A Very Old Final Exam 1 a Find the parametric equations of the line through 1 2 0 parallel to the intersection of the planes x 2y 3z 6 and x 6y 2 S r t 1 2 0 t 18 3 4 works There are many other parametrizations of this line of course b For what choice of c will the normal to the plane 4x 2y cz 2 be perpendicular to the line x y z c2 3t 4t 5 t S c 20 2 For the curve traced by x t y t z t 3 sin at 3 cos at 2at where a is a constant find the velocity vector the acceleration vector the tangential component of acceleration and the curvature 2 r 3a cos at 3a sin at 2a ddt2 r 3a2 sin at 3a2 cos at 0 d dt d2 r dt2 0 so the tangential component of acceleration is zero and 3 13 Notice that the curvature does not depend on a That is a useful check curvature does not depend on how fast you trace the curve S d r dt 3 In this problem f x y x y x2 y 2 when x y 0 0 and f 0 0 is not defined Sketch the level curves f 2 f 0 and f 2 What can you say about lim x y 0 0 f x y S f 2 is a circle centered at 1 4 1 4 f 2 is a circle centered at 1 4 1 4 and f 0 is just the line y x Both circles and the line go through the origin 0 0 though the origin is not on any of the level sets because f is not defined there Since you can approach 0 0 along any one of the level sets and f takes different values on them lim x y 0 0 f x y does not exist 4 a Find the directional derivative of f x y 3x2 4y 2 at 2 1 in the direction of a tangent vector to the level curve of g x y x2 y 2 through that point S 4 5 b Find the equation of the tangent plane to the surface defined by z 4 xy y 4 1 at 3 1 1 S x y 4z 2 5 Find and classify the critical points of the function f x y x4 12xy 2y 2 S Saddle at 0 0 local minima at 3 9 and 3 9 2 r d r 6 Suppose ddt2 r t e2t 2et and d dt 0 0 2 What is dt t and what is the r 2t t length of the curve traced by r t as t goes from 1 to 1 S d dt 1 2e 2e and the length of the curve is 2 12 e2 e 2 7 Find the point x y in the plane such that the sum of the squares of the distances from x y to 0 0 0 1 1 1 and 2 0 is a minimum S x y 3 4 1 2 8 On a steep hillside a railroad track is laid so the rate of climb rise run is 1 10 2 Suppose that the hillside is the graph of f x y and fx x0 y0 2 and fy x0 y0 0 What are the possible directions of the track at x0 y0 S 1 20 399 20 9 Suppose that the curve t y t z t lies in the intersection of the surfaces 4 1 2 2 3 7 F x y z 6 and G x y z 10 If y 4 1 z 4 1 F and G 4 1 2 3 1 1 find y 4 and z 4 S y 4 19 4 z 4 7 4 10 Use a Lagrange multiplier to find the level surface of F x y z 2x2 2xy z 2 which is tangent to the plane x y z 3 S The level surface F x y z 9 is tangent to the plane x y z 3 at 3 3 3
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