mathematics for electrical engineering and computing - m. attenborough

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Mathematics for Electrical
Engineering and Computing
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Mathematics for
Electrical Engineering
and Computing
Mary Attenborough
AMSTERDAM BOSTON LONDON HEIDELBERG NEW YORK
OXFORD PARIS SAN DIEGO SAN FRANCISCO
SINGAPORE SYDNEY TOKYO
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Newnes
An imprint of Elsevier
Linacre House, Jordan Hill, Oxford OX2 8DP
200 Wheeler Road, Burlington MA 01803
First published 2003
Copyright © 2003, Mary Attenborough. All rights reserved
The right of Mary Attenborough to be identified as the author of this work
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Contents
Preface xi
Acknowledgements xii
Part 1 Sets, functions, and calculus
1 Sets and functions 3
1.1 Introduction 3
1.2 Sets 4
1.3 Operations on sets 5
1.4 Relations and functions 7

1.5 Combining functions 17
1.6 Summary 23
1.7 Exercises 24
2 Functions and their graphs 26
2.1 Introduction 26
2.2 The straight line: y = mx + c 26
2.3 The quadratic function: y = ax
2
+ bx +c 32
2.4 The function y = 1/x 33
2.5 The functions y = a
x
33
2.6 Graph sketching using simple
transformations 35
2.7 The modulus function, y =|x| or
y = abs(x) 41
2.8 Symmetry of functions and their graphs 42
2.9 Solving inequalities 43
2.10 Using graphs to find an expression for the function
from experimental data 50
2.11 Summary 54
2.12 Exercises 55
3 Problem solving and the art of the convincing
argument 57
3.1 Introduction 57
3.2 Describing a problem in mathematical
language 59
3.3 Propositions and predicates 61
3.4 Operations on propositions and predicates 62

3.5 Equivalence 64
3.6 Implication 67
3.7 Making sweeping statements 70
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vi Contents
3.8 Other applications of predicates 72
3.9 Summary 73
3.10 Exercises 74
4 Boolean algebra 76
4.1 Introduction 76
4.2 Algebra 76
4.3 Boolean algebras 77
4.4 Digital circuits 81
4.5 Summary 86
4.6 Exercises 86
5 Trigonometric functions and waves 88
5.1 Introduction 88
5.2 Trigonometric functions and radians 88
5.3 Graphs and important properties 91
5.4 Wave functions of time and distance 97
5.5 Trigonometric identities 103
5.6 Superposition 107
5.7 Inverse trigonometric functions 109
5.8 Solving the trigonometric equations sin x = a,
cos x = a, tan x = a 110
5.9 Summary 111
5.10 Exercises 113
6 Differentiation 116
6.1 Introduction 116

6.2 The average rate of change and the gradient of a
chord 117
6.3 The derivative function 118
6.4 Some common derivatives 120
6.5 Finding the derivative of combinations of
functions 122
6.6 Applications of differentiation 128
6.7 Summary 130
6.9 Exercises 131
7 Integration 132
7.1 Introduction 132
7.2 Integration 132
7.3 Finding integrals 133
7.4 Applications of integration 145
7.5 The definite integral 147
7.6 The mean value and r.m.s. value 155
7.7 Numerical Methods of Integration 156
7.8 Summary 159
7.9 Exercises 160
8 The exponential function 162
8.1 Introduction 162
8.2 Exponential growth and decay 162
8.3 The exponential function y = e
t
166
8.4 The hyperbolic functions 173
8.5 More differentiation and integration 180
8.6 Summary 186
8.7 Exercises 187
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Contents vii
9 Vectors 188
9.1 Introduction 188
9.2 Vectors and vector quantities 189
9.3 Addition and subtraction of vectors 191
9.4 Magnitude and direction of a 2D vector – polar
co-ordinates 192
9.5 Application of vectors to represent waves
(phasors) 195
9.6 Multiplication of a vector by a scalar and unit
vectors 197
9.7 Basis vectors 198
9.8 Products of vectors 198
9.9 Vector equation of a line 202
9.10 Summary 203
9.12 Exercises 205
10 Complex numbers 206
10.1 Introduction 206
10.2 Phasor rotation by π/2 206
10.3 Complex numbers and operations 207
10.4 Solution of quadratic equations 212
10.5 Polar form of a complex number 215
10.6 Applications of complex numbers to AC linear
circuits 218
10.7 Circular motion 219
10.8 The importance of being exponential 226
10.9 Summary 232
10.10 Exercises 235
11 Maxima and minima and sketching functions 237

