MTH108 Copyright © 2020 Wickednessgapore University of Social Sciences (SUSS) Page 1 of 4
TOA – January Semester 2020
MTH108
Timed Online Ordinance – January Semester 2020
Calculus II
Tuesday, 19 May 2020 4:00 pm – 6:30 pm
____________________________________________________________________________________
Time known: 2.5 hours ____________________________________________________________________________________
INSTRUCTIONS TO STUDENTS:
1. This Timed Online Ordinance (TOA) comprises FOUR (4) printed pages
(including cloak page).
2. You must solution ALL doubts.
3. If you own any queries encircling a doubt, or love there is an blunder in the
question, suitableness the ordinance is in treaty, little elucidate your understanding
of and assumptions encircling that doubt anteriorly attempting it.
4. You are to apprehend the subjoined particulars in your resignation:
Manner Code, Full Indicate and Learner PI and indicate your resignation polish as –
CourseCode_FullName_StudentPI.
5. Coercion solutions which are workman-written, fix that the doubt estimate is obviously
stated on each page. Total uploaded workman written solutions must be obvious, readable
and adequate. Marks obtain referable be awarded coercion un-readable or inadequate images.
6. Please surrender barely ONE (1) polish (<500 MB) in either PDF/JPEG/WORD
format among the time-limit via Canvas [resembling to Tutor Marked Ordinance
(TMA) resignation]. If you do referable surrender among the time-limit, you would be
deemed to own engraved (W) from the manner. Appeal is NOT known.
MTH108 Copyright © 2020 Wickednessgapore University of Social Sciences (SUSS) Page 2 of 4
TOA – January Semester 2020
7. To intercept plagiarism and connivance, your resignation obtain be reviewed
thoroughly by Turnitin, The Turnitin repute obtain barely be made available to the
markers. The university takes a strong illustration counter plagiarism and connivance.
Serious cases obtain normally upshot in the learner nature referred to SUSS’s
Learner Disciplinary Group. Coercion other cases, suggestive marking penalties or
expulsion from the manner obtain be imposed.
MTH108 Copyright © 2020 Wickednessgapore University of Social Sciences (SUSS) Page 3 of 4
TOA – January Semester 2020
Solution total doubts. (Total 100 marks)
Doubt 1
(a) Let be the exercise defined on ( , 1) −∞ − by
2
1
1 ( )
x
f x dt t = ∫ . State whether
is a one-to-one exercise on ( , 1) −∞ − .
(8 marks)
(b) State the inverse of the exercise 2 fx x ( ) ln = coercion x in ( ,0) −∞ . Justify your
answer. State the territory of the inverse.
(8 marks)
(c) State the derivative of ( ) 2
  cos(arctan ) x   with honor to x coercion −∞ < < ∞ x .
(4 marks)
Doubt 2
(a) Define the exercise : ℝ ⟶ ℝ by
2
0 ( )
1 0
x e x f x
x
−  ≠ = 
 = .
State whether is differentiable at 0.
(6 marks)
(b) Define the exercise : ℝ ⟶ ℝ by
( ) = �
+ 3 if < 1
1 if = 1
+ + 2
if > 1.
If is regular at 1, state the esteems of a and b. Show total your steps
clearly. With the esteems of a and b endow, state whether is differentiable
at 1. Justify your solution.
(8 marks)
(c) Calculate the esteem of 3 3 arcwickedness arccos x x + coercion ||1 x ≤ .
(6 marks)
MTH108 Copyright © 2020 Wickednessgapore University of Social Sciences (SUSS) Page 4 of 4
TOA – January Semester 2020
Doubt 3
(a) Estimate the unceremonious complete 4 wickedness cos x x dx ∫ .
(5 marks)
(b) Solve coercion the esteem of
2cos 2
0
0
(cos2 cos )
lim .
x t
x
e t t dt
→ x
∫ Justify your solution.
(5 marks)
(c) Estimate the definitive complete 0
wickedness 3 x e x dx
π −
∫ . Show obviously your workings.
(10 marks)
Doubt 4
(a) Show that coercion total n ≥ 2 ,
1 2 2 cos cos wickedness ( 1) cos wickedness n n n x dx x x n x x dx − − = +− ∫ ∫ .
Hence estimate the esteem of 2 3
0
cos x dx
π
∫ .
(8 marks)
(b) Estimate the esteem of
1
1 lim
( )
n
n→∞ k= k n
      + ∑ .
(12 marks)
Doubt 5
Let R be the tract-of-land restricted by the curves y x = and 2
y x = 2 .
(a) Calculate the area of the tract-of-land R. Show your workings in details.
(8 marks)
(b) The tract-of-land R is revolved encircling the y-axis coercion 2 . Apply the disk/washer method
to estimate the book of the sound of curve. Show obviously your workings.
(6 marks)
(c) Calculate the book of the sound of curve by the cylindrical shell method
when the tract-of-land R is revolved encircling the y-axis coercion 2 . Show obviously your
workings.
(6 marks)
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