of the curve at the given point. 49. x = sin 2r, y = -cos 2t, z = 4t; (0, 50. x Inf, y = 2r, z=²; (0, 2, 1) 41. The graph of r(t) = (t - sin t, 1-2 cost, t) is shown in Figure 13.1.12(b). Where do you think the curvature is largest? Use a CAS to find and graph the curvature function. 57. Show that at every point on the curve For which values of t is the curvature largest? 12 1992 alsdoch Spe osqa riguondi 42. Use Theorem 10 to show that the curvature of a plane para- metric curve x = f(t), y = g(t) is r(t) = (e' cos t, e' sin t 51. Find equations of the osculating circles 9x² + 4y² = 36 at the points (2, 0) and ing calculator or computer to graph the osculating circles on the same screen. 52. Find equations of the osculating circles y=x² at the points (0, 0) and (1,1). c lating circles and the parabola on the sa 53. At what point on the curve x = 1, y = normal plane parallel to the plane 6x + CAS 54. Is there a point on the curve in Exercis osculating plane is parallel to the plane [Note: You will need a CAS for differe ing, and for computing a cross produc 56. Show that the osculating plane at ever ohl or work wud r(t) = (+2, 1-²) is the same conclude about the curve? 55. Find equations of the normal and oscu curve of intersection of the parabolic a z = x² at the point (1, 1, 1). the angle between the unit tangent ve the same. Then show that the same re unit normal and binormal vectors. part 58. The rectifying plane of a curve at a p- contains the vectors T and B at that p odifying plane of the curve r(t) = sin ti the point (√2/2, √√2/2, 1). 59. Show that the curvature K is related t normal vectors by the equation dT ds = KN 60. Show that the curvature of a plane cu where is the angle between T and of inclination of the tangent line. (TH definition of curvature is consistent plane curves given in Exercise 10.2.e
Transcribed Image Text:and N(t). 34-35 Use a graphing calculator or computer to graph both the curve and its curvature function K(x) on the same screen. Is the graph of k what you would expect? 34. y=x¹2x² or (b) E 36-37 Plot the space curve and its curvature function k(t). Comment on how the curvature reflects the shape of the curve. 36. r(t)= (1 - sin t, 1 cos 1, 4 cos(1/2)), 0≤t≤ 8T (te', e, √21), 37. r(t) = 38. 38-39 Two graphs, a and b, are shown. One is a curve y = f(x) and the other is the graph of its curvature function y = x(x). Identify each curve and explain your choices. 39. a FROUNUBOO0 b 35. y=x2 -5=1=5 221 (99) 15/ sqola zuoninoo and bublin K = 40. (a) Graph the curve r(t) = (sin 3t, sin 21, sin 3t). At how many points on the curve does it appear that the curva- ture has a local or absolute maximum? (b) Use a CAS to find and graph the curvature function.i Does this graph confirm your conclusion from part (a)? a |xÿ - yx | [x² + y²13/2 y = b sin wt m 2 (0 mabing where the dots indicate derivatives with respect to t. in (1) ibolov ori ai timil at bris dignel 43-45 Use the formula in Exercise 42 to find the curvature. 43. x = t², y = t³ 44. x = a cos wt, 45. x = e' cos t, y e' sin t SECTION 13.3 Arc Length and Cur 47-48 Find the vectors T, N, and B at the gi 47. r(t) = (1²,1³,1), (1,3,1) 48. r(t) = (cos t, sin r, In cos t), (1,0,0) 46. Consider the curvature at x = 0 for each member of the family of functions f(x) = e. For which members is K (0) largest? out of Joogaon dirw sometib to saada 49-50 Find equations of the normal plane an of the curve at the given point. 49. x = sin 2r, y = -cos 2t, z = 4t; (0, 50. x Inf, y = 2r, z=²; (0, 2, 1) 41. The graph of r(t) = (t - sin t, 1-2 cost, t) is shown in Figure 13.1.12(b). Where do you think the curvature is largest? Use a CAS to find and graph the curvature function. 57. Show that at every point on the curve For which values of t is the curvature largest? 12 1992 alsdoch Spe osqa riguondi 42. Use Theorem 10 to show that the curvature of a plane para- metric curve x = f(t), y = g(t) is r(t) = (e' cos t, e' sin t 51. Find equations of the osculating circles 9x² + 4y² = 36 at the points (2, 0) and ing calculator or computer to graph the osculating circles on the same screen. 52. Find equations of the osculating circles y=x² at the points (0, 0) and (1,1). c lating circles and the parabola on the sa 53. At what point on the curve x = 1, y = normal plane parallel to the plane 6x + CAS 54. Is there a point on the curve in Exercis osculating plane is parallel to the plane [Note: You will need a CAS for differe ing, and for computing a cross produc 56. Show that the osculating plane at ever ohl or work wud r(t) = (+2, 1-²) is the same conclude about the curve? 55. Find equations of the normal and oscu curve of intersection of the parabolic a z = x² at the point (1, 1, 1). the angle between the unit tangent ve the same. Then show that the same re unit normal and binormal vectors. part 58. The rectifying plane of a curve at a p- contains the vectors T and B at that p odifying plane of the curve r(t) = sin ti the point (√2/2, √√2/2, 1). 59. Show that the curvature K is related t normal vectors by the equation dT ds = KN 60. Show that the curvature of a plane cu where is the angle between T and of inclination of the tangent line. (TH definition of curvature is consistent plane curves given in Exercise 10.2.e
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