Final_MAT1332_Fall_2021
.pdf
keyboard_arrow_up
School
University of Ottawa *
*We aren’t endorsed by this school
Course
1332
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
14
Uploaded by aleclafrance88
Universit´e d’Ottawa
University of Ottawa
1
Facult´e des sciences
Math´ematiques et statistique
Faculty of Science
Mathematics and Statistics
613–562–5864
613–562–5776
www.uOttawa.ca
STEM 336
Ottawa
ON K1N 6N5
Marker’s use only:
Question
Marks
1-5 (/14)
6 (/5)
7 (/5)
8 (/5)
9 (/10)
10 (/13)
Total (/52)
MAT1332 A: Instructor Catalin Rada
Final Examination
December , 2021
Duration:
3 hours for writting the exam
+ 15 minutes, the time to upload in Brightspace.
Family name:
First name:
Student number :
•
You have 3 hours to complete this exam. AFTER 3 hours HOURS STOP
WRITITNG AND SCAN, AND UPLOAD YOUR EXAM IN BRIGHTSPACE.
Please do not wait until the last minute to do this. When you are fi
nished, scan the pages into a single document and upload it in the ”Assign-
ments” tab on Brightspace (the same place you found this). You may use a
scanner or your phone or any other device.
•
SHOW ALL WORK. MARKS ARE INDICATED FOR EACH EXERCISE.
•
SHOW ALL YOUR WORK! We shall Mark Your Paper Only If It is In
Brightspace! In Assignments!
Very important:
WE ACCEPT ONLY BRIGHTSPACE SUBMISIONS. ONLY ONE FILE.
ONLY ONE SUBMISSION.
RESPECT THE DUE DATE, AND UPLOAD
IN TIME:
SIGNATURE
.................................................................................................
Universit´e d’Ottawa
University of Ottawa
2
1. (2 points) (a)
Find the equation of the tangent plane to the graph of
f
(
x, y
) =
e
sin(
x
)
+
e
x
+
e
y
+
x
+
y
+ 1 + sin(2
x
) + cos(6
x
) at the point: (0
,
0).
Solution:
(b)
Compute the linearization at (0
,
1) and then use it to approximate
g
(
-
0
.
1
,
1
.
1) if
g
(
x, y
) =
y
ln(
x
2
+
x
+ 1) +
e
xy
+ 3
x
+
y
+ 1.
Solution:
2. (2 points)
(a)
If
z
= 4 + 3
i
and
v
=
-
2
-
i
, then find
z
-
2
i
v
+ 1 + 2
i
. Express answer in
a
+
bi
form.
Solution:
(b)
If
z
= 2
⇥
cos(
-
⇡
12
) +
i
sin(
-
⇡
12
⇤
, find
z
6
. Express answer in
a
+
bi
form.
Solution:
Universit´e d’Ottawa
University of Ottawa
3
3. (2 points)
Consider the following integral:
Z
1
0
e
-
66
x
x dx.
(a)
Why is this integral improper?
Answer:
(b)
Find the value of this improper integral, if it converges.
Solution:
Universit´e d’Ottawa
University of Ottawa
4
4. (3 points)
Given that
y
(0) = 2
e
, solve:
dy
dx
=
e
2
x
ye
e
2
x
Solution:
5.
a) (
2 points
) Find and draw the domain of
f
(
x, y
) =
p
y
ln(
-
y
+
x
). Indicate if boundaries
are included or not.
Universit´e d’Ottawa
University of Ottawa
5
b) (
1 point
) Find the range of
g
(
x, y
) = 66
p
-
6
x
+ 5
y
+ 7
-
8
c) (
2 points
) On the graph below, sketch two level curves of the function
k
(
x, y
) =
p
-
3
x
+ 3
y
2
+ 3
.
Label each level curve with the height of the corresponding part of the graph
of
f
.
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help