MATH 1001 D IFFERENTIAL C ALCULUS

THE UNIVERSITY OF SYDNEY

MATH 1001 D IFFERENTIAL C ALCULUS

Semester 1



Assignment



2014



This assignment is due on Tuesday 15th April at 4pm. It should be posted in the locked collection

boxes on the verandah of Carslaw Level 3, which are located at the end of the verandah closest to

Eastern Avenue. (Note: Don’t use the locked collection boxes near the pyramids on Carslaw Level 3,

nor the open pigeonholes.) Please do not post your assignment before 15th April, since the boxes are

also used for the collection of assignments in other units.

A cover sheet (obtained from the MATH1001 website) must be completed, signed and stapled to

the front of the assignment. Your assignment must be then be stapled inside a manilla folder, on

the front of which you should write the initial of your family name as a LARGE letter. Assignments

which do not comply with the guidelines for submission of written work (Junior Mathematics Handbook page 26) may be returned unmarked. The School of Mathematics and Statistics encourages

collaboration between students when working on problems, but students should write up their own

version of the solutions.

There are 25 marks available for this assignment, as given below, and this assignment is worth 5%

of your nal assessment for this course. You need to show all working, as there are marks allocated

in each question for working, and one of the points of an assignment is to develop your skills in

communicating your mathematical ideas.



√

1. Let z = −1 + i and w = 2 − 2 3i.

(a) Evaluate Im(z ).

(b) Calculate |w| + z .

¯



(1 mark)

(1 mark)



(c) Convert z and w to polar form, and hence calculate zw.

(d) Hence, or otherwise, calculate (zw)6 giving your nal answer in

simplest Cartesian form.

(e) Find all 4th roots of w. Leave answers in polar form with principal argument.



2. (a) Find all the roots of p(z ) = z 3 − 5z 2 + 11z − 15,

given that 1 − 2i is a root.

2



(b) Find all solutions of z for z + 4¯ + 4 = 0 where z ∈ C.

z

3. Suppose f (x) = ln (1 − x) and g (x) = e−x , nd the natural domain and

corresponding range of g ◦ f .



4. Let g (x, y ) = 2 − x2 − 2y 2 .

(a) State the natural domain of g .



(3 marks)

(1 mark)

(2 marks)



(2 marks)

(3 marks)



(3 marks)



(1 mark)



(b) Sketch the level curves of g (x, y ) for c = 0, c = 1, c = 2.



(3 marks)



(c) Find the equation of the tangent plane to the surface z = g (x, y ) at

the point (1, −1).



(2 marks)



5. Given a parametric curve x = 2 sin t, y = sin (2t), where t takes all values in R.

(a) Find an implicit equation of this curve by eliminating t.

(b) Sketch this parametric curve in the xy -plane.



(2 marks)

(1 mark)