math 1720 final
pdf
keyboard_arrow_up
School
University of Windsor *
*We aren’t endorsed by this school
Course
1720
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
10
Uploaded by UltraSeaLionMaster403
UNIVERSITY OF WINDSOR
DEPARTMENT OF MATHEMATICS AND STATISTICS
Di
ff
erential Calculus MATH 1720/1760
Final Exam
Wednesday, December 15, 2021, 12:00 pm - 2:30 pm
Last name
:
First name
:
Student No.
:
Instructions
:
Download this exam to avoid any problem with internet while you are solving it.
You can print this exam and write your solutions on it, or you can work on your own paper
and put just the question number next to your solutions. In both cases make sure you have
written your name and ID on your papers. Try to keep the questions in the same order when
you scan your solutions.
This exam has 10 questions. You have 150 minutes (2 hours and 30 minutes).
Read carefully and answer
all
questions.
Show all your work to receive full credit
.
Give
exact answers
not the decimal approximations unless a question asks to give answer
in decimal approximations.
Only non-programmable and non-graphic calculators are allowed.
You must stop writing at 2:30 pm and must upload a single pdf file (not a Zip
file) with your solved exam by 3:00 pm on Blackboard
. Make sure the pdf is clear
and your solutions can be seen clearly to avoid any confusion during marking. If your exam
cannot be read, it will not be marked. Better to use pen or dark pencil if possible.
You can
submit your exam solutions once
(you can save draft and check that it is fine before you
submit). Make sure to check your submission is submitted and not in progress (you should
see a yellow icon in My Grades if it is submitted). You are responsible for your submission,
this means
it is your responsibility to make sure your submission is on time and
accurate
(the file is correct and includes all pages, pages on email will not be accepted).
Late
submissions will not be accepted
.
Code of Conduct:
This open book examination must be written without collaboration.
While writing the exam you must not communicate with any other person and you must not
consult the internet or any material other than the course textbook, lectures, your course
notes, assignments, and any material posted on the course Blackboard pages. If there is any
evidence of academic misconduct, we will file an allegation of misconduct according to Bylaw
31.
All rights reserved. This assessment is the intellectual property of the author and may not be repro-
duced in any manner whatsoever without the express written permission of the author.
Downloaded by ze md (mmhm7548@gmail.com)
2
1.
(17)
Evaluate the following limits:
(a) lim
x
→
0
x
+ tan
x
3
x
+ 4
x
2
(b) lim
x
→
7
-
x
2
-
5
x
-
14
x
2
-
49
(Do not use L’Hospital rule in part(b))
(c) lim
x
→∞
e
-
3 sin
x
x
2
(d) lim
x
→
1
+
sin(
π
x
) ln (
x
-
1)
Downloaded by ze md (mmhm7548@gmail.com)
3
2.
(5)
For what value(s) of the constant
c
is the function
f
continuous on (
-∞
,
∞
)
f
(
x
) =
c
2
-
(
x
+ 1)
c,
if
x >
1
(2
x
+ 1)
c
-
x
2
-
5
,
if
x
≤
1
3.
(6)
Use the definition of derivative to find the derivative of
f
(
x
) =
√
3
x
-
7
Downloaded by ze md (mmhm7548@gmail.com)
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
4
4.
(9)
Consider the equation
e
x
+ 2
x
= 0
(a) Use the Intermediate Value Theorem to show that the equation has at least one real root.
(b) Use Rolle’s theorem to show that the equation has at most one real root.
Downloaded by ze md (mmhm7548@gmail.com)
5
5.
(15)
For each of the following, find
y
. You do not need to simplify your answer.
(a)
y
=
2
x
+
√
x
6
x
2
+ sin
-
1
(3
x
)
(b) sec(
y
) + 5
x
3
y
2
= 9
y
(c)
y
= (2
x
+ 1)
cos
x
Downloaded by ze md (mmhm7548@gmail.com)
6
6.
(5)
Use linear approximation to estimate
3
√
27
.
05. (round answer to 5 decimal places)
7.
(7)
A television camera is positioned 8 km from a helicopter landing pad. The helicopter is rising
vertically with a speed of 2 km/h. If the television camera is always kept aimed directly at the
helicopter, how fast should the camera’s angle of elevation be changing when the helicopter is
at 3 km altitude? (The angle of elevation is the angle between the ground and the camera’s
line of sight.)
Downloaded by ze md (mmhm7548@gmail.com)
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
7
8.
(9)
Find the largest possible area of a rectangle inscribed in the region bounded by the half-
hyperbola
y
2
-
x
2
= 4,
y >
0, and the line
y
= 8, where one side of the rectangle lies on the
line
y
= 8 and two vertices of the rectangle are on the half-hyperbola
y
2
-
x
2
= 4 (see one of
the possible inscribed rectangles in the picture below). (round answer to 2 decimal places)
x
y
0
y
2
-
x
2
= 4,
y >
0
y
= 8
Downloaded by ze md (mmhm7548@gmail.com)
8
9.
(19)
Let
f
(
x
) =
x
2
e
-
x
(a) (1 Mark) State the domain of
f
.
(b) (2 Marks) Find the
x
and
y
intercepts (if any).
(c) (5 Marks) Find the intervals of increase/decrease.
(d) (3 Marks) Determine the points at which
f
has a local maximum/local minimum(if any).
Downloaded by ze md (mmhm7548@gmail.com)
9
(e) (6 Marks) Find the intervals of concavity.
(f) (2 Marks) Determine the inflection point(s) (if any). (round answer to 3 decimal places)
Downloaded by ze md (mmhm7548@gmail.com)
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
10
10.
(8)
Let
A
be the area of the region that lies under the graph of
f
(
x
) =
x
3
+ 4
x
between
x
= 0
and
x
= 2. Find expression for
A
as a limit using right endpoints. Evaluate the limit
without
using the Fundamental Theorem of Calculus. (No marks will be given for finding the area by
any other method)
Downloaded by ze md (mmhm7548@gmail.com)