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程序代写案例-MATH1012
时间：2021-05-13
HKUST
MATH1012 Calculus IA
Midterm Examination (White Version) Name:
25th Oct 2015 Student ID:
14:15-15:30 Lecture Section:
Directions:
• This is a closed book examination. No Calculator is allowed in this examination.
• DO NOT open the exam until instructed to do so.
• Turn off all phones and pagers, and remove headphones. All electronic devices should be
kept in a bag away from your body.
• Write your name, ID number, and Lecture Section in the space provided above, and also in
the Multiple Choice Item Answer Sheet provided.
• DO NOT use any of your own scratch paper, and do not take any scratch paper away from
the examination venue.
• When instructed to open the exam, please check that you have 7 pages of questions in
addition to the cover page. Two blank pages attached at the end can be used as scratch
paper.
• Answer all questions. Show an appropriate amount of work for each short or long problem.
If you do not show enough work, you will get only partial credit.
• You may write on the backside of the pages, but if you use the backside, clearly indicate that
you have done so.
• Cheating is a serious violation of the HKUST Academic Code. Students caught
cheating will receive a zero score for the examination, and will also be subjected
to further penalties imposed by the University.
Please read the following statement and sign your signature.
I have neither given nor received any unautho-
rized aid during this examination. The answers
submitted are my own work.
I understand that sanctions will be imposed, if I
am found to have violated the University’s regu-
lations governing academic integrity.
Student’s Signature :
Question No. Points Out of
Q. 1 1
Q. 2-9 16
Q. 12 9
Q. 13 9
Q. 14 15
Total Points 50
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
1Some useful formula:
sin(x+ y) = sinx cos y + cosx sin y
cos(x+ y) = cosx cos y − sinx sin y
tan(x+ y) =
tanx+ tan y
1− tanx tan y
Part I: Answer all of the following multiple choice questions.
• Mark your MC answers to the boxes in the Multiple Choice Item Answer Sheet provided.
• Do not forget to write your name and mark your student ID number on the Multiple Choice Item
Answer Sheet.
• Mark only one answer for each MC question. Multiple answers entered for each single MC question
will result in a 2 point deduction.
Write also your MC question answers in the following boxes for back up use only. The
grading will be based on the answers you mark on the MC item answer sheet.
Question 1 2 3 4 5 6 7 8 9 Total
Answer
Each of the MC questions except Q1 is worth 2 points. Q1 is worth 1 point. No partial
credit.
1. What is the color version of your midterm exam paper?
(a) Green (b) Orange (c) White (d) Yellow (e) None of the previous
2. A student attends school by cycling with speed 3 km per hour for 30 minutes, then he parks his
bike in a lot and walk with speed 1 km per hour for 10 minutes. What is the average speed (in km
per hour) of this student in his trip?
(a) 5/6 (b) 1 (c) 5/3 (d) 2 (e) 5/2
solution: This student has traveled for 3× 12 + 1× 16 = 53 km in 23 hour. Thus, his average speed is
5
3
3
2 =
5
2 km per hour.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
23. Compare the magnitudes of the numbers
1
2
,
1√
2
, sin
pi
5
and cos
pi
5
.
(a)
1
2
< sin
pi
5
<
1√
2
< cos
pi
5
(b) sin
pi
5
<
1
2
<
1√
2
< cos
pi
5
(c)
1
2
<
1√
2
< sin
pi
5
< cos
pi
5
(d) cos
pi
5
<
1
2
< sin
pi
5
<
1√
2
(e)
1
2
< cos
pi
5
<
1√
2
< sin
pi
5
solution: Since pi6 <
pi
5 <
pi
4 <
3pi
10 , sin of these angles follow the same order.
4. Evaluate lim
x→−∞
sin sinx
x
(a) −∞ (b) −1 (c) 0 (d) 1 (e) +∞
solution: Since | sin z| ≤ 1 for all z, we apply sandwich to 1x ≤ sin sinxx ≤ − 1x .
