Name:
Key
Math 251-01 Exam 2
Spring 2025
Monday, March 31st, 2025
[2 points] Put your name on the test above, put a check next to your discussion section in the following chart, and read and sign the honor pledge
below:
Ahmet Aydin 9am (Section 02)
Fadlulah Omitogun 9am (Section 03)
Ahmet Aydin 8am (Section 04)
Student Academic Conduct Policy: As per university policy, by enrolling in this course, each student assumes the responsibilities of an
active participant in UMBC’s scholarly community in which everyone’s academic work and behavior are held to the highest standards of
honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong.
Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. To read the full
student academic conduct policy, consult the UMBC student handbook, the faculty handbook, or the UMBC policies section of the UMBC
directory.
I agree not to engage in academic misconduct. I further agree not to tolerate misconduct among other students.
Signature:
Question Possible Score
points
0
2
1
8
2
8
3
8
4
8
5
8
6
8
7
16
8
14
9
14
10
16
TOTAL
100
Instructions:
• Read each problem carefully.
• Mark your answers clearly.
• Show all your work on these sheets.
• We will not accept answers without
justification.
• This exam has 5 pages, and 10 questions. Please make sure that all pages
are included.
• You may not use books, notes or calculators.
• You have 50 minutes to complete this
exam.
Good luck!
Math 251-01 Exam 2
Spring 2025
Page 2 of 5
of 24
27 2 97
2xy + 3y3 at the point
Question 1. [8 points] Find the directional derivative of f (x, y) = 2x2
(2, 1) in the direction of the vector h4, 3i.
Due
(a) 29
29
(b) p
5
(c) 5
(d) 25
(e) Some other answer.
112 D410,5
I
s s
Question 2. [8 points] Use an appropriate linear approximation to estimate the value of
⇣
⌘2
(3.02)2 3(0.97) . Simplify your answer.
Bas
j.ge
zzfis
(a) 35.88 (b) 36.36 (c) 37.44 (d) 38.52 (e) Some other answer.
f 31
xis
36
fx3,11 x 3 fy 3 1
72 x 3
L 3.02 0.97
1
36 y 1
36 72 3.02 3 360.97 1
36 72 02 36 03 36 1.44 108 38.52
Question 3. [8 points] Use implicit di↵erentiation to find
2y2 z
3xy2 = 3 at the point ( 1, 1, 2).
Find
(a) 7
1
(b)
7
36
xeniz 3 5
1
(c)
7
(d) 7
(e) Some other answer.
@z
for the surface given by xz2 +
@y
E
7
s
Math 251-01 Exam 2
Spring 2025
Page 3 of 5
Question 4. [8 points] Express the equation y = 3 in polar coordinates.
3csc
(a) r = 3 sec ✓
(b) r = 3 cos ✓
(c) r = 3 csc ✓
(d) r = 3 sin ✓
(e) Some other answer.
Question 5. [8 points] Let f (x, y) = x3 3xy + y3 . f has a critical point at (1, 1). What does
the second derivative test say about f at this critical point?
3
D fxxfy fx
36xy
11111
(a) f has a local maximum at (0, 0).
(b) f has a local minimum at (0, 0).
(c) f has a saddle point at (0, 0).
(d) The second derivative test gives no conclusion.
Question 6. [8 points] Which of the following correctly gives the bounds for
where D is the triangular region with corners (0, 0), (2, 2), and (4, 0)?
(a)
Z 2Z 4
(c)
Z 2Z 4 x
0
0
0
x
(x + 2y) dxdy
(x + 2y) dxdy
(e) None of the other answers.
(b)
Z 2Z 4 y
0
y
(x + 2y) dxdy
!
D
(x + 2y) dA,
fiffiii
(d) More than one of the previous answers.
3
1953770
Math 251-01 Exam 2
Spring 2025
Page 4 of 5
Question 7. [16 points] Find the absolute maximum and minimum of f (x, y) = x2 +y2 subject
to the constraint (x 1)2 + y2 = 1.
If
2x 2y
2
x 2
glx
1
21 12 7
Tg
Of
77g
2x 7 2 2
iii
iii
2,0
bit pts 10,0
iii
"
Question 8. [14 points] Set up and simplify
is the region contained between the curves y = x
D
xy2 dA in rectangular coordinates, where D
1 and y = x2
x 1 x 4
ririii.im
F
Fifi
y x2 4
1
3
4
4x + 3.
Math 251-01 Exam 2
Spring 2025
"
Question 9. [14 points] Set up and simplify
Page 5 of 5
x
dA in polar coordinates, where D
x2 + y2
is the region above the y-axis, below the line y = x, and below the curve x2 + y2 = 2.
6
D
p
fit
nerds
firmore
Question 10. [16 points] Evaluate the integral
of integration.
Z 2Z 1
0
x/2
cos(x2 ) dy dx by changing the order
My
viridian
j
as 2 dy
2yus y ldy
4
tianan
sinu
sin
sino
2
du 2ydy