Australian Student Number: National
University
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2017 MATH1116 — Advanced Mathematics and Applications 2 Book A — Analysis
Exam Duration: 180 minutes. Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• One A4 page with hand written notes on both sides. (This A4 page is to cover both Analysis and Algebra.)
• 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:
• 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 rst 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
(a) Suppose that K = ( (t) : t 2 f0, 2 g), where : f0, 2 g ! R2 is dened by (t) = ⇣2cos2t,2sin2t⌘.
(i) Is 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 dened by
x( ) = p2 cos ,sin , and set C = (x ( ) : 2 [0, ln(1117)]) .
(i) Find the length |C| of C. Show your working. Write your solution here
MATH1116 — Book A, Page 2 of 15
Recall that x : [0, ln(1117)] ! R2 is given by
and that C = (x ( ) : 2 [0, ln(1117)]) .
(ii) Find the arc length parametrisation of C . 4 pts
Write your solution here
x( ) = p2 cos ,sin ,
MATH1116 — Book A, Page 3 of 15
Question 2
Let :R2 !Rbegivenby
(x, ) = 3×2 4x 2 12 + 16x .
(a) Findr (x, ),for(x, )2R2. Write your solution here
(b) Let = ⇣p1 , p1 ⌘. Evaluate @f (0,0), justifying your calculation.
Write your solution here
MATH1116 — Book A, Page 4 of 15
Recallthat :R2 !Risgivenby
(x, ) = 3×2 4x 2 12 + 16x ,
as on the previous page.
(c) Find the critical points of , and say what type each critical point is. 3 pts
Write your solution here
MATH1116 — Book A, Page 5 of 15
Recallthat :R2 !Risgivenby
(x, ) = 3×2 4x 2 12 + 16x ,
as on the previous two pages.
(d) Restrict the domain of to be the closed square [0, 10] ⇥ [0, 10] .
Find the point in [0, 10] ⇥ [0, 10] at which achieves its maximum value. Write your solution here
MATH1116 — Book A, Page 6 of 15
Question 3
Let f : R2 ! R be given by
(a) Find the rst partial derivatives of f at (x, ) , (0,0).
Write your solution here
x (x ) 2 2
if (x, ) , (0,0); if (x, ) = (0,0).
[As a check: your answers should evaluate to
@f(0, )= ,for ,0 and @f(x,0)=x,forx,0.] @x @
MATH1116 — Book A, Page 7 of 15
Recallthat f :R2 !R isgivenby
if(x, ),(0,0); if (x, ) = (0,0),
MATH1116 — Book A, Page 8 of 15
Recallthat f :R2 !R isgivenby
f(x, ) = the same as in parts (a) and (b).
@ @x Write your solution here
x (x ) 2 2
if (x, ) , (0,0); if (x, ) = (0,0),
(c) Notingthat f = and f = ,calculate f (0,0) and f (0,0). x @x x @ x x
(d) What can you conclude about f based on your answer to part (c)? Write your solution here
MATH1116 — Book A, Page 9 of 15
Question 4 7 pts
For which values of x does the power series X1 (x 4)n converge? n=2 n ln(n)
Justify your answer with reference to relevant convergence tests.
Write your solution here
MATH1116 — Book A, Page 10 of 15
Extra space for previous question
MATH1116 — Book A, Page 11 of 15
Question 5
Find the value of the sum of the series
[Hints: yoRu may use the power series expansion ex = X1 xn , for x 2 R. 1 n=0 n!
Evaluate 0 xex dx , referring to relevant theory to justify your calculations.] Write your solution here
n=0 n!(n + 2)
MATH1116 — Book A, Page 12 of 15