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DIFFERENTIAL CALCULUS



FOR



BEGINNERS.



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.a.



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DIFFEKENTIAL CALCULUS



FOB



BEGINNEES.



BT



JOSEPH EDWARDS, M.A.

FOBMBBLY FELLOW OF SIDNEY SUSBXX COLLEGE, CAMBBIDOE.



Hondo n:
MACMILLAN AND CO., Ltd.

NEW YORK: MACMILLAN & CO.
1896



[All rights reserved.}



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1/




First Ediiion 1893. Reprinted 1896.



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PKEFACE.

The present small volume is intended to form a
sound introduction to a study of the Diflferential Cal-
culus suitable for the beginner. It does not therefore
aitn at completeness, but rather at the omission of all
portions which are usually considered best left for a
later reading. At the same time it has been con-
structed to include those parts of the subject pre-
scribed in Schedule I. of the Regulations for the
Mathematical Tripos Examination for the reading of
students for Mathematical Honours in the University
of Cambridge.

Particular attention has been given to the examples
which are freely interspersed throughout the text. For
the most part they are of the simplest kind, requiring
but little analytical skill. Yet it is hoped they will
prove sufficient to give practice in the processes they
are intended to illustrate.

It is assumed that in commencing to work at the
Diflferential Calculus the student possesses a fair know-



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VI PREFACE.

ledge of Algebra as far as the Exponential and Logarith-
mic Theorems ; of Trigonometry as feur as Demoivre's
Theorem, and of the rudiments of Cartesian Geometry
as far as the equations of the several Conic Sections in
their simplest forms.

Being to some extent an abbreviation of my larger
Treatise my acknowledgments are due to the same
authorities as there mentioned. My thanks are also
due to several Mends for useftil suggestions with regard
to the desirable scope of the book.

Any suggestions for its improvement or for its
better adaptation to the requirements of junior
students, or lists of errata, will be gratefully received.



JOSEPH EDWARDS.



80, Cambbzdob Gabdbns,

NOBTH EBNSINaTON, W.

December, 1892.



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CONTENTS.

CHAPTER I.
LnaTZNO Yalubs. Eubmbntaby Undbtbbmined Fobms.

PAOB

Definitions 1 — 6

Four important Limits 6 — 7

Undetermined forms: method of procedure 8 — 11

CHAPTER IL

DiFFEBBBTlATIOH FBOM THB DEFINITION.

Tangent to a Curve. Definition. Equation . 18 — 14

Differential Coefficient 14—17

A Rate-measurer 18—20

CHAPTER in.

FUNDAMBNTAL PBOPOSITIONS.

Constant. Sum. Product. Quotient .... 22—25

Function of a function 27—29



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vm CONTENTS,

CHAPTER IV.
Standabd Fobms.

PAGE

Algebraic, Exponential, Logarithmic Forms 31 — 33

Direct and Inverse Trigonometric Fmictions . 34 — 38

Table of Results 39—40

Logarithmic Differentiation 40 — 41

Partial Differentiation 44 — 47



CHAPTER V.

SnOOESSIYE DiFFEBENTIATION.

Standard results and processes 52 — 54

Use of Partial Fractions and Demoivre's Theorem . . 55 — 56

Leibnitz's Theorem 67 — 60

Note on Partial Fractions 61—63



CHAPTER VI.

Expansions.

Elementary Methods 66—68

Taylor's Theorem . 69—70

Maclaurin's Theorem 71—73

By the Formation of a Differential Equation . . 73 — 77

Differentiation or Integration of a Known Series . 77 — ^79



CHAPTER Vn.

Infinitesimals.

.llness. Infinitesimals
)mall Arc from Chord



83—86
87



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CONTENTS. ix

CHAPTER VIIL
Tangents and Nobsials.

PAGE

Equations of Tangent and Normal 90 — 96

Cartesian Snbtangent, Subnormal, etc 97 — 99

Values of 3- , -=- , etc 100—101

Polar Co-ordinates, Subtangent, Subnormal, Angle between

Radius Vector and Tangent, etc 102—106

Pedal Equation 106—107

Maximum number of Tangents or Normals from a given

point 108—110

CHAPTER IX.

Asymptotes.

Methods of finding Asymptotes of a Curve in Cartesians . 115 — 125

Polar Asymptotes 126 — 129

CHAPTER X.

Cubvatube.

Expressions for the Radius of Curvature. . 132 — 150

Co-ordinates of Centre of Curvature .... 152 — 153

Contact 153—168

CHAPTER XL

Envelopes.

