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Database System Concepts
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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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).
Transcribed Image Text: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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