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Calculus II

MTH108 Copyright © 2020 Singapore University of Social Sciences (SUSS) Page 1 of 4
TOA – January Semester 2020
MTH108
Timed Online Assignment – January Semester 2020
Calculus II
Tuesday, 19 May 2020 4:00 pm – 6:30 pm
____________________________________________________________________________________
Time allowed: 2.5 hours ____________________________________________________________________________________
INSTRUCTIONS TO STUDENTS:
1. This Timed Online Assignment (TOA) comprises FOUR (4) printed pages
(including cover page).
2. You must answer ALL questions.
3. If you have any queries about a question, or believe there is an error in the
question, while the assignment is in session, briefly explain your understanding
of and assumptions about that question before attempting it.
4. You are to include the following particulars in your submission:
Course Code, Full Name and Student PI and name your submission file as –
CourseCode_FullName_StudentPI.
5. For answers which are hand-written, ensure that the question number is clearly
stated on each page. All uploaded hand written answers must be clear, readable
and complete. Marks will not be awarded for un-readable or incomplete images.
6. I need help writing my essay – research paper submit only ONE (1) file (<500 MB) in either PDF/JPEG/WORD
format within the time-limit via Canvas [similar to Tutor Marked Assignment
(TMA) submission]. If you do not submit within the time-limit, you would be
deemed to have withdrawn (W) from the course. Appeal is NOT allowed.
MTH108 Copyright © 2020 Singapore University of Social Sciences (SUSS) Page 2 of 4
TOA – January Semester 2020
7. To prevent plagiarism and collusion, your submission will be reviewed
thoroughly by Turnitin, The Turnitin report will only be made available to the
markers. The university takes a tough stance against plagiarism and collusion.
Serious cases will normally result in the student being referred to SUSS’s
Student Disciplinary Group. For other cases, significant marking penalties or
expulsion from the course will be imposed.
MTH108 Copyright © 2020 Singapore University of Social Sciences (SUSS) Page 3 of 4
TOA – January Semester 2020
Answer all questions. (Total 100 marks)
Question 1
(a) Let be the function defined on ( , 1) −∞ − by
2
1
1 ( )
x
f x dt t = ∫ . Determine whether
is a one-to-one function on ( , 1) −∞ − .
(8 marks)
(b) Determine the inverse of the function 2 fx x ( ) ln = for x in ( ,0) −∞ . Justify your
answer. State the domain of the inverse.
(8 marks)
(c) Determine the derivative of ( ) 2
  cos(arctan ) x   with respect to x for −∞ < < ∞ x .
(4 marks)
Question 2
(a) Define the function : ℝ ⟶ ℝ by
2
0 ( )
1 0
x e x f x
x
−  ≠ = 
 = .
Determine whether is differentiable at 0.
(6 marks)
(b) Define the function : ℝ ⟶ ℝ by
( ) = �
+ 3 if < 1
1 if = 1
+ + 2
if > 1.
If is continuous at 1, determine the values of a and b. Show all your steps
clearly. With the values of a and b found, determine whether is differentiable
at 1. Justify your answer.
(8 marks)
(c) Calculate the value of 3 3 arcsin arccos x x + for ||1 x ≤ .
(6 marks)
MTH108 Copyright © 2020 Singapore University of Social Sciences (SUSS) Page 4 of 4
TOA – January Semester 2020
Question 3
(a) Compute the indefinite integral 4 sin cos x x dx ∫ .
(5 marks)
(b) Solve for the value of
2cos 2
0
0
(cos2 cos )
lim .
x t
x
e t t dt
→ x
∫ Justify your answer.
(5 marks)
(c) Compute the definite integral 0
sin 3 x e x dx
π −
∫ . Show clearly your workings.
(10 marks)
Question 4
(a) Show that for all n ≥ 2 ,
1 2 2 cos cos sin ( 1) cos sin n n n x dx x x n x x dx − − = +− ∫ ∫ .
Hence compute the value of 2 3
0
cos x dx
π
∫ .
(8 marks)
(b) Compute the value of
1
1 lim
( )
n
n→∞ k= k n
      + ∑ .
(12 marks)
Question 5
Let R be the region bounded by the curves y x = and 2
y x = 2 .
(a) Calculate the area of the region R. Show your workings in details.
(8 marks)
(b) The region R is revolved about the y-axis for 2 . Apply the disk/washer method
to compute the volume of the solid of revolution. Show clearly your workings.
(6 marks)
(c) Calculate the volume of the solid of revolution by the cylindrical shell method
when the region R is revolved about the y-axis for 2 . Show clearly your
workings.
(6 marks)
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