MATH 261 EXAM I FALL 2015 NAME: CSU ID: Problem Points 1 12 2 18 3 18 4 16 5 18 6 18 Total 100 SECTION NUMBER: You may NOT use calculators or any references. Show work to receive full credit. GOOD LUCK !!! Score 1. (12 pts; 3 pts per part) Let u = h2, 1, 3i, v = h1, −1, −1i, w = h1, 0, −1i, z = h3, 4i. Compute each of the following or write “invalid operation.” (a) u + v (b) w · z (c) 7 + u z (d) |z| 2. (18 pts; 6 pts per part) Consider the line L given by L : r(t) = h3t + 1, t + 2, 2t − 1i and the two planes P1 and P2 given by P1 : x + y − z = 6 P2 : 2x − 2y + 2z = 2. (a) Does the line L intersect the plane P1 ? If so, find the point of intersection. If not, explain why. (b) Find the angle between P1 and P2 . You should leave your answer as the arccosine of some number. (c) Find the distance between the origin and the line L. 3. (18 pts; 6 pts per part) For parts (a) and (b), consider the two planes P3 and P4 given by P3 : x + y + z = 1 P4 : 3x + y − z = 5. (a) Find a vector parallel to the line of intersection of planes P3 and P4 . (b) Find a point on the line of intersection of planes P3 and P4 . (c) Given point (2, 4, 6) on the line L and vector h1, 3, 5i parallel to L, write parametric equations for the line. 4. (16 pts; 8 pts per part) Given a(t) = h12t2 + 2, 6t − cos(t), et i, v(0) = h1, 0, −5i, and r(0) = h3, −3, 0i, find the following: (a) Find v(t). (b) Find r(t). 5. (18 pts; 6 pts per part) The three parts of this problem are independent. In other words, these are three distinct problems that do not rely on each other. (a) Suppose r(t) = (3 cos t)i + 4tj + (3 sin t)k. Compute v(t) and T(t). 3 4 , 0, (b) Suppose that a = h2, 1, 1i and T = at some value of t. Compute 5 5 aT and aN at that value of t. π π π π (c) Find N , given that a = h2, 4, −2i, T = h1, 0, 0i, aT = 2, and 2 2 2 2 π = −2. aN 2 6. (18 pts; 9 pts per part) Evaluate the limit or show that the limit does not exist. Proper work must be shown. (a) (b) lim (x,y)→(0, π2 ) sin(y) cos(y) yex x2 y 2 (x,y)→(0,0) x4 + 3y 4 lim