Need calculus expert to help with my real time exam of 2.5 hour

MTH108_JAN_2020_TOA.pdf

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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 If you do not submit within the time-limit, you would be deemed to have withdrawn (W) from the course.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.

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.

Answer all questions. (Total 100 marks) Question 1

(a) Let 𝑓𝑓 be the function defined on ( , 1)−∞ − by 2

1 ( )

f x dt t

= ∫ . Determine whether 𝑓𝑓 is a one-to-one function on ( , 1)−∞ − .

(8 marks) (b) Determine the inverse of the function 2( ) lnf x x= for x in ( , 0)−∞ . Justify your

answer. State the domain of the inverse. (8 marks)

2 cos(arctan )x   with respect to x for .x−∞ < < ∞

(4 marks)

Question 2 (a) Define the function 𝑓𝑓: ℝ ⟶ ℝ by

0 ( )

1 0

xe x f x

− ≠ = 

= .

Determine whether 𝑓𝑓 is differentiable at 0.

(6 marks)

(b) Define the function 𝑔𝑔: ℝ ⟶ ℝ by

𝑔𝑔(𝑥𝑥) = � 𝑎𝑎 + 𝑥𝑥3 if 𝑥𝑥 < 1

1 if 𝑥𝑥 = 1 𝑒𝑒𝑏𝑏+𝑥𝑥+𝑥𝑥

2 if 𝑥𝑥 > 1.

If 𝑔𝑔 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 𝑔𝑔 is differentiable at 1. Justify your answer.

(8 marks)

Question 3 (a) Compute the indefinite integral 4sin cosx x dx∫ .

(5 marks)

(b) Solve for the value of 2 cos 2

0 0

(cos 2 cos ) lim .

x t

e t t dt

x→ ∫

Justify your answer.

(5 marks)

sin 3xe x dx π −∫ . Show clearly your workings.

(10 marks)

Question 4 (a) Show that for all 2n ≥ ,

1 2 2cos cos sin ( 1) cos sinn n nx dx x x n x x dx− −= + −∫ ∫ .

Hence compute the value of 32 0

cos x dx π

∫ . (8 marks)

(b) Compute the value of 1

1 lim

( )

n k k n→∞ =

   +  ∑ .

(12 marks)

Question 5

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

(6 marks)

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1. This Timed Online Assignment (TOA) comprises FOUR (4) printed pages (including cover page).