Find the curvature of the curve y = x at the point (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
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Question
13,19
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:
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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