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A curve C in the plane is defined by the parametric equations: x=t². = x=f²+1, y=²-1 i. Find the length of C from t=0 to 1=2 ii. Find the curvature of C at t= 1

A curve C in the plane is defined by the parametric equations: x=t². = x=f²+1, y=²-1 i. Find the length of C from t=0 to 1=2 ii. Find the curvature of C at t= 1

Advanced Engineering Mathematics
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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A curve C in the plane is defined by the parametric equations: x=t² i. Find the length of C from t=0 to 1-2 ii. Find the curvature of C at t = 1 b. The vector function r(t)=sin 2ti-cos2tj+t√√5k determines a curve C in space. i. Find the unit tangent vector T and the principal unit normal N ii. Determine the curvature of C at time t x=f² +4y=2²=1 iii. Determine the tangential and normal component of the acceleration vector. 5. Let f(x,y)=xln(x/y) + xy² a. Calculate f, and f b. Determine the directional derivative off at the point (2, 2) in the direction of the vector a=i-2j af Ət c. Suppose that x = ste' and y=2se. Calculate d. Determine an equation for the tangent plane to the surface == f(x, y) at the point (2, 2, 8) on the surface. 6. Let F(x, y, z)=2xy² +2y=²+2x²z. a. Determine the maximum directional derivative of F at the point (1, -1, 1). b. Find the directional derivative off at the point (-2, 1, -1) in the direction parallel to the line x=34t, y=2-t, z=3t. c. Determine symmetric equations for the normal line to the level surface F(x, y, z)=-2 at the points (-1, 2, 1). d. Suppose the x=²+1, y=2t, z=r. Calculate dF dt
Transcribed Image Text:A curve C in the plane is defined by the parametric equations: x=t² i. Find the length of C from t=0 to 1-2 ii. Find the curvature of C at t = 1 b. The vector function r(t)=sin 2ti-cos2tj+t√√5k determines a curve C in space. i. Find the unit tangent vector T and the principal unit normal N ii. Determine the curvature of C at time t x=f² +4y=2²=1 iii. Determine the tangential and normal component of the acceleration vector. 5. Let f(x,y)=xln(x/y) + xy² a. Calculate f, and f b. Determine the directional derivative off at the point (2, 2) in the direction of the vector a=i-2j af Ət c. Suppose that x = ste' and y=2se. Calculate d. Determine an equation for the tangent plane to the surface == f(x, y) at the point (2, 2, 8) on the surface. 6. Let F(x, y, z)=2xy² +2y=²+2x²z. a. Determine the maximum directional derivative of F at the point (1, -1, 1). b. Find the directional derivative off at the point (-2, 1, -1) in the direction parallel to the line x=34t, y=2-t, z=3t. c. Determine symmetric equations for the normal line to the level surface F(x, y, z)=-2 at the points (-1, 2, 1). d. Suppose the x=²+1, y=2t, z=r. Calculate dF dt
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