5. Let f(x) = x3 for all x. L is the line tangent to the graph of f at the point (a, a3), which is not the
origin. At which point, besides (a, a3), the graph of f intersects L?
(a) (−3a,−27a3) (b) (−2a,−8a3) (c) (0, 0) (d) (2a, 8a3) (e) (3a, 27a3)
solution: f ′(a) = 3a2 and so L is defined by y − a3 = 3a2(x− a). To find the intersection of L and
the graph of f , we solve x3 − a3 = 3a2(x− a) to get roots a, a and −2a.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
36. Let f(x) = x5 + x + 1 for all x. Which of the following interval contains a root of the equation
f(x) = 0?
(a) (−∞,−1] (b) [−1, 0] (c) [0, 1] (d) [1,+∞) (e) none of the above
solution: Since f(−1) = −1 < 0 and f(0) = 1 > 0, the equation f(x) = 0 has a root in (−1, 0) by
the intermediate value theorem.
7. The population of a city will increase exponentially to three times as much after every 10 years.
How long does it take (in years) in order that the population of this city increases to twice as much?
(a)
ln 2
ln 3
(b)
ln 3
ln 2
(c) 10
ln 2
ln 3
(d) 10
ln 3
ln 2
(e) 10
solution: For each t, the population in this city after t years is 3t/10 times of the initial population.
So this problem is about solving the equation 3T/10 = 2.
8. Compute tan
pi
8
.
(a)
1√
2
(b)
1
2
(c)
√
2− 1 (d) √2 (e) √2 + 1
solution: By the double angle formula of tan,
2 tan pi8
1− tan2 pi8
= tan pi4 = 1. Solve this equation to yield
tan pi8 =
√
2− 1 or −√2− 1. The latter one is impossible as tan pi8 is positive.
9. Let f(x) =
2x
x+ 1
for all x 6= −1 and g(x) = 3x− 2
x
for all x 6= 0. Find all roots of the equation
f−1 ◦ g ◦ f(x) = x.
(a) −1 and −2 (b) −1 and 2 (c) 1 only (d) 2 only (e) 1 and 2
solution: The only roots of the equation g(x) = x are 1 and 2. On the other hand, the only root of
the equation f(x) = 1 is 1 while the equation f(x) = 2 has no root. Therefore, the only root of the
given equation is 1.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
4Part II: Answer each of the following questions. Give your reasoning for full credits
Evaluate the following limits if they exist.
(a) lim
x→0
x
x+ sinx
[3 pts]
solution: Since
x
x+ sinx
=
1
1 + sinxx
for x 6= 0,
and lim
x→0
sinx
x
exists and is 1,
we see that lim
x→0
x
x+ sinx
exists and is 1/(1 + 1) = 1/2.
Ans. 1/2
(b) lim
x→+∞
√
4x4 + 1 + 1
x2
[3 pts]
solution: Since
√
4x4 + 1 + 1
x2
=
√
4 +
1
x4
+
1
x2
for x > 0
and lim
x→+∞
1
x
= 0,
we see that lim
x→+∞
√
4x4 + 1 + 1
x2
exists and is
√
4 = 2.
Ans.2
(c) lim
x→0
2x+ |x|
x+ 2|x| [3 pts]
solution: Since lim
x→0+
2x+ |x|
x+ 2|x| = limx→0+
2x+ x
x+ 2x
= 1,
and lim
x→0−
2x+ |x|
x+ 2|x| = limx→0+
2x− x
x− 2x = −1,
we see that lim
x→0+
2x+ |x|
x+ 2|x| does not exist.
Ans.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
510. ([9 pts])
(a) Evaluate f ′(1) if f(x) = (1 +
√
x)2 − (1−√x)2 for all x > 0. [3 pts]
solution: For every x > 0, we rewrite and see that f(x) = 4
√
x.
Thus, f ′(1) = 2/
√
1 = 2.