Method of finding an Envelope 162 — 168

CHAPTER Xn.
Associated Loci.

Pedal Curves 178 179

Inversion 180—181

Polar Redprooal 182

Involutes and Evolutes 188—187



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X CONTENTS.

CHAPTER Xm.
Maxtma and Mnma.

PAOE

Blementary Algebndoal and Qeometrioal Methods . . 192 — 196
The General Problem 197—206

CHAPTER XIV.

Undbtbbmined Fobms.

Applications of the Diffexential Calcnlns to the several

forms 210—219

CHAPTER XV.

Limitations of Taylob's Theobem.

Continuity 222—224

Lagrange-formnla for Remainder after n terms of Taylor's

Series 225—230



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DIFFEKENTIAL CALCULUS.



CHAPTER I.

LIMITING VALUES. ELEMENTARY UNDETERMINED
FORMS.

1. Object of the Differential Calculus. When
an increasing or decreasing quantity is made the subject
of mathematical treatment, it often becomes necessary
to estimate its rate of growth. It is our principal object
to describe the method to be employed and to exhibit
applications of the processes described.

2. Explanation of Terms. The frequently re-
curring terms "Constant," "Variable," "Function," will
be imderstood from the following example :

Let the student imagine a triangle of which two sides x, y are
unknown but of which the angle (A) included between those sides is
known. The area (A) is expressed by

A=4^ysinii.
The quantity ^ is a "constant " for by hypothesis it retains the same
value, though the sides x and jr may change in length while the
triangle is under observation. The quantities x^ y and A are there-
fore called variables. A, whose value depends upon those of x and y^
is called the dependent variable ; x and y^ whose values may be any
whatever,, and may either or both take up any values which may be
assigned to them, are called independent variables.

The quantity A whose value thus depends upon those of x, y and
A is said to be & function of x, y and A,

E. D. C. ^ 1



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2 DIFFERENTIAL CALCULUS.

3. Definitions. We are thus led to the following
definitions :

(a) A Constant w a quantity which, during any set
of mathematical operations, retains the same value.

(b) A Variable is a quantity which, during any set of
mathematical operations, does not retain the same value
but is capable of assuming different values.

(c) An Independent Variable is one which mjay
take up any arbitrary value that may be assigned to it

(d) A Dependent Variable is one which assumes
its value in consequence of some second variable or system
of variables taking up any set of arbitrary values iJiat
mxxy be assigned to them.

(e) When one quantity depends upon another or upon
a system of others in such a mjanner as to a^ssume a
definite value when a system of definite values is given to
the others it is called a function of those others.

4. Notation. The usual notation to express that
one variable y is a function of another x is

y=f{x) ovy^F{x) or y = 4>{x).

Occasionally the brackets are dispensed with when no
confusion can thereby arise. Thus^ may be sometimes
written for f{x). If u be an unknown function of
several variables x, y, z, we may express the fact by the
equation u =f(x, y, z).

6. It has become conventional to use the letters
a, b, c... a, fi,y... from the be^nning of the alphabet
to denote constants and to retam later letters, such as
u, V, w, X, y, z and the Greek letters f , i;, f for variables.

6. Limiting Values. The following illustrations
will explain the meaning of the term "Limiting
Value":



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LIMITING VALUES, 3

(1) We say *S = g, by which we mean that by taking enough sixes
we can make *666... differ by as little as we please from §.

(2) The limit of r- when x is indefinitely diminished is 3.

For the difference between =- and 3 is — ^ , and by diminishing

x-\-L a;4-i

X indefinitely this difference can he made less than any assignable
quantity however small.

The expression can also be written ^ , which shews that if x

l+-

X

be increased indefinitely it can be made to continually approach and
to differ by less than any assignable quantity from 2, which is there-
fore its limit in that case.

It is useful to adopt the notation Ltx=a to denote
the words "the Limit when a; = a of."

»., , 2aj-|-3 2a; + 3

(3) If an equilateral polygon be inscribed in any closed curve and
the sides of the polygon be decreased indefinitely, and at the same
time their number be increased indefinitely, the polygon continually
approximates to the form of the curve, and ultimately differs from it
in area by less than any assignable magnitude^ and the curve is said to
be the limit of the polygon inscribed in it.

7. We thus arrive at the following general defini-
tion:

Def. The Limit of a function for an assigned valuta
of the independent variable is that value from which the
function may be made to differ by less than any assign-
able quantity however smalT by making the independent
variable approach sufficiently near its assigned value.

