MATH15062022PRACTICETEST1
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York University *
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Course
1506
Subject
Mathematics
Date
Feb 20, 2024
Type
Pages
15
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INSTRUCTIONS
1. Page 1 and the bubble sheet on the back of it, on which you will record your answers, are not
attached to the questions booklet. Hand in page 1 at the end of the test
.
2. Completely and accurately fill out the ID information on both sides of page 1. If you do not put
the email address that is associated with your eClass account, the TAs may not be able to assign
your midterm to your account and you will get a mark of zero.
3. You may use a dark pen or pencil to fill in the bubbles as you select your answers. If you use pen,
you will not be able to change your answers once filled in.
4. There will be no extra time given to fill in your answers at the end of the midterm, so be certain
to fill out the answers as you complete the midterm.
5. You may erase an answer if you change your mind, but make sure you completely erase your
previous answer before filling in another bubble.
6. Only the bubble sheet on the back of page 1 will be graded.
Anything you write on the other
pages of the test will not be assessed.
Copyright 2022
c A. McEachern
3
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1. Consider the graph
If you attempted to solve the system of linear equations that represents these two lines using
Gaussian elimination, the bottom row in the augmented matrix would be
A) (0 0
|
1)
B) (0 0
|
-
1)
C) (0 0
|
2)
D) (0 0
|
4)
E)
None of the above.
2. A system of linear equations with three variables and two equations has at least one solution. The
equations are not multiples of one another. What would this solution look like if we plotted it
out?
A) It would be a plane in two dimensions.
B) It would be a point in three dimensions.
C) It would be a plane in three dimensions.
D)
It would be a line in three dimensions.
E) None of the above.
Copyright 2022
c A. McEachern
4
3. Given the system of linear equations
a
-
b
+
c
= 1
-
2
a
-
3
b
-
4
c
=
-
9
-
a
+ 4
b
+
c
= 4
-
3
a
+
c
=
-
2
determine if a solution exists. If it does, what is the solution?
A)
(1
,
1
,
1)
B) (2
,
2
,
1)
C) (
-
2
,
2
,
8)
D) (
-
1
,
-
1
,
1)
E) No unique solution.
4. Given the system of linear equations
x
1
-
x
2
-
x
3
= 1
2
x
1
-
3
x
2
+
x
3
= 7
determine if a solution exists. If it does, what is the solution?
A) Let t be a variable such that t goes from negative to positive infinity. The solution set is
(4t - 3, 3t - 4, t).
B) Let t be a variable such that t goes from 0 to positive infinity. The solution set is
(4t - 3, 3t - 4, t).
C) Let t be a variable such that t goes from negative to positive infinity. The solution set is
(3t - 4, 4t - 3, t).
D) Let t be a variable such that t goes from negative to positive infinity. The solution set is
(t, 4t - 3, 3t - 4).
E)
None of the above.
Copyright 2022
c A. McEachern
5
5. Consider the image below.
Let the curve be the graph of
f
(
x
). On what interval would
f
(
x
) have an inverse?
A) (
-
4
,
0)
∪
(0
,
4)
B) (
-∞
,
0]
∪
[3
,
∞
)
C) [0
,
3]
D)
(
-∞
,
0)
∪
(3
,
∞
)
E) None of the above.
Copyright 2022
c A. McEachern
6
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In the following image,
g
(
x
) is represented by the graph. Use the image to answer questions 6, 7,
8, and 9.
6. What is
lim
x
→-
2
-
g
(
x
)?
A) -2
B)
0
C) 2
D) Does not exist
E) None of the above.
7. What is
lim
x
→-
2
+
g
(
x
)?
A) -2
B) 0
C)
2
D) Does not exist
E) None of the above.
8. What is lim
x
→
2
g
(
x
)?
A) -2
B) 0
C) 2
D)
Does not exist
E) None of the above.
9. What is lim
x
→
1
g
(
x
)?
A) -2
B) 0
C) 2
D) Does not exist
E)
None of the above.
Copyright 2022
c A. McEachern
7
Consider the image below.
There are two functions represented. The left image is the graph of the function
f
(
x
), and the
right image is the graph of the function
g
(
x
). Use these images to answer questions 10, 11, 12,
and 13.
10. Using the limit laws and the graphs above, evaluate
lim
x
→-
3
+
(
f
(
x
) +
g
(
x
))
A) -2
B)
0
C) 2
D) 4
E) None of the above.
11. Using the limit laws and the graphs above, evaluate
lim
x
→-
3
-
(
f
(
x
)
-
3
g
(
x
))
A) -2
B) 0
C)
6
D) 8
E) None of the above.
12. Using the limit laws and the graphs above, evaluate
lim
x
→-
5
2+
g
(
x
)
f
(
x
)
A) -2
B) 0
C) 2
D) 4
E)
None of the above.
Copyright 2022
c A. McEachern
8
13. Using the limit laws and the graphs above, evaluate lim
x
→
1
(
f
(
x
))
2
A) -2
B) 0
C) 2
D)
4
E) None of the above.
14. Consider the image below that has a graph of a continuous function.
and select the statement that is
false
.
A)
The Intermediate Value Theorem guarantees that the function has a root at A.
B) The Intermediate Value Theorem does not guarantee that the function has a root at A.
C) The Intermediate Value Theorem guarantees that the function has a root at B.
D) The Intermediate Value Theorem guarantees that the function does not have a root at C.
E) None of the above.