Ans. 2
(b) Evaluate g′(1) if g(x) = (1− x)(1 + x2)(1 + x4)(1 + x6) for all x [3 pts]
solution: For every x, let p(x) = 1− x and q(x) = (1 + x2)(1 + x4)(1 + x6).
Then, p(1) = 0, p′(1) = −1 and q(1) = 8.
So, g′(1) = p(1)q′(1) + p′(1)q(1) = −8.
Ans. -8
(c) Evaluate h′(1) if h(x) =
e3x√
x
for all x > 0. [3 pts]
solution: For every x,
h′(x) =
3e3x√
x
− e
3x
2
√
x3
=
(6x− 1)e3x
2
√
x3
.
So h′(1) = 5e3/2.
Ans. 5e3/2
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
611. ([15 pts])
(a) Sketch the graph of f if f(x) = e−x for all x, and find all horizontal asymptotes of the graph
of f by revealing the appropriate limits. [3 pts]
lim
x→+∞ e
−x = 0 and lim
x→−∞ e
−x = 0 does not exist, so the x-axis is the only horizontal asymptote
of the graph of f .
(b) Sketch the graph of g if g(x) = 1− e−|x| for all x. [3 pts]
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
7(c) Sketch the graph of h if h(x) =
x
|x| [1− e
−|x|] for all x 6= 0, and find all horizontal asymptotes
of the graph of h by revealing the appropriate limits. [4 pts]
lim
x→+∞ 1 − e
−x = 1 and lim
x→−∞−(1 − e
x) = −1, so the lines defined by y = ±1 are the only
horizontal asymptotes of the graph of h.
(d) Define h(0) in order that h becomes a continuous function. [5 pts]
Since lim
x→0+
h(x) = lim
x→0+
1− e−x = 0,
and lim
x→0−
h(x) = lim
x→0+
−(1− ex) = 0 also,
we see that lim
x→0
h(x) exists and is 0. We define h(0) = 0 in order that h is continuous.
Ans.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
学霸联盟
MATH1012 Calculus IA
Midterm Examination (White Version) Name:
25th Oct 2015 Student ID:
14:15-15:30 Lecture Section:
Directions:
• This is a closed book examination. No Calculator is allowed in this examination.
• DO NOT open the exam until instructed to do so.
• Turn off all phones and pagers, and remove headphones. All electronic devices should be
kept in a bag away from your body.
• Write your name, ID number, and Lecture Section in the space provided above, and also in
the Multiple Choice Item Answer Sheet provided.
• DO NOT use any of your own scratch paper, and do not take any scratch paper away from
the examination venue.
• When instructed to open the exam, please check that you have 7 pages of questions in
addition to the cover page. Two blank pages attached at the end can be used as scratch
paper.
• Answer all questions. Show an appropriate amount of work for each short or long problem.
If you do not show enough work, you will get only partial credit.
• You may write on the backside of the pages, but if you use the backside, clearly indicate that
you have done so.
• Cheating is a serious violation of the HKUST Academic Code. Students caught
cheating will receive a zero score for the examination, and will also be subjected
to further penalties imposed by the University.
Please read the following statement and sign your signature.
I have neither given nor received any unautho-
rized aid during this examination. The answers
submitted are my own work.
I understand that sanctions will be imposed, if I
am found to have violated the University’s regu-
lations governing academic integrity.
Student’s Signature :
Question No. Points Out of
Q. 1 1
Q. 2-9 16
Q. 12 9
Q. 13 9
Q. 14 15
Total Points 50
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
1Some useful formula:
sin(x+ y) = sinx cos y + cosx sin y
cos(x+ y) = cosx cos y − sinx sin y
tan(x+ y) =
tanx+ tan y
1− tanx tan y
Part I: Answer all of the following multiple choice questions.
• Mark your MC answers to the boxes in the Multiple Choice Item Answer Sheet provided.
• Do not forget to write your name and mark your student ID number on the Multiple Choice Item
Answer Sheet.
• Mark only one answer for each MC question. Multiple answers entered for each single MC question
will result in a 2 point deduction.
