(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
HKUST
MATH1013 Calculus IB
Final Examination (Green Version)
Name:
19th Dec 2015
12:00-15:00
Student ID:
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.
• DO NOT use any of your own scratch paper, and do not take any scratch paper away from
the examination venue.
10
• When instructed to open the exam, please check that you have
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 unauthorized aid during this examination. The answers
submitted are my own work.
Question No.
I understand that sanctions will be imposed, if I
am found to have violated the University’s regulations governing academic integrity.
Q. 1-18
54
Q. 19
10
Q. 20
12
Q. 21
12
Q. 22
12
Total Points
100
Student’s Signature :
Points
Out of
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
1
Part I: Answer all of the following multiple choice questions.
Enter your MC answers to the boxes below.
Question
1
2
3
4
5
6
7
8
11
12
13
14
15
16
17
18
9
10
Answer
Question
Answer
Each of the MC questions is worth 3 points.
1. No point will be given to you for this question if you do not choose the color version of your exam
correctly or do not mark your ID number correctly in the MC answer form (Check your
ID marking in the MC form right now!!).
(a) Green
(b) Orange
2. Find the one-sided limit lim
x→1−
(a) −2
(c) White
(d) Yellow
(e) None of the previous
(x − 1)(3x − 1)
.
|x2 − 1|
(b) −1
(c) 0
(d) 1
(e) 2
3. Find the value of k that makes the following function continuous on −∞ < x < π2 .
(
3x2 + k cos 2x if x ≤ 0
f (x) =
tan x
x
if 0 < x < π2
x + 3e
(a) 1
(b) 2
(d) 4
(c)
(e) 5
3
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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4. Find the value of k such that the following function is differentiable at x = 0.
(
4x + 2x2 sin x1 if x < 0
f (x) =
x cos x + kx
if x ≥ 0
(a) k = −1
5. Let f (x) =
(a) −1
(b) k = 0
(c) k = 1
(d) k = 2
(e) k = 3
ex cos x
. Find the derivative of f at x = 0; i.e., f ′ (0) .
1 + 2x2
(b) 1
(c) −2
(d) 2
(e)
2
3
6. Find the slope of the tangent line to the curve defined by the equation y ln x + 2x2 ey = 2x at the
point (1, 0).
(a)
−2
(b) −1
(c) 1
(d) 2
(e) 3
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
3
7. If
d
f (6x) = 18x2 + 4, find f ′ (2).
dx
(a) −1
(b) 1
(c) −2
(d) 2
(e) 3
8. To find an approximate root of the equation x2 − 1 + sin x = 0 by Newton’s Method, the iteration
formula is:
(a) xn+1 = xn −
x2n − 1 + sin x
2xn + cos xn
(b) xn+1 = xn −
2xn − cos xn
x2n − 1 + sin x
(c) xn+1 = xn −
2x2n + sin x
2xn − cos xn
(d) xn+1 = xn −
2xn − cos xn
x2n − 1 + sin x
(e) xn+1 = xn −
x2n − 1 + sin x
2xn − sin xn
9. Find the limit lim
x→∞
(a) 0
x+3
x+2
x
(b) e−1
.
(c) e
3
(d) e 2
(e)
∞
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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10. How many inflection points does f have if its derivative function is
f ′ (x) = x2 (x − 4)4 ?
(a) 0
(b) 1
(c)
2
(d) 3
(e) 4
11. Given the graph of the derivative function f ′ of f as shown below, how many local minima does f
have?
y
y = f ′ (x)
x
(a) 2
(b) 3
(c) 4
(d) 5
(e) none
12. A particle is moving along the curve y = sin[π(1 + x2 )] such that the x coordinate (in meters) of
dx
the particle increases at a rate of
= 4 m/s. If D is the distance between the particle and the
dt
origin, what is the rate of change of D with respect to time when x = 3 m?
y
(x, y)
D
x
1
(a) 2 m/s
(b) 3 m/s
(c) 4 m/s
2
3
(d) 5 m/s
(e) 6 m/s
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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13. Which of the following gives antiderivatives of
(a)
x sin x
+C
(1 + x)2
(d)
x sin x + cos x + sin x
?
(1 + x)2
−x cos2 x
+C
1+x
(b)
− cos x
+C
1+x
(e)
(c)
x sin x cos x
+C
(1 + x)2
sin x
+C
1+x
14. The acceleration function of a particle moving along the s-axis is given by
a(t) = 20 sin t + 6 cos t .
