ELEMENTS

DIFFERENTIAL CALCULUS.

ED G A R W”

. B A S S ,

Colonel Un i ted S ta tes A r my, R et i r ed Pr of essor of Ma t/i ema t ics in tke

U. 5 . M i l ita ry A ea a’ehzy Apr i l 1 7 , 1 878, to O ctoéer 7 , 1 898 .

S E C O ND E D I T I O N ,

F IR ST TH O USAND .

NEW YO RK

J O H N W I L EY S O N S .

LO NDO N : CHAPMAN &'

HALL , L IMI TED.

Copyright , 1 90 1 ,

EDGAR W. BASS°

RO BERT DRUMMO ND , ELECTRO TYPER AND PR INTER , NEW YO RK.

PREFACE.

T H I S text-book has been prepared for the use of the

Cadets of the U . 8 . MilitaryAcademy who begin the sub

jec t with a knowledge of the elements of Algebra,Geometry,

and T rigonometry which ranges from fair to excellent .

T he time al lotted to the subj ec t ( ten and one hal f weeks) ,and the requirements of the subsequent courses , espec ial lyMechanic s

, O rdnance and Gunnery, and Engineering, limitand determine the scope of the work.

My experience leads me to the belief that the more rig

orous and comprehensive method of infinitesimals is suitab le only for a treatise

,and not for a text-book intended

for beginners .At the same time I believe that anypresentation of the

sub j ect,no matter how elementary, should in no manner

prejudice the student against any established method. O n

the contrary, it should, I think, endeavor to lead him to an

understanding of the relations between those in general use,

and, above all,it should aim to construct the best possible

ground work for the subsequent studyof. the subj ec t treatedin the most rigorous and extended form.

T he principle of interchange of infinitesimals,which con

sis ts in replacing one infinitesimal by another when unity

iv PREFA CE.

is the limit of their ratio, has been used to overcome the

difficulty encountered by beginners in the determinationof the limit s of ratios of infinitesimals .

T o the O fficers of the U . S . Armywho have taught thesubjec t with me, and in many cases to my pupil s , I am

greatly indebted for much valuable assistance.

T o CaptainWm. Croz ierf L ieut . J. A. Lundeen,L ieut . H .

H . Ludlow ,and Lieut. F . Mc Intyre I am under ob ligations

for many demonstrations and solutions .As sociate Profes sor W. P. Edgerton has been my col lab

orator throughout the work , and to him much c redit is due

for numerous demonstrations , improvements , and sugges

I have added a list of the works of authors wh ich I havefreely consulted in the preparation of this book , for the

purpose of acknow ledging my indeb tedness tothem,and

for the benefi t of students who may des1re to extend theirknow ledge of the subj ec t.

EDGAR W. BASS.

WEST PO INT , N. Y June 1 5, 1 896.

PREFACE TO THE SECO ND EDITIO N.

F O R the correc tions and changes made in this edition Iam indeb ted to the Department ofMathematic s

, U . S. M. A.

My thanks are espec ial ly due to Professor W. P. Edgerton,

Assoc iate Professor Chas . P. Echol s, L ieut. George B .

B lakely, and L ieut . F . W. Coe .

EDGAR W. B ASS.

524 F I FTH AVENUE, NEW Y O RK CITY ,

L I ST O F AUTHO RS WHO SEWO RKS HAVE BEEN

CO NSULTED IN THE PREPARAT I ONO F T H I SB O OK.

AMER ICAN.

Church,R ice and Johnson,Byerly,T aylor,

ENGL I SH .

T odhunter,Wil liamson

,

Edwards,*

FRENCH .

Harnack’

s Introduction to the Calculus , translated fromthe German by Cathcart . is a rigorous treatment of the

subjec t .

T reatises

Greenhill,*

Price,

*

Haddon, Examples.

CO NTENTS.

D I F FERENT IAL CALCULUS.

INTRO DUCT I O N.

DEF INI TI ONS . N O TA TI ON AND F UNDAMEN TAL PRINCIPLES

CHAPTER I . CO NSTANTS, VAR IABLES, AND FUNCTI O NS .

I I . PRINCI PLES O F L IMI TS .

I I I . RATE O F CHANGE O F A

PART 1.

DI F FEREN TIAL S AND D I F FEREN TIA TI ON .

IV. THE D I FFERENTIAL AND D I FFERENTIAL CO EF

F I CIENT

V. D I FFERENTIATI O N O F FUNCTI O NS .

VI . SUCCESSIVE

VI I . IMPL ICIT FUNCTI O NS AND DI FFERENT IAL EQUA

VI I I . CHANGE O F THE INDEPENDENT VARIABLE .

PART I I .

ANAL Y TI C APPL [ CA TI ONS .

IX. LIMITS O F FUNCTI O NS WH ICH ASSUME INDETER

MINATE F O RMS .

X . DEVELO PMENTSXI . MAX IMUM AND MINIMUM STATES. .

viii CON TEN TS .

PART I I I .

GEOMETR I C APPL I CA TI ONS .

PAGE

CHAPTER XI I . TANGENTS AND NO RMALS .

XI I I . ASYMPTO TES . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

X IV. DI RECTI O N O F CURVATURE. SINGULAR PO INTS . .

XV. CURVATURE O F CURVES

XVI . INVO LUTES AND EVO LUTES .

XVI I . O RDER S O F CO NTACT O F CURVES AND O SCU

LATI NG LINES

XVI I I . ENVEL O PESXIX . CURVE

XX. APPLI CATI O NS To SURFACES .

DIFFERENTI AL CALCULUS.

INTR O DUCT IO N .

DEF INI T I ONS , N O TA TI ON AND F UNDAMEN TAL

PR INCIPLES .

C H A PT ER I .

CO NSTANTS,VAR IAB LES AND FUNCT IO NS .

I . In the Calculus quantities are divided into two

general c lasses, cemtarzz

‘

s and w r z’

aéles.

A Constant is a quantity that has, or is supposed to

h ave,an absolute or relative fixed value .

A Variable is a quantity that is, or is supposed to be,

continual ly changing in value.

I n general,constants are represented by the fi rst let ters

of the alphabet, and variables by the las t but they Shouldnot

,therefore, be confused with the known and unknown

quantities of Algebra.

T he same quantity may sometimes be either a variableo r a constant

,depending upon the c ircumstances under

which it is considered. T hus,in the equation of a curve

,

the coordinates of its points are variables ; but in the

2 D I F FEREN T I AL CAL CUL US .

equation of a tangent to the curve,the coordinates of the

point of tangency are general ly treated as constants . I t is,

therefore, necessary to determine from the c ircumstances ,or obj ec t in View

,wh ich quant ities are to be regarded as

variables, and wh ich as cons tants,in each discussion.

I n general , any or al l of the quantities represented byletters in any mathematic al expression or equation may

have definite values assigned to them,and be regarded as

constants or theymaybe considered as changing in value ,and treated as variables . T hus

,in the express ion 4 7rr r is

a constant if we suppose it to represent the radius of a par

ticular sphere but if r is considered as changing in value,

it will be a Variable . I n the first case, 4 7rr

2

is a cons tant,and measures the surface of a particular sphere ; but whenr is variable

, 4 7172

is al so variab le,and represents the surface

of any sphere no matter how much it may increase or

L. 1I11 lIl lS ll .

I t should not be unders tood, however, that we may in

all cases treat quantities as constants or variables at pleasure without affecting the charac ter of the magnitude rep

resented by the expres sion or equat ion. F or example , 72'

is general ly assumed to represent the ratio of the Ci rcumference of anycirc le to its diameter, which ratio is invariable .

I f a different value be ass igned to 7x,the expres s ion 4 7 7

2

wil l not measure the surface of a sph ere whose radius is r .

I n some cases variation in a quantity changes the di

mensions of the magnitude represented by the expression

or equation ; in others it c hanges the position only ; and

again it may change the charac ter of the magnitude. T hus,

if we suppose R to vary in the equation

(x 6 02

+ (y — fi f = R i

CON S TAN T S , VAR I AB LES AND F UN CT I ON S . 3

we shall have a series of c irc les differing in Siz e ; but by

changing a or fl and not R the position only will beaffec ted.

By changing 52with in positive limits , the equation

R22 an:

represents different el lipses , but negativevalues for I 2 cause the equation to represent hyperbolas . I ngeneral

,however, cons tants are supposed to have fixed

values in the same expres s ion, unles s for a particular discussion it is otherwise s tated.

FUNCT I O NS .

2 . A quantity is a f unction of another quantitywhenits Value depends upon that of the second quantity. T hus

,

4d x is a func tion of 4 , a , and x . I n general , anymathemat

ical expression wh ich contains a quantity is a func tion of

that quantity. I f,however, a quantity disappears from an

expression by reduction or simplifi cation the expression isnot a func tion of that quantity. T hus , x

2 —l (e x) (e x ) ,( ix/bx ,

and tan x cot x ,are not func tions of x .

3 . A fonction of a sing le w r ioole is one whose valuedepends upon that of a single variable and varies

'

with it.T hus

,

4x2

/ ( I x4

) , 2px ,log (a x ) , sec 90

,

In which n is the onlyvariable, are functions of a singlevariable.

Any func tion of a single variab le is also a variable, andvaries simul taneously with the variable.

4 . The relation between a function of a variable and its

variab le is one of mutual dependence . Any change in the

value of one causes a dependent vari ation In that o f the

other . Either may, therefore . be regarded as a func tion o f

4 D I F FEREN T I AL CAL CUL US .

the other ; and they are cal led inner se functions. T hus,if

x passes from the value 2 to 3 , the func tion 2x2

wil l varyfrom 8 to 1 8 and conversely, A

“

wil l increase from 2 to 3, if

2 902

changes from 8 to I 8 . I f x be again increased the same

amount, that is from 3 to 4 , the func tion wil l increase from1 8 to 32 . Similarly, with o ther func tions we shal l find that

,

in general , equal changes in the variables do not give equalchanges in the corresponding func tions .

I t is therefore necessary, in referring to a change in a

func tion cor responding to a change in the variab le,to con

sider the states from wh ich and to wh ich the func t ion and

variab le change , as wel l as the amount of change in each .

With that unders tanding, cor r esponding changes in a func

tion and its variable are mutually dependent .

the equation of a curve,the ordinate o f any point is a

func tion of the absc issa, and the absc issa is the inversefunc tion of the ordinate .

T he func tion is considered as dependent , and the vari

able as independent ; for which reason,the latter is cal led

t/ze independent w riooie,or simplyMe w r ioole.

R epresent ing a func tion of x,as x

3

,byy,

we have y x3

;

0 03

solvmg W i th respec t to x,we have x a form express

ing directly x as a func tion of y.

T he differenc e in form in the following important exam

ples of direc t and invers e func tions should be observed.

H aving, y x”

,

y = a +m

y2 : ax ; x z y/cz.

ox

; x logay.

CON S TAN T S , VAR IAB LES AND F UN CT I ON S .

5 . A state of a function cor responding to a value '

or ex

pres sion for the variable is a result obtained by substitutingthe value or expression for the variable in the func tion .

T hus,

-oo,

—1 6cz

,—

2 cz,

are the states of the func tion z ax3corresponding, respec

tively, to the values or expressions for x ,

—w,

are the states of the function sin gocorresponding, respectively, to the expressions or values of go,

fl, afl/z ,

A func tion of a variable has an unlimited number of

states . I t may have equal states corresponding to differentvalues of the variab le and it may have two or more statescorresponding to the same value of the variable. T hus

,

s and I, H MS , I 3 and 5, g i t

/Zr, 2 5 and 1 3 ,

are the s tates of the func tion 2 x I : l: corresponding,respec tively, to the values of x ,

T rigonometric func tions have equal states for all anglesdiffering by any entire multiple of

I n connec tion with any s tate of a func tion correspondingto any value of the variable , it is frequently necessary toconsider another s tate of the function, wh ich results from

6 D I F FEREN T I AL CAL CUL US .

increasing the value of the variable corresponding to the

first state by some convenient arbitrary amount In orderto dis tinguish between these two states of the func tion

,the

first is designated as a pr imitive state, and the other as its

new or second state.

Any arbitrary amount by which the variab le is increasedfrom any assumed value is cal led an increment of tne var i

able. I t is general ly represented by the letter It, or It , or byA written before the variable as

,A x

,read increment of

90

L et x’ represent any particular value of x

,and It

,or A x

’

,

its inc rement then wil l mac and 2 a (x’

n)2

, or

2 a (oc’

A x’

)2

,represent, respec tively, the primitive and

new states of the func tion zax2

,corresponding to x

’and its

inc rement It , or A x’. T he general expression 2 a (x It)

2

represents the second state of any primitive state of the

func tion mod

,and from it we obtain the second state corre

sponding to any par ticular primitive state by subs titutingthe proper value of x .

6 . A function of two or more oar iaoles is one wh ich de

pends upon two or more variables and varies with each .

T hus ,

as siny, 3) log or,

are functions of x andy and

x ”Ty ‘ l" 3 } ” l" tan Z Sin2

(x2

y) ’ +y2

+ log Z,

are func tions of x , y and 2 . Each variable is independentof the others . Particular values or expres sions may be

* Increment as here used is in an algebraic sense, and includes a

decrement , wh ich is a negative increment . In general , an assumed

increment of a variable is regarded as positive .

CON S TAN T S , VAR I AB LES AND F UN CT I ON S . 7

assigned to one or more of the vari ab les,and the result dis

cussed as a func tion of the remaining variable s . A func

tion of two or more variab les possesses all of its properties

as a func tion of each variable . By subs tituting in the

func tion 2 x2

+y,any assumed value for y,

as 5, the result2 x

2

5 is a func tion of a single variable .

7 . A quantity is a func tion of the sum of two va r iaéles

when every operation indicated upon either variab le inc ludes

the sum of the two . T hus ,

Sin (x ziz y) , 10 g (90 Sky) ,

and all algebraic express ions wh ich may be written in the

+ H,

in wh ich A ,B

,etc .

,are constants , are functions of the sum

of the two variables x and i y.

—Jf x

x —i—y’x Sin (30 “

JO?

are not func tions of the sum of x andy.

Sin (cc2

i f ) , A (ac2

3 log (ac2

i f ) , iii

/Ax“ i y

?

) + 7a ,

are func tions of the sum of the two variables x “

and i y”

,

but not of the sum of x and i y.

Ni t/5 + ayz

) , 2 Vllog (oi/5+ ay

2 —3c) ,

are functions of the sum of the two variables t W and of .

I n any func t ion of the sum of two variables, a single

variable may be subs tituted for the sum,and the original

func tion expressed as a func tion of the new variable .

8 D I F FEREN T I AL CAL CUL US .

T hus,2 may be substituted for (x +y) in the func tion

a (.rz giving the func tion in the form az”

. In a similar

manner we may write

tan (a: — y) tan z,ax ‘ l‘ fi’ 2aVlog (oc

~—y) z en/ log 3 ;

but it must be remembered that z in the new form is a

func tion of the two variables x and y.

8 . A state of a func tion of two or more variables , corresponding to a set of values or expressions for the variab les ,I S the result obtained by substituting those values or ex

pressions for the corresponding variables . T hus ,

2 0,

are states of the func tion 4x + 3y 2 corresponding , re

spec tively, to the values or expressions for x and y,

4 ) 8) (O ) I ) , ( 2 ;

are s tates of the function tan (x +y) corresponding, re

spec tively to the values or expressions for x and y,

M9 ) . (o , fl/2 )

Any func tion, in wh ich all of the variab les are indepen

dent, is a variable, and has an unlimited number of s tates .

9 . A func tion of several variables may be equal to some

constant value or expression in which c ase one of the

variab les is dependent upon the others . T hus, the first

member of the equation 2x z 7 is a func tion of the

two variables , x and y but x and y are mutual lydependent.

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I O D I F F EREN T I AL CAL CUL US .

when all of its terms are of the same degree with respect to

A linear func tion is a homogeneous function of the fi rstdegree .

1 1 . A Transcendental func tion is one that cannot be

expressed definitely by the ordinary operations of Algebra.

In general , a transcendental func tion may be expressedalgebraically by an infinite series .T ransc endental func tions inc lude exponential, log ar itkmic,

tr igonometr ic, inver se tr igonometr ic, Icyperoolic and inverse

Izyperool ic func tions .

An Exponentia l func tion is one w ith a variable , or an in

commensurab le cons tant,exponent ; as

,

(a/x I )x

,z ex

A Logarithmic func tion is one that contains a logarithmof a variable ; as,

log (a +y) , zax x/log x .

*

A Trigonometric func tion is one that involves the sme,or cosine

, or tangent, etc .,of a variable angle as

,

cot x , sec 2x2

, (ao— Sin x)/x3

, versin290.

An Inverse Trigonometric func tion is one that containsan angle regarded as a func tion of a variab le s ine

,or cosine

,

or tangent, etc .

Sinr f

y,tan

“ 1

y,read the angle whose Sine is y whose

tangent isy are symbol s used to denote such func tions .Having given y versin x

,then x versin-

y ; and if

u cos y,then y cos

—“u, etc .

Napierian logar ithms w il l a lways be cons idered unless some o therbase , as a

,is indicated by loga .

CO N S TAN T S . VAR I AB LES A IVD F UN CT I ON S .I I

1 2 . Hyperbolic Functions . F rom T rigonometry, we

r : x

cos x I

2

Placing V i,and subs tituting x i for x ,

we have

Sin x i i (xx"

x7

x2

x4

x6

0

: tCo0 0 8m I +2+

4+

F rom Algebra,we have

e"

- I + x +z +x3

+ + etc . ;

e-x

z I etc .

II ence,

x2

x4

x6

I

2etcos

e" x x x

6

x “ I" etc .

T herefore,

Sin xi cos xi : (ex

cos x i is real,and is cal led the Izyperool ic cosine of 90. I t

is general lywritten cosh x . T he real fac tor in the Sine of

x i I S c al led the nyperéolic sine of x, and is written Sinh x .

Sinh x cosh x (ex

1 2 D I F FEREN T I AL CAL CUL US .

F rom which ,cosh 2 x S inh 2 x z I .

Comparing with x2 —

y2

I,we see that cosh x and Sinh

x may be represented by the coordinates of points of an

equilateral hyperbola, referred to rec tangular coordinate

axes, with the origin at its centre. Hence the name hyperholic functions . T he functions Sinh x and cosh x are not

periodic func tions for real values of x,but increase with x

indefinitely.

By analogy with trigonometric func tions,

other hyper;

bolic functions are defined, and written, as

tanh x

coth x

sech x

cosech x

I t fol lows that

tanhQx sech2 1 coth 2 x cosech 2 x .

1 3 . InverseHyperbol ic Functions.

we have x Sinh“ l

y.

F romy Sinh x : (ex

we find exz yd: VI +y

”

;

Writingy Sinh x,

henc e,Sinh

“ l

y x log Similarly,

y cosh x (ex

e“ x

)/ 2 gives cosh“ l

y= log (y : l: Vy2

y tanh x gives tanb"y 5 10 g : i ii;

CON S TAN T S , VAR IAB LES AND F UN CT I ON S .

—x

y coth x(

xgives coth “ l

y«5 log

y sech x gives sech“ y log

2

ex—l- e

yz cosech x= _xgives cosech ‘ l

y loge”

e

14 . Expl icit and Impl icit Functions — When a func tion

is expressed direc tly in terms o f its variable or variables,it

is an explicit func tion otherwise it is an implicit func tion.

