The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1021: Calculus of One Variable Semester 1, 2022
Lecturer: Mary Myerscough
This individual assignment is due by 11:59pm Thursday 17 March 2022, via
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Copyright c© 2022 The University of Sydney 1
1. Explain why
(a) e−2−ipi = − 1
e2
.
(b)
∣∣∣∣ (6i− 8)4(√2i)6(3 + 4i)3
∣∣∣∣ = 10.
2. (a) The set A is illustrated below.
0 1 2 3 4 5−1−2−3−4−5
(i) Write A using interval notation.
(ii) Write A in two different ways, using set notation, (i.e. in the form
{x ∈ . . . | . . .}).
(b) The set B is illustrated below.
0 1 2 3 4 5−1−2−3−4−5
(i) Write set B as a list, for example, {−1, 0, 1, 2, 3}.
(ii) Write B using set notation of the form {x ∈ . . . | . . .}.
3. Consider the polynomial equation z3 − 8 = 0. Solve this equation using two different
methods and show that both methods produce the same set of solutions. (Useful fact:
a3 − b3 = (a− b)(a2 + ab + b2).)
4. (a) Let z = x + iy where x = Re z, y = Im z and so x and y ∈ R). Express the
perpendicular distance from z to the real axis in terms of x and/or y. (Hint: draw
a diagram.)
(b) Consider a parabola, drawn on a plane. The perpendicular distance of each point
on the parabola from a given line (the directrix) is equal to its distance from a
given point (the focus). Sketch the set P = {z ∈ C | | Im z| = |z − i|} in the
complex plane. Indicate the directrix and focus of the resulting parabola and show
the point where the parabola intersects the imaginary axis.
(c) By setting z = x + iy where x and y ∈ R, derive an equation for the parabola in
the previous part in terms of the real and imaginary parts of z.
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