8. Undetermined forms. When a function in-
volves the independent variable in such a manner that
for a certain assigned value of that variable its value
cannot be found by simply substituting that value of the
variable, the function is said to take an undetermined
form.

1—2

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4 DIFFERENTIAL CALCULUS.

One of the commonest cases occurring is that of a
fraction whose numerator and denominator both vanish
for the value of the variable referred to.

Let the student imarine a triangle whose sides are
made of a material capaDle of shrinking indefinitely till
they are smaller than any conceivable quantity. To fix
the ideas suppose it to be originally a triangle whose
sides are 3, 4 and 5 inches long, and suppose that the
shrinkage is uniform. As the shrinkage proceeds the
sides retain the same mutual ratio and may at any
instant be written 3m, 4m, 5m and the angles remain
unaltered. It thus appears that though each of these
sides is ultimately immeasurably small, and to all prac*
tical purposes zero, they still retain the same mutual ratio
3:4:5 which they had before the shrinkage began.

These considerations should convince the student
that the vltimate ratio of two vanishing quantities is not
necessarily zero or vmity.

a? — a?

9. Consider the firaction : what is its value

x — a

when ^ = a ? Both numerator and denominator vanish

when X is put = a. But it would be incorrect to assume

that the fi:uction therefore takes the value unity. It is

equally incorrect to suppose the value to be zero for the

reason that its numerator is evanescent ; or that it is

infinite since its denominator is evanescent, as the

beginner is often fallaciously led to believe. If we wish

to evaluate this expression we must neoer put x actually

equal to a. We may however put a? = a + A where h is

anything other than zero.

a^ — a?

Thus =2a + A,

x — a

and it is now apparent that by making A indefinitely
small (so that the value of x is made to approach inde-
finitely closely to its assigned value a) we may make the
expression differ from 2a by less than any assignable



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EXAMPLES. 5

quantity. Therefore 2a is the limiting value of the
given fraction.

10. Two functions of the same independent variable
are said to be ultimately equal when as the independent
variable approaches indefinitely near its assigned value
the limit of their ratio is unity.

Thus Lte^o —z~ — 1 ^7 trigonometry,

and therefore when an angle is indefinitely diminished
its sine and its circular measure are ultimately equal.

EXAMPLES.

1. Find the limit when a?=0 of ^ ,

(a) when y= mo?,
(6) wheny=^/a,
(c) wheny=aa;*+5.

.2. "FindLt y , (i) whena?=0, (ii) whena?=oo.

3. Find Z^aj.a ; ^^»-a ; ^«-aT3 «•

007 + -

4. i^d the limit of m i , (i) when d;=0, (ii) when 0?= oo .

ca? + -

• 6. The opposite angles of a cyclic quadrilateral are supple-
mentary. "What does this proposition Mcome in the limit when
two angular points coincide ?

•c* — Cic* + 1 1«P — 6

7. Evaluate the fraction — =—-5 — ^ - ^ — \ — s for the values

a?=ao,3,2,l,i, J,0, -00.

8. Evaluate Lt^.^^ and X^^^o '^^"^^'"^



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6 DIFFERENTIAL CALCULUS.

IL Timr Importaiit Limits. The following
timit« are important :

a) Xf#-o^-=l; i<#-oco8tf=l,

(II) le,.i^^ = n,

(III) Ltg^ * ( ^ "*■ ~ ) ~ ^' where 6 is the base of the

^ Napierian logarithms,

12« (I) The limits (I) can be found in any standard
text-book on Plane Trigonometry.

13, (II) Toproveifa^.i^— j-=n. Leta?=l+2^. Then

when X approaches unity z approaches zero. Hence we
can consider -g: to be less than 1, and we may therefore
Apply the Binomial to the expansion of (1 + zy what-
ever n may be.

r^ , a?** - 1 , (1+^)^—1

Thus Lt^^i • — -i = Ltg^o^^^^^-^

x—l z

nCn—l) .
- Ti ^ •



14 (III) To prove X««.«(l+^Y = e.
log,y = a?log,^l + -j.



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FOUR IMPORTANT LlMlTti, 7

Now X is about to become infinitely large, and there-
fore - may be throughout regarded as less than unity,

iXj

and we may expand by the Logarithmic Theorem.



Thus log« y — x



= 1 — X [a convergent series].

Thus when x becomes infinitely large
Lt \ogey = 1,
and Lty = e,

i.e. Lt^:^Jl + -^ =e.

a* — 1
16. (IV) To prove Ltg=.o = log«a.

