MAT 1330 Practice Midterm 1
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Toronto Metropolitan University *
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Course
1330
Subject
Mathematics
Date
Feb 20, 2024
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Uploaded by DrBravery13210
Calculus I for the Life Sciences MAT 1330E
Midterm 1
October 2, 2019
Prof. Andrew Wagner
You must
sign below
to confirm that you have read, understand, and will follow these
instructions
:
•
This is a
75-minute
closed-book exam; no notes are allowed. You may use a calculator if it
has no differentiation, integration, or graphing capabilities. Otherwise, no electronics, no notes,
scrap paper, cell phones, smartwatches or related devices of any kind are permitted. All such
devices, including cell phones,
must be stored in your bag under your desk or at the
front or back of the room for the duration of the exam
.
•
The exam consists of 10 questions on 9 pages. If you need additional space for rough work, use
the backs of the pages.
Do not detach any of the pages.
•
Questions 1 through 6 are multiple choice, worth a total of 6 points.
Record your answers
to the multiple choice questions in the table in the middle of page 2.
•
Questions 7–10 are long answer, worth points as indicated.
You must show your work, your
work must be legible and well-justified, and you must record your answers in the
spaces provided.
LAST NAME:
First name:
Student Number:
Signature:
Please circle
the DGD section where you would like to have your test returned to you:
DGD 1
DGD 2
DGD 3
FRI 8:30 – 9:50
WED 10:00 – 11:20
WED 13:00 – 14:20
VNR 1095
VNR 3075
MRN 021
MAT 1330 Midterm 1
2
For grading purposes only:
Question
1–6
7
8
9
10
Total
Max
6
2
2
5
6
21
Marks
Please enter your multiple choice answers in the boxes below.
Question
1
2
3
4
5
6
Answer
MAT 1330 Midterm 1
3
1. (1 point)
What is the domain of the function
f
(
x
) =
ln(4
-
x
2
)
√
1
-
x
?
A.
all
x
B.
0
< x
≤
1
C.
-
1
< x <
1
D.
-
2
< x <
1
E.
-
2
< x <
2
F.
x <
-
2
Your answer:
2. (1 point)
Which of the following functions is the inverse of
f
(
x
) =
x
-
5
x
+ 2
?
A.
f
-
1
(
x
) =
x
+ 5
2
-
x
B.
f
-
1
(
x
) =
2
x
1
-
x
C.
f
-
1
(
x
) =
2
x
+ 5
1
-
x
D.
f
-
1
(
x
) =
5
x
+ 2
x
-
1
E.
f
-
1
(
x
) =
x
-
1
1
-
5
x
F.
None of the previous answers.
Your answer:
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MAT 1330 Midterm 1
4
3. (1 point)
Which of the following is equal to
ln(
√
x
2
+ 4
x
)
-
1
2
ln(
x
)
on its domain?
A.
ln(
√
x
+ 4)
B.
p
ln(
x
+ 4)
C.
ln(
x
+ 2)
D.
ln(2) + ln(
√
x
)
E.
ln(4) + ln(
x
)
F.
ln(
x
+ 4)
Your answer:
4. (1 point)
Which of the following expressions is equal to
(
x
12
+ 7)
1
/
3
x
2
for
x
6
= 0 ?
A.
x
10
+
7
x
2
B.
3
r
x
6
+
7
x
6
C.
3
r
x
4
+
7
x
8
D.
x
2
+
3
√
7
x
2
E.
x
7
+
7
x
5
F.
3
√
x
6
+ 7
Your answer:
MAT 1330 Midterm 1
5
5.
(1 point)
Given that
a
is a positive constant, what is the value of the following limit?
lim
x
→
a
x
2
-
a
2
√
x
-
√
a
A.
4
a
2
√
a
B.
∞
C.
0
D.
4
a
E.
2
√
a
F.
4
a
√
a
Your answer:
6. (1 point)
A study on a particular forest has tracked the forest’s area over several decades.
Over the course of this study, scientists observed that the forest’s area had a natural growth
rate of 2% each decade. Furthermore, an average of 30 km
2
of the forest was cut down each
decade. If
S
t
represents the forest’s area
t
decades after the study began (measured in km
2
),
which of the following DTDS models the dynamics of this forest?
A.
S
t
+1
= 0
.
98
S
t
-
30
B.
S
t
+1
= 1
.
02
S
t
+ 30
C.
S
t
+1
= 1
.
02
S
t
-
10
D.
S
t
+1
= 0
.
02
S
t
+ 30
E.
S
t
+1
= 1
.
02
S
t
-
30
F.
S
t
+1
= 0
.
98
S
t
+ 30
Your answer:
MAT 1330 Midterm 1
6
Long-answer questions:
You must show your work, your work must be legible
and well-justified, and you must record your answers in the spaces provided.
7. (2 points)
Evaluate the following limit, if it exists.
You must use algebraic methods
and justify your answer mathematically to earn credit for this question.
lim
x
→
5
x
2
-
9
x
+ 20
|
x
-
5
|
Your answer:
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MAT 1330 Midterm 1
7
8. (2 points)
Find all solutions of the following inequality.
Show your work and justify
your reasoning.
x
+ 5
x
+ 3
<
x
-
2
x
Your answer:
MAT 1330 Midterm 1
8
9.
(1+1+2+1=5 points)
The DTDS governing the growth of a population’s density is
given by
x
t
+1
= 0
.
85
x
t
+ 6
where
t
is in years, and
x
t
denotes the density (in number of individuals per km
2
) at time
t
.
In this model, a repopulation strategy has been implemented to ensure that the population’s
density increases by
c
= 6 individuals per km
2
each year.
The growth rate factor 0
.
85 was
determined following several years of study.
(a)
Find the fixed point
x
*
of this DTDS. Show your work.
Fixed point:
(b)
Give the general solution to this DTDS if the initial density is
x
0
= 10 individuals/km
2
.
General solution:
=
(c)
If the initial density is
x
0
= 10 individuals/km
2
, find the number of whole years it will
take for the density to reach at least 35 individuals/km
2
. Show your work!
Minimum number of whole years required:
(d)
If we would like this population’s fixed point to be
x
*
= 50 individuals/km
2
(instead of
what we found in part (a)), by what density
c
(in individuals per km
2
) should the repopulation
strategy aim to increase this population’s density each year?
c
=
MAT 1330 Midterm 1
9
10. (1+2+2+1=6 points)
Consider the following non-linear DTDS:
x
t
+1
=
2
.
5
x
2
t
3 +
x
t
(a)
Give the updating function for this DTDS:
f
(
x
) =
(b)
Find, algebraically, all fixed points
x
*
for this DTDS.
Show your work!
Fixed points:
(c)
The graph of
y
=
f
(
x
) is drawn below.
Label the axes
, identify the
fixed points
of
the DTDS on the graph, then
draw a cobweb
for at least three iterations, starting with an
initial population density of
x
0
= 1
.
5.
Use your student card as a straight edge.
(d)
Which of the fixed points of this DTDS are
stable
?
Briefly justify your answer for
each fixed point below.
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