T hus, in the equations

y 2 x”

33 . y tan2x,y y = 10 g W e

“

. r f (x.

y is an explic it func tion of the variab les in the secondmembers

,and in the equations

ai

r?

52x” W ”

. ii

10 g x”

. J’2

r”

fly, x )= 0.

y is an implicit func tion of x .

T he relation between an implicit func tion and its variab les is sometimes expressed by two equations . T hus

,

y 3u, u22 : Vx

,in which y is an implic it function of x .

y and J’

are forms expressingy as an implic it func tion o f x .

1 5 . Increasing and Decreasing Functions - A func tion

that inc reases when a variab le inc reases,and decreases

when that variable decreases , is ah increasing func ti on of

that variable. T hus,2x

, 7x3

,ax

s

/o, tan x,are inc reas

ing func tions of x .

A func tion that decreases wh en a variable inc reases,and

inc reases when that variab le decreases , is a dec reasing

I4 D I FFEREN T I AL CAL CUL US .

function of that variab le. T hus,I / x , (c x)

3

,b/ax

”

,cot x

,

are decreasing functions of x .

F unc tions are sometimes inc reasing for certain values of

the variab le, and decreasing for oth ers . T hus, (c or)

"

an increasing"function for all values of x greater than c ;

but dec reasing for all values of x less than c. zax2 is an

inc reasing func tion when x is positive, and decreasing when

x is negative . T he positive value oty i V” is an

increasing func tion for Values of x from r to 0,but de

c reasing for values of x from o to r . T he negativevalue of y is a decreas ing func tion for negative values of x ,

and increasing for positive values of x . Sin x, cos x ,sec x

,

vers x ,are increasing func tions for some values of the vari

able and decreasing for others .16 . Continuous and Discont inuous Functions — A func

tion 18 continuous between states corresponding to any two

values of a variable when it has a real state for everyinter

mediate value of the variable, and as the difference between

any two intermediate values of the variable approachesz ero , the difference between the corresponding states al so

approaches z ero. O therwise a func tion is discontinuous

between the states considered.

T he varying heigh t of a growing plant is a continuous

func tion of time.

I f for any value of the variab le a func tion is unlimited ,imaginary or changes from one state to ano ther Without

passing through all intermediate states,the continuity of

the func tion I S broken at the corresponding s tate.

:t i/s is cont inuous between states c orresponding to

x z o and x

: lz t/a Va2

x2

is continuous between s tates correspond

ing to x :

CON S TAN T S , VAR IAB LES AN D F UN CT I ON S .

a2

is continuous between states correspond

x z — oo and x : a and x :

but is discontinuous between s tates corresponding to

a and x z a .

Sin x,cos x

,ex

,and all entire algebraic func tions are

always continuous .A continuous func tion in passing from anyassumed state

to another mus t pass through all s tates intermediate to

those assumed but it may have intermediate states greateror less than the states as sumed. T hus

, the func tion

Vr ” x2 is continuous between the states 0 and r/ 2 V3,wh ich correspond to x r and x r/ 2 but it is greaterwhen x z 0 than either of the states consideredA func tion always continuous changes its Sign only by

passing through z ero ; but a discontinuous func tion maychange its sign without passing th rough z ero .

Unless otherwise s tated, func tions wil l be regarded as

continuous in the vic inity o f s tates under consideration.

1 7 . Functional Notation.—A func tion of any quantity,

as x,is general ly represented thus , f (x ) , read function of

x . O ther forms are also used ; as, F (x ) ,

Ax ) , one) , Mr ) . W ) .

T hus , ax/ ( I + x ) may be represented byf (x ) . T he f,or exterior symbol , is cal led the func tional symbol , or sym

bol of operation. I t represents the O perations involved inany particular func tion. T hus; having f (x ) ax/ ( I x ) ,

f indicates that x is to be multiplied by a,and that the

produc t is to be divided by I x . I ts significance re

mains unchanged th roughout the same discussion or subj ec t,

and placed before the parenthesis enc losing anyother quan

16 D I F FEREN T I AL CAL CUL US .

tity it indicates that the quantity enc losed is to be sub

j ec ted to the same operations that x is in the expression

ax/ ( I x ) . T hus,

In order to represent different func tions of the same

quantity the func tional symbol only is changed. T hus,if

F (x ) is selec ted to represent 2 Vi ce,then some other fort) “

as or etc .

,should be taken to denote 4cx 2x

D ifferent func tions of different quantities are representedby different symbol s with in and without the parentheses .

and respec tively.

A func tion of x2

is written f (x2

) or F (xi

) , etc .,and the

square of a func tion of x is designated byf (x )2

or

etc .

When the quantity is represented by a simple symbol ,the notation f x , ai r , i , ¢x

3

,etc .

,I S frequently used.

3c i/mT22 may be expressed as a func tion of my

2

by some

form,as f (my

2

) or etc .

H aving represented an?by f (z ) , and 3cVaz

2

by F (azQ

) ,

we may write 3c1: F (ae

2

)

Having ax

and bViz—x

we write

8bVET — 3MVII

z dbVa ”

k

wh ich 5 Vax

An express ion containing several different func tions o f a

variable,as z ax

2

log x 3 Sin x,may be considered as

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18 D I F FEREN T I AL CAL CUL US .

I iavmg

$01) M2

¢ (x J’) 2 4(x + 3 0

2c(x +r )

I/I (ax3) 4(ax

3)2

c(ax3) , ¢ (S in 9) z 4 SIO

QB c S in 6.

F (x) l ix + r ) an y ax >< ay F (x) F (v)

F (ry i ii )"

For“

(ax

) , w)" M y) .

f ( limit x) log ( l imit x ) .

f n ( limit or) al imit x

.

I f ¢ (z)ii

i/G, ¢(x) sax ,and F (w)

7L n TlM/ S“ V2" ( 5“ W)

I f q) (x , y)= 2x+ siay,and

-

2 then 2x + s iny.

y,z ) 7ax

2yz , and F (y) 2 and ¢ (x ) a

x and INA) 22 ,

then th (F [f (x , z )] 2 2ai/ ( 7ax 2yzp

18 . Lines are c lassed as algebraic or transcendentalaccording as their equations involve algebraic func tions

only or contain transcendental func tions .Anyportion of any line may be cons idered as generated

by the continuous motion o f a point. T he law o f its motion

determines the nature and c lass of the line generated .

Let s represent the length of a varying portion o f any

line in the coordinate plane X V,of wh ich the equatlon In

x and y is given. s depends upon the coordinates of its

variable extremities , and varies with each ; but the equation

of the line establishes a dependence between these coordi

nates . Henc e, s is a func tio'

n o f one independent variableonly.

I f the line is in space , its two equations estab lish a de

CON S TAN T S , VAR IAB LES AND F UN CT I ON S .

pendence between the three coordinates of its extremities,

so that one only is independent .T he same result will fol low if a sys tem of polar coordi

nates is used.

1 9 . Convex ity and Concavity.— T he side of an arc of

any curve upon wh ich adj acent tangents , in general , inter

sect is cal led the convex,and the other

,or that upon which

adj acent normal s intersec t , the concave side. A curve , atany point, is said to be convex towards the convex Side

and concave in the opposite direct ion.

2 0 . Graphic Representative of a Function of a Single

Variable.

able may be expressed by the equation formed by plac ingT he relation between any func tion and its vari

the func tion equal to a symbol . T hus, plac ing f x equal to

y, we have y = f x ,which expres ses the relations betweeny

and x,and therefore between the func tion f x and its

variable x . y= f x is also the equation of a locus,the

coordinates of whose points bear the same relations to eachother as those existing between the corresponding states ofthe func tion and variable. T herefore

,by constructing

,as

in Analytic Geometry, anyp oint of this locus,its ordinate

wil l represent graphical ly the state of the func tion corre

sponding to the state of the variab le similarly representedby its abscissa . T he locus thus determined is cal led the

g rap/c of the function. I t is important to notice that it is

2 0 D I F FEREN T I AL CAL CUL US .

the ordinate of the graph , not the graph itsel f, that represents the func tion.

T o il lustrate, let the line AB be the

graph of f x . T hen the ordinate RA is the

graph ic representative of f x ,corresponding

to a value of x represented by OP. Sim

ilarly, P’B represents f x when x O R

’

T he ordinates PM and PM 'of the

graph III QM’

represent two different

s tates of the func tion corresponding to

the same value of the variable , § 5.

T he ordinates PM ,RN

,and S 0 of

the graph MN O represent equal statesof the function correspondingto different values of the vari

able, 5.

T he graph of a func tionwhich is of the fi rst degree withrespect to its variab le is a

righ t line, otherwise not .

T he graph of a continuous func tion is a continuous line.

2 1 . Surfaces .-Any portion of any surface may be con

sidered as generated by the continuous motion of a line.

T he form of the line and the law of its motion determine

the nature and c las s of the surface generated.

2 2 . L et u represent the area of a varying portion of the

surface generated by the continuousmotion of the ordinate of any givenl ine In the plane X V.

u depends upon the coordinates ofthe variable extremities of th at portiono f the given line wh ich limits it, and varies with each but

”

CON S TAN T S , VAR IAB LES A IVD F UN CTI ONS .

the equation of the given line es tab lishes a dependence between these coordinates . Hence

,u is a func tion of but one

independent variable.

2 3 . L et r : f (o) be the polar equation of any planecurve

,as DM

,referred to the

pole F ,and the righ t line PS .

L et u represent the area of a

varying portion of the surface,generated by the radius vec torrevolving about the pole. u willchange with 7) and r ; but 7, and r

are mutual lydependent. Hence ,u is a func tion of but one independent variable.

24 . L et any line in the plane X V,as AM ,

revolve about

the axis of X . I t wil l generate a sur

face of revolution.

T he same surface may be generatedby the c ircumference of a circ le

, whosecentre moves along the axis X

,with its

plane perpendicular to it ; and whose radius changes withthe absc issa of the centre of the c irc le

,so as to always

equal the corresponding ordinate of the curve AM T he

radius of the generating circumference is, th erefore, a func

tion o f the absc i ssa of its c entre . Hence,the generating

c ircumference, and any varying z one of the surface generated as described, is a function of but one independent

variab le .

2 5 . T he area of any surface with two independent variable dimensions is a func tion of two independent variables .F or example, -the area of any rec tangle with variab le Sides,

parallel respec tively to the coordinate axes X and Y,is a

func tion of the two independent variab les x andy.

D I F FEREN T I AL CAL CUL US .

26 . Having any surface,as A TL

,let AB CD 2 u be a

portion inc luded between the coordinate planes XZ,YZ

,

and the planes D OR and BPS, paral lel to them respec

tively. Let OP x and O s be independent varia

bles . u will depend upon x, y,

and 2 ; but the equation of

the surface makes a dependent upon x andy. Hence,u is

a function of but two independent variables . Similarly, itmay be Shown that any varying portion o f the surfaceinc luded between any four planes , paral lel two and two

,to

the coordinate planes XZ and YZ,is a func tion of but two

independent variables .2 7 . Graphic Representative of a Function of Two

Variables .-Placing anyfunc tion of two variables

,asf (x y) ,

equal to 2,we have z f (x ,y) wh ich expresses the rela

tions between the function and its variables .

T he locus whose equation is z is cal led the

graph ic surface of f (x ,y) , for the reason that the ordinate

CON S TAN T S , VAR I AB LES AND F UN CT I ON S .

of any of its point s wil l represent graph ical ly the s tate of

the func tion corresponding to the states of the variablesSimilarly represented by their respec tive absc issas .I t is important to notice that it is the ordinate of the

graph ic surfac e that represents the func tion, and not a

portion of the surface as in the c ase desc ribed in 2 6 .

T he graph ic surface of a func tion wh ich is of the firstdegree with respec t to each of two variables is a plane ,otherwise not.

2 8 . Volumes .

s idered as generated by the continuous motion of a surface.

Any portion of any volume may be con

T he form of the surface and the law of its motion deter

mine the nature and c lass of the vo lume.

2 9 . L et any plane surface inc luded between any line inthe plane X V,

as A I11 ,and the axis of X be revolved about

X . I t wil l generate a volume of revo=

lution . T he same volume may be gen

erated by the c irc le, whose centre

moves along the axis X ,with its plane

perpendicular to it ; and whose radiuschanges with the absc issa of the centre of the c irc le

,so as

to always equal the corresponding ordinate of the curve

AM . T he radius of the generating c irc le is,therefore

,a

func tion of the absc issa of its centre. Hence,the generat

ing c irc le,and anyvarying segment of the volume generated

as described, is a func tion of but one independent variable.

I t is important to notice in th is case, that the generatingsurfac e is limited by the ordinates PA and F

’M,corre

sponding to the extremities of the limiting curve,wh ich

ordinates are perpendicular to the axis of revolution.

30 . Having any volume,aS

‘ A TL,bounded by a surface

whose equation is given, and the coordinate planes, let

2 4 D I FFEREN T I AL CAL CUL US .

AB CD O IV V be a portion inc luded between the coor

dinate planes XZ ,YZ

,and let the planes D OR and B PS

be paral lel to them respec tively.

Let O R r - x and O Q Z y be independent variables .

V will depend upon x, y and Z ; but the equation of the

surface makes z dependent upon x and y. Hence,V is a

function of but two independent variables.

In a Simi lar manner it may be Shown that any varying

portion of the volume inc luded between any four planes ,parallel two and two

,to the coordinate planes X2 and YZ

,

is a func tion of but two independent variables .

3 1 . Any volume with th ree independent variable dimensions is a func tion of three independent variab les . F or

example, the volume of any paral lelopipedon with variableedges parallel , respec tively, to the coordinate axes X ,

Y

and Z,is a func tion of x

, y and 3 ; all of Wh ich are inde

pendent

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D I F FEREIVT IAL CAL CUL US .

An infinite cannot be a limit . T hus,

is a form indicating that as x approaches z ero, I /x is nu

limited.

A tangent to any curve is a limiting pos ition of a secantth rough the point of tangency, under the law that one or

more of its points of intersec tion with the curve approachcoinc idence with the point of tangency.

In some cases,due to the form of the func tion or to the

law of change,the variable c an never become equal to its

limit . T hus,

Hm. I I .

x

1

I,for all values of x .

The c ircumference of a c irc le is the limit of the perim

eter of an insc ribed regular polygon as the number of itssides is continual ly inc reased . T he radius is the limit ofthe apothem,

and the c irc le that of the polygon, under thesame law .

An incommensurable number is the limit of its suc cessivecommensurable approximating values . T hus

, the terms of

the series etc .

,taken in order

,are approach

ing V3 as a l imit.

In all cases,whether a variable becomes equal or not to

its limit,the important property is that their difference is

an infinitesimal .An infinitesimal is not necessarilya smal l quantity in any

sense. I ts essence lies in its power of dec reasing numeri

PR I IVCI F LES O F L IM I T S . 27

c ally in other words , in having z ero as a l imit,and not in

any smal l value that it may have I t is frequently definedas an infinitely small quantity that is not

, however, itsSignificanc e as h ere used .

In representing infinitesimals by geometric figures theyshould be drawn of convenient Siz e ; and it is useless tostrain the imagination in vain efforts to conceive of the

appearance’

of the figure when the infinitesimals decreasebeyond our perceptive facul ties . U sually one or two auxiliary figures representing the magnitudes at one or two of

their states under the law give all the as sistance that canbederived from figures .I n all cases

,when referring to the limit of a variab le

,it is

necessary to give the law for the limit depends not onlyupon the variab le, but al so upon the law by which it

changes . Under a law,a determinate variable has but one

limit but it may have different l imits under different laws .An important consequence of the definition of a l imit is

that if two variables , in approaching limits under a law,

have their corresponding values always equal , their limitswil l be q I al . T hus

,for all values of x ,

we have

hence

limit(a

2x ) 2 lim [a x]—) d

33 . A'

var iable wbicb, in approacbing a limit,ultimately bas

and retains a constant sign cannot bane a limit wit/c a contrary

F or suppose f (x ) becomes and remains positive, and thatlimit f (x ) C. F rom the definition of a limit

, f (x )may be made to differ from C by a value numerical ly

28 D IFFEREN TIAL CAL CUL US .

I"less than C . I t would therefore become negative, which iscontrary to the hypothesis . In a Similar manner

,it may be

shown that a variable always negative cannot have a positive l imit .

34 . If t/ze dzfi'

erence between tbc cor responding values ofany two var iables, approacbing limits, is an infinitesimal , tbeTa iables bane tbc same l imit.*

Let U and V represent any two variables giving

U V z d , or U : V—l—d,

in wh ich 6 is an infinitesimal.Let C be the limit of U, then U : C e

,in wh ich 6 is

an infinitesimal .Substituting we have

C or C V 2 6 + e,

the second member of which is an infinitesimal . Hence, Cis the l imit of V.

35 . Tbe l imit of tbe sum of anyfinite number of var iables

is tbe sum of tbeir limits.

L et U,V,W

,etc . ,

represent any variab les,and A

,

B,C,etc .

,their respective limits then

U i -s A — E,

W : C

”i which 6

,d,09

,etc .

,are infinitesimals.

Adding the corresponding members we have

II — B — c + d— co -l—etc .

I n o rder to avo id the frequent repet ition of the express ion under

the law ,it w il l be assumed , unless o therw ise stated, that the changes

in a l l the variab les cons idered together , o r in the same"discussion,

are due to one and'

the same law ; that all variab les and the ir functionsare cont inuous between a ll states considered ,

and that they havel imits under the law .

PR I NCIPLES O F L IMI TS . 2 9

II ence,

limit [ U — V—I : A

limU limV lim [V etc .

36 . In general,tIze limit of tbcproduct of any twovar iables

is Meproduct of tbeir limi ts .

L et U and V represent any two variab les having A and

B,respec tively, as limits .T h en U = A e and V Z B — 5

, in wh ich 6 and O‘

are infinitesimals. Multiplying member by member, we

UV : AR B e A6 es,

limit [ UV] AB 2 limit U limit V.

I t fol lows that, in general , t/ze l imit of anypower or root

of any var iable is tbc cor responding power or root of its limit.

T hus,limit U “

( limit and l imit U 5 2 ( limit U ) 5,Having ax N ,

x and N approach corresponding limits

together ; hence al im x lim N 2 lim a

x

,and lim x

log lim N . Al so,since x 2 log N ,

we have lim x lim

log N . T herefore log lim N : lim log N .

37 . I n general , tbc l imit of Me quotient of any var iables is

tbc quotient of tbeir limits .

With the Same notation as in § 36, we have

lim U Alim [UV

‘ I

] lim U [lim V]" I

lim V B

When B o,and A : l: o

,U/ V is unlimited.

When B : o A,the princ iple fail s to determine the

l imit wh ich by definition is determinate.

38 I t follows from 35, 36 , 37, that . in general , t/ze

30 D I F FEREN T I AL CAL CUL US .

limit of anyfunction of any var iables is ibe samef unction oftbeir respective l imits.

T hus , in general ,

limit f ( U V, . U,lim V,

and to obtain the limit of any func tion of variab les we,in

general , substitute for each variable its limit.

