mata30-test1-practice-1
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University of Toronto *
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Course
30
Subject
Mathematics
Date
Jan 9, 2024
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MATA30 Test1 Practice 1
Calculus I for Physical Sciences (University of Toronto)
Studocu is not sponsored or endorsed by any college or university
MATA30 Test1 Practice 1
Calculus I for Physical Sciences (University of Toronto)
Downloaded by Flora Feng (flora.feng1@student.tdsb.on.ca)
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University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA30 - Term Test 1 - Practice Test 1
Date: Saturday, February 11, 2023 from 17:00 - 18:30
Signature:
•
Time: 90 minutes
•
Write your solutions in this booklet (only those pages with a QR code will be graded).
•
Use the back of each page for
rough
work.
•
This is a closed-book test. No aids are allowed for this midterm other than the provided
formula sheet. Calculators and the use of personal electronic or communication devices
is prohibited.
•
This test has 11 pages with the last page being blank.
•
There are 9 problems with the number of points indicated by each problem.
•
The total number of points possible on this test is 50.
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO.
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MATA30 - Term Test 1 - Practice Test 1
Page 2
1. (4 points) (a) Give a definition for the logarithmic function log
a
(
x
).
(b) Describe what the notation lim
x
→
a
f
(
x
) =
L
means.
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MATA30 - Term Test 1 - Practice Test 1
Page 3
2. (5 points) (a) Find a formula for the inverse of
f
(
x
) =
1
-
x
1 +
x
.
(b) What does your answer say about the graph of
f
(
x
) (graphically)?
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MATA30 - Term Test 1 - Practice Test 1
Page 4
3. (7 points) Find the natural domain of the function
f
(
x
) =
radicalbigg
x
4
-
1
x
-
3
+ sin(
e
x
) + ln
parenleftbigg
2
x
+ 1
2
parenrightbigg
.
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MATA30 - Term Test 1 - Practice Test 1
Page 5
4. (6 points) Explain why
sin
-
1
(sin (9
π/
4)) = sin
-
1
(sin(3
π/
4))
is true by computing a value for both sides.
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MATA30 - Term Test 1 - Practice Test 1
Page 6
5. (6 points) Compute the following limits.
(a) lim
x
→
4
-
9
-
x
2
|
1
-
x
|
(4
-
x
)
.
(b) Compute lim
x
→
1
√
x
-
1
x
2
-
1
.
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MATA30 - Term Test 1 - Practice Test 1
Page 7
6. (6 points) Compute lim
x
→
0
+
x
sin(ln(
x
)). Show all your work.
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MATA30 - Term Test 1 - Practice Test 1
Page 8
7. (6 points) Compute the following limit (or show that it does not exist):
L
=
lim
x
→-∞
√
x
2
+ 5
x
4
-
1
1
-
3
x
2
-
x
.
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MATA30 - Term Test 1 - Practice Test 1
Page 9
8. (6 points) The graph of a function
f
(
x
) is shown below.
(a) Evaluate
L
= lim
x
→
5
f
(
x
) + lim
x
→
3
-
(
f
(0)
f
(
x
)).
(b) Find a value of
c
for which lim
x
→
c
+
f
(
x
) = 2. If no such
c
exists, write ”No such point”.
(c) Evaluate lim
x
→
3
f
(
x
), if it exists.
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MATA30 - Term Test 1 - Practice Test 1
Page 10
9. (4 points) (a) Give an example of a function
f
(
x
) that contains logarithms and has its
domain equal to (4
,
5)
∪
(5
,
∞
). Justify your answer.
(b) Is it possible for a function
f
(
x
) to satisfy lim
x
→
0
f
(
x
) = 1 and have
f
(0) undefined?
Justify your answer.
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MATA30 - Term Test 1 - Practice Test 1
Page 11
(Page intentionally left blank in case extra space is needed for solutions)
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