Math22lO Midterm 2(11.9.12.1-12.8)
Spring, 2015
uid number:
Instructor: Kelly MacArthur
Special number:
(J
Instructions:
• Please show all of your work as partial cred
it will be given where appropriate, and there
may be no credit given for problems where there
is no work shown.
•
All answers should be completely simplified
, unless otherwise stated.
•
There are no calculators or any sort of electronic
s allowed on this exam. Make sure all
cell phones are put away and out of sight. If
you have a cell phone out at any point, for
any reason, you will receive a zero on this exam
.
•
You will be given an opportunity to ask clarifying
questions about the instructions at
exactly 8:40 a.m. (for a couple minutes). The
questions will be answered for the entire
class. After that, no further questions will be allow
ed, for any reason.
•
You must show us your U of U student ID card
when finished with the exam.
•
The exam key will be posted on Canvas by noon
.
•
Somewhere on this exam, you may need to
know
o
o
1/8=0.125
D(arctanx)=
1
1+x
2
•
You may ask for scratch paper. You may use
NO other scratch paper. Please transfer all
finished work onto the proper page in the test
for us to grade there. We will j grade
the work on the scratch page.
•
You are allowed to use one 8.5x1 1 inch piece
of paper with notes for your reference
during the exam.
(This exam totals 100 points, not including the
extra credit problem.)
STUDENT—PLEASE DO NOT WRITE BELOW THIS
LINE. THIS TABLE IS TO BE USED FOR GRADIN
G.
j--
1 &2
3
4&5
+——1
8
9
EC
Total Percentage:
I
1. (12 pis) Find the directional derivative of f(x, y, 3
+4xyz
y
2
z)=x
—
yz at
p=(3,0, I) in the direction of a=i—2j+2k
0—0> Of-I —
f (3o )
.
4JZL2_
.,V
Lz)+
(3i
5(ztl) +
Lc)
22
_
2:2-f
Answer:
2. (10 pts) Use the total differential
from P(-4, 0.25) to Q(-4.O1, 0.26) for
dz to approximate the change in z as (x, y) moves
z= arctan(xy)
-O2q
)[
-)
Answer:
2
0
Cr ol)
—g,o--iic
>
L
CD
D
I
tJ
+
-L
+
--j
+
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4
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C
6. (10 pts) Given
questions.
2
w(x,y)
—
3
xy-i-2y
=2x
x
=
3
u—v
,
y=uv
—
LA)
8w
(b) Find
when
u=—3
,
v=—2
.
_L1)-—I
-3L—J’
.
f2
i_
-
Answer:
7. (10 pts) Find the limit, if it exists. (Show all your reasoning.)
3x—4y
urn
(x,y)—.(O,O) 2x+y
x o;
j2
2
-p:
‘1)
1X+
Answer :
5
,
answer these
8w
that you need to use for this problem.
8u
(a) Write out the chain rule for
x
.
>
4
09
-
CD
D
4i
—
CD
CD
C)
0
Lfl
0
-4-
CD
-4-
a
C-)
0
0
-‘
0
Q5
D
cA
-
<
CD
0
cic
0
-,
0—
(p
-
(2
r
ii
cc
f-c
D
0
D
D
-40
0
C)
0
0
ci
D
0
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C
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CD
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0
C-)
0
3
0
CDI
ci”
C)
0-I-
I!
CD
3
y)=x
9. (13 pts) Find all critical and saddle points for
2 2
+y
3x + 2y +4
f (x,
Determine whether each point is a minimum, maximum or saddle point.
=
X’Y
-O
1
(JV’0i
e)
.:=)
c
L°
z.
(-)
0
-i)
I
12(?)_[i2->0
C4-
C2-1)
(\fLY
‘
-cw).-(>2
0
H-3Lo)
2
’
3
L2)’’
Critical point(s) (Specify whether they’re mm, max or saddle.):
Note: Give FULL 3-d points.
(_Q-L )
(2)d)-(
7
—
s”AAQ.
Extra Credit: (5 pts) Answer each of these short answer questions.
1. Circle all of the following statements that are true. (lt
s possible more than one of these
t
statements is correct.)
Jhe vector <f(x, y) f
,(x, y)
1
z=f(x,y) atthe point(x,y).
,
,
—1> is perpendicular to the surlace
(b) The vector <f(x, y) f(x, y)> is perpendicular to the surface
z—f(x, y) at the point (x, y).
,
ljhe vector <f(x,y), f(x, y)> is perpendicular to the level curve of the
ace z—f(x, y) at the point (x, y).
2. Which of the foVowing two statements is true?
(circle one)
(a) 1ff is continuous at (x, y), then it’s differentiable there.
f is differentiable at (x, y), then it’s continuous there.
3. If
Iinif(y,y)L then
y—O
urn
f(x,y)=—L
(x,y)—’(O,O)
True
4. The level curves for
or
(circle one)
zg25
2
-y
—2x are circles.
True
or
ccircie one)
If your answer is false, then state what shape the level curves ore:
a
2±?
8
2-SE