Complex Numbers
1 Introduction
Complex numbers arouse feelings of both awe and suspicion for many when
they come across them for the first time. However those who work with com-
plex numbers simply regard them as a natural extension of the real numbers,
m

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CALCULUS I
Extras
Paul Dawkins
Calculus I
Table of Contents
Preface . ii
Extras . 2
Introduction . 3
Proof of Various Limit Properties . 4
Proof of Various Derivative Facts/Formulas/Properties . 15
Proof of Trig Limits . 28
Proofs of Derivative Applicatio

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Page 1 of 20
INVESTIGATION INTO PROPOSED HOUSING
DEVELOPMENT IN ESTUARY POINT TO
DETERMINE MOST SUITABLE DEVELOPMENT
PLAN
PREPARED FOR: Mr Robbie Byrd
Municipal manager
Estuary point
PREEPARED BY: development researching team
Nosihle Gwala
Phathu Siboiboi

MAM 1006H
Resource Book
2015
21nd Semester
Please bring this Book to all
Tutorials
I I I a I I I I I I I I l I - n A n I I I I I I I I I - I I I I I I I l u - 1 I I I I I I I Eiﬁiﬁi 7
Informatioh for Smdents
"Diary fer Terms 3 and 4
Tutorials
Further No

1 4+8i
Sl+i 9 1—21'
e) 2.2 , Where
10. Now that you know how to add and multiply complex numbers you can study the
properties of these Operations. ls addition commutative? Is multiplication associative? Does
the distributive law work for complex numbers

5i
MAM 1006H Taylor and Maclaurin Series / Expansions / Polynomials
H
Introduction:
In this section we use infinite polynomial series (also called power series) to help as approximate
familiar functions. Taylor and Maclaurin series provide us a way of app

A Statement of the Principle of Mathematical Induction:
Let n be a positive integer, and let PM) he a “statement” (or proposition)
about ii.
If (A) we can prove thatP( 1) is true, and
(B) if we assume that P(k) is true for any It 2 1, and
(C) then can pro

is determined if you know the angleﬁ and the distance 7".
Im
Re.
This idea motivates the two deﬁnitions (modulus and argument) of this
section.
4.2 The‘modulus
Deﬁnition The modulus of the complex number 2 = a + 52', denoted by I2]
is deﬁned by
lzl = «a?

,-
(=>) If e; = 6‘” with z =-a + hi and w = c + 032', then by (a) of the same
theorem we have a? 2 e5, and so a = c. Norar
eh1 = e‘IE1 :> cosb+isinb= cosd+isind
so 5 = d + 27m and hence the result.
A last observation for this section: the famous equation

~Tutorial 28
Determinants, Rotation Matrices
l.Evaluate: .
1 2 7 11 2 3 x o 0 0 a 0
(a)—i —21 (b)2 41 (0):; y 0 (d)0 O b
8 1 11 1 213 b c z c 0 O
b
2. Prove the following properties/laws/theorerns for any 2 x 2 matrix A, where A = {a j .
(a) [A] = M1.
(

3?
MAM 1006B Solving Systems of Linear Equations
How can we solve a system of n linear equations in n unknowns, Ax = .5?
Basically there are 4 methods:
By substitution (school method) me very eenﬁistng if there are mere than 3 01“ .4,
equatiens.
b. By us

10. If it is given that 2 + 1' is a solution of the equation 423 —1922 + 322 '15 = 0, ﬁnd the only
real solution to the equation.
1 1. Solve the equation 23 = —1 + $3} a and plot the solutions on a circle in an Argancl diagram.
What is the angle between t

INF1002FClassExercise1
2013
INF 1002F Introduction to Excel
Formatting, Formulas, & Cell Referencing
LEARNING OBJECTIVE:
Be able to interpret, Manipulate & Transform data in Excel
Case 1
1. Download the file Excel Class Exercise 1 -2012.xlsx from Vula and

Hello,
Welcome to your Wave 2 excel assignment.
This file is yours. Do not use any other file. Work in this file and submit it on Vula by 12 noon on Saturday 5 October under the Assignment tab.
Please fill in your name and student number below:
Name
Stude

Item
Almond Tart
Apple & Rhubarb Crumble
Apple Crumble
Apple Crumble Slice
Apple Crumble Tart
Apple Pie
Apple Pie Slice
Apple Turnover - Plain or Filled
Apricot Crumble Slice
Baileys Cheese Cake
Bakewell Tart
Banana Baked Cheesecake
Banana Cake
Banana Cak

Examples of how to cite according to the Author-date style:
An electronic journal:
Aird, A. 2001. E-commerce in higher education: can we afford to do
nothing? Ariadne. 26. (Online). Available:
http:/www.ariadne.ac.uk/issue26/e-commerce/ (2003, November 25

Question
4 points
1
Save
Each of the following, except one, is a condition that characterizes
a perfectly competitive labor market. Which is the exception?
Workers appear identical to firms.
Workers receive wages that are above their marginal
revenue prod

ECO110H and ECO110S Examination January/February 2005
UNIVERSITY OF CAPE TOWN
SCHOOL OF ECONOMICS
ECO110H and ECO110S
SUPPLEMENTARY EXAMINATION
JANUARY/FEBRUARY 2005
TIME: 3 HOURS
MARKS: 180
SECTIONS
A
TOPIC
MCQs
QUESTIONS
1-30
B
Structured
Questions
1-5

ECO110H and ECO110S Examination November 2004
UNIVERSITY OF CAPE TOWN
SCHOOL OF ECONOMICS
ECO110H and ECO110S
EXAMINATION
NOVEMBER 2004
TIME: 3 HOURS
MARKS: 180
SECTION
A
TOPIC
MCQs
QUESTIONS
1-30
B
Structured
Questions
1-5
MARKS
60
(60 minutes)
120
(120

Proof of the Chain Rule
Given two functions f and g where g is dierentiable at the point x
and f is dierentiable at the point g (x) = y, we want to compute the
derivative of the composite function f (g (x) at the point x. In other
words, we want to compu

BUS2010S
TUTORIAL 1
Marketing campaign: Nandos marketing campaign
Lifes too short for bad food (one of the recent campains)
Why have I chosen the Nandos marketing campaign? :
Nandos has managed to stay relevant over the past 10 years with thought provokin