Table Of ContentACADEMIC PRESS INTERNATIONAL EDITION
An Introduction to
ANALYTIC GEOMETRY
AND CALCULUS
A. C. Burdette
UNIVERSITY OF CALIFORNIA
DAVIS, CALIFORNIA
MF) ACADEMIC PRESS New York London
ACADEMIC PRESS INTERNATIONAL EDITION
This edition not for sale in the United States of America
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To my wife Emily
Preface
The addition of another calculus textbook to the large number already in
existence calls for some justification. In this case it is very simple. It was
written to satisfy an unfulfilled need for a textbook suitable for an existing
course in the University of California, Davis. Since it is plausible that our
requirements are not too different from those of others, it is hoped that this
book will prove to be a useful contribution to the field.
The general goals of the book may best be defined by giving a brief descrip
tion of the course for which it was designed. It is a one-year course, meeting
three times a week and presupposing a preparation equivalent to three
semesters of high-school algebra and one semester of trigonometry. It is
expected to provide the student not only with a reasonable degree of under
standing of the basic concepts, but also with a working knowledge of the
elementary operations of calculus. The presentation is at the freshman level
but it serves students of all ranks. Graduate students in various fields find it
particularly useful for making up mathematical deficiencies. For many, it is
their terminal course in mathematics.
Those looking for a completely novel treatment of calculus will be dis
appointed. It is true that the mean value theorem plays the central role it
should, but seldom does; that the definite integral stands on its own, not as a
mere adjunct to the antiderivative; that the choice of subject matter and the
level at which it is presented are carefully coordinated with the objectives to
be served ; and that there is a wider than usual range of applications. On the
whole, however, this is a traditional treatment of the subject. This is true, in
particular, of the notation and the terminology, in the belief that this best
serves the needs of the students involved. For example, the nonmathematics
major, reading in his own field, will find the mathematical models of his
problems stated in traditional notation, not the highly sophisticated notation
of the modern mathematician.
In general, I have held to the conviction that the historical road of develop
ment of a discipline is the simplest path to its understanding. This philosophy
explains why, for example, I have defined the logarithm function as the
inverse of the exponential function, rather than the more direct, and perhaps
more logical, definition as an integral. It is my opinion that this approach is
more meaningful to the beginning student in spite of its disadvantages. These
disadvantages worry him not at all at this point.
No attempt has been made to eliminate the need for an instructor, although
there are numerous examples and, hopefully, the student can use the text to
amplify and clarify classroom discussion. This book is intended to serve as
a basic core of material upon which an instructor can develop a course with a
considerable degree of latitude. For example, in Chapter 4 limits are defined, a
vii
vìii PREFACE
few proofs of simple cases are given, and then the general theorems are stated
without proof. The teacher who wishes to give a more rigorous course can
elaborate the work here by presenting proofs of these theorems, while the one
who wishes to give a more intuitive course can amplify the motivating
examples and soft-pedal the ε, ^-definitions.
The analytic geometry has been reduced to bare essentials in order to have
more time to devote to calculus. Even so, there is probably somewhat more
material than can be covered in the allotted time. Chapters 12, 13, 14 are
independent of each other so parts or all of these chapters may be omitted
without affecting the others. This permits a certain amount of flexibility to
adjust to local needs. At Davis, many of our students expect to take physical
chemistry so we cover Chapter 13 completely. On the other hand, we omit
Chapter 14 entirely and include only parts of Chapter 12.
The results of many informal discussions with my colleagues are to be
found in these pages. I thank them for this contribution. Also, I am deeply
indebted to Professor S. Saslaw of the United States Naval Academy who
read the entire manuscript in its original form and whose suggestions have
substantially improved the finished product. Special thanks are due Professor
E. B. Roessler, Chairman of the Department of Mathematics, University of
California, Davis, for permission to use a preliminary edition of this material
in the classroom.
Finally, I wish to acknowledge the valuable assistance rendered by the
editorial staff of Academic Press in bringing this project to completion.
December, 1967 A. C. BURDETTE
Davis, California
Chapter 1
THE COORDINATE SYSTEM-
FUNDAMENTAL RELATIONS
l-l. Introduction
This chapter, together with the next two, will be devoted to the task of develop
ing a modest geometric background from the algebraic point of view. This is
usually referred to as analytic geometry in contrast to synthetic geometry
already studied in high school or elsewhere. Analytic geometry will serve us
well as a basic tool as we proceed with our study of calculus.
I-2. Directed Lines
In elementary geometry a line segment is a portion of a line defined by the
two points which mark its extremities. If the two end points are designated by
A and B, we label a line segment AB or BA, making no distinction between
the two notations. When the processes of algebra are applied to the problems
of geometry, as they are in analytic geometry, we find it useful to define
directed lines.
DEFINITION l-l. A directed line is a line on which a positive direction has
been assigned.
A line segment measured in the positive direction is considered positive and
the same portion of the line measured in the opposite direction is considered
negative. The positive direction may be assigned arbitrarily although, for
consistency of notation and convenience in interpreting results, there are some
situations in which certain positive directions are used ordinarily. For
example, on horizontal and vertical lines the positive directions are usually
taken to the right and upward, respectively.
DEFINITION I-2. On a directed line, BA = -AB.
The following theorem regarding directed line segments will be useful in
later work.
