MATH 1001 D IFFERENTIAL C ALCULUS

THE UNIVERSITY OF SYDNEY MATH1001 DIFFERENTIAL CALCULUS

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 Hand- book 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 final 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). (1 mark)

(b) Calculate |w| + z̄. (1 mark) (c) Convert z and w to polar form, and hence calculate zw. (3 marks)

(d) Hence, or otherwise, calculate (zw)6 giving your final answer in simplest Cartesian form. (1 mark)

(e) Find all 4th roots of w. Leave answers in polar form with principal argument. (2 marks)

2. (a) Find all the roots of p(z) = z3 − 5z2 + 11z − 15, given that 1 − 2i is a root. (2 marks)

(b) Find all solutions of z for z2 + 4z̄ + 4 = 0 where z ∈ C. (3 marks)

3. Suppose f (x) = ln (1 − x) and g(x) = e−x, find the natural domain and corresponding range of g ◦ f . (3 marks)

4. Let g(x, y) = 2 − x2 − 2y2. (a) State the natural domain of g. (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. (2 marks)

(b) Sketch this parametric curve in the xy-plane. (1 mark)