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Midterm 1: Practice Exam Math 2210-003 Fall 2015 Warnings: (1) Please bring your student ID with you during midterm exams. (2) No references or calculators can be used for midterm exams. Practice Exam Z 3 xydx along the curve x = t − 1, y = t + 1. (1) Calculate 1 (2) Find the arc length of this curve: x = t3/2 , y = t3/2 , z = t; 2 ≤ t ≤ 4 (3) Two vectors are given: u =< 1, −3, 4 > and v =< −2, 1, −1 > (a) Calculate the angle between the two vectors; (b) The projection of u onto v and that of v onto u; (c) Suppose u is orthogonal to v +cw, where w =< 0, 0, 1 >. Determine the scalar c. (4) Find the plane through (2, 4, 3) and is parallel to: (a) xy plane; (b) the plane x − y + 6z = 213 (5) Find the equation of the plane through (1, 3, −2) that is perpendicular to both x − 4y + 2z = 0 and 2x − 3y + z = −1. (6) Find the distance: (a) from the point (1, 0, −3) to the plane 3x − y + z = 4 (b) between two parallel planes x − 2y + z = 3 and x − 2y + z = 5 (7) The position vector: µµZ r(t) = t ¶ µZ t ¶ ¶ 2/3 e dx , sin πθdθ , t x 1 π is given. Calculate the following quantities at t = t1 = 1: (a) The velocity v(t); (b) The acceleration a(t); (c) The speed s. (8) Find the symmetric equation of the line of intersection of the given pair of planes: 4x + 3y − 7z = 1 and 10x + 6y − 5z = 10. (9) Find the equation of the line through (5, −7, 2) and perpendicular to both < 2, 1, −3 > and < 5, 4, −1 >. (10) Find the equations of the line through (2, 4, −5) that is parallel to the plane 3x + y − 2z = 5 and perpendicular to the line x+8 y−5 z−1 = = 2 3 −1 (11) Find the equation of the plane containing the line x = 3t, y = 1 + t, z = 2t and parallel to the intersection of the planes 2x − y + z = 0 and y+z+1=0 (12) (a) Turn ρ = 2 cos φ into Cylindrical coordinates; (b) Turn r2 cos 2θ = z into Cartesian coordinates. (13) Prove that this limit doesn’t exist. x2 − 6y (x,y)→(0,0) 3x2 − y lim (14) A function w = f (x, y, z) = x2 + sin (xyz) + z+1 is given. x2 + 1 (a) In what direction does the function w increases most rapidly at any point (x, y, z). (b) Calculate the directional derivative in the direction of (0, −1, 0).