Write also your MC question answers in the following boxes for back up use only. The
grading will be based on the answers you mark on the MC item answer sheet.
Question 1 2 3 4 5 6 7 8 9 Total
Answer
Each of the MC questions except Q1 is worth 2 points. Q1 is worth 1 point. No partial
credit.
1. What is the color version of your midterm exam paper?
(a) Green (b) Orange (c) White (d) Yellow (e) None of the previous
2. A student attends school by cycling with speed 3 km per hour for 30 minutes, then he parks his
bike in a lot and walk with speed 1 km per hour for 10 minutes. What is the average speed (in km
per hour) of this student in his trip?
(a) 5/6 (b) 1 (c) 5/3 (d) 2 (e) 5/2
solution: This student has traveled for 3× 12 + 1× 16 = 53 km in 23 hour. Thus, his average speed is
5
3
3
2 =
5
2 km per hour.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
23. Compare the magnitudes of the numbers
1
2
,
1√
2
, sin
pi
5
and cos
pi
5
.
(a)
1
2
< sin
pi
5
<
1√
2
< cos
pi
5
(b) sin
pi
5
<
1
2
<
1√
2
< cos
pi
5
(c)
1
2
<
1√
2
< sin
pi
5
< cos
pi
5
(d) cos
pi
5
<
1
2
< sin
pi
5
<
1√
2
(e)
1
2
< cos
pi
5
<
1√
2
< sin
pi
5
solution: Since pi6 <
pi
5 <
pi
4 <
3pi
10 , sin of these angles follow the same order.
4. Evaluate lim
x→−∞
sin sinx
x
(a) −∞ (b) −1 (c) 0 (d) 1 (e) +∞
solution: Since | sin z| ≤ 1 for all z, we apply sandwich to 1x ≤ sin sinxx ≤ − 1x .
5. Let f(x) = x3 for all x. L is the line tangent to the graph of f at the point (a, a3), which is not the
origin. At which point, besides (a, a3), the graph of f intersects L?
(a) (−3a,−27a3) (b) (−2a,−8a3) (c) (0, 0) (d) (2a, 8a3) (e) (3a, 27a3)
solution: f ′(a) = 3a2 and so L is defined by y − a3 = 3a2(x− a). To find the intersection of L and
the graph of f , we solve x3 − a3 = 3a2(x− a) to get roots a, a and −2a.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
36. Let f(x) = x5 + x + 1 for all x. Which of the following interval contains a root of the equation
f(x) = 0?
(a) (−∞,−1] (b) [−1, 0] (c) [0, 1] (d) [1,+∞) (e) none of the above
solution: Since f(−1) = −1 < 0 and f(0) = 1 > 0, the equation f(x) = 0 has a root in (−1, 0) by
the intermediate value theorem.
7. The population of a city will increase exponentially to three times as much after every 10 years.
How long does it take (in years) in order that the population of this city increases to twice as much?
(a)
ln 2
ln 3
(b)
ln 3
ln 2
(c) 10
ln 2
ln 3
(d) 10
ln 3
ln 2
(e) 10
solution: For each t, the population in this city after t years is 3t/10 times of the initial population.
So this problem is about solving the equation 3T/10 = 2.
8. Compute tan
pi
8
.
(a)
1√
2
(b)
1
2
(c)
√
2− 1 (d) √2 (e) √2 + 1
solution: By the double angle formula of tan,
2 tan pi8
1− tan2 pi8
= tan pi4 = 1. Solve this equation to yield
tan pi8 =
√
2− 1 or −√2− 1. The latter one is impossible as tan pi8 is positive.
9. Let f(x) =
2x
x+ 1
for all x 6= −1 and g(x) = 3x− 2
x
for all x 6= 0. Find all roots of the equation
f−1 ◦ g ◦ f(x) = x.