11.1 Introduction 237
11.2 Stationary points, local maxima and
minima 237
11.3 Graph sketching by analysing the function
behaviour 244
11.4 Summary 251
11.5 Exercises 252
12 Sequences and series 254
12.1 Introduction 254
12.2 Sequences and series definitions 254
12.3 Arithmetic progression 259
12.4 Geometric progression 262
12.5 Pascal’s triangle and the binomial series 267
12.6 Power series 272
12.7 Limits and convergence 282
12.8 Newton–Raphson method for solving
equations 283
12.9 Summary 287
12.10 Exercises 289
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viii Contents
Part 2 Systems
13 Systems of linear equations, matrices, and
determinants 295
13.1 Introduction 295
13.2 Matrices 295
13.3 Transformations 306
13.4 Systems of equations 314
13.5 Gauss elimination 324

13.6 The inverse and determinant of a 3 × 3
matrix 330
13.7 Eigenvectors and eigenvalues 335
13.8 Least squares data fitting 338
13.9 Summary 342
13.10 Exercises 343
14 Differential equations and difference equations 346
14.1 Introduction 346
14.2 Modelling simple systems 347
14.3 Ordinary differential equations 352
14.4 Solving first-order LTI systems 358
14.5 Solution of a second-order LTI systems 363
14.6 Solving systems of differential equations 372
14.7 Difference equations 376
14.8 Summary 378
14.9 Exercises 380
15 Laplace and z transforms 382
15.1 Introduction 382
15.2 The Laplace transform – definition 382
15.3 The unit step function and the (impulse) delta
function 384
15.4 Laplace transforms of simple functions and
properties of the transform 386
15.5 Solving linear differential equations with constant
coefficients 394
15.6 Laplace transforms and systems theory 397
15.7 z transforms 403
15.8 Solving linear difference equations with constant
coefficients using z transforms 408
15.9 z transforms and systems theory 411

15.10 Summary 414
15.11 Exercises 415
16 Fourier series 418
16.1 Introduction 418
16.2 Periodic Functions 418
16.3 Sine and cosine series 419
16.4 Fourier series of symmetric periodic
functions 424
16.5 Amplitude and phase representation of a Fourier
series 426
16.6 Fourier series in complex form 428
16.7 Summary 430
16.8 Exercises 431
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Contents ix
Part 3 Functions of more than one variable
17 Functions of more than one variable 435
17.1 Introduction 435
17.2 Functions of two variables – surfaces 435
17.3 Partial differentiation 436
17.4 Changing variables – the chain rule 438
17.5 The total derivative along a path 440
17.6 Higher-order partial derivatives 443
17.7 Summary 444
17.8 Exercises 445
18 Vector calculus 446
18.1 Introduction 446
18.2 The gradient of a scalar field 446
18.3 Differentiating vector fields 449

18.4 The scalar line integral 451
18.5 Surface integrals 454
18.6 Summary 456
18.7 Exercises 457
Part 4
Graph and language theory
19 Graph theory 461
19.1 Introduction 461
19.2 Definitions 461
19.3 Matrix representation of a graph 465
19.4 Trees 465
19.5 The shortest path problem 468
19.6 Networks and maximum flow 471
19.7 State transition diagrams 474
19.8 Summary 476
19.9 Exercises 477
20 Language theory 479
20.1 Introduction 479
20.2 Languages and grammars 480
20.3 Derivations and derivation trees 483
20.4 Extended Backus-Naur Form (EBNF) 485
20.5 Extensible markup language (XML) 487
20.6 Summary 489
20.7 Exercises 489
Part 5 Probability and statistics
21 Probability and statistics 493
21.1 Introduction 493
21.2 Population and sample, representation of data, mean,
variance and standard deviation 494
21.3 Random systems and probability 501

21.4 Addition law of probability 505
21.5 Repeated trials, outcomes, and
probabilities 508
21.6 Repeated trials and probability trees 508
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x Contents
21.7 Conditional probability and probability
trees 511
21.8 Application of the probability laws to the probability
of failure of an electrical circuit 514
21.9 Statistical modelling 516
21.10 The normal distribution 517
21.11 The exponential distribution 521
21.12 The binomial distribution 524
21.13 The Poisson distribution 526
21.14 Summary 528
21.15 Exercises 531
Answers to exercises 533
Index 542
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Preface
This book is based on my notes from lectures to students of electrical, elec-
tronic, and computer engineering at South Bank University. It presents
a first year degree/diploma course in engineering mathematics with an
emphasis on important concepts, such as algebraic structure, symme-
tries, linearity, and inverse problems, clearly presented in an accessible
style. It encompasses the requirements, not only of students with a good
maths grounding, but also of those who, with enthusiasm and motiva-