39 . Exceptions to the above general rule arise,and are

indicated by the occurrence of some indeterminate form,as

w

o/o , 0 00,00 00

,1

T o il lustrate, having f (x ) 2 (x2

1 ) (x the generalrule gives limit f (x ) whereas we find

f ( z ) 3, f ( L S) z: f ( L I ) f ( I ' O I )

and the nearer we take x to I,the nearer will f (x) approach

to 2 . By taking x suffi c iently near to 1, f (x) may be made

to differ from 2 by a number less numerically than any as

sumed number however small . Hence, 2 is 32 ) the limitof f (x ) as —> I . I t Should al so be ob served that 2

,con

sidered with the States of f (x ) which immediately precedeand fol low it, conforms to the law of continuity.

T 0 il lustrate a failing c ase geometrical ly, let the curve

B MM '

be the graph of a func tion, Take any State,as

PR IN CIPLES O F L IM I T S . 3 I

PM corresponding to x O R,and increase x by F F

’rep

resented by A x . D raw the ordinate F’M ’

and the secantMM ’

. T hrough M draw M O’

paral lel to X . Q’M '

, de

noted by Ay, will represent the increment of the func tioncorresponding to A x .

Ay/ A x tan Q’MM ’

wil l be the ratio

of the inc rement of the func tion to the corresponding inc rement of the variable.

At M draw M T tangent to the curve . T hen, under thelaw that A x approaches z ero

,the secant MM '

wil l approach coinc idence with the tangent M T

,and the angle

Q’MM will approach the angle Q

’

I II T,or its equal XH T

,

as a limit .

I I ence

limit ( Ay/ Ax) lim. tan Q’MM '

tan XI I T,

A x»»—>o

whereas the general rule gives as a result.

We observe from the above illus tration that tbc limit of

tbe ra tio of any increment of anyfunction of a sing le var iable

to tbc cor responding increment of tbc var iable,under tbe law

tnot tbc increment of tbc var iable approacbes zero,is equal to

tbc tangent of tbe ang le made w itb t/ze ax is of abscissas by a

tangent, to tire g rapb of tbe f unction, at tbc point correspond

ing to tire state considered.

WhenM ’coincides with M the secant may have any one

of an infinite number of positions other than that of the

tangent line M T,for the only condition then imposed is

that it shal l pass through 111 .

T herefore, wh ile limit ( Ay/ A x ) is definite , and equal toAnon—m

the tangent of the angle that the tangent line at M makes

32 D I F FEREN T I AL CAL CUL US .

with X ,limit Ay/limit A x indicates that the tangent

of the angle which the secant makes with X becomes indeterminate whenM ’

coinc ides with ML imit ( Ay/ A x ) is, therefore, one of the manyvalues thatlimit Ay/limit A x may have under the law.

T he exceptional cases , in general , require transformation

in order that fac tors common to the numerator and de

nominator may be cancel led , or from which the limit may

otherwise be determined. T hey are of the h ighest importance , for the D ifferential Calculus, as it wil l be seen

,is

based upon the limit of the ratio of the increment of the

func tion to the corresponding increment of the variableunder the law that the increment of the variable vanishes.

T he remainder of this chapter will , therefore, be devoted tocertain important exceptional cases and methods.

T his formula is deduced in Algebra for all commensu

rable values of m. Since 32 ) any incommensurablenumber is the limit of its suc cessive commensurab le ap

proximating values , the formula holds true when m is

incommensurable.

T hat is, as x» o is the limit of the ratio of ax/ x to its

preceding value ; consequently 0 is the limit of a /]x .

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34 D I F FEREN T I AL CA LCUL US .

T hus,

lim[C—l U] lim [C —l V ] ; lim [CU ] = lim [CV]

l-im C U lim CV; lim

T here fore,in searcbing f or tbc limit of anyf unction under

a law , we may r eplace anyvar iable enter ing it by anotlzer var i

able, provided tbat

,under tbe same law

,unity is tlce limit of

t/ze r a tio of tbe two var iables interc/zang ed. T he great advantage in so doing arises when it enables us to determinethe required limit more readily. T hus , in the las t exampleabove we may be able to determine the limit of V/W morereadily than that of U/W.

I n making the above substitution it is important to notice

that the limits onlyare equal , and that corresponding valuesof the quantities interchanged, in general , are not equal toeach other .

The privilege of replacing one variab le by anoth er underthe conditions desc ribed

,so facilitates the determination

of limits in certain exceptional cases , that it is important todetermine under what circumstances the limit o f the ratioof two variables is equal to unity.

45 . In genera l when lim U = lim V,

lim [U/ V ] lim U/lim V : I . § 37.

T hat is , in general , unity is the limit of the ratio of any

two variables when,under the same law , theyhave the same

limit,or

,what is equivalent , when the difference between

their corresponding values is an infinitesimal .I f

,however

,lim U : lim V o , it does not fol low that

lim [U/ V] z I Such c ases require Spec ial inves ti

gation,and the fol lowing are se lec ted on account of their

subsequent importance.

PR IN CIPLES O F L IM I T S . 35

46 . T ake any plane surface, as PMM’P’

,inc luded be

tween any arc,as MM ’

,the ordinates O f its extremities

and the axis of X .

T hrough M andM Y 0»

respec tively,drawM Qand M ’

Q paral lel toX

,and complete the

rec tangle M QM’

Q'

.

Let y PM,and y

' P'M '.

T hen as PP’A x

,0,we ultimately have

PQM’P’

Zand —>y.

l imit PQM’P’

PM QIPI

limit I .

—>o

Hence (S44)

limit PMM?”

—> O A 30

T herefore

I f the coordinate axes make an angle (9 with each other,

limit PMM 'P'

—>o A x

MPM

y sin HA xlim 2 y Sin

the surface generated by the radiusvec tor PM r

,revolving about

P,as a pole , from any assumed

position,as PM

,to any other

,as

R PM’

. L et A v represent the

corresponding angle MPM ’.

With P as a centre,and the rad i i

PM and PM ’

, describe the arc sM O

’

and M ’R respec tively.

36 D I FFEREN T IA L CA L CUL US .

T hen,as A v 0

,we ul timately have, in any case,

area RPM ’

2 area MPM ’

2 area MPQ’

,

limit [area RPM area MPQ’

]—>o

Hence, [area MPM’

/area MPQ’

] 1 .

T herefore 44)

area MPMlim lim

A ?)

Let PMM ’P’

T hen as A x 0,we

ul timately have

Vol . gen. by Vo l g en. by Vol . gen . byPQM

IPI PMM IPI PM Q

IPI

limi t Vol . gen by Vol . gen. byA xon PQM

’P’ PM Q'P' 1m

limit Vol . gen. by Vol . gen. bH ence

’ ->o PMM 'P' PM Q’P'

T herefore 44)

Vol . gen. byPMM ’PL

A xlim

figure as described in §46 ,and Me d

’

Q’the cor

responding rec tangle.

R evolve the entire

figure about X .

PRINCIPLES O F L IM I T S . 37

49 . Let MNM ’N ’be a portion of any Surface inc luded

between the coOrdinate planes ZX ,ZY and the two planes

N ’SE and NPD paral lel to them respec tively.

Denote the corresponding vo lume MNM ’N ’

,OPF S

by V. Cons truc t the paral lelopipedons OPF S -0 1W and

OPF S-FM ’

,and represent their volumes by P and P

’

respectively. Let OP ‘

2 It,O S I R

,O F 2 1

.0M 3

,

and F M ’2’

T hen as—> o

,or what is equivalent, as o

,

whence z’

-> z,V wil l , in any case

,ultimately be, and

remain, between P and P’

.

11m“[P/P

'

j £212 I .—>o

Hence,limit

[ V/P] I,

and 44)—>o

limit V/rec t. OPF S ] lim [ID

/file] lim [elite/lib] z .—>o

a8 D I FFEREN TI AL CAL CUL US .a

50 . L et MNM ’N ’

, denoted by S ,be a portion of any

surface inc luded between the coO rdinate planes ZX,ZY

and the two planes N’SE and NPD paral lel to them

respec tively.

L et OP lz,and O S b.

At M draw the tangents MB and MB ’to the curves

MN and MN ’respec tive ly, complete the paral lelogram

and denote it by T . T is the portion of the

tangent plane to the surface at M inc luded between the

planes which limit S . L et flequal the angle wh ich T

makes with X V, giving T cos fl2

“

OPF S . I nsc ribedin S ,

conceive an auxiliary surface, composed of n planetriangles the sum of wh ich , as w il l have S as a

limit and such that the sum of their proj ec tions upon X Vwil l equal OPF S . The two triangles MN ’M ’

and MN I II ’

in the figure illus trate a set fulfilling the Conditions . L et t,

t’

, etc .

,represent the areas of the triangles and 9, etc .

,

t h e: un c l e : w h ic h t h e ir n l a n e q r e q n e o t iv e lv m ake. w ith X V .

PM NCJP‘

LE’

S O F L IM I TS. 39

T hen O F F S Et cos 0 T cosfl,

Et cos fi/ T cos [5 I .

As n »—> cc , S remaining constant , each triangle is an

infinitesimal,and we have

limit[S/Et] 1 .—> oo

T he same effec t and result fo llows if S is made infini

tesimal and 72 remains constant . Hence,under the law

that it and k vanish , or, what is equivalent, that O F ,repre

sented by l , is infinitesimal,we have

—>o

Under the same law, flis the common limit of 0, etc .

H ence ( 1 )limit

[2 1 cos fi/ T cosfl] lim [2 t/ T ] I .

->o

T herefore (a) .1imit

[S/ T ]A» ) o

um [me] um [at/m ewe]

I cos [i

5 1 . Unity is tne limit of tlze ratio of an ang le to its sine,of

an ang le to its tangent, and of tne tangent of an ang le to its

sine, as tne ang le approae/zes z ero.

L et O CM z given in radiansbe any angle less than 7t/a ; thentan gb sin and as —> o

,

we have always

tan (b

l imit tan 915sin qb

4o D I F FEREN T I AL CALCULUS

Hence, since ¢/sin 975 is always between tan ¢/sin gb and

unity, we have11mit

Similarly, since 1 qb/tan 915 sin ¢/tan 925, we have

$532 qb] 3: I .

52 . Unity is tne l inzit of tae ratio of any are of any curve

to its elzora’

,as tae are approae/zes z ero.

L et s denote the length of any arc of any curve,and

conceive it to be divided into n equal parts , the consecutive

points of division being joined by chords forming an in

sc ribed broken line whose length is designated byp.

T hen kml t[s/fi] r .

n»—> oo

Under the above law the equal arc s of s vary inverselywith n ; hence the same effec t and result may be caused byretaining any fixed value for n

,and making s approach

z ero. Hence,limit—>o [s/p] I

’

and if nall

eg ohrc/chord] I .

53 . Unity is t/ze limit of the ratio of tile surf ace generated

oy any are of any carve under a law to tnat generated by its

elzora’as tne are approaenes z er o.

Let s,p and n denote, respec tively, the same quantities

as in the last artic le, and conceive s and?to move together,

under a law ,so as to generate two surfaces represented by

S and P respec tively.

As n»—> oo,any s tate of s without regard to form is the

limit of the corresponding state ofp,and S is the limit ofP.

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EI F FEREN TJAL CAL CUL US .

Hence, since 52 )limit

[s/e] I, we have 44)

limit[o/s] I .—) O SM O

T herefore 44)

limit 5 5 A se/cos Q IAme-9 0 A x A x cos Q

'M T"

55 . L et be the polar equation of any planecurve as AMM ’

,referred to the right line PD , and

Let AM s be any portion of the curve,and PM r

the radius vec tor corresponding to MR egarding s as a func tion of a 1 8 ) let 7) be increased

by A 7} . T he arc MM ’will be the correspond

ing increment of s. D raw M Q’

perpendicular to PM’

,

and denote PM ’by r". T hen 44, 52 ) we have

l imit arc MM ’ch. MM ’ M Q

’2

Q’M

11m — 11m 2A v»)—>o A 7) A 2) ( A ?)

PR IN CIPLES OF L iM ] 17‘

s . 43

Hence 5r)

We al so have

tan PMD tan Q’M ’M lim

limr cos A 71 r

Ir r

Hence,l imi t

tan PMD AW H O A 7)

wh ich,substituted in ( I ) , gives

I f the radius vector PM coincides with the normal tothe curve at M

,we have

tan PMD z oolimit

[arc 71]A

56. Let any plane figure, as

PM JV’P'

,inc luded between any

plane arc,as MM the ordi

nates of its extremities and the

ax i s of X ,be revolved about X .

T hen 44 , 53)

limit Sur gen. by arc MM ’

[3303—50 A xSur. gen. by ch. MM ’

A x

7r (y —y'

) A se/cos Q’MM ’

lim

11mcos Q

’M T°

4 4 D I F FEREN T IAL CAL CUL US .

CHAPTER I I I .

RATE O F CHANGE O F A FUNCT IO N.

57 . A func tion changes uniformly with respect to a variab le when from each state all increments of the variab leare direc tly proportional to the corresponding incrementsof the func tion.

I t fol lows that from all states equal increments of the

func tion correspond to equal inc rements of the variab le ;al so that the ratio of any inc rement of the func tion to the

corresponding increment of the variable is constant.

T hus,having zax ,

increase x by any amount denoted by

it,then za (x n) 2ax z ali wil l be the corresponding

inc rement of w oe. I t varies directlywith ll , it is the samefor al l states of the function, and 2a/z/lz 2 a is a constant .Hence , 2ax changes uniformly with respect to x .

L et f x be any uniformly varying func tion, and it any in

c rement of x . f (x -Jr—lz) - f oc will be the corresponding

inc rement of the func tion, and

b) 1 : constant A.

Henc e f (x n) 2 Ag

in which to : 0,gives f (n) Ale

T herefore, f x 2 Ax

RA TE O F CHAN GE O F A F UN CT I ON . 4 5

Hence, all functions wh ich change uniformlywith respec tto a variab le are algeoraie, and of the first deg ree withrespect to that variable, and all alg eoraie func tions o f the

first deg ree with respect to a variable change uniformlywith that variable.

The graphs of such functions are righ t lines , and any

func tion whose graph is a righ t l ine changes uniformlywith respect to the variable.

T o il lustrate, the righ t line AD is the graph of a func tion.

Consider anystate, as that represented bythe ordinate

Inc rease the corresponding value of the variable , repre

sented by O P,by any inc rements

,as PP’

and PP Q'

B

and S”C wil l represent the corresponding inc rements of

the func tion, and the similar triangles AO'

B and AS C

AQ’AS

”

Q’

B S”C.

T hat is, the corresponding inc rements of the variable and

func tion are proportional .By giving to x OP any equal increments

,as PP’

,

P’P”

,in succession,

the corresponding increments

D I F FEREN T I AL CAL CUL US .

of the func tion, Q

'

B, Q C, and Q 29

,are equal to each

other.

I t is al so evident that the ratio of any inc rement of thefunc tion to the corresponding inc rement of the variab le, as

or S"O/PP or Q is constant.

58 . Having the func tion 2 x ,

x : 2 30 :

95 :

Hence, the function 2x increases two units while the

variable increases one ; in other words,twice as fast.

Having the func tion 5x ,

x

x :

T herefore, the function 5x changes fives times as fast asthe variable .

In general , different functions change,with respect to

the i r variab les , with different degrees of rapidity.

T lze measure of Me relative deg ree of rapidity of enange ofa f nnetion and its var ial7le at any state is talled tne rate ofelzang e of tbefunction, zoitfi respeet toflee var iable, cor respond

ing to til e state.

A rate of change of a func tion with respect to a var l able,

corresponding to a state,is an answer to the question : At

the state cons idered,how many times as fast as the variable

is the function changing P

RA TE O F CHAN GE O F A F UN CT I ON . 47

59 . Since,from all states, any uniformlyvarying func tion

receives equal inc rements for equal inc rements of the

variable , its rate, from state to state,is constant

,and equal to

tbc ratio of any increment of tbcfunction to t/ze cor r espondingincrement of tlze var iable.

T hus, rate of 2x

[300 + 72) 2 ] [390 + 2 ]n

3] [5x 3]n

e Rate of 330 + 2 7 :

Rate of 5x 3

I t fol lows that the rate of any uniformlyvarying function is equal to the tangent of the angle wh ich its graphmakes with the axis of the variable

,al so that the produc t

of the rate by‘

any inc rement of the variable is equal to thecorresponding inc rement of the function.

60 . I t fol lows from 57 that algebraic func tions not ofthe first degree with respec t to the variable

,and all

transcendental func tions , are, with respec t to the variable,

non-uniformly varying func tions . T hus,having ax

’

,in

crease x by any amount as lz. 2 ax/z + abz

,wh ich varies

with x , is the corresponding increment of ax2. Al so the

ratio ( 2 anb 2 : z ax all z gb(x ,lt) . H ence

,aoc

2

does not change uniformly with respec t to x .

inwh ich n is not equal to I, then [f (x b )

"

¢ (oc, it) , and f (x ) is therefore a non-uni

formly varying func tion with respec t to x .

T he graphs of such functl ons are curves,and any func

tion whose graph is a curve does not change uniformlywith respec t to the variab le.

48 D I F FEREN T I AL CALCUL US .

T o il lustrate,the curve MN T is the graph of a func

tion. Increase,in. succession

,

any value of ac,as OP’

,by any

amounts , as P’P”and P’

F .

OB /S T is the

ratio of the inc rements of the

func tion.

func tion

with x .

61 . In the function 2x“

,

2 90223 2 .

Wh ich shows that at different states the func tion 2x2

has

different rates wi th respec t to x .

A non-uniformly varying func tion,in general

,receives

”

unequal increments for equal inc rements of the variable .

I t follows that its rate varies from s tate to state .

Any particular rate is,therefore, designated as the rate

corresponding to a particular state .

I f a func tion has two or more states corresponding to any

value of the variable , each state w ill have a rate .

I f a func tion has equal states for different values of the

variab le , it may have a different rate at each ; in wh ich

case it is necessary to indicate the v alue of the variablecorresponding to the s tate considered.

variab le,

and it differs from

QN /S T ,wh ich is the ratio of

the corresponding inc rements of

the ordinate of MN T , and

by it , do not change uniformly

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so D I F FEREN T JA L CAL CUL US .

and [f (x it) rate of f (oc 0b ) . ( I )

This relation wil l always exis t as fan—m,

Hencek) rate of f ee. ( 2 )

T hat is,tbc rate of cbange of anyf unction zoitb respect to a

var iable,cor r esponding to any state

,is equal to tire l imit of tbc

ratio of any increment of tbefunction, from tbc state consid

ered to tbc cor responding increment of tbc var iable, under t/ze

law t/zat tbe increment of tbc var iable approaclces zero.

T he above princ iple enab les us to find the rate of any

func tion with respec t to a variable,corresponding to any

state , by the fol lowing general ruleGive to tbc var iable any var iable increment

,andfrom tbc

cor r esponding state of tbe function subtract tbc pr imitive.

D ivide t/ze remainder by fi re incr ement of tbc var iable,and

determine tlze limit of tlzis ratio,under tbc law tlzat tbe incre

ment of tbc var iable approacbes z ero. I n tlze result substitute

t/ze value of tbe var iable cor responding to t/ze state.

I t should be observed that a rate,determined by the

above method,is equal to a limit of a ratio of two infinitesi

mals,which limit is determinate and that it is not equal to

the ratio of their limits . 39 .

