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MTH108 Copyright © 2019 Singapore University of Social Sciences (SUSS) Page 1 of 4 Examination – July Semester 2019

MTH108

Examination – July Semester 2019 Calculus II Tuesday, 19 November 2019 4:00 pm - 6:00 pm ____________________________________________________________________________________

Time allowed: 2 hours

____________________________________________________________________________________

INSTRUCTIONS TO STUDENTS:

1. This examination contains FIVE (5) questions and comprises FOUR (4) printed pages (including cover page).

2. You must answer ALL questions.

3. All answers must be written in the answer book. Marks will only be awarded if

FULL working is shown.

4. This is an open-book examination. At the end of the examination Please ensure that you have written your examination number on each answer book used. Failure to do so will mean that your work cannot be identified. If you have used more than one answer book, please tie them together with the string provided.

THE UNIVERSITY RESERVES THE RIGHT NOT TO MARK YOUR SCRIPT IF YOU FAIL TO FOLLOW THESE INSTRUCTIONS.

MTH108 Copyright © 2019 Singapore University of Social Sciences (SUSS) Page 2 of 4 Examination – July Semester 2019

Answer all questions. (Total 100 marks) Question 1

(a) Let f be the function defined on ( , )−∞ ∞ by 3

0

( ) t x

f x e dt= ∫ . Construct an argument to determine whether f is a one to one function on ( , )−∞ ∞ .

(8 marks) (b) Determine the inverse of the function

3

( ) xf x e= − for all x in ( , )−∞ ∞ . Justify your answer. State the domain of the inverse.

(8 marks) (c) Determine the derivative of 2 2arccos arcsinx x+ with respect to x for 0 1x≤ ≤ .

(4 marks)

Question 2 (a) Let f be the function defined on ( , )−∞ ∞ by

| | 0 ( )

1 0

xe x f x

x

− ≠ = 

= .

Construct an argument to determine whether f is differentiable at 0.

(6 marks)

(b) Let ( 1)

. 2 sin 2 x x

y x

+ =

+ Find

dy dx

in terms of x for 0x > . Hence compute the value

of when . 2

dy x

dx π

=

(8 marks)

(c) Differentiate ( ) 2

cos(arcsin )x   with respect to x for 1 1x− ≤ ≤ . (6 marks)

Question 3

(a) Find the indefinite integral 21 x dx−∫ for 1 1x− ≤ ≤ .

(6 marks)

MTH108 Copyright © 2019 Singapore University of Social Sciences (SUSS) Page 3 of 4 Examination – July Semester 2019

(b) Given that 4 3 24 sin cos sin 3 sinx dx x x x dx= − +∫ ∫ , compute the definite

integral 4 0

sin x dx π

∫ . (8 marks)

(c) Solve for the value of

2 cos 2

0 0

(cos ) lim .

x t

x

e t dt

x→ ∫

Show clearly your workings.

(6 marks)

Question 4

(a) Evaluate 2 3

8 0 1

x dx

x+∫ . (6 marks)

(b) Compute the value of 1

1 lim sin .

n

n k

k n n

π →∞

=

      ∑

(8 marks) (c) Let f be a continuous function defined on [0, 8] such that

2 3 2

1 2 0

( ) 5, ( ) 1, ( ) 8f x dx f x dx f x dx= = =∫ ∫ ∫ .

Calculate the values of

(i) 1

0

( )f x dx∫ ; (2 marks)

(ii) 0

1

( )f x dx∫ ; (2 marks)

(iii) 3

1

( )f x dx∫ . (2 marks)

MTH108 Copyright © 2019 Singapore University of Social Sciences (SUSS) Page 4 of 4 Examination – July Semester 2019

Question 5 Let R be the region bounded by the curves 2y x= and 2x y= . (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 .π 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 .π Show clearly your workings.

(6 marks)

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