tion, can make up the necessary knowledge. Engineering applications
are integrated at each opportunity. Situations where a computer should
be used to perform calculations are indicated and ‘hand’ calculations
are encouraged only in order to illustrate methods and important special
cases. Algorithmic procedures are discussed with reference to their effi-
ciency and convergence, with a presentation appropriate to someone new
to computational methods.
Developments in the fields of engineering, particularly the extensive
use of computers and microprocessors, have changed the necessary sub-
ject emphasis within mathematics. This has meant incorporating areas
such as Boolean algebra, graph and language theory, and logic into
the content. A particular area of interest is digital signal processing,
with applications as diverse as medical, control and structural engineer-
ing, non-destructive testing, and geophysics. An important consideration
when writing this book was to give more prominence to the treatment
of discrete functions (sequences), solutions of difference equations and z
transforms, and also to contextualize the mathematics within a systems
approach to engineering problems.
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Acknowledgements
I should like to thank my former colleagues in the School of
Electrical, Electronic and Computer Engineering at South Bank
University who supported and encouraged me with my attempts to
re-think approaches to the teaching of engineering mathematics.
I should like to thank all the reviewers for their comments and the
editorial and production staff at Elsevier Science.
Many friends have helped out along the way, by discussing ideas or
reading chapters. Above all Gabrielle Sinnadurai who checked the orig-
inal manuscript of Engineering Mathematics Exposed, wrote the major

part of the solutions manual and came to the rescue again by reading
some of the new material in this publication. My partner Michael has
given unstinting support throughout and without him I would never have
found the energy.
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Part 1
Sets, functions,
and calculus
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1
Sets and functions
1.1 Introduction
Finding relationships between quantities is of central importance in
engineering. For instance, we know that given a simple circuit with a
1000  resistance then the relationship between current and voltage is
given by Ohm’s law, I = V/1000. For any value of the voltage V we can
give an associated value of I. This relationship means that I is a function
of V . From this simple idea there are many other questions that need
clarifying, some of which are:
1. Are all values of V permitted? For instance, a very high value of the
voltage could change the nature of the material in the resistor and the
expression would no longer hold.
2. Supposing the voltage V is the equivalent voltage found from con-
sidering a larger network. Then V is itself a function of other voltage
values in the network (see Figure 1.1). How can we combine the func-
tions to get the relationship between this current we are interested in

and the actual voltages in the network?
3. Supposing we know the voltage in the circuit and would like to know
the associated current. Given the function that defines how current
depends on the voltage can we find a function that defines how the
voltage depends on the current? In the case where I = V/1000, it is
clear that V = 1000I . This is called the inverse function.
Another reason exists for better understanding of the nature of func-
tions. In Chapters 5 and 6, we shall study differentiation and integration.
This looks at the way that functions change. A good understanding of
functions and how to combine them will help considerably in those
chapters.
The values that are permitted as inputs to a function are grouped
together. A collection of objects is called a set. The idea of a set is very
simple, but studying sets can help not only in understanding functions
but also help to understand the properties of logic circuits, as discussed
in Chapter 10.
Figure 1.1 The voltage V is
an equivalent voltage found
by considering the combined
effect of circuit elements in
the rest of the network.
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4 Sets and functions
1.2 Sets
A set is a collection of objects, called elements, in which the order is not
important and an object cannot appear twice in the same set.
Example 1.1 Explicit definitions of sets, that is, where each element is
listed, are:
A ={a, b, c}

B ={3, 4, 6, 7, 8, 9}
C ={Linda, Raka, Sue, Joe, Nigel, Mary}
a ∈ A means ‘a is an element of A’ or ‘a belongs to A’; therefore in the
above examples:
3 ∈ B
Linda ∈ C
The universal set is the set of all objects we are interested in and will
depend on the problem under consideration. It is represented by E .
The empty set (or null set) is the set with no elements. It is represented
by ∅ or {}.
Sets can be represented diagrammatically – generally as circular
shapes. The universal set is represented as a rectangle. These are called
Venn diagrams.
Example 1.2
E ={a, b, c, d, e, f, g},A={a, b, c},B={d, e}
This can be shown as in Figure 1.2.
Figure 1.2 A Venn diagram
of the sets E =
{a, b, c, d, e, f, g},A ={a, b, c},
and B ={d,e}.
We shall mainly be concerned with sets of numbers as these are more
often used as inputs to functions.
Some important sets of numbers are (where ‘ ’ means continue in
the same manner):
The set of natural numbers N ={1, 2, 3, 4, 5, }
The set of integers Z ={ −3, −2, −1, 0, 1, 2, 3 }
The set of rationals (which includes fractional numbers) Q
The set of reals (all the numbers necessary to represent points on a
line) R
Sets can also be defined using some rule, instead of explicitly.