T o il lus trate,let the curve AB

”B

’in the figure on page

.5 1 be the graph of f x ,and let PA be the state at which theirate is required.

I I

f (x +IZ) f x Q B

tan

wvhich is the rate of t he function whose graph .is the secant

.AB

RA TE O F CHANGE O F A F UN CT I ON . 51

When It PP

I I I I

f (x f

}

? f x Qpfi" tan Q AB

",

which is the rate of the function whose graph is the secantAB

As —> o,the above ratio is always the rate of a func tion

whose graph is a secant approach ing the tangent A T as a

limit . Hence, tbc limit of tbe ratio is the rate of the function whose graph is the tangent A T ,

and is equal to thetangent of the angle Q

’A T .

T hat is, Me limit of tbc ratio of any incr ement of any

f unction Of a sing le var iable to tbc cor r esponding increment

of tbc var iable, under t/ze lazv t/zat tne increment of tbc var i

able approacbes zer o, is equal to tbc tangent of tbc ang le made

zvitb tbc axis of abscissas by a tang ent at tbc cor responding

point of tbc g rap/t of tbcfunction.

T he same resul t was obtained in 39 . T he angle wh ichthe tangent A T makes withX is c alled tbc inclination, and

the numerical value of its tangent is called tbc slope of the

graph at MT he rate of any func tion at any state is therefore the

same as that of a. uniformlyvarying func tion Whose graph

52 D I F FEREN T I AL CAL CUL US .

is the tangent to the graph of the given func tion at the

point corresponding to the s tate considered .

T h is agrees w ith previous conceptions and definitions,

for the direc tion of the motion of the point generating thecurve at any position is along the tangent at the point, andthe ordinate of the curve representing the state consideredis changing at the same rate as the ordinate of the corre

sponding tangent .

I t fol lows that the produc t of the rate of a non-uniformlyvarying func tion by an inc rement of the variable is not, in

general , equal to the corresponding increment of the func

tion.

63 . Lety PA represent any state of anyincreasing function of x ; and y

’

the new state corresponding to an incre

ment PP’

it , of the variable. (y’ —

y)/b wil l be the ratio

of the increment of the func tion to that of the variable. I f

It is assumed suffic ientlysmal l,th is ratio wil l be positive and

remain so as ba—eo . 1 5.

Hencelimit

[ (y'

tan XEA is positive.—>o

L et y PIA

.represent any state of a decreasing func tion

and y'its new state due to an increment of the variable

equal to P1P

,

’it . T hen (y

’-

y)/b wil l be negative, if b

is small enough , and wil l remain so as b 0 .

Hencelimit

[(y’

4 tan XE,A is negative,—>o

RA TE O F CHANGE OE A F UN CT I ON . 53

T herefore, tbc rate cor responding to any state of an incr eas

ing function is positive, and of a decreasing function is neg ative.

I t fol lows t/tat a function is an increasing one wben its

rate is positive, and a decreasing one ref/ten its rate is negative.

When,for any state of a func tion,

as the one representedbyPA or P

’A

’

, the function is neither inc reasing nor de

P X.

creasing,its rate is z ero

,and the tangent at the correspond

ing point of its graph is paral lel to the axis of x .

I f for any state of a function,as the one represented by

P” A ”

,the rate is unlimited, the corresponding tangent to

its graph is perpendicular to the axis of x .

EXERCI SES .

F ind the rate of change of each of the following functions

l imit za(x + bl — Zax

I . zax .

k2a .

limit (“V -i f»?

n»—>o n

3 . ax2+ bx . Ans . =2ax+b

5 . mar”. Ans . 4ax . 6. x

3.

54 D I F FEREN T I AL CAL CUL US .

7. 4x4. Ans . 16x3

°

3 + x

'

I O . H ow is the ordinate of a parabola, corresponding tox 3 , changing with respec t to the absc is sa ?

Rate o f

y= i ==WEZL

—¢ V9 L = i V§

Ans . : l:

1 1 . Same corresponding to focus Ans. 1 .

1 2 . F ind the absc issa of the point , of the parabolay2

4x , where the ordinate is changing twice as fast as the

abscissa. Ans. x1 3 . At the vertex of a parabola, how is the ordinate

changing as compared with the absc issa1 4 . F ind the rate of change of the absc issa of a parabola

with respec t to the ordinate ? Ans . y/p : l: Vz oc/p.

1 5. F ind the coordinates of the point of the parabola

y22 See

,where the abscissa is changing twice as fast as the

ordinate . Ans. (8 ,1 6. F ind the rate of change of the ordinate of the right

line 2y 3x 1 2,with respec t to the absc issa. Ans. 3/ 2

1 7. A point moves from the origin so that y always inc reases times as fast as x ; find the equation of the line

generated .

5/4 tan of ang le line makes with X . Ans. 4y 5x .

RA TE O F CHANGE O F A F UN CTI ON . 55

1 8 . F ind the s lope of the graph of ziz t/ 1 2x when x :

Ans.

1 9. F ind the absc issa of the point of the graph of V2pnwhen the slope is 1 . Ans. x z p/ z .

2 0 . F ind the angles which the lines y2

8x ,and 3y

2x 8,make with each other at their inter sec tions

Ans. 1 1°

1 8’

and 7 7 30

2 1 F ind the angles which the lines y2

4x ,and 2y

2,make with each other at their intersec tions.

Ans. 1 0°

and 33°

64 . A function of two or more variab les is a uniformlyvarying function with respect to all of its variables whenit changes uniformly with respect to each . I t fol lows

57) that all uniformly varying func tions are algebraic,

and of the first degree with respec t to each variable, and all

algebraic functions of the firs t degree with respect to eachvariable are uniformly varying func tions .Let u Ax By C2 etc . ,

in which A,B

,C,etc .,

are constants , be any uniformlyvarying function. I ncreasethe variables x , y, 2 , etc .

,respec tively, by any increments , as

It,b,l,etc .

,giving a new state,

u’ =

,

A(x /z) 8 0 + e) C(e+ 1) etc .

it Alt B k Cl etc . ,is the corresponding in

crement of the func tion. I t is independent of the s tate of

the func tion, dependent upon the increments of the variables ,and is equal to the sum of the inc rements due to the

inc rease of each variable separately.

65 . A uniformly varying func tion of two variab les issome particular case of the general expression Ax B y

C, in which A ,

B and C are constants . I ts graphic surface

56 O LEEEEEN TLAL CAL CUL US .

is a plane and any func tion of two variables whosegraphic surface is a plane I S a uniformly varying func tion.

T o il lus trate, take any ordinate,as NM

,of any plane, as

MLE . T h rough N A] pass the planes MN B and MNP

paral lel , respec tively, to ZX and ZY,intersec ting the glven

plane in the l ines ME and ML . AssumeN B as the incre

ment of x ,and NP as the increment of y. Complete the

paral lelogram N S , the parallelopipedon N Q,and in the

given plane the parallelogram M L TH . Produce PK to H,

S Q to T,and PA to L . X I -I is the increment of MN due

to the inc rement,NE

, of or: alone , and AL is its inc rement

due solely to NP, the inc rement of y. Q T is the entire

increment of MN due to the increase of both variablestogether.Draw HD paral lel to X Q,

and draw AD ; it will be paral lel to L T , because it is paral lel to M 17

,which is parallel

to L T. Hence,D T AL .

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58 D I FFEREN T IAL CAL CUL US .

66 . The Calculus is that branch of mathematic s in wh ichmeasurements , relations , and properties of func tions and

their states are determined from their rates of change . I t

is generally separated into two parts, cal led,respec tively,

D ifferential and Integral Calculus .Differential Calculus embraces the deduc tions and ap

plications of the rates of func tions.

DIFFERENTIAL CALCULUS.

PART I .

D I F FEREN TIALS AND D I F FEREN TIA TI ON .

CHAPTER IV.

THE DIF FERENT IAL AND DI FFERENT IAL CO EFF ICIENT .

FUNCT I O NS O F A S INGLE VAR IABLE.

67 . An arbitrary amount of change as sumed for the independent variable is cal led tlce dif

ferential of tbc var iable.

I t is represented bywriting the letter d before the symbolfor the variab le ; thus doc, read differential of ac

,denotes

the differential of x .

I t is always as sumed as positive, and remains constantthroughout

i

the same discus sion unles s otherwise stated.

68 . T he differential of a func tion of a single variab le istbc cbange taat tbc f unction would undergo from any state

,

were it to retain : its rate at titat state, wltile t/ze var iable

cbanged by its dzj’erential .

T he differential of a func tion is denoted bywriting the

letter d before the func tion or its symbol .

60 ETEEEEEN TLAL CAL CUL US .

T hus , d 2 ax3

,read differential of w as

,indicates the

differential of the func tion 2 ax3

.

H aving y2 : log Vao

d

,we write dy

“

2 d elog Van2

Z?)

doc denotes the differential of y regarded as a func tion9C

o f cc ; and y is a symbol for the differential of the inMy

verse func tion that is, of x regarded as a function of y.

T he differential of a func tion wh ich varies uniformlyw ith its variable is equal to the change in the func tion cor

responding to that assumed for the

variable,because its rate is con

stant .

T hus,let PA be any state of the

uniformly varying func tion whose

graph is the right line AB . As

sume PB = doc. T hen QB z dy, the corresponding changein the func tion

,is the differential of the func tion.

T he differential of a func tion wh ich does not vary uniformly with its variab le is not

,in general , equal to the cor

responding change in the func tion,because its rate varies

but it is equal to the corresponding change of a func tion

having a constant rate equal to that of the given func tionat the state cons idered or

,in other words , it is the change

that the func tion would undergo were it to continue to

change from any state, as it is cbang ing at tbat state,uni

f ormly with a change in the variab le equal to its differential .T hus , let PA be any state of a given function whose

graph is the curve AM Assume PB 2 doc.

QM is the corresponding change in the func tion ; but

QB ,the corresponding change in the function represented

D I F FEREN T I AL— D I F FEEEN T I AL COEF F I CIEN T .6 1

by the ordinate of the righ t line AB drawn tangent toAM at A

,is the differential of

the given func tion corresponding to the state PA . T he func

tion whose graph is AB has a

c onstant rate equal to that of the

given func tion at PA,and Q B is

the change that the given func

tion would undergo were it to

continue to change from the state PA,as it is cbang ing at

tbal state, uniformly with a change in oc equal to doc.

The differential of a func tion wh ich does no t vary uni

formly with its variable may

be less than the correspond

ing ch ange in the func tion.

T hus, QB , Q lM

,is the dif

ferential o f the func tion rep

resented by the ordinate of

the curve AM,correspond

ing to PA .

A train of cars inmotion affords a famil iar examp.e of a differen

tial of a function.

Suppose that a train o f cars starts from the station A , and moves

in the direction AE w ith a cont inuouslv increas ing speed . L et or de

no te the var iable distance o i the train from A a t any ins tant it w il lbe a function o f the time , represented by t , during wh ich the train hasmoved , g iving xSuppose the train to have arrived at B ,

for wh ich po int x AB .

Let B D represent the dis tance that the tra in w il l ac tually run in the

next unit o f t ime , say one second ,w ith its ra te constantly increas ing .

Le t B C represent the distance tha t the tra in would run if it we re

to move from B w ith its rate at tha t po int unchanged ,in a second ,

62 E LEEEEEN TLAL CAL CUL US .

T hen w il l the distance B C represent the differential of x regarded as

a function of t, corresponding to the s ta te or AB ; and one second

w il l be the differential of the variab le .

69 . F rom the definition of a differential of a func tion,

and from 59, it follows that a differential of a func t ion isthe produc t of two fac tors , one of which is t/ze rate ofelzang e of tbefunction at the s tate considered , and the otheris the assumed diff erential of tbc var iable. Hence

, the dif

ferential of any given func tion may be determined by finding its rate

,by the general rule, § 6 2 , and multiplying it by

the differential of the variab le. T hus,having the func tion

2x2

,we find

[z2

2x2

l lml t 2 05[

Z4x rate corresp. to any state .

->o

4xdoc is, therefore, a general expressmn for the differen

t ial of 2x 2

,and is written d z oc

2

4xdx .

I ts value corresponding to anyparticular state is obtained

by subs t ituting the value of the variab le corresponding to

the state thus,for ao 2

,we have (d 8doc.

70 . Since, in the expression for the differential of a function,

tbe rate of cbange of tlze function is the c oeffic ient ofthe differential of the variable, it is, in general , called the

“dif

ferential coefiicient, and may be determined by the

general rule, 6 2 .

T he differential Of a functi on IS therefore equal to the

produc t of the differential coefi cient by the dzf erentz

'

al oftbe var iable.

I t fol lows that the differential coefficient is the quotientof the differential of the func tion by the differential of thevariable. T hus , having d z oo

2

4xdoc, 4x is the differentialcoeffic ient. In general having and representing

O LEEEEEN TLAL— D LEEEEEN T I AL COEF F I CIEN T . 63

its d i fferential by dy or dx,its differential coefficient is

The expressions and dy’

/dx’are used to de

note the particular value of dy/dx corresponding to x r x’

andy 1 :y'.

T hus,havingy 2x

”

,then dy/dx 4x ,

and 8 .

H aving y in wh ich y is any func tion of any vari

ab le x,let y

’denote the new state of the func tion corre

sponding to any increment of the variable, as It or A x,and

let Ay y'

y represent the corresponding increment of

y. T hen 6 2 )

limit f (x b) f (x)lim limit Ar a

’

y

fl »)—) O ll AxW O A x dx

Since the increment of the variab le,represented by b or

A x,varies , it may happen that lc A x dx . I t is exceed

ingly important to observe,however

,that the correspond

ing value of y’

y or Ay is not, in general , equal to dy ;for that would give

k It z dx s dx

which , in general , is impossib le, since dy/dx is not a valueof the ratio (y

’but is its limit under the law that

b vanishes .I f

,however

,the func tion changes uniformly with respec t

to the variable (y’ —

y)/lz wil l be constant for all values ofit andy y wil l be equal to dy when it is equal todx .

7 1 . T he fol lowing are important fac ts in regard to a dif

ferential c oefficient

64 ELEEEEEN TLAL CAL CUL US .

I t is z ero for a cons tant quantity. In other words , a

constant has no differential coeffic ient .I t is constant for any func tion wh ich varies uniformly.

I t varies from state to s tate for any func tion wh ich doesnot vary uniformly.

I n general , therefore, it is a func tion of the variable, andhas a differential .I t is positive for an increasing func tion, and negative for

a decreasing one . § 63 .

I t may have values from — oo to 00

Having represented a func tion by the ordinate of a curve,

the differential coeffic ient is equal to the tangent of the

angle made with the axis of absc issas , by a tangent to the

curve at the point corresponding to the s tate considered.

6 2 .

T hus, assuming PR 2 dx,the differential coeffic ient of

the func tion whose graph is the curve AM,at the state

PA ,is

dy/dx tan XEA tan QAB .

I t should be noticed that dy/dx is independent of the

value assumed for the differential of the variab le ; for ifPR

’

dx,then Q

’D dy, and we have, as before, dy/dx

In this il lustration the func tion is an increasing one , its

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66 D TEEEEEN TLA L CAL CUL US .

72 . T he fol lowing fac ts concerning a differential of afunction should now be apparent

I t is z ero for a cons tant.I t is constant for any function which varies uniformly.

I t is a function of the variab le for any func tion wh ichdoes not vary uniformly ; and in such cases it has a differ

ential.

I ts value depends upon that of the differential coeffic ientand that assumed for the differential of the variab le .

I t may have values from 00 to “ i 00 .

I t wil l be numerically greater or less than the differentialcoeffic ient depending upon whether the differential of thevariab le is assumed greater or less than unity.

I t has the same sign as its differential coefficient,and

therefore is positive for an increasing func tion and negative

for a decreasing one.

F unc tions wh ich are equal in all their successive s tateshave their corresponding differential s equal .

73 . Tbe differential coefi cient of anyfunction is equal to

til e reciprocal of tile cor responding difi erential coefi cient of its

inversefunction.

L et y= f (x) ( 1 ) and x F (y) ( 2 ) be any direct

and inverse func tions . L et A x and Ay represent , respec tively, any set of corresponding inc rements of x and y in

I t follows 4) that theywill represent a set of the same in

and we have

Ay/A x

Hence 70 )

dy 1 1

dx A x/ Ay dx/dy'

T o illustrate, let the func tion y be represented by the

D JEEEEEN T I AL—D LEEEEEN T JAL COEEELCJEN T .67

ordinate of the curve AM . Assume PB,and from

the figure we have, corresponding to the state PA,

tan QAB .

T he inverse func tion wil l d y

be represented by the ab

scissa of the curve AM re

garded as a func tion of the

ordinate ; and assuming dy

XL,we have for the state XA ,

corresponding to A ,

[TE/AH dx/dy tan EAT] . EAH =90

°

QAB .

I I ence,

I

tan QAB cot EAHtan F AI -I

;

I t should be observed that,in general , dy in the first

member of the above equation is not the same as dy in the

second ; for the first is the differential of y as a func t ion,

and the second is a differential of y as the independent

variable. T he same remarks apply to dx , in the two mem

bers , taken in reverse order.

T he figure illus trates the differences referred to.

74 . Tbe differential of tbc sum 0f anyfinite number Of f unc~

tions is equal to tire sum of t/zeir diff er entials.

Let y v s w etc .,in wh ich v, s, w ,

etc .

,are func

tions of any variable, as x . Increasing x by A x,we have

y+ Ay= v+ Av — (s + A s) + m Aw + etc .

VVhence

Ay= Av A s —l A w + etc . ,

68 D JEEEEEN TLAL CAL CUL US .

and 70 )

liml t A ?) A S A 70

AWE-W A x A x+

xetc

dx dx+

T herefore

d(v

I t fol lows that tbc differential of tbe sum of any finite

number of f unctions and constants is equal to tile dzf erential

of tbc sum of tbefunctions. T hus,C being constant

,

d[f (x ) C] : df (x ) .

If cor responding dzj’erentials are equal it does not follow

tbat t/zefunctionsfrom w/zic/t t/zey wer e der ived ar e equal .

75 . Tbe differential of tbe product of any number of func

tions is equal to tlze sum of tbeproducts of tbe diff erential ofeacbfunction by all of tile otber functions.

Let v z yz be the produc t of any two functions of any

variable, as x ,then

77 + A v : (y—j —j- Z . Ay+y . A z —j—Ay. A Z,

whence A 0 2 2 . Ay+y . A z -l A y . A 3 ; and (§ 7o )

limitAxw o A x

11m z +y A y

zdy/dx ydz/dx .

T herefore, dyz Zdy yaz .

T o illustrate,let O NPM be a state of a rec tangle w ith a

R variable diagonal represented by

x . T wo adjac ent sides,denoted

by z and y respec tively, wil l befunctions of x

,and ye will be the

variable area of the rec tangle.

Assume dx : PP,and complete the rec tangles Ps dz

D LEEEEEN T IAL—D JEEEEEN T IAL COEEELCTEN T .69

and PS 2 zdy. T hen,since dyz yds zdy, we have

d (rect. OP) rect. PT rec t. PS,which is the amount

of change required by the definition of a differential . § 68.

