Apple from the iPod to the iPad
A Case Study in Corporate Strategy
Second Edition 2012
Dr John Ashcroft PhD BSc.(Econ) CBIM, FRSA
Foreword
John Ashcroft
Contents
Apple from the iPod to the iPad
This is the case study of Apple in the digital age. The great

Maclaurin and Taylor
Series
16.5
Introduction
In this Section we examine how functions may be expressed in terms of power series. This is an
extremely useful way of expressing a function since (as we shall see) we can then replace complicated
functions in

Contents
29
Integral Vector
Calculus
29.1 Line Integrals Involving Vectors
2
29.2 Surface and Volume Integrals
34
29.3 Integral Vector Theorems
54
Learning outcomes
In this Workbook you will learn how to integrate functions involving vectors. You will lea

Contents
19
Differential Equations
19.1 Modelling with Differential Equations
2
19.2 First Order Differential Equations
11
19.3 Second Order Differential Equations
30
19.4 Applications of Differential Equations
51
Learning outcomes
In this Workbook you wi

Multiple Integrals
over Non-rectangular
Regions
27.2
Introduction
In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We
now see how to extend this to non-rectangular regions.
In this Section we introduce funct

Contents
28
Differential
Vector Calculus
28.1 Background to Vector Calculus
2
28.2 Differential Vector Calculus
17
28.3 Orthogonal Curvilinear Coordinates
37
Learning outcomes
In this Workbook you will learn about scalar and vector fields and how physical

Contents
27
Multiple Integration
27.1 Introduction to Surface Integrals
2
27.2 Multiple Integrals over Non-rectangular Regions
20
27.3 Volume Integrals
41
27.4 Changing Coordinates
66
Learning outcomes
In this Workbook you will learn to integrate a functi

Differential Vector
Calculus
28.2
Introduction
A vector field or a scalar field can be differentiated with respect to position in three ways to produce
another vector field or scalar field. This Section studies the three derivatives, that is: (i) the gra

Engineering 1B (H1034)
Week 3
1. 16.1 - Sequences and Series
You should know:
(a) the difference between a sequence and a series,
(b) the notation for the value of a term in a sequence or a series, for example an is
the value of the nth term,
P
(c) the su

Engineering 1B (H1034)
Week 11
1
29.1 - Line integrals
You should be able to calculate line integrals over scalar fields of the form
r
2
p
dy
2
2
where ds may be written as ds = dx + dy , or ds = 1 + dx
.
R
C
F (x, y)ds,
R
You should be able to calculate

Engineering 1B (H1034)
Week 9
1
27.3 - Volume integrals
You should know how to evaluate triple integrals in Cartesian, Cylindrical and Spherical coordinates.
Their volume integrals are, respectively:
Z bZ dZ f
Z 2Z aZ c
Z Z aZ
dzdydx,
rdzdrd,
a
c
e
0
0
b

Engineering 1B (H1034)
Week 4
1. 16.5 - Maclaurin and Taylor Series
You should know:
(a) the general Taylor Series formula,
(b) the difference between a general Maclaurin and a general Taylor series formula
(the Maclaurin formula is exactly the same as th

Engineering 1B (H1034)
Week 1
1
14.4 - Lengths of curves and surfaces of revolution
(From the previous HELM section) Learn and know how to use the volume of revolution formula for y = f (x):
Z b
V =
(f (x)2 dx.
a
Learn and know how to use the arc length

Engineering 1B (H1034)
Week 2
1
15.2 - Calculating centres of mass
Know the meaning of centre of mass and the definition of moment.
Learn and know how to use the formula for calculating the centre of mass of individual
masses located on an axis, for exa

Engineering 1B (H1034)
Week 5
1. 19.1 - Differential Equations (d.e.s)
You should:
(a) Understand what is meant by a differential equation, its degree, and which variables represent the unknown (dependent) function and the given (independent)
2
t
function

Engineering 1B (H1034)
Week 10
1
28.1 - Background to vector calculus
Make sure you are secure in using partial differentiation with two or more variables
(see Eng 1A notes), and in the concept of vectors and partial vector differentiation.
You should k

Second Order
Differential Equations
19.3
Introduction
In this Section we start to learn how to solve second order differential equations of a particular type:
those that are linear and have constant coefficients. Such equations are used widely in the mode

Integral Vector
Theorems
29.3
Introduction
Various theorems exist equating integrals involving vectors. You have already met the fundamental
theorem of line integrals. Those involving line, surface and volume integrals are introduced here.
They are the mu

Ewan J. Gunn
e-mail: ejg55@cam.ac.uk
An Experimental Study of Loss
Sources in a Fan Operating With
Continuous Inlet Stagnation
Pressure Distortion
Sarah E. Tooze
Cesare A. Hall
Yann Colin
Whittle Laboratory,
University of Cambridge,
1 JJ Thomson Avenue,
C

Lengths of Curves and
Surfaces of Revolution
14.4
Introduction
Integration can be used to find the length of a curve and the area of the surface generated when a
curve is rotated around an axis. In this Section we state and use formulae for doing this.
P

First Order
Differential Equations
19.2
Introduction
Separation of variables is a technique commonly used to solve first order ordinary differential
equations. It is so-called because we rearrange the equation to be solved such that all terms involving
t

Contents
15
Applications of
Integration 2
15.1 Integration of Vectors
2
15.2 Calculating Centres of Mass
5
15.3 Moment of Inertia
24
Learning outcomes
In this Workbook you will learn to interpret an integral as the limit of a sum. You will learn
how to ap

Surface and
Volume Integrals
29.2
Introduction
A vector or scalar field - including one formed from a vector derivative (div, grad or curl) - can be
integrated over a surface or volume. This Section shows how to carry out such operations.
Prerequisites
Be

Calculating
Centres of Mass
15.2
Introduction
In this Section we show how the idea of integration as the limit of a sum can be used to find the
centre of mass of an object such as a thin plate (like a sheet of metal). Such a plate is also known
as a lami

Contents
16
Sequences and Series
16.1 Sequences and Series
2
16.2 Infinite Series
13
16.3 The Binomial Series
26
16.4 Power Series
32
16.5 Maclaurin and Taylor Series
40
Learning outcomes
In this Workbook you will learn about sequences and series. You wil

Power Series
16.4
Introduction
In this Section we consider power series. These are examples of infinite series where each term
contains a variable, x, raised to a positive integer power. We use the ratio test to obtain the radius
of convergence R, of the

16.2
Infinite Series
Introduction
We extend the concept of a finite series, met in Section 16.1, to the situation in which the number
of terms increase without bound. We define what is meant by an infinite series being convergent
by considering the parti

Contents
22
Eigenvalues and
Eigenvectors
22.1 Basic Concepts
2
22.2 Applications of Eigenvalues and Eigenvectors
18
22.3 Repeated Eigenvalues and Symmetric Matrices
30
22.4 Numerical Determination of Eigenvalues and Eigenvectors
46
Learning outcomes
In th

Moment of Inertia
15.3
Introduction
In this Section we show how integration is used to calculate moments of inertia. These are essential
for an understanding of the dynamics of rotating bodies such as flywheels.
Prerequisites
Before starting this Section

Volume Integrals
27.3
Introduction
In the previous two Sections, surface integrals (or double integrals) were introduced i.e. functions
were integrated with respect to one variable and then with respect to another variable. It is often
useful in engineer