Exam (elaborations) MATH 32A 2018 Final Solutions

Name: ID: Signature: To get credit for a problem, you must show all of your reasoning and calculations. No calculators may be used. Box your final answer. Please write only on the front. You may use left-over space on a page as extra space: clearly label your work with the problem number. If you cannot find a vector that you need for a later part of a problem, you may use the vector h1, 2, 3i. Circle your section: Section: Tuesday: Thursday: TA: 2A 2B Frederick Vu 2C 2D Nicholas Boschert 2E 2F Victoria Kala Problem Possible Points Problem Possible Points 1 24 9 15 2 10 10 20 3 10 11 15 4 20 12 10 5 10 13 15 6 10 14 15 7 20 15 10 8 10 16 25 Total 249 MATH 32a Final Exam December 12, 2018 1. (2 points each) True/False! Circle the appropriate answer. No justification is needed here. (1) For any three vectors ~v, ~w, and ~u, ~v⇥(~u⇥~w) = (~v⇥~u)⇥~w. True False (2) For any two vectors ~v and ~u, ~v ⇥ ~u = −~u ⇥~v. True False (3) A sum of two or more continuous functions is continuous. True False (4) For any two vectors ~u and ~v, ||~u ⇥~v|| = ||~u|| · ||~v|| · cos , where is the angle between ~u and ~v. True False (5) For any function f(x, y), if for all m, lim x!0 f(x,mx) = 0, then lim (x,y)!(0,0) f(x, y) = 0. True False (6) Any unit vector ~u can be written as ~u = hcos , sin i for some True False (7) The cross product of two unit vectors is always unit vector. True False (8)The set of points {(x, y) | 0 x2 + y2  4} is bounded. True False (9) If ~rf(0, 0) = h1, 1i, then there is exactly one direction ~u for which D~uf(0, 0) = 1. True False (10) A continuous function on a closed and bounded region has an absolute maximum and minimum. True False (11) A set that is closed is also necessarily bounded. True False (12) If a level curve intersects itself at a point so that there are two distinct tangent directions, then the point is a critical point True False 1 OO O O O O O O O MATH 32a Final Exam December 12, 2018 2. (10 points) Find the volume of the parallelepiped spanned by h1, 1, 1i, h−1, 2,−1i, and h−1, 0, 1i. 3. (10 points) Find the plane containing the line ~r(t) = h1+2t,4+t,−3+ti and the point (4,−1, 9). 2 It Htt itiniiitai p 2 2 2 F o 41,4 3 the nectar from this to p is 3 5,12 n 2,1 I x 3 5,12 It Is 12T 3J lot 315 5T 24J ME 21J 1315 X l y 4 2 3 O or 17 x i 2l y 4 13 2t3 MATH 32a Final Exam December 12, 2018 4. (20 points) Find @z @x and @z @y for (a) xy + xz + yz = 1 (b)