Math 211, Section 03, Final Exam, 2014

Math 211, Section 03, Fall 2014 Final Exam, Monday, December 15, 2014 Instructions: Do all twelve numbered problems. If you wish, you may also attempt the three optional bonus questions. Show all work, including scratch work. Little or no credit may be awarded, even when your answer is correct, if you fail to follow instructions for a problem or fail to justify your answer. If your answer for a given problem is a sum of fractions with different denominators, you may leave it that way. Otherwise, simplify your answers whenever possible. If you have time, check your answers. WRITE LEGIBLY. NO CALCULATORS. 1. (12 points) A particle is travelling in such a way that its velocity vector at time t is √ 0 2 given by ~r (t) = ht , 2t, 1i. (1a) How far does the particle travel from time t = 1 to time t = 2? (1b) What is the curvature of the path the particle traces out at the point it passes through at time t = 2?  3 3  2x + y if (x, y) 6= (0, 0) 2. (18 points) Let f (x, y) = 2x2 + y 2  0 if (x, y) = (0, 0). (2a) Compute fx (0, 0) and fy (0, 0). (2b) Compute D~u f (0, 0), where ~u is the unit vector pointing in the direction of h1, −1i. (2c) Based on your answers to parts (a) and (b), explain (briefly) why f cannot be differentiable at (0, 0). 3. (18 points) Find and classify (as local minimum, local maximum, or saddle point) every critical point of the function f (x, y) = x2 y − 3x2 − 6y 2 + 2. 4. (12 points) Find the point on the ellipse x2 +6y 2 +3xy = 40 with the largest x-coordinate. 5. (20 points) Find the volume of the region that is inside the sphere x2 + y 2 + z 2 = 9 and also inside the cylinder x2 + y 2 = 3x. (Note that the cylinder is centered around a vertical line that is not the z-axis.) 6. (20 points) Let E be the solid lying • inside the sphere x2 + y 2 + z 2 = 9, 2 • outside the sphere xp + y 2 + z 2 = 1, • below the cone z = x2 + y 2 , and • inZthe Z Z first octant. Compute y dV . E 7. (20 points) Let E be the solid bounded by the surfaces y = x + z = 4. Compute the volume of E. √ x, x = 2y, z = 4, and 2 8. (15 points) Let C be the quarter of the circle x2 + Z y = 9 in the second quadrant, i.e., the quarter-circle arc from (0, 3) to (−3, 0). Compute x2 y ds. C 9. (15 points) Let C be the boundary of the triangle with vertices (0,Z0), (1, 0), and (1, 2), F~ · d~r. oriented counterclockwise. Let F~ (x, y) = h3y 2 , x2 y + cos8 yi. Compute C 10. (15 points) Let C be the line segment from (1, 0, −1) to (0, −2, 2). Z Compute 2y dx − 3xy dy + z 2 dz. C 11. (15 points) Let F~ (x, y) = h2xy + 6x2 , x2 − y 3 i. (a) Show that F~ is conservative by finding a potential function f (x, y) for F~ . 2 (b) Let C be the Z curve parametrized by ~r(t) = ht(t − 2), t (t − 3)i, for 1 ≤ t ≤ 3. Compute F~ · d~r. C 12. (20 points) Let F~ (x, y, z) = hy 2 , z, xi, and let S be the part of the paraboloid 2 z = 3x2 +Z3y Z in the first octant and below the plane z = 3, oriented downward. ~ Compute F~ · dS. S OPTIONAL BONUS A. (2 points) Let C be the circle (y − 2)2 + z 2 = 1 in the yz-plane, and let S be the (surface of the) torus formed by rotating C around the z-axis. Compute the surface area of S. OPTIONAL BONUS B. (2 points) Let C be the portion of the circle x2 + y 2 = 1 in the first quadrant, oriented counterclockwise, i.e., running from Z (1, 0) to (0, 1). 3 2 2 2 2 Let F~ (x, y) = h2xy cos(x ), x + 3y sin(x )i. Compute F~ · d~r. C OPTIONAL BONUS C. (1 point) On December 6, 2014, there was a special runoff election to decide a Senate race for a certain US State. Which one of the 50 states was it?