(a) −1 and −2 (b) −1 and 2 (c) 1 only (d) 2 only (e) 1 and 2
solution: The only roots of the equation g(x) = x are 1 and 2. On the other hand, the only root of
the equation f(x) = 1 is 1 while the equation f(x) = 2 has no root. Therefore, the only root of the
given equation is 1.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
4Part II: Answer each of the following questions. Give your reasoning for full credits
Evaluate the following limits if they exist.
(a) lim
x→0
x
x+ sinx
[3 pts]
solution: Since
x
x+ sinx
=
1
1 + sinxx
for x 6= 0,
and lim
x→0
sinx
x
exists and is 1,
we see that lim
x→0
x
x+ sinx
exists and is 1/(1 + 1) = 1/2.
Ans. 1/2
(b) lim
x→+∞
√
4x4 + 1 + 1
x2
[3 pts]
solution: Since
√
4x4 + 1 + 1
x2
=
√
4 +
1
x4
+
1
x2
for x > 0
and lim
x→+∞
1
x
= 0,
we see that lim
x→+∞
√
4x4 + 1 + 1
x2
exists and is
√
4 = 2.
Ans.2
(c) lim
x→0
2x+ |x|
x+ 2|x| [3 pts]
solution: Since lim
x→0+
2x+ |x|
x+ 2|x| = limx→0+
2x+ x
x+ 2x
= 1,
and lim
x→0−
2x+ |x|
x+ 2|x| = limx→0+
2x− x
x− 2x = −1,
we see that lim
x→0+
2x+ |x|
x+ 2|x| does not exist.
Ans.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
510. ([9 pts])
(a) Evaluate f ′(1) if f(x) = (1 +
√
x)2 − (1−√x)2 for all x > 0. [3 pts]
solution: For every x > 0, we rewrite and see that f(x) = 4
√
x.
Thus, f ′(1) = 2/
√
1 = 2.
Ans. 2
(b) Evaluate g′(1) if g(x) = (1− x)(1 + x2)(1 + x4)(1 + x6) for all x [3 pts]
solution: For every x, let p(x) = 1− x and q(x) = (1 + x2)(1 + x4)(1 + x6).
Then, p(1) = 0, p′(1) = −1 and q(1) = 8.
So, g′(1) = p(1)q′(1) + p′(1)q(1) = −8.
Ans. -8
(c) Evaluate h′(1) if h(x) =
e3x√
x
for all x > 0. [3 pts]
solution: For every x,
h′(x) =
3e3x√
x
− e
3x
2
√
x3
=
(6x− 1)e3x
2
√
x3
.
So h′(1) = 5e3/2.
Ans. 5e3/2
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
611. ([15 pts])
(a) Sketch the graph of f if f(x) = e−x for all x, and find all horizontal asymptotes of the graph
of f by revealing the appropriate limits. [3 pts]
lim
x→+∞ e
−x = 0 and lim
x→−∞ e
−x = 0 does not exist, so the x-axis is the only horizontal asymptote
of the graph of f .
(b) Sketch the graph of g if g(x) = 1− e−|x| for all x. [3 pts]
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
7(c) Sketch the graph of h if h(x) =
x
|x| [1− e
−|x|] for all x 6= 0, and find all horizontal asymptotes
of the graph of h by revealing the appropriate limits. [4 pts]
lim
x→+∞ 1 − e
−x = 1 and lim
x→−∞−(1 − e
x) = −1, so the lines defined by y = ±1 are the only
horizontal asymptotes of the graph of h.
(d) Define h(0) in order that h becomes a continuous function. [5 pts]
Since lim
x→0+
h(x) = lim
x→0+
1− e−x = 0,
and lim
x→0−
h(x) = lim
x→0+
−(1− ex) = 0 also,
we see that lim
x→0
h(x) exists and is 0. We define h(0) = 0 in order that h is continuous.
Ans.
(MATH1012)[2015](f)midterm~=0izepju^_64297.pdf downloaded by shhoaj from http://petergao.net/ustpastpaper/down.php?course=MATH1012&id=2 at 2021-05-13 18:36:13. Academic use within HKUST only.
学霸联盟