X

Assume the expansion for a* viz.

a*= 1 + a? logett + 2| (logea)'* 4- . . . ,
which is shewn in Algebra to be a convergent seines.

Hence = log^a H-a? ^ ^, + ...

X ^ 2!

= log^a +a? X [a convergent series].

And the limit of the right-hand side, when x is in-
definitely diminished, is clearly log«a.



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8 DIFFERENTIAL CALCULUS,

X6. Method of procedure. The rule for evaluating

a function which takes the undetermined form - when

the independent variable x ultimately coincides with its
assigned value a is as, follows : — -.

Put x=a-\-h and expand both numerator and deno-
minator of the fraction. It will now become apparent
that the reason why both numerator and denominator
ultimately vanish is that some power of A is a common
factor of each. This should now be divided out. Finally
let h diminish indefinitely so that x becomes ultimately
a, and the true limiting value of the function will be
clear.

In the particular case in which a? is to become zero
the expansion of numerator and denominator in powers
of X should be at once proceeded with without any pre-
liminary substitution for x.

In the case in which a? is to become infinite, put
1

y

so that when x becomes oo , y becomes 0.

Several other undetetmined forms occur, viz. x oo ,

— , 00 — 00 , 0^ 00 ®, 1 *, but they may be made to depend

upon the form ^ by special artifices.

The method thus indicated will be best understood
by examining the mode of solution of the following
examples : —

This is of the form k if we put a?=l. Therefore we put x=l-\-h
and e|xpand. We thus obtain

^7_2x6+l_ (l + fe)7>2(l+/i)g+l

^^-^ V - 3x8+ 2 "■'^^*=* (1 + A)8 - 3 (1 + /i)2+ 2



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=!•«»-



'»-o-



V^DETERMINED FORMS. 9

■* (1 + 8^+3/1^+. ..)-8(l + 2/i+/i«) + 2
-3^+/t8+...
3^+...



- ^«»'«-3+...



■el-'-

Xt will be seen from this example that in the process of expansion
it is only necessary in general to retain a few of the lowest powers ofh,

a* -6*
Ex. 2, Find Lt^^o .

Here nnmerator and denominator both vanish if x be put equal to 0.
We therefore expand a' and b* by the exponential theorem. , > Hence

a* -6*
Ltg^o — z —



=Lt



|l+a ; logea+|j(log,a)'+..| - |l+glog,5 + ^(log,5)'+..|

X

=X«.»o |log.a-log.6 + |^(i^|»-log.6|«) + ..|



=log.a-log.6=log,^.

Ex. 3. Find
Since



-.-.(^1



tan X _ 1 sing
~i cosx' X '



, _^ tana: ,
wehaye -Lt,„o = !•

X

Hence the form assumed by ( — — j is an undetermined form 1*"
when we put x=0.

Expand sin x and cos d; in powers of x. This gives
1
I'«.-o(



■C^T-"-




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10 DIFFERENTIAL CALCULUS.

(X* \«i

1 +— +higher powers of « J






1



-".-.(■-¥)-■



where 2 is a series in ascending powers of x whose first term (and
therefore whose limit when a;=0) is unity. Henoe

1

Ex. 4. FiTid Lt,«xa;i-».

This expression is of the undetermined form 1"".
Put l-«=y,

and therefore, if a; = 1, ^ = ;

therefore Limit requiredsLe^.o (1 - yY^e"^ (Art. 14).

I
Ex.6. Xt.„«a;(a«-.l).

This is of the undetermined form ao x 0.
Put x^-,

y

therefore, if 2= ao , ^ =0»

o^-l
and Limit required=Xey»jo =log«a (Art. 15).

17. The following Algebraical and Trigonometrical series are
added, as they are wanted for immediate use. They should be learnt
thoroughly.

(l+x)«=l + n.+'Lg^:r'+ "<"-^j<;3-"> :^-^

a.= l+.log.a+^^^ + f^(^%

''=»+'+S+S+



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EXAMPLES. 11



j:* «• X*
loge(l+r)=a: - + - - -^ +

l0ge(l-*)=-«-^-3-;|-

tan-»«=«-^ + 5-



0<)8h«[which=^^±^ = l+^ + ||+

siiih*[wliich=^^^=*+?J + J+

EXAMPLES.