I t fol lows that t/ie differential of tbc product of a function

and a constant is equal to t/zeproduct of tbe constant and tbc

dif erential of tbc f unction. T hus,C being cons tant ,

dCf (x ) Cdf (x ) .

Let vsu be the produc t of anythree func tions of the samevariable. Place vs 2 r , giving vsu 2

"

D ifferentiating, we have dvsu dru rdu udr,inwhich

dr vds sdo. Hence,by subs t itution.

dvsu vsdu vuds sudv ( 1 )

In a similar manner the principle may be estab lished forany number of functions .Dividing each member of ( 1 ) byvsu, we have

dvsu da ds dv

vsu u s v’

Similarly, it may be shown that tbc differential of tbe

product of any number of functions divided by tbeir productis equal to tbc sum of tbc quotients of tbc diff erential of eac/z

function by t/zefunction itself .

EXAMPLES.

d[(a+on A] (e x ) d<a+ x ) <e+ x ) are+ x ) (a 6+2x)a’x

a [3(c en see . are an no x ) (a eve .

Ji ha d—9 0x ] “ la ’ i‘

(a + x )x a + x+d

76 . Tbe diff erential of a quotient of twofunctions is equalto tlze denominator into tiie diyf erential of tii e numerator

,

70 D I F FEPEJVT I AL CAL CUL US .

minus tbe numerator into tbe di/f erential of tbe denominator ,divided by tbe square of tbe denominator .

Let y v/s, in which v and s are functions of anyvariable. T hen v sy, and

do sdy yds (s y vds)/s,

whence dy (sdv VdS)/Sz

.

C being a constant, we have

d(C/S) Gds/s2

,and d(V/C) dV/C.

EXAMPLES.

+ eu d oe/(x+ xr. don) stifle/x”

d'

l I l/(x I )] (x9

2x I )?

d(x/3) doc/3. a’

(2x/5a) = zan/so.

77 . Tbc differential coefi cient of y regarded as afunction

of x is equal to tbe product of tbc dif erential coefi cient of yregarded as a function of u

,by tbc diff erential coefi cient of u

regarded as a function of x .

Having and u 7 let A x,A u and Ay

be corresponding increments of x,u and y,

respec tively.

T hen 4) A n is the same in both cases,and

Ay/ A x Hence,

limit—>o

and 70 ) dy/dx (dy/du) X

Similarly, havingy f (a) , n

dy/ds (dy/da) X

and the same form holds truethe intermediate functions .

[A y/ A x] lim [Ay/ A u] lim

(du/dx)

whatever be the number of

D I F FEREN T JAL— D I F FEA’EN T IAL COEF F I CI EN T . 7 1

Having y and x we maywrite u 2

and dy/dx= (dy/du) but 73) du/dx=

fi ence,

dy/dx

T hat is, t/ze dif erentia l coefi cient of y regarded as a f unc

tion of x is equal to tbc quotient of tbe di/f erential coef ficient

of y regarded as a function of u,byMe dif erential coefi cient

of x regarded as a function of u.

EXAMPLES.

I . y auQ, u bx . 2ab9x .

dy dy/du dx

4 . y u9,

x 3n, 2 s2. dy/a

’s

du/ds dy

dz/dsXda

6; J’ 7/ 2 7} z F (s) ’ Z F 1 (x) °

dy dy/da dv/ds dz

dx cl iff/dotXdz/a

’

sXdx

'

7= x + n.

5, y u Z

Then

I , hence

72 D I F FEREN T I AL CAL CUL US .

CHAPTER V.

DI FFERENT IAT I O N O F FUNCT I O NS .

FUNCT I O NS O F A S INGLE VAR IABLE.

78 . T he differential of any func tion of a single variablemay be determined by applying the general rule , 70 ,

6 2,

and mul tiplying the resul t by the differential of the variab lebut by applying the general rule, 70 , 6 2

,to a general

representative of any particular kind of func tion,there wil l

result a par ticular form,or rule

, f or differentiating sucb

functions,which is general ly used in prac tice .

79 . Tbe dijf er ential of any power of anyfunction wit/t a

constant exponent is equal to t/zeproduct of tlce exponent of tbc

power , tbefunction w it/t its exponent diminis/zed by unity, and

tbe dijf erential of tbc function.

L et y x”

,in which x is any variable and n is any

constant. T hen,increasing x by b , we have 70 )

ctr limit (90 x"

Adx—>o

Plac ing x b s,whence it s x

, and as It o,

s x,we have 40 )

and dx “ nx"“ 1dx .

S

Having y”

,in which y is any func tion of any variable,

as x,we have 77)

dyn

/a’x (Ja

n

/dot)

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74 D I F FEREN T I AL CAL CUL US .

26 . a’

(2x9)2

rox sa’x .

2 8 . d (3x )’ 2

2 ( 3x )—3d( 3x) 6(3x )

- 3dx .I

s ew e?

) d(a?

(a?

d [(x3—l a) ( 3x

2 b)] ( 1 5x4

3bx2 6ax)dx .

d [x 3/m x2

)3] 3x

2dx/( I x2

)5 l2

.

d [a x 3x2

4x3

] ( 1 6x 1 2x2

)dx .

d [( I x2

)( 1 x3) ] 2 ( 2x ar

2

5x4

)a’x

d (a berm) ” z : bmn(a—1dx .

daze onV1 + x9] w a

s"d [x re x F ox/(a

?

d ( i + x ) dx/ 2 ( 1 4 h e.

38. of Va ”

+ 959

xdx/ var+ x

2.

were + e >u (se sw) (a oxide .

eter

/(x on

] ner ve /( r one

.

d[I / VI x9] xa

’x/( I x

9)3’2

.

d [x/ VI x] z ( 2 x)dx/2 ( 1

d[x/ V1 x2] doc/( 1

d [x3/( 1 x

iii /2] 3x

9dx/( 1

d [ Vi + x/ Vi x ] z: doc/( I x ) Vii—555.

open/( i 2nx2n“ 1dx/( I

d[( r x )/ VI x ] z ( 1

d [x/(x Vi x9) ] dx/ vi A (o Vi

d 3/(2a x ) (3a x ) Vx

-

dx/(za

d ( Va

d [ax/(x Va x a x/(x Va sar)2 Va 4 x

x’z/a ) dx/x

2Va x

9.

D I F FEREN T IA T I ON O F F UN CT I ONS . 75

d [x/( I x ) r ner ve /< 1 on“

.

d l(a ‘ i ” ml/c VT ] (Mm-"Sawx/MV959

d [xQ/(a x

3)9] ” (A 2x

3)dx/(a x

3)3.

a’x (a

‘2x?) Va x

2z (a

4 m xey/

e.

d [x3’2( 1 x l

l ’Q/( I x )"2] “

2 (3+2x 3x9) a

’x/2 ( r

58 . on+ x ) V1 x z ( I 3x )dx/2 Vi

59 . d [ V1

-

x_

] — dx/2 ( 1 —Vo—c) Vx — x

2

6 I . u z : [a b/ Vg -l (c2

d(a—b/ [b/zx VE+2 (e—ar)- f ( 2x )/3]dx

[b/2x Vx Inc/go?

) a’ 32 _ 2 2

2x 4/x c — lx 4 446 x )

62 . In the parabo la y2 9x , find the rate of y w ith respect to x

when x z 4 . What value w il l x have when rate oty equals that of x ?When rate of y is the greates t ? When the least?

dy/dx : l: 3/2 Mx . z: 3:

dy/dx : l: 107 : I g ives x _

dy/dx is the greatest when x o,and the least when x

63. F ind the s lope of the curve y z : 1: M9 or2 when x 2 ,

values of x and y when the slope is I .

Ans . ziz x’

: t VI Ey' z iF V4 . 5

64 . F ind the ang le wh ich the curve y x/ ( I x9) makes w ith if at

the ir po int of intersec tion.

65 F ind the ang le at wh ich the curvesyQ

1 0 x and x2+y

9 144

in tersect . Ans . 7 1°

0’

58

76 D I F FEF EN T I AL CAL CUL US .

66 F ind the value of x at the po int Where the s lope ofy?

4x3 is

Ans .

67. At what rate does the vo lume of a cube change with respect to

the leng th of an edge ? Ans . 3 (edge)?

68. F ind the ang les that a tangent to the curve x? 6y

2+ 3y

—l 1,

at the point (8 , makes with the axis X . Ans . tan

69 . F ind the rate of change of ( Vx 3/ax'2) when x 3.

Ans . 2/9a .

70 . F ind the rate of change of the ordinate of a circ le with respect

to the ab scissa . Ans .:Fx/i

'zx”

.

7 1 . F ind the rate of change o f the ordinate o f an e l lipse w ith re~

spect to the abscissa. Ans .:Fbx/a Va? x

2.

80 . l e dzf erential of tbc logar it/zm of anyfunction is

equal to tbc dijf erential of tbc functiondivided byMe funct ion.

Lety log x ,then l oa § 4 2 )

a’

y limit loe(x n) log x

log 1

- lim

1 1

t log 1 +

Hence, d log K dx/x .

Since log, e M , it follows that d log . x M ,

Having y we write (877)

D I F FEREN T I A T I ON[

O F F UN CT I ON S . 77

'

d logy d log yxdy d y

dx dy dx y dx’

Hence, d log y dy/y.

EXAMPLES.

I . a’

log em 2 . d log sin x (d sin x )/sin x .

3. a’log x

2a’

xQ/x

'22xdx/x

22dx/x .

d log Vx : dt/o-

l /V;

5. d log xn a’xn/xn nxn

- ldx/xn ndx/x .

6 . a’log we

l owe —I / 1/x2 - I xa

’x/(x

9I ) .

7. a’ log [( 1 + x l/( I x )] x )l

x9) .

8 . a[leg log x] d log x/log x x dx/x log x .

9 . d [x log x] xd log or log xa’x ( I log x )dx .

m. d log or

x3) one Len/(x x

3) (ex sense/(x? an.

1 1 . d( log m(log x )m“ l d log x m( log x )m

1 2 . d( I / log x") d log xn/ ( log x

”

)2

na’x/x ( log xn)2.

af log on x) (ix/< 1

d log + x z 2dx/3x%( 1 xi ) .

d x’m ( log x )n [m( log x )

n n( log x )n ' 1]xm

- 1dx.

a’logWm /we

I ] a ux/(as

d log [( 1 VI x2W] fix/x VI x

”.

d log van/< 1 v’

xn dx/(x x )We

“

.

d log [x V-T'

i doc/Vac? 1 .

20 . d logx 4/x

'2 —l—I

78 D I F FEREN T I AL CAL CUL US .

Vx Q —l- IzT — x — 2dx

2 1 . d logvx2+ e

2+ x vx r+ a

2 2 . d logV

-

I

—TFZ—l- VI — x

+ x VI — x

23 . d logVVI + x + 36

V1 + x9 —x VI —l—x

’

d log VET-To?) xdx/(a2

d log Va? x2

xdx/(a2+ x

2

) .

d (log l(a one nor/2a) AAA

d log (x i 4/x2t a

g) : i dx/ it/x

‘2: l: a

”.

d[log n times)

30 . d log [log (a r: —l bx” ) log (a+

31 . u ( log o . Put log xn z : y, then 77)

da d(log xn)m mu( log

32 . F ind the rate of change of a logarithm in the common system

with respect to the number. Ans . M 1 o/number .

33. F ind the s lope of the curve y loga x at the point ( 1 , o) .

Ans . Ma .

8 1 . Tbe differential of any exponential function w it/t a

constant base is equal to tice pr oduct of tbefunction, tbe logar itbm of tbe base, and tbc dif

ferential of tbc exponent .

L et then (sm. § 43)

fb' _ 1imit a - da’ I

dx 1m)» 0a”log a.

Hence,

a“ log adx.

D I F F EREN T I A T I ON O F F UN CT I O IVS . 79

Having at,in whichy f (x ) , we have 77)

ay log a X

T herefore at log ady.

I t fol lows that dc? oydy.

EXAMPLES.

aa3z log a dx 2 2 : 2 6,a log a dx .

2 . dalog x a

log x log a d log x

3 . dam” av” log a d

a

awdx“ —l xnda"c1

6 . de e” doc/x

2.

7. d [(eac e- w)/2 ] z : (em

8 . c iao?” a)/z ]

9 . dew log x ( I /x log x )ewdx .

10 . d [(ew 2ewdx/ (ew

I I . d log [(e5c I )] 2 exdx/ (e2a’ I ) .

1 2 . d log (ex e‘ w) (ex e e

‘ w) .

1 3 . d log (ex e—w) r: (eac e

‘ w) .

14 . a f(a “c ~l~ I ll log a

1 5. d(aw—l x )9

2 (a~’U x ) (aw log a —l 1 )dx .

16 . dex( 1 x3) e

ff/( 1 3x

2x3)dx .

1 7. a[(cf° e' w)] 4dx /(e9c

1 8 . [ex( 1 x )

19 . dx ”

( 1 x )” noon—1G —f 2x )dx .

alog a: log a dxfx.

“V5: log a dx/2 Vo

—cl

a log a doe/x“.

awxn ‘ 1(x log a 12W.

80 D I FFEREN T I AL CAL CUL US .

2 0 . When x 0 , find the inc l ination of the curve y 10 56 to X

Ans . 66°

3 1'

30

82 . Logarithmic Differentiation. T he differentiationof an exponential func tion, or one involving a produc t or

quotient, is frequently simplified by first taking the Na

pierian logarithm of the func tion.

T hus, let u = yz , in whichy and z are functions of thesame variab le.

and 80. s75)

Hence. (in dy”

zaz ‘ I

a’

r T"logya

’z.

wh ich is the sum of the differentials obtained by app lying

first the rule in 79 , then that in 8 1 .

EXAMPLES.

— i~ 10 g (x — 2 ) — % 10 g (x

dx 7962 sox 97

2 3 x 3 1 2 (x —1 ) (x— 2 ) (x — 3)

(x2

(7x2

sox 97)dx

1 2 (x 2 )7/4(x 3)I°/3

2 . a’xx

xx

( 1 log x )dx .

xx( 1 log x )dx .

x

xx

xx

( log2x log x I/x )dx

5 . d l x ( I x ) r : (2 + 36

)3/2

( 7x2 —l 30 x 97)a

’x

2 )7’4(x

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82 D I FFEREN T I AL CAL CUL‘

US .

Similarly, by applying the general rule, § 70 , and the

principle in § 77, the differential of any trigonometric function may be determined but it is perhaps simpler to makeuse of the relations existing between the functions .

(1 cos X d sin (rt/ 2 x ) cos x )d( rt/ 2 x)sinx dx .

dxd tanx a

sm xsec

2x dx ( 1 tan

2x )dx .

cos x cos x

d cot x d tan ( 7t/a x ) dx/sin2x cosec

2x dx

( 1 cot2x )dx .

(1 sec X d( I /cos x ) sin x dx/cos2

x tan x sec x dx .

(1 cosec x d sec ( rt/ 2 x ) cot x cosec x dx

d vers x d( r cos x ) sin x dx .

dcoversx : d yers (n/z x ) cos x dx.

Inorder to il lustrate the formulas for the differential s of thesine and cosine of any angle

,

let ACB 2 gb be any given

angle . Assume B CN : d¢ ,

and with any radius,as CO

R,describe an arc

,

OMN . T hen

PM/R sin a) ,

O A CP/R cos go, a rc MN Pd¢ .

T he definition of a differential in th is case, requires

that sin qb and cos gb, retaining their rates at the states

corresponding to gb ACB ,shal l continue to change from

those states wh ile to increases by the angle B CN dgb.

D I F FEREN T I A T I ON O F F UN CT I ON S . 83

Draw the tangent line to the arc at M,and lay off M T

equal to the arc MN q fi. T hrough T draw TQ parallel to MP

,and through M draw M Q paral lel to O C.

T hen Q T and M Q are,respec tively, the changes that

the lines PM and CP would undergo were they to continue

to change with the rates they have when qb ACB,while

(b increases by dqb.

Hence, Q T/E,and M Q/E are

,respec tively, the

changes that sin T and cos qb would undergo under thesame requirements .The angle M TQ 1

“

cb. Hence,

Q T=M T cos (b l-P cos gbdqb, and Q T /Pz d sin 49:

M Q=M T sin ¢ =E s inmap, and M Q/d cos (p:

Similarly the formulas for the differential s of the othertrigonometric func tions may be il lustrated.

R egarding the right linesPM

,CP

,O E

, O’B

, etc .,as Q

func tions of the variable angle

75, we have

dPM z a’If sin $ 2 11?CO S ¢ d¢ . dCPZ dE cos qb z

— F sin ¢ d¢ .

dOE z dP tan dO’B z dP co t q> _ -Pa’qb/s in

2go.

dCE dB sec 4) dCB dB cosec q)

- R tan 90 sec obo’

gb .—R cot (b cosec (pdgb .

dOP d}?vers (b 1?s in ¢ d¢ . dO’

d B covers qbz— [ i’ cos qbdqb .

I t is important to notice the difference between the dif

ferentials of the above lines , wh ich depend upon the radius

84 D I F FEREN T I A L CAL CUL US .

of the c irc le used, and the differentials of the trigonometricfunc tions which do not depe

n d up on any radius or circ le.

10

m

a

g

m

a

s»

1 0 .

I I .

33.

34 .

35

EXAMPLES.

a’sin x 2x cos x

2 dx . 1 2 . d eos x2

2x s in x9 dx .

d s in? x z 2 s in x cos x dx . 1 3 . a’

s in3 x 3 s in x co s x dx .

d cos’x 2 cos x s in x dx . 14 . d cos

3x 3 cos

ax sin x dx .

d tan2 x r : 2 tan x dx,/ cos2 x . 1 5 . d tan3 x z 3 tan

2x a

’x/cos

2x .

a’c0 1

9x — 2 cot x dx/s in

2x . 16 . d co t3 x —

3 x .

d sec2x z 2 sec

2x tan x dx . 1 7 . d sec

3x z : 3 sec

3x tanxdx .

d cosec9 x : — 2 cosec2x co t x dx . 1 8. d cosec

3x : —3 cosec

3cot x dx .

d vers x : 2 vers x s in x dx . 19 . d vers3 x =3 vers”sin x dx .

d covers2x : 2 covers x cos x dx . 2 0 . a’cos x

3 —3x

Qs in x

3 dx .

d tan x 2 2xa’x/cos

2x 2 . 2 1 . d covers

3x : — 3 covers

9x cos x a

’x .

a’eSil l ”3

eS in “7

cos x dx . 2 2 . d log cos x tan x dx .

d s in2 x 2 z 2 sin xza’

s in x2z: 4x sin x

2cos x

2 dx .

t an 2x sec22x dx/ i/ tan 255.

d cos ( 1/x ) ( l/ JCQ

) s in

d tan2 x2

4x tan x2 dx/cos

2x’2

d cos mx s in mx d(mx ) m sinmx dx.

. d s in 3x z z cos 3x d 3x 3 cos 3x dx . .

d s in2 2x 4 sin 2x cos 2x dx .

d s innax na s inn—lax cos ax dx .

d tann x z n tann- l

x dx/cos2x .

d( tan x x ) tan2x dx .

d [(x sin x cos x )/2 ] sin2 x dx .

a’tan x sec x sec2 x tan2 x ) sec x dx.

l /xd tan a 5 6 0

2a log a dx/x

”.

