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. Please 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deemed to have withdrawn (W) from the course.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)

—– END OF PAPER —–

## About The Author: Admin

More posts by admin