Example 1.3 Define the set A explicitly where E = N and
A ={x |x<3}.
Solution The A ={x |x<3}is read as ‘A is the set of elements x, such
that x is less than 3’. Therefore, as the universal set is the set of natural
numbers, A ={1, 2}
Example 1.4 E = days of the week and A ={x |x is after
Thursday and before Sunday}. Then A ={Friday, Saturday}.
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Sets and functions 5
Subsets
We may wish to refer to only a part of some set. This is said to be a subset
of the original set.
A ⊆ B is read as ‘A is a subset of B’ and it means that every element
of A is an element of B.
Example 1.5
E = N
A ={1, 2, 3},B={1, 2, 3, 4, 5}
Then A ⊆ B
Note the following points:
All sets must be subsets of the universal set, that is, A ⊆ E and
B ⊆ E
A set is a subset of itself, that is, A ⊆ A
If A ⊆ B and B ⊆ A, then A = B
Proper subsets
A ⊂ B is read as ‘A is a proper subset of B’ and means that A is a subset
of B but A is not equal to B. Hence, A ⊂ B and simultaneously B ⊂ A
are impossible.
Figure 1.3 A Venn diagram
of a proper subset of B:

A ⊂ B.
A proper subset can be shown on a Venn diagram as in Figure 1.3.
1.3 Operations
on sets
In Chapter 1 of the background Mathematics notes available on the com-
panion website for this book, we study the rules obeyed by numbers
when using operations like negation, multiplication, and addition. Sets
can be combined in various ways using set operations. Sets and their
operations form a Boolean Algebra which we look at in greater detail
in Chapter 4, particularly its application to digital design. The most
important set operations are as given in this section.
Complement
¯
AorA

represents the complement of the set A. The complement of A is
the set of everything in the universal set which is not in A, this is pictured
in Figure 1.4.
Figure 1.4 The shaded area
is the complement, A

,ofthe
set A.
Example 1.6
E = N
A ={x | x>5}
then A

={1, 2, 3, 4, 5}
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6 Sets and functions
Figure 1.5 A ={x |x < 5} and A

={x|x  5}.
Figure 1.6 The
shaded area
represents the
intersection of A
and B.
Figure 1.7 The
intersection of two
sets {1, 2, 4}∩
{1, 5, 6}={1}.
Figure 1.8 The
intersection of two sets:
{a, b, c, d, e}∩
{a, b, c, d, e, f, g, h, i, j}=
{a, b, c, d, e}.
Figure 1.9 The
intersection of the
two sets:
{−3, −2, −1}∩
{1, 2}=∅, the empty
set, as they have no
elements in common.
Example 1.7 The universal set is the set of real numbers represented
by a real number line.
If A is the set of numbers less than 5, A ={x |x<5} then A


is the
set of numbers greater than or equal to 5. A

={x |x  5}. These sets
are shown in Figure 1.5.
Intersection
A ∩ B represents the intersection of the sets A and B. The intersection
contains those elements that are in A and also in B, this can be represented
as in Figure 1.6 and examples are given in Figures 1.7–1.10.
Note the following important points:
If A ⊆ B then A ∩B = A. This is the situation in the example given
in Figure 1.8.
If A and B have no elements in common then A ∩ B =∅and they
are called disjoint. This is the situation given in the example in
Figure 1.9. Two sets which are known to be disjoint can be shown
on the Venn diagram as in Figure 1.10.
Figure 1.10 Disjoint sets A
and B.
Union
A ∪B represents the union of A and B, that is, the set containing elements
which are in A or B or in both A and B. On a Venn diagram, the union can
be shown as in Figure 1.11 and examples are given in Figures 1.12–1.15.
Note the following important points:
If A ⊆ B, then A ∪ B = B. This is the situation in the example
given in Figure 1.13.
The union of any set with its complement gives the universal set, that
is, A ∪A

