49-50 Find equations of the normal plane and osculating plane of the curve at the given point.f 49. x = sin 2t, y = −cos 2t, z = 4t; (0, 1,2m)
49-50 Find equations of the normal plane and osculating plane of the curve at the given point.f 49. x = sin 2t, y = −cos 2t, z = 4t; (0, 1,2m)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
49
![47-48 Find the vectors T, N, and at the given point.
47. r(t) = (t², 3t³, t), (1, 3, 1)
48. r(t) = (cos t, sin t, In cos t), (1,0,0)
49-50 Find equations of the normal plane and osculating plane
of the curve at the given point.
49. x = sin 2t, y = -cos 2t, z = 4t; (0, 1, 2π)
50. x = ln t, y = 2t, z = t²; (0, 2, 1)
51. Find equations of the osculating circles of the ellipse
9x² + 4y² = 36 at the points (2, 0) and (0, 3). Use & graph-
ing calculator or computer to graph the ellipse and bond
osculating circles on the same screen.
52. Find equations of the osculating circles of the parabola
y = x² at the points (0, 0) and (1,1). Graph both oscu-
lating circles and the parabola on the same screen.
53. At what point on the curve x = t³, y = 3t, z = t is the
normal plane parallel to the plane 6x + 6y - 8z = 1?
CAS 54. Is there a point on the curve in Exercise 53 where the
osculating plane is parallel to the plane x + y + z = 1?
[Note: You will need a CAS for differentiating, for simplify-
ing, and for computing a cross product.]
55. Find equations of the normal and osculating planes of the
y² and
curve of intersection of the parabolic cylinders x
z = x² at the point (1, 1, 1).
the ni moito M
56. Show that the osculating plane at every point on the curve
(₁+2 1-
1²) is the same plane. What can you
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Transcribed Image Text:47-48 Find the vectors T, N, and at the given point.
47. r(t) = (t², 3t³, t), (1, 3, 1)
48. r(t) = (cos t, sin t, In cos t), (1,0,0)
49-50 Find equations of the normal plane and osculating plane
of the curve at the given point.
49. x = sin 2t, y = -cos 2t, z = 4t; (0, 1, 2π)
50. x = ln t, y = 2t, z = t²; (0, 2, 1)
51. Find equations of the osculating circles of the ellipse
9x² + 4y² = 36 at the points (2, 0) and (0, 3). Use & graph-
ing calculator or computer to graph the ellipse and bond
osculating circles on the same screen.
52. Find equations of the osculating circles of the parabola
y = x² at the points (0, 0) and (1,1). Graph both oscu-
lating circles and the parabola on the same screen.
53. At what point on the curve x = t³, y = 3t, z = t is the
normal plane parallel to the plane 6x + 6y - 8z = 1?
CAS 54. Is there a point on the curve in Exercise 53 where the
osculating plane is parallel to the plane x + y + z = 1?
[Note: You will need a CAS for differentiating, for simplify-
ing, and for computing a cross product.]
55. Find equations of the normal and osculating planes of the
y² and
curve of intersection of the parabolic cylinders x
z = x² at the point (1, 1, 1).
the ni moito M
56. Show that the osculating plane at every point on the curve
(₁+2 1-
1²) is the same plane. What can you
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