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 18 02  Multivariable Calculus
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MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Practice Exam 1 A z Problem 1 15 points A unit cube lies in the rst octant with a vertex at the origin see gure O a Express the vectors OQ a diagonal of the cube and OR joining O to the center of a face in terms of k x b Find the cosine of the angle between OQ and OR Q R y Problem 2 10 points Compute 3 cos t 3 sin t t k The motion of a point P is given by the position vector R the velocity and the speed of P Problem 3 15 points 10 5 1 3 2 1 a b 1 1 5 nd a and b a Let A 2 0 1 then det A 2 and A 1 2 2 1 1 0 2 2 6 x 1 b Solve the system A X B where X y and B 2 z 1 c In the matrix A replace the entry 2 in the upper right corner by c Find a value of c for which the resulting matrix M is not invertible For this value of c the system M X 0 has other solutions than the obvious one X 0 nd such a solution by using vector operations Hint call U V and W the three rows of M and observe that M X 0 if and only if X is orthogonal to the vectors U V and W Problem 4 15 points The top extremity of a ladder of length L rests against a vertical wall while the bottom is being pulled away Find parametric equations for the midpoint P of the ladder using as parameter the angle between the ladder and the horizontal ground P Problem 5 25 points 10 5 10 a Find the area of the space triangle with vertices P0 2 1 0 P1 1 0 1 P2 2 1 1 b Find the equation of the plane containing the three points P0 P1 P2 1 1 1 and passing c Find the intersection of this plane with the line parallel to the vector V through the point S 1 0 0 Problem 6 20 points 5 5 10 be the position vector of a path Give a simple intrinsic formula x t y t z t k a Let R d for R R in vector notation not using coordinates dt has constant length then R and V are perpendicular b Show that if R be the acceleration still assuming that R has constant length and using
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