= E , the universal set. This is pictured in Figure 1.15.
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Sets and functions 7
Figure 1.11 The
shaded area
represents to union
of sets A and B.
Figure 1.12 The
union of two sets:
{1, 2, 4}∪{1, 5, 6}=
{1, 2, 4, 5, 6}.
Figure 1.13 The
union of two sets:
{a, b, c, d, e}∪
{a, b, c, d, e, f, g, h, i, j}=
{a, b, c, d, e, f, g, h, i, j}.
Figure 1.14 The
union of the two sets:
{−3, −2, −1}∪
{1, 2}=
{−3, −2, −1, 1, 2}.
Figure 1.15 The shaded
area represents the union of a
set with its complement giving
the universal set.
Cardinality of a finite set
The number of elements in a set is called the cardinality of the set and is
written as n(A) or |A|.
Example 1.8
n(∅) = 0, n({2}) = 1, n({a, b}) = 2
For finite sets, the cardinality must be a natural number.

Example 1.9 In a survey, 100 people were students and 720 owned a
video recorder; 794 people owned a video recorder or were students. How
many students owned a video recorder?
E ={x |x is a person included in the survey}
Setting S ={x |x is a student} and V ={x |x owns a video recorder},
we can solve this problem using a Venn diagram as in Figure 1.16.
Figure 1.16 S is the set of
students in a survey and V is
the set of people who own a
video. The numbers in the
sets give the cardinality of the
sets, n(S) = 100, n(S ∪V) =
794, n(V) = 720,
n(S ∩ V) = x.
x is the number of students who own a video recorder. From the diagram
we get
100 −x +x +720 −x = 794
⇔ 820 −x = 794
⇔ x = 26
Therefore, 26 students own a video recorder.
1.4 Relations
and functions
Relations
A relation is a way of pairing up members of two sets. This is just like
the idea of family relations. For instance, a child can be paired with its
mother, brothers can be paired with sisters, etc. A relation is such that it
may not always be possible to find a suitable partner for each element in
the first set whereas sometimes there will be more than one. For instance,
if we try to pair every boy with his sister there will be some boys who have
no sisters and some boys who have several. This is pictured in Figure 1.17.

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8 Sets and functions
Figure 1.17 The relation
boy → sister. Some boys
have more than one sister
and some have none at all.
Functions
Functions are relations where the pairing is always possible. Functions
are like mathematical machines. For each input value there is always
exactly one output value.
Figure 1.18 An arrow
diagram of the function
y = 1/x.
Calculators output function values. For instance, input 2 into a cal-
culator, press 1/x and the calculator will display the number 0.5. The
output value is called the image of the input value. The set of input values
is called the domain and the set containing all the images is called the
codomain.
The function y = 1/x is displayed in Figure 1.18 using arrows to link
input values with output values.
Functions can be represented by letters. If the function of the above
example is given the letter f to represent it then we can write
f : x →
1
x
This can be read as ‘f is the function which when input a value for x gives
the output value 1/x’ . Another way of giving the same information is:
f(x)=
1

x
f(x)represents the image of x under the function f and is read as ‘f of x’.
It does not mean the same as f times x.
f(x) = 1/x means ‘the image of x under the function f is given by
1/x’ but is usually read as ‘f of x equals 1/x’.
Even more simply, we usually use the letter y to represent the output
value, the image, and x to represent the input value. The function is
therefore summed up by y = 1/x.
x is a variable because it can take any value from the set of values in
the domain. y is also a variable but its value is fixed once x is known.
So x is called the independent variable and y is called the dependent
variable.
The letters used to define a function are not important. y = 1/x is the
same as z = 1/t is the same as p = 1/q provided that the same input
values (for x, t,orq) are allowed in each case.
More examples of functions are given in arrow diagrams in
Figures 1.19(a) and 1.20(a). Functions are more usually drawn using
a graph, rather than by using an arrow diagram. To get the graph the
codomain is moved to be at right angles to the domain and input and
output values are marked by a point at the position (x, y). Graphs are
given in Figures 1.19(b) and 1.20(b).
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Sets and functions 9
Continuous functions and discrete
functions applied to signals
Functions of particular interest to engineers are either functions of a real
number or functions of an integer. The function given in Figure 1.19 is
an example of a real function and the function given in Figure 1.20 is an
example of a function of an integer, also called a discrete function.