D I F FEREN T I A T I ON O F F UN CT I O NS .

36. d [( I tan x )/sec x ] z ( cos x s in x )dx .

37. d [s in nx/cos” x] n cos (n 1 )x x .

38 . d tan VI x ( sec VI x )2dx/2 V1 —x .

39 . s inV; cosVb-

c

-

dx/4 Vx sinV5}:

40 . d (eos x )sin “7

(cos x )Sin “7

(cos x log cos x sin2 x/cos x )dx .

4 1 . d s in ( sin x ) cos x cos ( sin x )a’x .

42 . d cos ( sin x ) cos x sin ( sin x )dx .

43. d sin ( log nx ) cos ( log

44. di“ a:

xSin x

( cos x log x sin x/x )dx .

d(s in ax )b ab ( sin ax ) cos (bx

45.(C0 5 bx ) “ (cos bx )

a+1

46. d s in3 x cos x sin2 x ( 3 4 s in2 x )dx.

47. d cos5x sin2 x ( 2 cos

2x 5 sin

2x ) cos

4as sin x dx .

48. d [( s in x cos x cos x 2dx/( sin x cos

49 . d(sin x )“ mx

(sin x )tan a:

( 1 sec2x log sin x )dx .

50 De termine the manner in wh ich the sine of an ang le varies withthe ang le .

The rate of change of sin x is 70 ) d sin x/dx cos x , from whichwe see that as qb increases from 0 to 7t/2 , the rate is but diminishing ; hence , the s ine increases, but its increments decrease .

F rom 7c/2 to 713 the rate is and diminish ing ; hence , the sine di

minishes and its decrements increase numerical ly.

F rom a to the rate is and increas ing . T hat is , the sine

decreases , but its decrements diminish numerical ly.

F rom to 2 77: the rate is and increasing . That is , the sine

inc rea s e s , and its increments increase .

In a s imilar manner the c ircumstances of change of each trigono.

metric function w ith respect to the ang le may be determined .

51 . Assuming dx 7t/4, w e have ,corresponding to x z : 7r/é,

d sin x : d cos x = d tan x :

8 3

86 D I F FEREN T IAL CAL CUL US .

a’

cot x d secn . d cosec x

52 . Corresponding to x 7t/4, we have

d s in x d cos x

2

d cot x d sec x d cosec x

53. What is the value of x when tan x is increasing tw ice as fast

as x ? Ans . 7r/4 .

54 . F ind the rate of change of the tangent , regarded as a function

of the sine of an ang le .

Since tan x : f x , and sin x Ex , we have 77)

d tan x/d sinx (d tan sin x/dx )

( I /cos2x )/cos x I /cos

3x .

In a s imilar manner the rate of change of any trigonometric func

tion regarded as a function of any o ther may be found .

55. F ind the s lope of the curve y siu x when x : o x

x n/2 . Ans. 1 ,

56 . When x 7r/3 , find the inclination of the curve y tanx to X .

Ans . 75°

57'

50

57 . F ind the angle which the curvesy sin x andy cos x make

w ith each other at the ir point of intersection. Ans . tan 124/ 2 .

I nver se T r igonometr ic F unctions.

L et to 2 : sin 1x ; then x z Q5 and dx/dgb cos gb.

d sin—lx/dx 1/cos to I / rl: VI x

2

.

*

The s ign depends upon that of cos a) . Formulas invo lving thedoub le s ign in th is article are g enera lly w ritten w ith the plus s ign

only, wh ich corresponds to ang les ending in the firs t quadrant.

D I F FEF EN T I A T I ON O F F UN CT I ON S . 87

By applying the principle in § 77, it may be shown that

d sin" 1

y dy/V1 y”

, in which y 2 : f x .

d cos—lX d(a/ 2 sin

“ 1x ) — dx/ VI X

“.

(1 tan—llx

Let qb tan"llx then

x tan go, and dx/d¢ 1 tan2

Hence d tan“ 1x/dx

d cot" 1X d( rt/z tan

“ lx ) dX/( I

d sec- 1 X dx/x Vx

2 — 1 .

Let 525 sec1x,

then x sec and dx/dgb sec (b tan ob. Hence

1d sec“

x 1 1

5 6 0 05 tan 95 sec gs ec2

go3

d cose0" 1X d( a/ 2 sec

" 1X) dX/XVX

2I .

d vers* 1 X dx/s x

2.

88 D I F FEF EN T I AL CAL CUL US .

(1 covers 1x d(a/ z vers “ 1

x ) dx/V2x

R egarding qb as a func tion of

the line PM , denoted byy, we

have qb sin Hence,

Similarly, having

CP : y, (p cos we have dob

OE : y, to tan we have do)

O’B : y, ob cot

—I i we have dob

we have dabrVJ/ 2 —182

CB =y, q) cosec we have dgbEdy

P0 =y,we have dqb

VzRy-e-y?

O'

Q : y,coversin we have dcp

EXAMPLES.

1 . d sin" l x

9 2xa’x/ VI x

4.

2 . d sin“ 1 2x VI 2dx/ VI x

2.

3 . a'

tan—1 [(a -l dx/ ( I 4. x

9) .

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90 D I F FEREN T I AL CAL CUL US .

30 ~ d tan- 1W1 £2

"

x ) fix/2 ( 1

3 1 47

0 0 52

) 2dx/1/

1 x2

.

1 o

32 . d esm x

esm x

dx/Vr x2.

33. d vers dx/x 1/ 2x I .

34 (fit VW A] z ydy/V2 77 —y“

‘

35. d in xxsm x

(s in_ 1

x/x + log x/vI x2)dx .

36 . y z 2 tan- 1

x )/ 1+ x . ( 1 x ) z :

Hence , x 2 cos y,and a

’

y doc/V1 x2.

37 r CO s 3x2

)/x3l Put (4 sech/x

32 .

T hen dy/da z 3/x 4/x2

I .

38. Wheny : 0 , andy 2 r , find the slope of the curve

x 2 r vers 4/ 2 ry—y2

.

Ans. co, and 0 .

MI SCELLANEO US EXAMPLES.

d s in x cosx dxI . d log sm x

s in x sin x taux'

d tan x

2 . d log tan xtan x cos x sm x sm 2x

x cos

3 . d log tan cosec x dx a’

log2 sm x cos x

dxa’10 tan4 g

sm ( 71:/2 x ) cos x

1 sin x

see x a’x d

2

logI

5. d

6 . dex cos x ex(cos x s in x )dx .

7. dxesm x

e8m x

x cos x )dx .

D I F FEREN T I A T I ON O F F UN CTI ONS . 9 1

8 . dxeCO S x

ecos x

x sin x )dx .

9 . d sin ( log x) cos ( log x )dx/x .

1 0 . u sinm x/cosn x . log u m log sin x n log cos x ,

CO S as

s in x

m s inmf"1 n sinm 1

I I . n log u x.

du u ( 1/x x2 ))dx ( I x x

1 2 . dx ” c” Sin";

nxn ' l

e” 3mx

13 . a’eCO S x

sin x ec"S x

( cos x

14. d [sin nx/sin”x ] n sin (n 1 )x

1 5. d cos log ( I /x ) sin log

16 . d cos sin x cos x s in sin x dx .

1 7. dex log x ex log

1 8. d log (xeCO S x

) ( I x sin x )dx/x .

1 9 . dx’”

( log x )’2

xm" l

( log x )’z— I

(m log x n)dx.

i i ( ‘

p’

y

20 . dx"elog x (n 1 )x

” —Ie1°g x

dx . 3O

Jl "

i

2 3. d s in tan x cos tan x dx/cos2x .

—j tan- l

2 5. d cos log sin x cot x sin log sin x dx .

26 . d sin 1/ s in x doc/ 2 cosec x .

2 7 . d log (x/ax ) log

2 8. d log s in—1

x r : dx/sin—l

x VI

92 D I F FEREN T I A L CAL CUL US .

29 . d sin_ 1

( tan x ) sec2x dye/VI tan

?x .

1x doc/cos

' lx x

230 . d log cos

3 1 . d log tan1x doc/ ( 1 + x

2

) tan" 1x .

32 . d tanT Ilog x dx/x [ 1 ( log

33 . d cos [a sin a sin (a cosec1x )dx/x Vx “

I .

34 . a’co t (cosec x cos x a

’x/( r sin2 x ) .

35. d sin—169 tan x ) 6’

xdx/ ( I x

”) 5

2 tan 4 "

Hyperbolic F unctions.

flex

e‘ x

)dx cosh X dx.

d cosh xt

z d—é—(ex

e' x

)

e‘ x

) dx sinh X dx.

(1 tanh X d(sinh x/co sh x ) sech?x dx.

d coth x d(cosh x/sinh x ) cosech2x dx.

(1 sech X d( I /cosh x ) sech X tanh X dx.

(1 cosech x d( 1/sinh x ) cosech x coth x dx.

EXAMPLES.

1 . c osh x sinh x dx/zvm 3. d log sinh x coth x dx .

2 . d log cosh x tanh x dx . 4 . d(x tanhx ) z : tanh“ x dx .

5. d [(sinh 2x )/4 —l x/2 ] cosh2 x dx.

6 . d [(sinh 2x )/4 x/2 ] s inh2 x dx .

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94 D I F FEREN T IAL CAL CUL US .

Hence

sech 2y 1 tanh ”y 1

Lety coth " 1 x ; then

x coth y,

d sech—1 x d log

and 85) dx/dy cosech2

y.

Lety sech " 1 x ; then

sech y,

Hence

d cosech-1 x z d log

and 85) dx/dy sech y tanhy.

1

sech y tanb y xVI

- dx

X XVX 2

+ I

L ety cosech“ l

x ; then

x cosech y,

Hence

and 85) dx/dy cosechy coth y.

D I F FEREN T I A T I ON O F F UN CT I ON S . 95

d cosech “ 1x 1

cosechy coth y 1

EXAMPLES.

a’

cosh"1 (x/a) dx/ Vx 2 a”.

2 . d s inh 2 : dx/ V52

4.

3 .z a dx/ (a?

d co th—1 (x/a ) a (IX/(622

5. d tan" l ( tanh x ) sech 2x dx .

6 . d [% tanb“ l x «lu g—tan

“ 1 x] d log z 5 tan—1

doc/( 1

7. d [3w t 1 x g- coth‘“ l x ] cot

- lx log

dx/(x4

I ) .

8 . d [éx 1/x2 1

§a2

1/x2

a2dx .

9 . d [%x et/a2 - x

2

%a2sinh Va ” x

2dx .

Geometr ic F unctions.

87 , Differential of an Arc of a Plane Curve— Let s

represent the length of a varying portion of any plane curve

in the plane X V. I t wil l be a function of one independent variableonly which we may take to

be ac.

Assume any point of the curve,

as M,and increase the correspond

ing value of x OP,by PP

’

A x . A s : MM ’will be the cor

responding increment of s, and Ay QM’,

96 D I F FEREN T IAL CAL CUL US .

T hen s 7c . s44)C“ M ”

lim

d dy /dx . Similarly Va’x2

Hence, ds 2 VdX2 —l dy2

.

*

The double sign is omitted because s may always be considered as

an increas ing function of x .

T hat is, tbe difi erential of an arc of a plane curve is equal

to tlte square root of tbc sum of tbc squares of tbc differentials

of tbc coordinates of its ex tremepoint.

I f s were to change from its state corresponding to any

point, as M ,with its rate at that state unchanged

,the

generatrix would move upon the tangent line at M ; hence,M T 2 4/dx

2dy

2

represents ds in direction and measure.

I n order to express ds in terms of a single variable and

its differential , find expressions for dy in terms of x and dx ,

or of dx in terms of y and a’

y, from the equation of the

curve,and subs titute them in the formula.

T hus,let s be an arc of the c irc le whose equation is

x2 —l—y

2

4 . Solving with respec t toy, and differentiating,we have

dy z :d x/ V4 x2

.

The square of the differential o f a variable represented bya sing leletter is generallyw ritten as indicated in the above formula , and is

s imi lar in form to the symbo l for the d ifferentia l of the square of the

variable . S imilarly, the nth power of dx is general ly written a’

x".

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98 D I F FEREN T I AL CAL CUL US .

->o

hence, ds VdX 2 dy2

dz 2

s may be a curve of single or of double curvature . T he

inc rement A s may or may not lie in the proj ec ting plane

of the chord MM ’I f not

,the proj ection of the chord

MM on the plane X V will change direc tion as A x ap

proaches z ero, but the above relations will not be affec ted

thereby.

L et cr’

, [3’

and y’

represent the angles made by the

chord MM ’with X ,

Y and Z,respec tively, then chord

MM ’

/ A x = 1/cos a’. L et a'

, B and y, respec tively,represent the corresponding angles made by the tangent at

M . T hen

[ 1/cos a’

] r/cos a,and dx ds cos 0c.

Similarly dy ds cos fl, and dz 2 ds cos y,in which x

, y,z

or s may be considered as the independent variab le .

89 . Differential of a Plane Area.—L et u represent the

area of the plane surfac e in

c luded between any varying por

tion of any plane curve,as AM

,

the ordinates of its extremities,

and the axis X .

D I F FEREN T I A T I ON O F F UN CT I ON S . 99

R egarding u as a function of x let x z OP’be ih

c reased byP’P” = A x . P’MM ” P”

wil l be the corresponding inc rement of u. H ence 70 ,

y,

wh ich gives duz ydx .

T hat I S,t/ze dif

ferential of a plane area is equal to tlze ordi

nate of tbe ex tremepoint of tbc bounding curve into tbc differ

ential of tbc abscissa .

T o illustrate , let u represent the y

area B AMP,and PR dx ; then

du ydx rect. PQ , wh ich ful

fi ls the requirements of the definition of a differentialSimilarly, it may be shown that

xdy is the differential of the plane area included betweenany arc

,the abscissas of its extremities , and the axis of Y.

I n c ase the coordinate axes are inc lined to each o therby an angle 9

,we have du : y sin d

,or du = x sin tidy.

I n order to express du in terms of x and dx ,sub stitute

fory,or dy,

its expression determined from the equation of

the bounding curve .

T hus,if a

H

y bz

xfl

aMb is the equation of the bounding

bcurve, we have y Va 2 x

2

; and daa

90 . Differential of a Surface of Revolution.-Let the

axis of X coinc ide with the axis of revolution ; and let

BM 2 s be any varying portion of the meridian curve

* I t is important to notice and remembe r tha t ydx is the diffe ren

tia l of a plane area bounded as de scribed ; and that it is no t , in

general , the differential of a plane area o the rwise bounded.

I O O D I F FEREN T I AL CAL CUL US .

generated by it w il l be the inc rement of the func tion u cor

responding to A x . Hence 70 ,

l imit sur . g en . by arc MM ’

-> O A x

Assume PR dx ; then R’T = dy,

M T ds,and cos

R’M T : dx/ds . Substituting this express ion for cos R

’M T

in above,we have

and du z rtyds 2 7cy1/dxi"+dy2 .

H ence, tbc diff erential of a surface of revolution is equal

to fi xproduct of tbc circum. of a circle perpendicular to t/ze

ax is and t/ze diff erential of tbc arc of tbc g enerating curve.

Similarly, it may be shown that 2 71x d 2dy

2

is the

differential of a surface of revolution generated by revolv

ing a plane curve about the axis of Y.

I n order to express du in terms of a single variab le and

its differential , find expressions for y and dy in terms o f x

and dx,or of dx in terms of y and dy,

from the equation o f

the generating curve and subs titute them in the formula .

T hus,ify

22 : 2px is the equation of the generating curve,

pdx

VE55we have y : 7 525. and dy Hence,

T hrough M draw the tangent M T ,

the ordinate MP,and the right

line MR ’

paral lel to X . Let u

represent the surface generated bys ; and regarding it as a func tion

of x let x OP be in

c reased by PP’ = A x . MM ’

A s wil l be the correspondingincrement of s ; and the surface

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10 2 D I F FEREN T I AL CAL CUL US .

ridian curve, we have dv 71 ( 2Rx x2

)dx ; or since dx

92 . Differential of anArc of a Plane Curve in Terms

of Polar Coordinates .—L et r f (v) be the polar equation

of any plane curve,as referred to the fixed right

line PD , and the pole P. Let B M : s be any varying

portion of the curve,and PM r the radius vector corre

sponding to M . R egarding s as a func tion of v let

v be inc reased byMPM’: A v. T he are MM ’

2 A s wil lbe the corresponding increment of s . With P as ac entre

and PM as a radius, describe the arc M Q

'

. D eno te PM ’

by r’

th en Q’M ’

r’

r wil l be the increment of r corresponding to A v. T hrough M draw the tangent M T

, and

the chords MM ’and IWQ

’. T hen 70 , 55) we have

ds limit are l l’

dr2

+dv AWE—m A v a

’v2

Hence dS Ve —l

D I F FEREN T IA T I O N O F F UN CT I ONS .

limit Q’M ’

—> O A 7}

I f the radius vec tor PM coinc ides with the normal tothe curve at M

,the corresponding tangent

M Q’

w il l coinc ide with M T ;

and 55)

ds limit are MM ’

Q7?)

—> O A v

giving ds 2 rdv.

I n th is c ase dr 0,because

the motion of the generatrix at

the point considered is perpendicular to the radius vec tor.An important example of this case is a c irc le with the

T hat is, tbc diyf erential ofan arc of a circle reg arded as a function of tbc cor responding

ang le at tbe centr e,is equal to its radius into tbc diff erential

of tbe ang le.

T o il lustrate,assume M CQ dv then wil l the arc M Q

rdu. T he direc tion of the motion of the generatrix at

any point is along the corresponding tangent to s ; h enc e,by laying off from M upon the tangent at that point a dis

tance M T ds rdv,we have ds represented in measure

and direc tion.

pole at its centre .

L et BM : s be any are

of a c irc le , and B CM = v

the subtended angle . T hen,since the radius is alwaysnormal to the arc

,we have

10 4 D I F FEREN T I AL CAL CUL US .

In order to represent graph ical ly the general case when

M the as sumed point, and MPM’ = dv. I f r were con

stant, as we have seen 111 the case of a c irc le,M T

’rdv

would be ds ; but, in general , ds is affec ted by a uniformchange in r

,in the direction PM

,equal to dr . T o deter

mine it we have

I l

r11m1t Q M

r tan T’M T .

At T’draw T

’T paral lel to PM ; then T

’T/T

’M ‘

tan T’M T T

’T/rdv. Hence

,dr/dv r T

’T/rdv

T’T/dv, and d T

’T . M T ds f: Vdr

2

ri dv

2

, there

fore, represents ds in measure and direc tion.

In order to express ds in terms of a single variable and

its differential,find expressions for r and dr in terms of v and

do,or an expres sion for dv in terms of r and dr

,from the

polar equation of the curve and substitute in the formula.

93 . Diff erent ial of a Plane Area in Terms of Polar

C06rdinates .— L et u represent the area of a varying portion

of the surface generated by the radius vec tor PM revolvingabout the pole P. R egarding u as a func tion of v

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10 6 D I F FEREN T IAL CAL CUL US .

T he point is then said to be moving unif ormly, or with uniform motion with respect to the origin.

I f the distance does not change uniformlywith the timethe point is said to be moving with var ied motion withrespect to the origin.

A train of c ars moves from a station with varied motionuntil it attains its greatest speed, after which its motion

along the track is uniform while it maintains that speed.