If the position of the particle at time t = 0 and t = 2π are s(0) = 4 and s(2π) = 20 respectively,
what is the position of the particle at t = π ? (All in SI units.)
(a) 8
(b) 15
(c) 18
15. Find the value of the definite integral
(a) 2 −
√
3
(b) 3 −
√
3
Z
(d) 21
(e) 24
1
2x3
dx.
x4 + 3
0
√
(c) 3 − 2 3
√
(d) 4 −
√
3
√
(e) 4 − 2 3
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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16. Suppose a continuous function f satisfies
Z
x
e +
x
f (t)dt = x2 + 3x .
0
Find the function value f (0).
(a) −1
(b) 1
(c) 2
(d) −2
(e) 3
17. By identifying the limit
1
1
1
1
√
lim
+√
+√
+ ··· + √
n→∞
3n2 + n
3n2 + 2n
3n2 + 3n
3n2 + n · n
as a definite integral or otherwise, find the limit.
√
√
√
(a) 2 − 2 3
(b) 3 3 − 2
(c) 4 − 2 3
√
√
(d) 2 3 − 2
18. Given the graph of the function y = f (x) as shown below, find
Z
4
−1
(e)
√
3 3−4
2
3 f (x) f ′ (x)dx.
y
y = f (x)
2
x
-4
4
-2
(a) −2
(b) 2
(c) 4
(d) 6
(e) 8
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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Part II: Answer each of the following questions.
19. [10 pts] The straight line y = 3x + 4 intersects the curve y = x2 at two points Q and R, and
P : (x, x2 ) is a point on the part of the curve y = x2 below the line segment QR.
(a) Find the coordinates of the points Q and R.
[2 pt]
y
R
S
Q
P : (x, x2 )
(b) Express the area A of the triangle P QR as a function of x by considering the triangles P QS
and P RS, or by other methods.
[4 pts]
(c) Find the coordinates of the point P such that the area of the triangle P QR is the largest
possible. Justify your answer for full credit.
[4 pts]
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20. [12 pts] Consider the function f (x) = x2 −
18
x−2 ,
with f ′ (x) = 2x +
18
(x−2)2 .
(a) Find the interval of increase of f . Show all your work for full credit.
[3 pts]
(b) Find the concave up interval of f . Show all your work for full credit.
[4 pts]
(c) Using appropriate scales on the axes in the given grid, sketch the graph of f together with all
its local maximum or minimum points, inflection point(s) and asymptote(s).
[5 pts]
y
x
(MATH1013)[2015](f)final~=pi37emu^_38523.pdf downloaded by hylawah from http://petergao.net/ustpastpaper/down.php?course=MATH1013&id=16 at 2022-12-05 15:12:33. Academic use within HKUST only.
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21. [12 pts]
Consider the functions y = cos x on the interval 0 ≤ x ≤ 1.
(a) Explain why cos x ≤ cos(x2 ) for 0 ≤ x ≤ 1.
(b) Show that
Z
π
6
0
[4 pts]
1
cos x2 dx ≥ .
2
[2 pts]
(c) Given the graph of the function y = cos(x2 ) on the interval 0 ≤ x ≤ 1 in the figure below,
[3 pts]
sketch the graph of the inverse function y = f −1 (x) in the same figure.
1
y
cos 1
x
1
(d) Which of the integrals
Z
1
0
cos x2 dx and
Z
1
f −1 (x)dx is larger? Why?
cos 1
[3 pts]
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22. ([12 pts]) Two functions g and f with continuous derivatives are given. Some function values of
these functions are shown in the following table:
x
g(x)
g ′ (x)
f (x)
f ′ (x)
0
2
1
2
2
2
0
−2
0
1
4
1
6
1
−1
Answer the following questions:
Z 2
Z 2
′
(a)
f (x)g(x)dx +
g′ (x)f (x)dx > 0 .
0
[4 pts]
0
The statement above is :
Brief reason:
True
False
(Circle your answer.)
(b) The graph of y = f (x) must have a horizontal tangent line.
The statement above is :
Brief reason:
(c) lim
x→2
True
False
(Circle your answer.)
g(x)
does not exist.
f (x)
The statement above is :
Brief reason:
[4 pts]
[4 pts]
True
False
(Circle your answer.)