Find the values of the following limits :

1
4. X<«-o • 5. ^«-i ^-^-^+1 •

fi 7^ ^+^^-2 o f/ arco8J?~log,(l+ ;r)
o. //^x-o ^ • ^* ^*«-o ^ •

jrg»-lo g,(l+j; ) , jT^sinorcos ^
10. x^s-o ^ • *^« -^^-o ^ •

_^ y^ ^VDT^x-x .^ r^ coshoT-cosa;
12. Ltx^o—n • 13. J^«-o •

_. -^ sin^^o: - y. sin'^x-sinha?



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12 DIFFERENTIAL CALCULUS.

2






16. Z^x.o-

1 1+^

17. Z«..o ;? . 18. ^^-^ ^^^i^^^i.^y

19. X...,?^^^. 20. i^..o(^)'.
1 1

2L i;«.-»(^ . 22. X«._,(^-j-j.

i 1

23. Z*._o(^- 24. X*._.^-^j .

1 ...

25. Z«a-o (^v®'^ ^)'- 26. Z<,«5(cosecar)*«*^



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CHAPTER II.

DIFFERENTIATION FROM THE DEFINITION.

18. Tangent of a Curve; Definition; Direction.

Let AB be an arc of a curve traced in the plane of the
paper, OX a fixed straight line in the same plane. Let




•P, Qy be two points on the curve; PMy QN, perpen-
diciJars on OX, and PR the perpendicular from P on
QN. Join P, Q, and let QP be produced to cut OX

at r.

When Q, travelling along tiie curve, approaches in-
definitely near to P, the limiting position of chord QP is
called the tangent at P. QR and PR both ultimately
vanish, but the limit of their ratio is in general finite;

{orLt^^LttsaiRPQ^LttSiXiXTP^tangentofthe

(angle which the ta/ngeni at P to the curve makes with
OX.



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14 DIFFERENTIAL CALCULUS,

If y:=s^(x) be the equation of the curve and x,
ic + h the abscissae of the points P, Q respectively ;
then MP = ^(x), NQ = «^(a? + A), JXQ = «^(a? + A) - (l>{x)
and PR = A.

Thus i,|| = Z..=o^^^:t^^^^>.

Hence, to draw the tangent at any point (x, y) on the
curve y = ^{x), we must draw a line through that point,
making with the axis of a; an angle whose tangent is

J. 4>{x-^h)-4>{x) ^

-^^A=0 T J

and if this limit be called m, the equation of the tangent
at P (x, y) will be

X, Y being the current co-ordinates of any point on the
tangent ; for the line represented by this equation goes
through the point {x, y), and makes with tne axis of x
an angle whose tangent is m.

19. Def. — ^Differential Coefficient.

Let <f>(x) denote any function of x, and (l>(x + h) the

same function of x + h; then Lth^o— r — ^-^ is

called the first derived function or differential
coefficient of tf> (x) with respect to x.

The operation of finding this limit is called differ-
eiitiating ^ (x).

20. Geometrical meaning. The geometrical
meaning of the above limit is indicated in the last article,
where it is shewn to be f A^ tangent of the angle yfr which
the tangent at any definite point {x, y) on the curve
y^<f>(ic) makes with the axis of x.

21. We can now find the differential coefficient of
any proposed function by investigating the value of the



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GEOMETRICAL MEANING. 15

above limit ; but it will be seen later on that, by means
of certain rules to be established in Chap. III. and a
knowledge of the diflferential coeflScients of certain
standard forms to be investigated in Chap. IV., we can
always avoid the labour of an ah initio evaluation.

a^
Ex. 1. Find from the definition the differential ooeffioient of — ,

where a is constant ; and the equation of the tangent to the cunre
Here 0W=^,



0(x + ;i) = V__i.,
=-^*»-o — r- — =i'<»-o — :; —



h no,



a '

The geometrical interpretation of this result is that, if a tangent

be drawn to the parabola ay=x^ at the point (x, y), it will be inclined

2x
to the axis of x at the angle tan~^ — .

a

The equation of the tangent is therefore

Ex. 2. Find from the definition the differential coefficient of
log, sin- , where a is a constant.

Here (a;) =log« sin - ,

and , . x-hh . . x
^/ . rx ^/ \ log, sin *-log,sm-

, X h X , h

, 8m-cos-+cos-sm-

y^ 1, a a a a

=£«... jj log.

sin-
a



=i«».oT log, f 1 +- cot - - higher powers of * j

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16 DIFFERENTIAL CALCULUS.

by substituting for sin - and cos - their expansions in powers of - |
, - cot - - higher powers of h

=-^'*-"— ^ — J

[by expanding the logarithm]

.=-cot-.
a a

t/ X

Hence the tangent at any point on the curve -= log, sin - is inclined


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