With uniform motion equal distances are passed over in

any equal portions of time, and with varied motion unequaldistances are pas sed over in equal portions of time.

L et s in both figures represent the variable distance fromany origin, as A , to a point moving on any line, as MN O ;

A s M;

N 05

and let t denote the number of units of time during whichthe point moves then s

I f f (t) is of the first degree with respec t to t, the dis tances wil l change uniformly ; o therwise the point approachesor rec edes from the origin with varied motion. § 57.

T he rate of change of s, regarded as a func tion of t

,cor

responding to any position of the moving point, is cal ledthe rate of motion of the moving point with respect to the

origin and since uniform motion causes s to change uni

formlywith t, the rate of motion,in such cases

,is cons tant.

§ 59.

D I F FEREN T I A T I ON O F F UN CT I ONS .10 7

I n varied motion the rate varies with t, and is thereforea func tion of t .

95 . I f the differential of the variab le is assumed equal tothe unit of the variable

,the differential -of a func tion and

the corresponding differential coeffic ient wil l have the samenumerical value.

T hus , if 2,and dx 1 inch

, we have dx : 2

inches . I n such cases the differential of the func tion ex

presses the rate in terms of the unit of the variable and

since it is more definite, it is frequently used instead of the

differential coeffic ient .

T o illustrate , let s denote any variable distance regardedas a func tion of time , giving

Assuming anycon

venient length to representthe unit of t, we may, by sub

stituting s for y and t for x

determine a line , asAM

,whose ordinate repre

sent s the given func tion.

ds3

dtdt Q represents

the change that s would undergo in one hour , from the staterepresented byPA ,

were it to retain its rate at that state

I f PR dt represents one hour,

and is more definite than the corresponding abstrac t valueof ds/dt.

96 . Velocity.—T he differential coefficient of the variable

distance from any origin to a point in motion, regarded as

a func tion of the time of the motion,is cal led the veloc ity

of the moving point with respec t to that origin.

10 8 D I F FEREN T IAL CAL CUL US .

R epresenting the variable distance by s,and the velocity

by v, we have v ds/dt.

F or‘

the reasons given above , veloc ity is measured by the

produc t of ds/dt and the distance as sumed to represent theunit of time.

T hat is , tbc measure of the velocity of a point in motion

at any instant, in any required direc tion,is tbc distance in

tbat direction tbat tbepoint would go in tbc next unit of time,

were it to retain its rate at tbat instant.

I t should be noticed that the distance referred to above,

and represented by s,may or

may not be estimated alongthe line or path upon wh ich thebodymoves . T hus , if a point

moves from A towards B ,and

the velocity at any point, as C,

in the direc t ionAB is required,

the dis tance s is estimated along the path described ; butif the rate or veloc ity with which a point, moving fromA to B

,is approaching D is required, s must represent the

variab le distance from the moving point to D ,in order

that ds/dt shal l be the required veloc ity.

Since velocity is a rate of motion, it is constant in uni

form motion, and a variab le func tion of time in variedmotion.

97 . Acceleration.—T he differential coefficient of velocity

regarded as a func tion of time is called acceleration. I t is

denoted by dv/a’t,in wh ich v represents veloc ity. Since

acc eleration is the veloc ity of a velocity, it is general lyexpressed in terms of the distance which represents theunit of time.

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I I O D I F FEREN T IAL CAL CUL US .

ac tual velocity of a point at tlze unit’s distance from the

vertex .

1 0 0 . Angular Acceleration.—T he differential coeffic ient

of angular veloc ity regarded as a func tion of time is c alledangular acceleration. I t is denoted by doa/dt, when on

represents angular velocity.

PRO BLEMS .

1 . T he side of a square increases uniformly 3 in. a

minute find the rate per minute o f its area when its sideis 6 in.

Le t x side of square in inches , and u 2 area x2; then dx/dt

3 in. min. and du/dx 2x . H ence a’

u/dt

3 X 2x : 6x sq. in. min. and 36 sq . in. min.

2 . T he radius of a c irc le increases uniformly . 0 1 in. per

second find the rate of its area when the radius is 1 in.

Le t r radius, and u area 7t r2

; then a’r/dt . 0 1 in. see .

and du/dr 2 rrr . Hence (5 du/dt . 0 1 X 2 7W . 0 2 7tr sq. in.

sec . ; and .0 2 71'

sq . in . s ee .

3 . F ind the rate of the radius when the area of a c irc le

inc reases uniformly at the rate of M r sq . in. see .

Ans. 1 in. see .

4 . T he radius of a sphere inc reases uniformly . 0 49 1 in.

sec . ; find the rate of its volume when the radius is ft .

Ans. 2 0 0 cu. in. sec .

5 . T he volume of a sphere inc reases uniformly 50 0 cu.

in . sec . ; when its radius inc reases at the rate of 2 in. sec .,

find the radius . Ans . in.

6 . T he area of a rec tangle inc reases uniformly 1 0 0 sq . in

min. I ts base and al titude are inc reasing at the rates of

3 and 7 in. per'

min. respec tively ; find the area when the

al titude is double the base . Ans. sq. in.

D I F FEREN T IA T I ON O F F UN CT I ON S . I I I

7. T he diameter of a c irc le increases uniformly 3 in.

sec . ; find the difference between the rates of the areas ofthe c irc le and its c ircumscribed square when the squareis 1 sq . ft. Ans. sq. in. sec .

A man 6 feet in h eigh t walks away from a ligh t 1 0 feetabove the ground at the rate of 3 mi. per hour . At whatrate is the end of his shadow moving

,and at what rate does

his shadow increase in length

Let x A IM dis tance from foot of l ight to man, y AB dis

tance from foo t of l igh t to end of shadow, and s MB leng th of

shadow . Let t number of hours.

Then we have dx/a’t z g mi. hr A L z ro ft . ; M C: 6 ft . ; ft. ;

and it is required to find a’

y/a’t and

T he s imilar triang les AB L and D CL g ive x : y z : 4 : 1 0 ; hence,y 5x/2 , and a

’

y/a’

x T herefo re 77)

a’

y/a’f mi. hr

x : s : : hence , s 3x/ 2 , and d’r/a

’x

dr/a’t X 3 z mi . hr.

A vessel sailing south at the rate of 8 mi. per hour is2 0 mi. north of a vessel sailing east at the rate of 1 0 mi. an

hour. At what rate are they separating at the time ? At

the end of 1 —5hrs . P At the end of 2 & hrs. ? When are theyneither separating from nor approach ing eac h other P

Rice and Johnson’

s Calculus .

1 1 2 D I F FEREN T IAL CAL CUL US .

Let t time in hours from the g iven epoch .

Let AB : y 20 —8z‘ distance of fi rs t sh ipfrom B C t hours after the g iven epoch .

Le t B Cr - l ot z distance of second sh ip fromEA at the same time .

Given,

6Required ,

I 0 + 1 64 ;

I mi

: o

T he fol lowing general outline of s teps may assis t the stu

dent in solving similar problemsD raw a figure representing the magnitudes and direc

tions under consideration ; and denote the variab le partsby the final let ters of the alphabet .

Write w ith t he proper symbols, all known data and

indicate the symbols for the required rates .F rom the relations between the magnitudes find an

expression for the func tion whose rate is required, in termsof the variable .

D ifferentiate and determine values or expres sions forthe required rates .In c ase an explic it func tion of a variab le cannot

found make use of the princ iples in 77.

2 : 2pz is the equation of

a parabola OM . A point startingfrom 0 moves along the curve in

such a manner that z 1 6. 1 z‘ 2

; in

wh ich 2 is expressed in feet, and z‘ in

seconds . F ind the rate of x withrespec t to z

‘

.

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1 14 D I F FEREN T I AL CAL CUL US .

rate of Q5, fi rst as a func tion of y,then as a func tion of tan

Q5. Explain the difference between the two results .Ans I /S ;

1 7. A train is running from A to B at the rate of 2 0 mi.

an hour. T he distance from A to C on a perpendicular toAB is 2 mi. F ind the rate of the angle at C inc ludedbetween CA and a right line from C to the train.

Le t qb variab le ang le at C, and y mi. from A to train.

T hen CA 2 mi. , and dy/a’t 20 mi. hr.

Jf a’

cby— CA tan ¢ . (1) - tan I

CAand

dy A

_

C2

+ y

5°

1 8 . F ind the rate of the surface and volume of a spherewhen its radius decreases at the rate of 2 ft. per minute .

Ans. 1 6 75 7"sq. ft . min. ; 8 7W2

cu. ft . min.

1 9 . A bal l of twine roll s along a floor in a right line at

the rate of 4 mi. per hour . O ne end of it is 30 feet above

the floor and is attached to the top of a pole. At what rate

is the bal l unwinding when it is 40 feet from the bottom of

the pole on the floor Ans. mi. hour .

2 0 . A ladder 2 0 ft . in length leans against a wall if the

bottom is drawn out at the rate of 2 ft . per second, at whatrate wil l the top descend when the bottom is 8 ft . from the

wal l ? Ans. in. sec .

2 1 . T he side of an equilateral triangle inc reases at the

rate of 2 in. per minute. F ind the rate of its altitude,and

the rate of its area when the side is 1 0 in.

Ans. in. mi. 1 0 V3sq . in. min.

D I F FEREN T IA T I O IV O F F UN CTI ON S . 1 1 5

2 2 . Two straigh t railways intersec t at an angle of

An engine approaches the intersection on one of the tracksat the rate of 2 5 mi . per hour, and on the other track an

engine is leaving it at the rate of 30 mi. per hour. At whatrate are the engines separating when each is 1 0 mi. fromthe intersection? Ans. mi. hr.

2 3 . A man walking on a horiz ontal plane approaches thefoot of a pole 60 ft. in heigh t, with a constant rate. Whenhe is 40 feet from the foot how wil l the rate with which heapproaches the top compare with that with which he ap

proaches the bottom? How far will he be from the footwhen he is approach ing it twice as fast as he is the top

Ans. 2 V1/ 1 3, V1 2 0 0 ft.

FUNCT I O NS O F TWO O R MO RE VAR IABLES.

1 0 1 . The Partial Differential of a Function of Two or

more Variables, with respect to one of the variables , is the

change that the function would undergo from any s tate,were it to retain its rate at that state

,with respec t to that

variab le , While that variable changed by its differential .The Total Differential of a Function. of TwoVariables

is the change that the func tion would undergo from any

state,were it to retain its rate at that state, with respec t to

each variable, while both variables changed by their differentials.

Any func tion of two variables wh ich changes uniformlywith each variab le has a constant rate with respect to each

,

and its form must be some particular case of the generalexpres sion Ax By C

R epresenting such a func tion by 2,we have

2 : Ax + By+ C.

1 16 D I F FEREN T I AL CAL CUL US .

I nc reasing oc and y by their differentials , and denotingthe corresponding new s tate of the func tion by z

’

,we have

2’

A (ac -l fix ) + B (y dr) C

Subtrac ting ( 1 ) from member frommember, we have

z Adoc d .

Since the func tion 3 changes uniform ly with respec t toeach variable, the total differential of it

, denoted by dz , is

equal to the corresponding change in the func tion.

T herefore, dz Adx d .

Adoc is the corresponding partial differential of the function 2 with respect to x and d is the same with respectto y.

Hence, Me loz‘

al dzj‘erenz

‘

ial of any f ancz‘

ion of izoo var i

ables, zo/zic/z clzanges uniformly w ick respect to caclz,is equal to

lice,

snnz of Me cor responding par l ial differentials.

T he total differential of any func tion of two variableswh ich does not vary uniformlywith each variab le is not

,in

in general , equal to the corresponding change in the func

tion,but it is equal to the corresponding change of a func

t ion having a constant rate with respec t to each variable,

equal to that of the given function at the state considered.

I n other words,the total differential is equal to Mal of a

function conic/z c/zanges unif ormly with each variab le,and

which has at the state considered its partial differential sequal to the corresponding partial differential s of the givenfunction.

Hence, l/ce total differential of any function of two var i

ables is equal toMe snnc of Me corresponding partial differen

Zia/5.

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D I F FEREN T I AL

T hrough NM pass the planes MN R and MNR paral lel ,respec t1vely, to ZX and Z Y. Let MC be the intersec

tion of the surface by the plane .MN R,and let M I I be the

tangent to it at M L et ME be the intersec tion of the sur

face by the plane MN R ,and let ML be its tangent at M .

ME and ML determine the tangent plane to the surface at

M .

As sume doc NR ,and dy NR . Complete the par

allelogram N S and the paral lelopipedon N O . Produce

RK,RA and SQ .

T hen, tanKMH : az/dx ,

and ICI I =

tan AML 2 : az/dy, and AL 2

D raw I I D paral lel to R Q . Connect A and D byAD ,

and draw L T paral lel to AD .

Q1) R H,

and

D I F FER EN TI A T I ON O F F UN CTI ON S .I I 9

L T is paral lel to AD ,which is paral lel toM I I . T there

fore lies in the tangent plane, andM T is the tangent toMB ,

the line of intersec tion of the surface by the plane MN S .

While oc and y are changing as assumed,the foot of the

corresponding ordinate passes with uniform motion from IVto S

,and Q T is the total differential required by definition.

As in the case of a func tion of a single variable, the ex

pressions az/dx and az/dy are,respec tively, independent of

doc and dy,but for any state of a func tion of two variables

there is no fixed total differential coefiicient. In the figure,

Q T/QM Z tan QM T = da/ d2

+ a’

y2

is the total rate of change of 2 , corresponding to the valuesassumed for doc and dy, but any change in the relative valueof doc and dy wil l change this total rate. F or any valuesassumed for doc and dy, the total rate of change of the func

tion, and therefore its total differential , wil l be the same as

that of t/ze ordinate of t/ce l ine cut out of the tangent planeby the plane through NM ,

whose trace on XY makes with

X an angle 2 tan“ 1dy/dx .

I t is now apparent that the function represented by the

ordinate of tire tang ent plane at M is the one witna constant

rate zoitnr espect to cac/z var iable, whose total differential forthe same values of doc and dy is the same as that of the

given func tion at the state corresponding to M .

dz/ Vdoc2

dy2

tan QM T is

~

the tangent of the anglewhich a tangent to MB makes with XY

,and 65) its

numerical value is the slope of the surface atM along MB .

F rom we write

dz/dx az/a’x

dz/dy az/dy

1 2 0 D I F FEREN T I A L CA L CUL US .

the first members of which depend upon the relative value

of doc and dy.

Draw T W paral lel to SR . T hen M W is the proj ec tionof M T upon the plane MN R ,

dz/dx tan KM W,and

dz /dy is the tangent of the angle which the proj ec tion of

M T upon the plane MNR makes w ith Y.

az/dx tan KM I I is the partial differential coeffic ientof the func tion with respec t to or only, and is independent

of doc and dy. I t is equal to the tangent of the angle whichthe intersec tion of the tangent plane, and anyplane paral lelto ZX ,

makes with X or the plane X Y. Whereas dz/dxtan KM W is dependent upon the quotient da/dy (Eq.

which has no definite value bec ause so and y are independ

ent, and their differential s arbitrary.

I f,however , bymeans of a second equation 0

,

x and y are related, thus mak ing one of them dependent

upon the other and determining dy/dx , then 2: becomes a

func tion of a single independent variable its graph becomesthe line of intersec tion of the surface z by the

cyl inder ¢ (x ,y) 0,dz/dx becomes the differential coeffi

c ient of z regarded as a func tion of x and is equal to tan 0,0 being the angle between the axis of X and the proj ec tionof the tangent to the graph on XZ.

Similarly both members of Eq. 1 may be divided by dt,

t being any independent variable , giving

dz/dt

T he values of both members wil l now depend upon the

values assumed 67) for dx ,dy, and dt.

But if we also write ¢ (x ,t) z 0 and ¢ (y,

t) z 0,thus

relating x and y to t,and determining dx/dt and dy/dt,

then z becomes a func t ion of one independent variable,and the firs t member becomes the differential coeffi c ient of

f

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1 2 2 D I F FEREN T I AL CAL CUL US .

R epresenting the func tion or byy,we have y 1: x

“.

D ifferentiating, we obtain

dyZ nx

n ' ldx,

and dy/doc

for thefir st differential and differential coeffic ient, respectively, of x

”.

D ifferentiating again,we have

dfy n(u and a’2

y/doc2

n(u

for the second differential and differential coeffic ient,respec tively, of or

".

da

y n(n 1 ) (n 4

da

y/doc3

n(u 1 ) (n 2 )x

for the tnird differential and differential coeffic ientrespec tively, of x

".

Similarly, the fourth,

fi fth , etc .,differential s may be

derived in suc cession.

I t should be observed that the symbol d’

y/dx2

,which

represents the second differential coeffic ient , is the differ

ential coeffic ient of the symbol dy/dx ,which represents the

fi rst differential coeffic ient .

Similarly,

Succes sive differential coeffic ients are al so cal led der ived

functions or der ivatives,and are frequently represented in

order by acc ents on the func tional letter.

S UCCES S I VE D I F FEREN T I A T I ON . 1 2 3

T hus,if f (x ) represents the primitive func tion, then

f (x) .

denote respec tively the first , second, third, etc ., deriva

tives of f (x ) .

O ther forms are al so used. T hus,havingy

represent the successive derived func tions in order.Each successive differential coefficient or derivative in

order is the rate of change of the immediately prec edingone

,and is an inc reasing or dec reasing func tion

of x ,ac cording as d

n

y/dx”is positive or negative. 71 .

L et y f (x ) 2 x”

,n being entire and positive.

T hen dy/doc f (x ) D ay2 noo

n" 1

D afy n(n 1 )x

etc .

dn

y/dx”

f”

(x ) D aff

y n(u 1 ) (n

fn+1(x ) 2 o .

I t should be observed that the symbolsD afy,

etc .

,serve only to represent expressions , and to

indicate thei r relations to the primitive func tion. I n the

above dz

y/doc2

denotes that n (u 1 )oc"" 2 is the second differ

ential coefficient of x ”.

T he differential of any order is the product of the cor

responding derivative and power of the differential of thevariab le.

( a)

represents a differential of the nth order,and d

n

y/dx”:

”

(x )

1 24 D I F FEREN T I A L CA L CUL US .

represent s a differential coeffic ient or derivative of the

nth order .D ividing both members of (a) by dx

n‘ l

,dx ” " 2

, etc . , in

succ es sion, we have, in order,

n - z

etc . etc .

r a w/er r

)etc .

F rom which we see that the produc t of a derivative of

any order by the first power of the differential of the

variab le is the fir st differential of the immediatelypreceding derivative , and its produc t by the second power of the

differential of the variable is the second differential of thesecond preceding derivative, etc . T he product of a deriv

a tive of the nth order by the (n 1 ) th power of the differential of the variable is the (n 1 ) th differential of the firstderivative.

EXAMPLES.

1 . y 2 : an4 dy 4ax

3dx, d2y I zax

’dxa,

d3y 24ax dx3,

2 ° f (x ) (a x ) 1 f (x ) 2 (a x )- 2

, f'

(x ) 2 (a

10 105) 1: n(a x )

- n- 1’

3. y sin x . n cos x , D 432)! sin x ,

cos x ,

T he exponent of the power of a func tion is diminished byunity at each differentiation and when entire and

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1 2 6 D I F FEREN T I AL CAL CUL US .

