MATH 32A 2018 Final Solutions
MATH 32a Final Exam December 12, 2018
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.
~ and ~u, ~v ⇥ (~u ⇥ w)
(1) For any three vectors ~v , w, ~ = (~v ⇥ ~u) ⇥ w.
~ True False
(2) For any two vectors ~v and ~u, ~v ⇥ ~u = ~u ⇥ ~v .
OO
True False
(3) A sum of two or more continuous functions is continuous.
O
True False
(4) For any two vectors ~u and ~v , True
O
False
||~u ⇥ ~v || = ||~u|| · ||~v || · cos ✓,
where ✓ is the angle between ~u and ~v .
(5) For any function f (x, y), if for all m, lim f (x, mx) = 0, then True
x!0 O
False
lim f (x, y) = 0.
(x,y)!(0,0)
(6) Any unit vector ~u can be written as ~u = hcos ✓, sin ✓i for
some ✓
O
True False
(7) The cross product of two unit vectors is always unit vector. True
O
False
(8)The set of points {(x, y) | 0 < x2 + y 2 4} is bounded.
O
True False
~ (0, 0) = h1, 1i, then there is exactly one direction ~u for
(9) If rf
which D~u f (0, 0) = 1.
True O
False
(10) A continuous function on a closed and bounded region has True False
an absolute maximum and minimum.
(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 True False
two distinct tangent directions, then the point is a critical point
1
, 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.
It Htt itiniiitai p
2 2 2
3. (10 points) Find the plane containing the line ~r(t) = h1 + 2t, 4 + t, 3 + ti and the point
(4, 1, 9).
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
l O or
X y 4 2 3
2
17 x i 2l 4 13 2t3
y
MATH 32a Final Exam December 12, 2018
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.
~ and ~u, ~v ⇥ (~u ⇥ w)
(1) For any three vectors ~v , w, ~ = (~v ⇥ ~u) ⇥ w.
~ True False
(2) For any two vectors ~v and ~u, ~v ⇥ ~u = ~u ⇥ ~v .
OO
True False
(3) A sum of two or more continuous functions is continuous.
O
True False
(4) For any two vectors ~u and ~v , True
O
False
||~u ⇥ ~v || = ||~u|| · ||~v || · cos ✓,
where ✓ is the angle between ~u and ~v .
(5) For any function f (x, y), if for all m, lim f (x, mx) = 0, then True
x!0 O
False
lim f (x, y) = 0.
(x,y)!(0,0)
(6) Any unit vector ~u can be written as ~u = hcos ✓, sin ✓i for
some ✓
O
True False
(7) The cross product of two unit vectors is always unit vector. True
O
False
(8)The set of points {(x, y) | 0 < x2 + y 2 4} is bounded.
O
True False
~ (0, 0) = h1, 1i, then there is exactly one direction ~u for
(9) If rf
which D~u f (0, 0) = 1.
True O
False
(10) A continuous function on a closed and bounded region has True False
an absolute maximum and minimum.
(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 True False
two distinct tangent directions, then the point is a critical point
1
, 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.
It Htt itiniiitai p
2 2 2
3. (10 points) Find the plane containing the line ~r(t) = h1 + 2t, 4 + t, 3 + ti and the point
(4, 1, 9).
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
l O or
X y 4 2 3
2
17 x i 2l 4 13 2t3
y
