Calculus and its Applications

541 Pages·1963·14.949 MB·English

by  P. Mainardi and H. Barkan (Auth.)

CALCULUS
and its
APPLICATIONS
By P. Mainardi and H. Barkan
Department of Mathematics
Newark College of Engineering
Newark, New Jersey
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PREFACE
Throughout this book on Calculus, emphasis has been
placed on the understanding of ideas. In keeping with
this goal we have frequently taken applications from
physics, engineering, economics and other areas, whenever
we felt that these applications would help clarify the
the mathematical concepts under study. We have not
hesitated, however, to employ mathematical rigor where
that rigor also contributed to increased understanding.
We have systematically used the "trapping" technique
in defining geometrical and physical entities that are
usually regarded as limits of sums. Its use in defining
length of curve and surface of revolution is believed to
be somewhat novel, as is the simplifying symbolism em­
ployed. Also, there may be some novelty in the treatment
of average value, where we give a more general definition
than is usual.
Our inclusion of the concepts of external and internal
parameters may serve to increase the studentfs appreciation
of parametric functions.
In the treatment of separable differential equations,
we stress with more detail than usual their special
ill
suitability in describing physical laws.
Among the many colleagues and friends who helped in
the preparation of this text, we would like to extend our
thanks particularly to Dona Hausser who typed the entire
manuscript, to Professor Robert Salamon who prepared
many of the drawings and to Murray Lieb who worked a
large number of problems and read the entire manuscript.
P. M.
H. B.
CHAPTER I
FUNDAMENTAL IDEAS
1.1 The Concept of Function
It is a commonplace that many of the facts of physics,
•geometry, biology, economics, etc., are expressible by
simple mathematical equations involving numerical quanti­
ties. The distance s traversed in the time t by a body
falling from rest in a vacuum is given by s • |gt^ where
g is a constant. In geometry we have for the volume V of
a sphere of radius r: V - ^ TTI^. The existence of such
relationships has lead mathematicians to define the con­
cept of function* Prior to 1950, the usual textbook
definition of function was given more or less as follows:
We consider the totality of numbers x which lie be­
tween two numbers a and b. The symbol x is regarded as
denoting any of the numbers in this interval* If to each
value of x in this interval there corresponds a single
definite value y, where x and y are connected by any rule
whatsoever, we say that y is a function of x and write
symbolically, y • f(x)« We call the x the independent
variable and the y the dependent variable*
More recent writers have objected to this definition
as too vague and imprecise. Consistent with the current
movement in mathematics toward increased abstraction, they
have formulated the following definition of function*
First we define an ordered pair of numbers as a pair in
which one is designated as the first and the other as the
second* Thus (2,3) is an ordered pair with 2 first and
3 second*
A set of ordered pairs is called a fraction if no two
different ordered pairs have the same first element* The
set of first elements is called the domain of the function,
and the set of second elements is called the range of the
function*
From this definition it follows that the set of
ordered pairs (2,7), (3,U), (7,7) constitutes a function*
The domain of the function is the set 2, 3, and 7# The
range of the function is the set 7,U«
I
2 Sec. LI
It is customary to call a symbol which represents an .
arbitrary element of the domain the independent variable.
A symbol which represents an arbitrary'element of ' tkie
range is called the dependent variable. If x is the in­
dependent variable and y is the dependent variable, and
f is a symbol standing for the function (the set of
ordered pairs), this information is represented symbol-
cally by y - fix). Similarly, the symbolical expression,
z * g(u)> implies that g stands for a set of ordered pairs;
u is the symbol denoting the set of first elements; and z
is the symbol denoting the set of second elements. Fur­
ther, none of the ordered pairs has the same first element
with differing second elements. This latter requirement
merely assures the function will be single-valued.
Further, note that the symbol f(x), read Rf of x,f,
stands for y, the dependent variable. The student should
note the difference in the meanings of f and f (x). Thus
in a specific instance, f(2) would denote the element in
the range of the dependent variable y corresponding to the
element 2 in the domain of the independent variable x.
Frequently, the correspondence between the independ­
ent variable x and the dependent variable y is established
by requiring that values of y be determined by specific
rules of operation on the corresponding values of x, as
for example in y • x2 - Ijx + 2 where x - 2, 3, U# Here
the dependent variable is denoted by both the symbol y
and the expression x2 - Ux + 2. The dependent variable
plays a dual role. It not only stands for the set of
numbers which constitute the range of the dependent varia­
ble, but it also tells us specifically how to compute the
values of the dependent variable y which correspond to
values of the independent variable x.
Further we will call * symbol used to represent an
arbitrary element of a set (even though no function is in­
volved) a variable. A symbol which is to be assigned one
and only one value during a discussion will be called a
constant. Thus we have
for x - 2, x2 - Ux • 2 - 22 - U(2) + 2 - -2;
for x - 3, x2 - Ux + 2 - 32 - U(3) • ? - - !;
for x - U, x2 - kx + 2 - U2 - U(U) + 2 - +2.
Sec. 1.2 3
We have seen that the dependent variable may be de­
noted* by a single letter, say y, or, for example, by the
more informative analytical expression x^ - Ux + 2. We
could of course have denoted y by the previously dis­
cussed bu£ less specific symbol f(x)* Thus, if we now
