Australian Student Number: National

University

Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2016 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 Algebra and Analysis sections are worth a total of 50 points each, with the value of each question as shown. It is recommended that you spend equal time on the Analysis and the Algebra papers.

• 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

Letx:[0,1]→R2 bedefinedbyx(t)=(t,t32) ∀t∈[0,1].

(a) Let t ∈ [0,1]. Prove that the curve {x(s) : s ∈ [0,t]} has length

8 ( ( 1 + 9 t ) 32 − 1 ) . 27 4

Write your solution here

(b) Supposethaty:[0, 8 ((13)32 −1)]→R2 isthearclengthparametrisationofthecurve 27 4

{x(t) : t ∈ [0,1]}. Compute y(12). 4 pts Write your solution here

MATH1116 — Book A, Page 2 of 10

Question 2

(a) Determine whether the series

Justify your answer using appropriate theorems / tests.

Write your solution here

∞ 2n+sinn

converges or diverges.

(b) Determine whether the series ∞ (−1)n3n converges or diverges. n=1 4n−1

Justify your answer using appropriate theorems / tests. Write your solution here

MATH1116 — Book A, Page 3 of 10

(c) Determine whether the series ne−n2 converges or diverges. 3 pts

Justify your answer using appropriate theorems / tests.

Write your solution here

MATH1116 — Book A, Page 4 of 10

Question 3

We consider a function f defined by a power series as follows:

(a) Determine the set of real numbers x for which 3n+2

n=0 For which x does the series converge absolutely?

For which x does the series converge conditionally? Write your solution here

∞ ( n + 1 ) x n

∞ ( n + 1 ) x n

converges.

MATH1116 — Book A, Page 5 of 10

(b) Provethat

Write your solution here

∞n9 (c) Provethat (n+1)−2 =

f(t)dt = 3−x −3,forsuitablevaluesofx. 3pts

3 25 Write your solution here

MATH1116 — Book A, Page 6 of 10

Question 4

Letд:R3 −→Rbegivenby

д(x,y,z) = x2y3z − 4yz . (a) Find∇д(x,y,z),for(x,y,z)∈R3.

Write your solution here

(b) Calculate the directional derivative of д in the direction of 1 , 1 , − 11 ,

evaluated at (2, 1, −1). Write your solution here

MATH1116 — Book A, Page 7 of 10

Question 5

Let f : R2 −→ R be given by

f ( x , y ) =

2 22 1 x +y sin√

(a) Find the first partial derivatives of f at (x,y) (0,0). Write your solution here

if(x,y)(0,0); if (x,y) = (0,0).

MATH1116 — Book A, Page 8 of 10

Recallthat f :R2 −→R isgivenby

2 22 1 x +y sin√

if(x,y)(0,0); if (x,y) = (0,0),

f ( x , y ) = 0

the same as in part (a).

(b) Find the first partial derivatives of f at (0, 0) .

Write your solution here

(c) Is f differentiable at (0, 0) ? Write your solution here

MATH1116 — Book A, Page 9 of 10

Question 6 9 pts

[Recall the following notation: if a function f : Rn −→ R is differentiable at x ∈ Rn , then dxf denotestheassociateddifferentialmap.]

Suppose that x ∈ Rn and that f , д : Rn −→ R are differentiable at x . Define h = f + д. Prove that h is differentiable at x , and that

Write your solution here

dxh = dxf +dxд.

MATH1116 — Book A, Page 10 of 10

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