Often, we are concerned with functions of time. A variable voltage
source can be described by giving the voltage as it depends on time, as also
can the current. Other examples are: the position of a moving robot arm,
the extension or compression of car shock absorbers and the heat emission
of a thermostatically controlled heating system. A voltage or current
varying with time can be used to control instrumentation or to convey
information. For this reason it is called a signal. Telecommunication
signals may be radio waves or voltages along a transmission line or light
signals along an optical fibre.
Time, t, can be represented by a real number, usually non-negative.
Time is usually taken to be positive because it is measured from some
reference instant, for example, when a circuit switch is closed. If time is
used to describe relative events then it can make sense to refer to negative
time. If lightning is seen 1 s before a thunderclap is heard then this can
be described by saying the lightning happened at −1 s or alternatively
that the thunderclap was heard at 1 s. In the two cases, the time origin
has been chosen differently. If time is taken to be continuous and rep-
resented by a real variable then functions of time will be continuous or
piecewise continuous. Examples of graphs of such functions are given in
Figure 1.21.
Figure 1.19 The function
y = 2x +1 where x can take
any real value (any number
on the number line). (a) is the
arrow diagram and (b) is the
graph.
Figure 1.20 The function
q = t −3 where t can take any
integer value (a) is the arrow
diagram and (b) is the graph.

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10 Sets and functions
Figure 1.21 Continuous and piecewise functions where time is represented by a real number > 0. (a) A
ramp function; (b) a wave (c) a square wave. (a) and (b) are continuous, while (c) is piecewise continuous.
A continuous function is one whose graph can be drawn without taking
your pen off the paper. A piecewise continuous function has continuous
bits with a limited number of jumps. In Figure 1.21, (a) and (b) are
continuous functions and (c) is a piecewise continuous function. If we
have a digital signal, then its values are only known at discrete moments
of time. Digital signals can be obtained by using an analog to digital
(A/D) convertor on an originally continuous signal. Digital signals are
represented by discrete functions as in Figure 1.22(a)–(c)
A digital signal has a sampling interval, T , which is the length of
time between successive values. A digital functions is represented by a
discrete function. For example, in Figure 1.22(a) the digital ramp can be
represented by the numbers
0, 1, 2, 3, 4, 5,
If the sample interval T is different from 1 then the values would be
0, T, 2T, 3T, 4T, 5T,
This is a discrete function also called a sequence. It can be represented
by the expression f(t) = t, where t = 0, 1, 2, 3, 4, 5, 6, or using the
sampling interval, T , g(n) = nT , where n = 0, 1, 2, 3, 4, 5, 6,
Yet another common way of representing a sequence is by using a
subscript on the letter representing the image, giving
f
n
= n, where n = 0, 1, 2, 3, 4, 5,
or, using the letter a for the image values,
a

n
= n, where n = 0, 1, 2, 3, 4, 5,
Substituting some values for n into the above gives
a
0
= 0, a
1
= 1, a
2
= 2, a
3
= 3,
As a sequence is a function of the natural numbers and zero (or if
negative input values are allowed, the integers) there is no need to specify
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“chap01” — 2003/6/8 — page 11 — #11
Sets and functions 11
Figure 1.22 Examples of
discrete functions. (a) A digital
ramp; (b) a digital wave; (c) a
digital square wave.
the input values and it is possible merely to list the output values in order.
Hence the ramp function can be expressed by 0, 1, 2, 3, 4, 5, 6,
Time sequences are often referred to as ‘series’. This terminology is
not usual in mathematics books, however, as the description ‘series’ is
reserved for describing the sum of a sequence. Sequences and series are
dealt with in more detail in Chapter 18.
Example 1.10 Plot the following analog signals over the values of t
given (t real):
(a) x = t

3
t  0
(b)
y =





0 t  3
t −33<t 5
2 t>5
(c) z =
1
t
2
t>0
Solution In each case, choose some values of t and calculate the function
values at those points. Plot the points and join them.
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12 Sets and functions
(a)
t 0 0.5 1 1.5 2 2.5 3 3.5
x = t
3
0 0.125 1 3.375 8 15.625 27 42.875
These values are plotted in Figure 1.23(a).
(b)
t 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

y 00 00 00.511.5 22 22 2
      
y = 0 y = t −3 y = 2
These values are plotted in Figure 1.23(b).
(c)
t 0.0001 0.001 0.01 0.1 1 10 100 1000 10000
z 10
8
10
6
10
4
100 1 0.01 10
−4
10
−6
10
−8
These values are plotted in Figure 1.23(c).
Figure 1.23 The
analog signals described in
Example 1.10.
(a) x = t
3
t  0
(b) y =






0 t  3
t −33< t  5
2 t > 5
(c) z = 1/t
2
t > 0
TLFeBOOK