计算机代写 MATH1116 — Advanced Mathematics and Applications 2 Book A — Analysis – cscodehelp代写

Australian Student Number: National
University
Mathematical Sciences Institute EXAMINATION: Semester 2 — End of Semester, 2019 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:
• Scribble Paper. • Formula Sheet.
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 marks for the questions in each book sum to 50.
• Unless speci￿cally noted otherwise, you must prove your answers.
• Please be neat. Illegible answers may not receive full credit.
Total / 50
Question 1
Let x : [0,10] ! R3 be de￿ned by x(t) = (3cost,3sint,4t).
(a) Find the arc length parameterisation of the curve C = (x (t ) : t 2 [0, 10]) . (b) Show that C has constant curvature.
Write your solution here
6 pts 2 pts
MATH1116 End of Semester Exam, 2019 — Book A, Page 2 of 12
Question 3 10 pts
21 Letf:R !Rbegivenbyf(x,)=(x)3.
(a) Evaluate @f (x,) at all points (x,) 2 R2 where it exists, and use symmetry to @x 2
evaluate @f (x,) at all points (x,) 2 R where it exists. @
(b) At which points (x,) 2 R2 is f di￿erentiable? Justify your answer. Write your solution here
MATH1116 End of Semester Exam, 2019 — Book A, Page 5 of 12
Extra space for previous question
MATH1116 End of Semester Exam, 2019 — Book A, Page 6 of 12
Question 4
(a) Determine whether the series X1 n2 converges or diverges. n=1 5n2 + 4n + 4
Write your solution here
(b) Determine whether the series X1 p 1 + n2 converges or diverges. n=1 n(1+n+n3)
Write your solution here
MATH1116 End of Semester Exam, 2019 — Book A, Page 7 of 12
Question 5 8 pts
Determine the set of real numbers x for which the power series X1 xn converges. For which x does the series converge absolutely? n=2 3nn ln n
For which x does the series converge conditionally? Write your solution here
MATH1116 End of Semester Exam, 2019 — Book A, Page 8 of 12
Extra space for previous question
MATH1116 End of Semester Exam, 2019 — Book A, Page 9 of 12
Question 6
Find a power series P1 anxn such that n=0
anx = 0 |t|dt, n=0
for all x 2 (1, 1) , or else explain why such a power series does not exist. Write your solution here
MATH1116 End of Semester Exam, 2019 — Book A, Page 10 of 12