55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and z = x² at the point (1, 1, 1).
55. Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y² and z = x² at the point (1, 1, 1).
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
55
![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 = t4 is the
normal plane parallel to the plane 6x + 6y - 8z = 1?
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
curve of intersection of the parabolic cylinders x = y² and
z = x² at the point (1, 1, 1).
DA
boley. 30502
56. Show that the osculating plane at every point on the curve
r(t) = (t + 2, 1 − t, 1²) is the same plane. What can you
conclude about the curve?
57. Show that at every point on the curve
ad
r(t) = (e¹ cos t, e' sin t, e¹)
Smoqquz
the angle between the unit tangent vector and the z-axis is
the same. Then show that the same result holds true for the
unit normal and binormal vectors.
58. The rectifying plane of a curve at a point is the plane that
contains the vectors T and B at that point. Find the recti-
fying plane of the curve r(t) = sin ti + cos tj + tan t k at
the point (√√2/2,√√2/2, 1).
59. Show that the curvature K is related to the tangent and
normal vectors by the equation
dT
KN
ramudit](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91bc99aa-940f-4bf1-9bff-5f4d70039c28%2F7821a435-f5b8-4309-ace8-4bafa6029a1e%2F14yi99v_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 = t4 is the
normal plane parallel to the plane 6x + 6y - 8z = 1?
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
curve of intersection of the parabolic cylinders x = y² and
z = x² at the point (1, 1, 1).
DA
boley. 30502
56. Show that the osculating plane at every point on the curve
r(t) = (t + 2, 1 − t, 1²) is the same plane. What can you
conclude about the curve?
57. Show that at every point on the curve
ad
r(t) = (e¹ cos t, e' sin t, e¹)
Smoqquz
the angle between the unit tangent vector and the z-axis is
the same. Then show that the same result holds true for the
unit normal and binormal vectors.
58. The rectifying plane of a curve at a point is the plane that
contains the vectors T and B at that point. Find the recti-
fying plane of the curve r(t) = sin ti + cos tj + tan t k at
the point (√√2/2,√√2/2, 1).
59. Show that the curvature K is related to the tangent and
normal vectors by the equation
dT
KN
ramudit
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