write f(x) • x^ - Ux + 2, we may view f(x) as a command
to perform certain operations on x which will yield the
corresponding value of y. The symbol f(2) thus may be
taken as a command to perform certain operations on the
number 2 which will provide the corresponding value of
the dependent variable, namely -2.
Before ending the discussion of function we should
point out that to Leibniz, who was the first to use the
word "function11 and to the mathematicians of the 18th
century, the function referred to a dependent variable
given by an analytical expression in the independent
variable, such as y • x^ - Ux + 2. In other words, in
Leibniz1 s definition the law of correspondence had to be
given by an analytical expression* The definition given
in this book places no such restriction on the law of
correspondence* Physicists found only too often that
their empirical data which exhibited a relationship be­
tween physical quantities resisted analytical formulation*
A.s the ideas they tried to describe became more complex,
the physicists were often at a loss to find analytic
functions which their empirical data would "fit" and
which would predict the correspondence between hitherto
unexamined values of the Variables* In the absence of
any analytic function which would describe an obvious
correspondence existing between the variables representing
physical quantities, the physicist turned to the more
general definition of function which places no restric­
tions on the form of the correspondence*
1*2 Exercises
1. Which of the following sets of ordered pairs constitute
functions?
a* (2,3), (2,U), (3,7).
b. (2,3), (3,3), (U,3).
-1
c
T
d. (0,1), (2,3), (U5).
f
\ Sec. L2
2. Indicate, where appropriate, the range and domain for each
of the sets in Exercise 1.
Some sets of ordered pairs (x,y) may be conveniently designated
by the notation C j j seme statement 1
x y
\ 9 I about x and y.J
This notation is taken to stand for the set of all ordered
pairs (x,y) for which ths statement about x and y is true. In
each of the Exercises 3 through 12, sketch in Cartesian
coordinates the sets indicated.
I
3. x,y) y = x and x £ 0^ .
h. x,y) | 7 * 4* + 2 and y > oj .
5. x,y) | y = Ux + ? and - • < x < <•> } .
6. { x,y) j y • Ux + 2 .and y - 3x + 2
and - oo < x< • } .
7. {
x,y) | — <» <x< oo and — • <y< 00 J
{
I
0. x,y) y "/x and x k 0 } .
{
9. ,y) | y = x 2 and y > x and x iO) .
10. ,y) I x2 + y2 < h} .
11. ,y) 1 x2 + y2 £ U and y £ -5 } .
12. ,y) I y - (x 2
- 2) } .
13. The table exhibits corresponding values X f(x)
of the variables x and f(x). By guessing, 2 h
formulate a simple analytic expression 3 9
for f(x) and use it to find the missing h 16
values of f(x). 5 25
6 7
7 7
Ik. Will f(x) = x + (x - 2)(x - 3)(x - U)(x - 5) satisfy the
set of values giver, in the table of exercise 1? Find the
Hissing entries using this rule of correspondence.
Sec. 1.2
15* Plot in the Cartesian coordinate system the values of
f(x) versus selected values of x, given
fx • 2 when xx ^- 0,
f(x) - 4 0
^ xr when x < 0.
Describe the domain and range*
16* Plot in Cartesian coordinates corresponding values of
g(x) versus selected values of x, given
, f2x + 3 when x ^ 0,
x
•s(x> a i
I U when x > 0.
Describe the domain and range.
17. Plot in Cartesian coordinates corresponding values of
f(t) versus selected values of t, given
t|_L_l when t / 1,
f(t)
2 when t • 1*
Describe the domain and range*
18. Given f (x) * x2 when x > 0,
g(x) • x when - « < x < + •
Do f and g represent the same function? Explain*
19. Given f(x) » |x| (The symbol |x| means the numeri­
cal or absolute value of x*)
fx when x A 0,
g ( x) " \-x when x < 0
#
Do f and g represent the same function? Explain*
20* Given a (t) - cos(-t) for all t,
0 (t) * cos t for all t.
Do ot and represent the same function? Explain*
21. Given f(x) - x2 for all x,
g(x) - (-x)2 for all x.
Do f and g represent the same function? Explain.
22. Given f(x) » x2 for -2 < x < 3,
g(x) • 3x for x > -1. What is the domain of
x for which f(x) • g(x) is defined?
o Sec. 1.3
23. Given <X(t) - t2 - U fcr t<5,
0(t) - t • 2 for t<3.
What is the domain of t for which fi^f is defined?
1.3 Introduction to the Limit Concept
Starting from vague sense impressions, man has had to
invent, perhaps over a period of thousands of generations,
such concepts a3 space, object, distance, time interval,
motion, and speed in order to acquire his existing store
of knowledge with which he tries to control his environ­
ment. It would not be an exaggeration to say as Conant
has said, "Science advances not by the accumulation of
new facts, but by the continuous development of new and
fruitful concepts.n
One of the most fruitful concepts invented by man is
that of instantaneous speed. However, we shall see that
this is a special case of a more general concept, namely,
that of limit.
Let us put ourselves in the position of an earlier
generation of men to whom the concept of speed had not
yet by the long process of usage and familiarity become
trivial and obvious. We might in this position of early
primitive man have become aware of the futility of
attempting to catch certain animals by direct oursuit.
We would have developed a vague concept of elusiveness
which included without distinction a conglomeration of
the ideas of swiftness, cnange of direction, change of
pace, and feinting, but excluded such ideas as physical
power and terrain conditions. With additional experience,
the need for a more refined concept might have asserted
itself. No doubt, some animals, relying almost exclusive­
ly on their fleetness, would have run in a fixed direction
from their pursuers. This could have led eventually to
the recognition of some men as eminent hunters because of
their ability to overtake these animals. Hence a
superior hunter might have evoked in the minds of his
tribesmen a dynamic image of rapidly moving feet, and of
diminishing distance between him and his quarry. At some
time we would have recognized that the pursuit of one
animal by another was vaguely similar to our own pursuit
of or by animals. Slowly over the ages distinctions be­
tween similar events would have become common sense