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
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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)