13 the arc of the curve with equations x = 1², y = t³, z = t^,0≤t≤ 3.8. Find the length of the curve r(t) = (213/2, cos 2t, sin 2t),0≤t≤ 1.9. The helix r(t) = cos ti+ sin tj + tk intersects the curver₂(t) = (1 + t)i + t2j + t³k at the point (1, 0, 0). Find theangle of intersection of these curves.10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost kwith 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,(c) the unit binormal vector, and(d) the curvature.12. Find the curvature of the ellipse x = 3 cos t, y = 4 sin t atthe points (3, 0) and (0, 4).13. Find the curvature of the curve y = x4 at the point (1, 1).14. Find an equation of the osculating circle of the curvey = x - x² at the origin. Graph both the curve and itsosculating circle.15. Find an equation of the osculating plane of the curvex = sin 2t, y = t, z = cos 2t at the point (0, 7, 1).

In this question we have to solve for finding the curvature of the curve y=x⁴ at point (1,1) Let's…

Engineering

Computer Science

13

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

the arc of the curve with equations x = 1², y = t³, z = t^,
0≤t≤ 3.
8. Find the length of the curve r(t) = (213/2, cos 2t, sin 2t),
0≤t≤ 1.
9. The helix r(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,
(c) the unit binormal vector, and
(d) the curvature.
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 = x4 at the point (1, 1).
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, y = t, z = cos 2t at the point (0, 7, 1).

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