Australian Student Number: National
University
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2018 MATH1116 — Advanced Mathematics and Applications 2 Book A — Analysis
Exam Duration: 180 minutes. Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• Unmarked English-to-foreign-language dictionary (no approval from MSI required). • No electronic aids are permitted e.g. laptops, phones, calculators.
Materials To Be Supplied To Students:
• Formula Sheets Booklet. • Scribble Paper.
Instructions To Students:
• Answer the Analysis questions in Book A, and the Algebra questions in Book B, in the spaces provided. If you run out of space, you may use the backs of pages, but please make a note on the front of the page that your solution is continued on the back.
• The value of each question is as shown. It is recommended that you spend equal time on the Analysis and the Algebra papers (50 points each).
• A good strategy is not to spend too much time on any question. Read them through first and attack them in the order that allows you to make the most progress.
• You must prove / justify your answers, unless explicitly told otherwise. Please be neat.
Total / 50
Question 1
π π 2 (a)SupposethatK= д(t):t∈ 0,2 ,whereд: 0,2 −→R isdefinedby
д(t) = 2cos2t,2sin2t.
(i) Is д the arc length parametrisation of K ? Answer:
[You do not need to justify your answer for Q1(a)(i).]
(ii) What is the (constant) curvature of K ? Answer: [You do not need to show any working for Q1(a)(ii).]
(b) Let x : [0, ln(1117)] −→ R2 be defined by
eθ x(θ) = √ cosθ,sinθ ,
andsetC= x(θ):θ∈[0,ln(1117)] .
(i) Find the length |C| of C. Show your working.
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 2 of 13
Recall that x : [0, ln(1117)] −→ R2 is given by
eθ x(θ) = √ cosθ,sinθ ,
andthatC= x(θ):θ∈[0,ln(1117)] .
(ii) Find the arc length parametrisation of C .
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 3 of 13
Question 2 10 pts
Let f : R2 −→ R be given by
f (x,y) = xye−(x2+y2)/2 .
(a) Find ∇f (x,y), for (x,y) ∈ R2. 2 pts
Write your solution here
(b) Find the equation for the best linear approximation L(x,y) to f (x,y), whose graph is tangent to the graph of f at (x,y) = (0,0). 1 pt
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 4 of 13
Recallthat f :R2 −→R isgivenby
f (x,y) = xye−(x2+y2)/2 ,
as on the previous page. The following holds:
f11(x,y) = xy(x2 − 3)e−(x2+y2)/2 ,
f12(x,y) = (1−x2)(1−y2)e−(x2+y2)/2 and f22(x,y) = xy(y2 − 3)e−(x2+y2)/2 .
[You may take the above information as given, but note that verifying one or two of them (on your scribble paper) is a way to double-check that your calculations in Part (a) are correct.]
(c) (i) Find the critical points of f , and say what type each critical point is. 5 pts (ii) Does f have absolute maximum and minimum values? Why? 2 pts
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 5 of 13
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book A, Page 6 of 13
Question 3
Let f : R2 −→ R be given by
f ( x , y ) =
x 4 + y 2
if (x,y) (0,0), if (x,y) = (0,0).
(a) Determine both partial derivatives of f for all values of (x,y) where they exist. (b) At which points (x,y) is f differentiable?
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 7 of 13
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book A, Page 8 of 13
Question 4 7 pts
2n (1 + √n) converge? Justify your answer with reference to relevant convergence tests.
Write your solution here
For which values of x does the power series
MATH1116 End of Semester Exam, 2018 — Book A, Page 9 of 13
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book A, Page 10 of 13
Question 5
Find the value of the sum of the series
n=0 n!(n + 2)
Write your solution here
[Hints: you may use the power series expansion e =
Evaluate 1 xex dx , referring to relevant theory to justify your calculations.]
n! , for x ∈ R.
MATH1116 End of Semester Exam, 2018 — Book A, Page 11 of 13
Question 6 8 pts
Let (an ) be a non-negative real valued sequence. Suppose that lim √n an = l, with l ∈ [0, 1) .
Prove that an converges. n=1
Write your solution here
MATH1116 End of Semester Exam, 2018 — Book A, Page 12 of 13
Extra space for previous question
MATH1116 End of Semester Exam, 2018 — Book A, Page 13 of 13