2 I. THE COORDINATE SYSTEM—FUNDAMENTAL RELATIONS
THEOREM l-l. If A, B, and C are any three points on a directed line,
AB + BC = AC. (1-1)
The truth of this theorem may be established by considering individually
the six possible cases of arrangement. For example, let the points be arranged
as indicated in Fig. 1-1, separated by the distances a ^ 0 and b ^ 0, and let
a b
—i 1 (-►
C A B
Fig. 1-1
the positive direction on the line be that of the arrow. Thus
AB = b, BC=-b-a, AC=-a
and (1-1) becomes
b + ( — b — a) = —a,
which verifies the theorem for this case. The other possibilities can be treated
in the same manner.
1-3. Cartesian Coordinates
The basis of the cartesian, or rectangular, coordinate system is a pair of
mutually perpendicular directed lines, X'X and Y'Y, on each of which a
number scale has been chosen with the zero point at their intersection, O.
This point is called the origin and the lines are called the x-axis and y-axis.
The x-axis is usually taken in a horizontal position and, in that case, it is
customary to take the positive directions on the two axes to the right and
upward. This will be our choice unless specifically stated to the contrary.
Let P be any point in the plane and drop perpendiculars from it to each of
the axes, thus determining the projections M and N of P on the x-axis and y-
axis, respectively (Fig. 1-2). The measure of the directed segment OM in the
units of the scale chosen on the x-axis is called the x-coordinate, or abscissa,
of P. Similarly, the measure of the directed segment ON in the units of the
scale chosen on the j-axis is called the y-coordinate, or ordinate, of P. Let the
measures of OM and ON be Xi and y respectively.! Then the position of
u
the point P with respect to the coordinate axes is described by the notation
(xi,j>i). As indicated, the abscissa is always written first in the parentheses.
The pair of numbers (x y^) is referred to collectively as the coordinates of P.
1?
When the coordinate axes have been selected and their corresponding
number scales chosen, then to each point in the plane corresponds a unique
pair of numbers and, conversely, to each pair of numbers corresponds a single
point. Thus a coordinate system attaches a numerical property to points and
t Hereafter, for simplicity of statement, we will often speak of a segment AB when
we mean the measure of AB. This should cause no confusion since the meaning
will be clear from the context.
1-3. CARTESIAN COORDINATES 3
Y
P(xi,y
N
-<p
O M '
γ·
Fig. 1-2
this makes it possible to apply the processes of algebra to the study of geo
metric problems. This is the distinguishing characteristic of analytic geometry.
There are many physical situations in which the quantity measured by x
is entirely different from that measured by y; for example, x and y may
represent time and temperature, respectively. If such situations are recorded
on a coordinate system, there is no need to use the same scale on the two axes
unless inferences are to be drawn from geometric considerations. However,
when geometric relationships such as angles between lines and distances
between points are involved, it is important that the same scale be used on
both axes. Since many physical applications of analytic geometry and calculus
hinge on geometric properties, it will be convenient for us to make a blanket
assumption on this subject:
Unless stated to the contrary, it is to be understood throughout this book that
the same scale is used on both coordinate axes.
The coordinate axes divide the plane into four quadrants. These are numbered
as shown in Fig. 1-3. Such a numbering is convenient for describing certain
general situations.
Y
II I
o
III IV
Fig. 1-3
4 I. THE COORDINATE SYSTEM—FUNDAMENTAL RELATIONS
1-4. Projections of a Line Segment on Horizontal and Vertical
Lines
Let M and M be the projections of Λ(·*ι>>Ί) and ^2(^5^2)» respectively
x 2
(Fig. 1-4), on the x-axis. The directed segment MM is said to be the pro
l 2
jection ofPP on the x-axis. Clearly, the projection of PP on any horizontal
l 2 l 2
line is equal to MM . Moreover, from (1-1)
i 2
MM = M O + OM = OM - ΟΜ.
i 2 x 2 2 γ
Ρι{Χι
Nì \
1
1
Mx O M2
Fig. 1-4
But by definition
OM = x OM = x ,
x l9 2 2
and we have
MM = x -x. (1-2)
l 2 2 l
Similarly, the projection of PP on any vertical line is equal to its projection
l 2
Λ^Λ^ on the >>-axis and
NN = N O + ON = ON - ON
l 2 x 2 2 u
or
N N = y-y. (1-3)
1 2 2 i
Example l-l. Given the two points A(3, —2), B( — 2, 5), find the projections
of AB on the coordinate axes.
Choose A as P and B as P and formulas (1-2) and (1-3) give the required
l 2
projections :
M M = (-2)-(3)=-5,
1 2
N JV = (5)-(-2) = 7.
1 2
Note that the opposite choice of P and P would change the sign of these
1 2
results. This is as it should be since this changes the direction.
1-5. MIDPOINT OF A LINE SEGMENT 5
1-5. Midpoint of a Line Segment
Let P(x, y) be the midpoint of the line segment whose end points are
P\(xu JiX Pi(x2 > y2)9 and let M, M M , respectively, be the projections of
u 2
these points on the x-axis (Fig. 1-5). Then, making use of (1-2), we have
P P _M M _x-x
X 1 l
¥F ~ MM ~ x - x '
2 2 2
and, since P is the midpoint of P\P >
2
PP
t
= 1.
PPy
P(x, yi)
2 2
P(x,y)\
Λ(Χι,^ι)
M, M O Mi
Fig. 1-5
Therefore
X X\
1,
X7. x
from which, solving for x, we obtain the formula
Χγ + X
2 (1-4)
Similarly, using the projections of P, P P on the >>-axis, we have
u 2
y 1 + y 2
y = —^—· (1-5)
Example 1-2. Determine the point P(x, y) which bisects the line segment
joining the points (2, —3), (4, 5).
From (1-4) and (1-5),
2+4 - 3 +5
* = — τ- = 3, }> = — = 1.
Thus the required point is (3, 1).