15. Consider the following function:
h
(
x
) =
3
x
+ 2
if
x < k
2
x
-
3
if
k
≤
x
≤
8
and find the value of
k
that makes
h
(
x
) continuous on the interval (
-∞
,
8)
A)
-5
B) 0
C)
5
2
D) 5
E) None of the above.
Copyright 2022
c A. McEachern
9
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16. Which value of x satisfies the following equation?
1
2
x
-
5 + 4 = 3
A) -8
B) 0
C) 4
D) 8
E)
None of the above.
17. If the function
f
(
x
) is defined on the set of points
(0
,
0)
,
(1
,
1)
,
(2
,
3)
,
(5
,
8)
,
(8
,
13)
,
(13
,
21)
,
(21
,
34)
what is
f
-
1
(21)?
A) 2
B) 8
C)
13
D) 34
E) None of the above.
18. It is always possible to restrict the domain of a function so that the function will have an inverse.
A)
True.
B) False.
Copyright 2022
c A. McEachern
10
19. Consider the graph of the function below
At what value of
x
is
f
(
x
) continous but not differentiable?
A) -1
B) 0
C) 1
D)
2
E) None of the above.
20. Given the function
f
(
x
) = 2
-
3
x
, if we were to find the derivative of
f
(
x
) using the definition
from first principles
f
0
(
x
) = lim
h
→
0
f
(
x
+
h
)
-
f
(
x
)
h
, what would the first line of our equation look like?
A)
f
0
(
x
) = lim
h
→
0
(2
-
3
x
+
h
)
-
(2
-
3
x
)
h
B)
f
0
(
x
) = lim
h
→
0
(2
-
3(
x
+
h
))
-
(2
-
3
x
)
h
C)
f
0
(
x
) = lim
h
→
0
(2
-
3(
x
+
h
))+(2
-
3
x
)
h
D)
f
0
(
x
) = lim
h
→
0
(2
-
3
x
+3
h
)
-
(2
-
3
x
)
h
E) None of the above.
Copyright 2022
c A. McEachern
11
21. Consider the following function
g
(
x
) =
x
if
x
6
= 3
4
if
x
= 3
Find and identify any discontinuities of
g
(
x
).
A)
g
(
x
) is continuous everywhere.
B)
g
(
x
) has an infinite discontinuity at
x
= 3.
C)
g
(
x
) has a jump discontinuity at
x
= 3.
D)
g
(
x
) has a removable discontinuity at
x
= 3.
E) None of the above.
22. Select the choice that makes the entire statement correct.
The slope of the secant line between the points (
x, f
(
x
)) and (
x
+
h, f
(
x
+
h
)) on a function
f
(
x
)
A) is equivalent to the instantaneous rate of change of the function
f
(
x
).
B) is equivalent to the average rate of change of the function
f
(
x
).
C)
is equivalent to the average rate of change of the function
f
(
x
) as
x
increases by
h
.
D) is equivalent to the instantaneous rate of change of the function
f
(
x
) as
x
increases by
h
.
E) None of the above.
23. What is the derivative of
f
(
x
) = 5
x
3
-
x
+ 1
A)
f
0
(
x
) = 15
x
2
B)
f
0
(
x
) =
5
x
4
4
-
x
2
2
+
x
C)
f
0
(
x
) = 15
x
+ 1
D)
f
0
(
x
) = 15
x
2
-
1
E) None of the above.
Copyright 2022
c A. McEachern
12
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24. What is the derivative of
h
(
x
) = (
x
2
+ 4)(
x
2
-
4)
A) 0
B)
h
0
(
x
) = 4
x
3
C)
h
0
(
x
) = 4
x
3
+ 16
x
D)
h
0
(
x
) = 4
x
3
+ 32
x
2
+ 16
E) None of the above.
25. What is the derivative of
h
(
x
) =
x
2
+ 4
x
2
-
4
A)
-
16
x
(
x
2
-
4)
2
B)
16
x
(
x
2
-
4)
2
C)
-
16
x
2
(
x
2
-
4)
2
D) 0
E) None of the above.
Copyright 2022
c A. McEachern
13
Use the following table to answer questions 26 and 27.
Assume that
f
(
x
) and
g
(
x
) are both
differentiable functions with values as given. Use the table to calculate the following derivatives.
26. Find
h
0
(1) if
h
(
x
) =
xf
(
x
) + 4
g
(
x
)
A) 38
B) 42
C) 46
D) 50
E)
None of the above.
27. Find
h
0
(3) if
h
(
x
) = 2
x
+
f
(
x
)
g
(
x
)
A) -36
B) -30
C) -24
D) -18
E)
None of the above.
Copyright 2022
c A. McEachern
14
Use the following graph of the functions
f
(
x
) and
g
(
x
) to answer questions 28, 29, and 30.
28. Let
h
(
x
) =
f
(
x
) +
g
(
x
). What is
h
0
(1)?
A) -2
B)
0
C) 1
D) 2
E) None of the above.
29. Let
h
(
x
) =
f
(
x
) +
g
(
x
). What is
h
0
(3)?
A) -2
B) 0
C) 1
D) 2
E)
None of the above.
30. Let
h
(
x
) =
f
(
x
) +
g
(
x
). What is
h
0
(4)?
A) -2
B) 0
C)
1
D) 2
E) None of the above.
THIS IS THE LAST PAGE OF THE MIDTERM.
Copyright 2022
c A. McEachern
15
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