1 5 . y s in x . d n

y/dx"

s in (n nut/2) .

16 . y z ( I -x ) . d ”

y/dx”z : 2 u/( I

1 7 . y tan x . d~°

jy/dx3z : 6 sec

4 x 4 sec2x .

1 8 . r Vzrx . d'Zy/n

’x 2 r /erxr'z —e2/r .

1 9 . f (x ) we x) . r iv (x ) xr.

2 0 . y emcos x . d ny/docn 2a./28 :13 cos (x nut/4) .

2 1 . y : : l: VR 2 — x 2 .

2 2 . f (x ) tan x—l sec x . cos x ( I

2 3 . y z oom

.

24 . f (x ) x3 log or.

2 5. x 3 sin“

y.

2 6 . y log sin 36 .

y 4/ sec 2x . a'Qy/a

’or2

3(sec 2x)“2

(sec za )”2

r a"gr/ctr ? 4a3/(a

2

y 2 sec x . d 2

y/dx2

2 sec3 x sec at ,

a’3y/dr

3sec x tan x (6 sec2 x I ) .

y xn- l log x . d ny/dx ” [n I /x .

31 . f (x ) r: (zar )a’b f ( 2 ) 4(a

32 . y tan—1 coty.

sin2 y,

d sin2 2 dy smyd

c

x

osy y5 111 2y sinzy.

d3y d(sin 2y s in?y) s in 2y 2 s iny cos y dy sin2 y cos 2y 2dy

dx3 dx a’

x

2 (siny) (s in 2y cos y cos 2y s in

2 s in3y s in 3y.

f (x )-2

d 2y/a'x2 x“ ( I 10 9

fi"(x )

g x )

a’4x /

a’ 4r (or -f —

r2)7’2

y 2! C0 51

5/0

S UCCES S I VE D I F FEREN T I A T I ON . 1 27

S imilarly, a’4y/doc

4| 3 sin4y sin 4y,

dny/dx” —l )n | n 1 s inny sin ny.

Since tan“ l

x 7t/2 tan" I

we have x )/a’xn 1 )n — 1 s inn y sin ny,

or d ” (tan* I

oc)/cz’ocn n I sin (n + x

9)

33 r air/fix (ex/a ay/dx

rz y/a

r.

34. J' Clix/(a

?x2) .

dy/dx“ i“ a

’Qy/a

’n2

2a2oc(oc

24. x

2)3

35 r : Cfl/z C7 + C dy/dx : ( 3+ C.

36. y 5+ c(x a)3/5

,

(if /“795 3C/5(x a

my/dx

2 6c/25(x a)7/5

37 19‘ s 15 ” f

el /xMg

, f x (296+ x“

39 f x'

x=c

"

40 . The re lation between the time,denoted by t, and the distance ,

represented by s , through wh ich a body,s tarting from rest, fa l ls in

a vacuum near the earth ’s surface ,is expressed very nearly by the

equation s 1: 16 . 1 t9; s be ing in fee t and t in seconds . Construct a table

g iving the entire distance fa llen th rough in I second ; in 2 seconds ;

in 3 seconds ; and in 4 seconds ; the dis tance passed over during eacho f the above seconds ; the velocityand acce leration at the end of each .

Ent ire D istance Distance eachin F eet . Second .

Veloci ty. Acceleration.

1 2 8 D I F FEREN T I AL CAL CUL US .

41 . Having s2

5t3,find the ve locity and acce leration when t z : 2

seconds ; t : 3 seconds . Ans . V¢=2 =—072 . Vt =3 =:

= 3V§/

42 . y=sin (m sin- lac) . dy/dx z m cos (rn s in

“ lx2

. Hence ,

ut2cos

2(m s in

" 1 x )

Differentiating again and dividing by 2dy , we have

oxen/ex “) afar/ex ) ma o.

sinh x . f x s inh x .

cosh x . f x cosh x .

1 0 4 ,Leibnitz ’s Theorem. Let y my

,in which u and n

are any functions of x then 75)r

“

u do/doc 7) du/dx ,

a’ i'

y/dac2

u d2

zr/doc 2 (du/doc) (du/dor) 7)

d”

? da d2'

o d2

u dogai

n”

(as

doc doc?

doc2

doc doc“

in which the numerical coeffic ients follow the law of thoseof the binomial formula. By a method similar to that usedin deduc ing that formula for positive entire exponents itmay be shown that

du dn‘ l

o n(7Z 1 ) din dn

“ 2o

”

dx dxn ‘ l doc

2

n(u —r —l 1 ) d

ru a

m" ”

7;”

am— l

u do

r darn—1

dx

EXAMPLES.

u ea”, du/dx d nu/dx” a

new .

d ”

y/dx”

aneawx .

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130 D I F FEREN T IAL CAL CUL US .

Similarly, we write

for the other partial derivatives of the second order.

Each partial derivative of the second order will , in gen

eral , admit o f differentiation with respec t to each variab le,and the differentiation may be continued, in general , to any

required order.T he notation adopted is as follows

dxn - r dy

r + 1 ,

wh ich represents the partial derivative of the (n 1 )thorder,

taken (n r ) times with respec t to x and (r -l 1 ) with re

spec t to y.

T he numerator indicates the number of differentiations ,and the denominator the order of the successive operationswith respec t to the variables .Bymultiplying the symbol for any partial derivative by

its denominator we obtain the symbol for the correspondingpartial differential . T hus

, am n

z represents

S UCCES S ] VE D IFFERENT I A T I ON . 13 1

a partial differential of the (rn n) order, taken tu times

with respect to x, and n times with respect toy.

HaVing Zl f (x )Z) ,

represent, respectively, the partial derivatives of the firstorder , each being, in general,a function of x

, y, and z . The

differentiation may, therefore, be continued, and the suc

cessive operations indicated by extending the notation usedabove . T hus , represents the partial derivative of the (nz n r )

thorder, taken t n times with

respect to x,n times with respec t to y, and r times with

respect to 2 .

In a similar manner, a partial derivative of anyorder of afunc tion of any number of variables may be obtained and

represented.

1 06 . Partial difl'

erentials and their cor responding der iva

tives are independent of tbe order of difi erentzation.

Assume z f (x , y) , and increase x byn, giving 70 )

az limit f (x + 12, y)doc

In th is expres sion increasey by k, and we have 70 )

limit f (x k, y—l—é ) —f ix . r+é l r )—f (x . r )]

n

132 D I F FEREN T IAL CAL CUL US .

Similarly, increasingy in the primitive function byR, we

write

limit (floor-l e) f (r .y)

L R

from which , increasing or by it , we obtain

—>o tan—m

f (x 6 . r é ) -f (x fr, r ) [flanr—l é) —f (x .r )]In:

wh ich compared with ( 1 ) gives

F rom which we have

Similarly, we derive

and, in general ,

Similarly, having a f (x ,y, it can be proved that

Ban/dz doc dy a

su/dydocdz , etc .

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134 D I F FEREN T I AL CAL CUL US .

sin (x y) . cos (x +y) .

14. z cos (x y) . cos (x y) . s in (x — y) .

cos (x — y) sin (ar—y) .

8 321 5. z = 10 g <x + y>.

dy. j 3 .

1 0 7 . Partial Differentials of a Surface.-Let A I L be

any surface, and AB CD u a portion of it inc luded be

tween the coordinate planes XZ,YZ

,and the planes D OR ,

B PS, paral lel to them respectively. F rom § 2 6 we have

r =flnnIncrease O R x by R R

’It,giving 70 )

/z

S UCCES S I VE D I F FEREN T IA T I ON .135

Now inc rease O Q = y by QQ'

A,giving 70 )

In wh ich

f (x Ar k)”

2 AEG-f. flow k) ABE ] ,

f (oc II,y) AEF D f (x , y) AB CD .

Hence,

k) —f (x ,y—é ) z AEGI AB I I I = B EGH .

f (x it,y) — f (x ,y) AEF D AB CD 1

“

B EEC.

— f (r , y—l- 12)

T herefore 50 )

a?” limit CF GH—> O

arudx dy/dx dy sec [3 dx dy.

1 0 8 . Partial Differentials of a Volume.—Let A TL

(figure 1 0 7) be any surface, and AB CD -ON va volumelimited by it , the th ree coordinate planes , and the planesD OR and B PS paral lel , respectively, to X2 and YZ. F rom

30 we have v = f (oc,y) .

By the method used in the last Artic le,considering the

corresponding volumes instead of the surfaces, we obtain

r-M) f (x , r

136 D I F FEREN T I AL CALCUL US .

I n which

f(36+ 72. Jf -I —f (x. r +k) 6 . r) —f (x . CF GH-NM

Hence

2 l imita 7)—>oCF GH NM

I VG= z ,doc dy ion-9 0

afivdx dy/dx dy zdx dy.

1 0 9 . Successive Total Differentiation.-Having z

§ 1 0 1 gives

in which the total differential of the first order dz , and the

az azcorresponding partial differential s c x and are

,in

general,functions of x andy.

Hence, the total differential of the second order , denoted

by d’z, is obtained by differentiating each term in the sec

ond member of ( 1 ) with respect to each variable, and

taking the sum of the partial differential s of the secondorder.

Differentiating Sida with respec t to each variable, we

Sim1larly, e dr d’

x55.

- fir“.

T herefore

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138 D I FFEREN TI AL CALCUL US

” " sdysetc.

I t is important to notice that in deriving eq. ( 2 ) from eq.

( 1 ) we write, in accordance with 1 0 6,

a zdocdy/dx dy a dydx/dydoc ;

and it fol lows that having any expression in the form R dx

Qdy, in order that it may be a total differential, it isneces sary and sufficient that

OP 822 8

22 aQ

dy doody dydx dx.

T his is known as Euler’s Test. I t determines whether ornot the given expression is the result of the complete differentiation with respec t to bot/c variables of some other ex

pres sion. T hus , having

2x doc —ydx

—l ady zydy,

we obtain

Ra fl ex -w) (x -l- zy

which shows that the given expression is a total differentialof some func tion of x andy.

Having an doc+ydx 2ydy, the test fail s and the expression is not a total differential of any func tion. By differentiating x

”

xy—1 y

“

, the student may confirm bothresults .T he successive total differential s of any function of any

number of variables fii ayflbe determined in a similar man

ner.

S UCCESS I VE D I F FEREN T I A TI ON . 139

T hus,having u f (x ,y, then

8°a

z 2 2 a ”2d a dx +

dy2dy +

dz 2dz

dx dy 2 dydz 2 dx dz .

dnu dx 57 d)»

Rule. s erentiate t/zefunction and obtain expressions f or

tire severa l par tial coefi ccents to tbc desired order . Substitute

t/cese in tkoproper f ormula f or their respective symbols.

EXAMPLES.

d 2z 6oen z1 2x9ydx dy 2x3dy

2.

(x2

r ) .

c.W W a

4<x 4 3x 2r2)a

’

r2

]

of 2z 2n 9 8xydx dy 2x°dy

9.

I zydx2 dy 1 2x dx dy

”

Extension of the symbo lic form (a ) , g 10 9.

140 D I FFEREN T IAL CAL CUL US .

(x9

your .

I

(x2

rT /d x?

612 3xy2dx3 -l 3y(2x

2 2)dx

i'dy

army2

-l zabdx dy

rear2

rive

”

diz z (6xy2

2y3)dx

21 2 (x

2

y xy9)dx dy (6x

’

y 2x3

)dy2.

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142 D IF FEREN T I A L CAL CUL US .

explicit function of y, y, and in general dy, being func tionsof x .

T hat is, the equation may be solved with respec t to y,

and y differentiated as an explicit func tion, or we maydiffer

entiate , regarding y as an impl icit function,and then solve

the resulting differential equation with respec t to dy.

T his princ iple is general ; for, having any equation con

taining two variables , x and y,it may be written

915 (my) ;

and regardingy as an implicit func t ion of x , each membermay be regarded as an expl icit func tion of x , equal to theo ther for all values of x ; therefore 72 ) their differential sare equal . I t is not essential that the value of the depend

ent vari able shal l be expressed in terms of the other,butit is nec es sary to remember wh ich is as sumed as the de

pendent and which the independent variable.

T he advantage of differentiating without solving with

respec t to the implic it func tion beforehand increases,in

general, with the degree of that func tion.

EXAMPLES .

I . ay2

x3+ bx 0 . dy/dx : .t ( 3x

2 b)/2 Va (x3 bx) .

2 . of

y2 b

'zx2

a2b2 . dy/clx b2x/a 2y.

3. (y a )2

4m. dy/dx i

4 . cos y z : a cos x . dy/dx tan x/tany.

5 . cos (x +y) 0 .

6 xy -

r"

afr/a’x (r

2xr 10 g

- xr 10 a x ) .

7 ( r ctr/ctr 2x : l: 5x372

2 .

8 . ay‘2

x3+ bx2 o. dy/dx z : l: (3x 2b)/2 Va (x b) .

IMPL I CI T F UN CT I ON S .143

9 ar’— x

3+ <b

zxw—o— u

dy x x3+ 1 2x -16)2 2 _10 . x 2x y 2x 8y 0 .

dx

I I . y9

dy s iny—y cos x

1 2 . y sm x

dx s iu x — x cos y'

1 3 . y3

3yx2

z : o. dy/dx : z 2x/(y x ) .

I 4 x/ —y+ 1 = o

By continuing the differentiation in a s imilar manner , express ions

for d /dx2, and derivatives of the h ighe r orders , may be obtained.

16 . x2/3+J/

Z/3 dr

y/07x

d ay/ex

? (var /3

)

1 7. x r vers Vary

aWeir (z r/r2. d i

’

y/dx"

r/yz.

1 1 1 . Having u = f (x , y) 0,and regarding y as an ime

plic it func tion of x,we write

Al so i t may be diffe rentiated w ith respect to x by d ifferent ia t ing

it as a func tion o f x and y (S and since y is a function of x

au/dy mus t be mul tipl ied by dy/dx (g

144 D I FFEREN T IAL CAL CUL US .

da an an dy

dx dx dx

an anwhence

EXAMPLES.

1 , u z ys

3yx2+ 2x

3:

6yx 6x2 3y2

3x9 dy

( ix

2 . I ii — 2 1mm2

r/(r — x )

a’

r W add”

4 2 2 2 z3 J’ 3a y 4a. xy a x

dx 3a2

y 2a2x

4 . x5

axs

y bx 2y2

y5

0 .

2 bx‘

y-l ax

3

a’

r arz

W e -l a2 3 2

05. y x 2ax 0dx

6 . y”

yo

u x2) 0 .

2

J’7. x4

zay3

3a2

2a2x2

a‘

we ail/W U

8 . (4r x3)?

(x 4>5<x o

r

dy 3x2

s(r 4l4

(x 3(x 4)5(x

«ix 4 8(4r x3

) 4(41I x3

)

D ifferentiating remembering that

d 8271

+a.” dy

dx dx2 dx By dx

’

82a dy

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D I F FEREN T I A L CAL CUL US

2 . x2

+y°

2bxy- a

2: 0

a’

r/a’x (x r).

d 2y/dx2

(b2

I )a2

/(y bx )3.

I f dy/dx o,d2

y/dx2

I /a VI b2

3. 2xy y2

a2

a"fr/fix

2

r (r zed/(I f x )3

dy/dx : l: (3a x ) VZ /(Za

d /d : l: 3a2

/(2a

Differentiating we obtain

8°

u dy3 d ay dx

82a 8

2

nd) »

3[may (ix

Similarly, it may be shown that

in wh ich the intermediate terms involve differential coeffic ients ofy w ith respec t to x of orders less than the nth.

1 1 2 . H aving any equation containing three variab les

D I F FEREN T IAL EQ UA T I ON S .I 47

x, y and 2

,one must be an implic it function of the other

two Each member may, therefore, be regarded as a

function of but two independent variables .T he differential of each member is equal to the sum of

its partial differential s ; and since the partial differential s ofthe two members are respec tively equal to each other, thetotal differential s are equal .I t is not necessary to express the implic it func tion or

dependent variab le in terms of the others, but it is always

important to distinguish it.

In a similar manner it may be shown that the total differentials of the members of any equation are equal , remembering that the number of independent variables is oneless than the number of variables .

D I FFERENT IAL EQUAT I O NS.

1 1 3 . An equation wh ich contains differential coefficientsis called a differential equation.

A differential equation obtained by one differentiationmay, in general , be differentiated again, giving a differentialequation of the second order

,and so on to differential equa

tions of the t/zird, etc orders .T hus , having y

22mxy x

2 z a”

,regarding x as the

independent variab le, differentiating , equating the results ,and reducing , we have

ydy mx dy mydx x dx 0,

or dy/dx (my mx) .

Differentiating again,

*we have

The importance of distinguish ing between the independent and

dependent variab les becomes apparent at the second operation of

d ifferentiation— as in above dx is a constant, whereas dy

148 D IF FEREN T I AL CAL CUL US .

dy2

+y d’

y mx d2

y 2mdx dy+ dx2

or, dividing by dx’

,

(3’ an

y/{57902

(if /doc2

2mdy/dx 1

T he order of a differential equation is the same as thatof the h ighest derivative it contains , and its deg ree is de

noted bythe greatest exponent of the derivative of the h ighest order in any term; provided that all such exponents areentire and positive. T hus ,

(dy/dx) a/x 0 is of the rst order and 2d degree.

di

y/dx“

0 is of the 2 d order and I st degree .

0 is of the nth

order and mtndegree.

5. agx =y(x

6 . y” bxs.

7. I“

2px -l

8 . y) .

EXAMPLES.

eli/sir2 d —2a

‘

/9r5

a2

(a x ) d‘t’

y 2a2

(x 2 0 )

(x a )3 dx 2 (x a ) 4

dr/dx zavy/(e2

4m) ,

d 2y,/de 2 M396

2

trai

l/ (x2

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1 50 D I F FEREN T I AL CAL CUL US .

In the first case both variables are independent, but in

the second only one. The difference becomes apparentat the second differentiation.

Equations derived by differentiating primitive equationsor other differential equations are cal led immediate dif

fer

ential equations.

1 14 . Differential equations also arise by combining suc

cessive immediate differential equations with each otherand the primitive equation in such a manner as to elimi

nate certain constants, or par ticular functions which enterthe primitive equation. T hus

,from

y ax b we obtain dy/dx a.

Eliminating a, dy/dx (y b)/x ,

which is independent of a .

Differentiating again, we have d2

y/dx2

0, which is in

dependent of both a and b.

As another example, take the equation of a circle

(x a)2

(y b)2 —i R “

.

Differentiating three times,we have

2 (x a)dx 2 (y b)dy 0 .

fix”

( if (r o.

2dr 42

r + dydb+ (y b)a’3

y= 0

'

Dividing (3) by member bymember, we have

(dx2

dy )/3dyd2

y dz

y/d3

y, which gives

D I FFEREN T I AL EQ UA T I ON S . 1 51

dx2

dy"

d2

y dy d”

y dfydx

2dx

2 3dx dx

2

dxs

which is independent of a, b and R .

And, in general , bydifferentiating an equation n timesweob tain n differential equations betweenwhich and the primitive equation n constants or particular functions may be

eliminated,giving a differential equation of the n

thorder.

EXAMPLES.

1 . Eliminate e” and sin x fromy z : ex sin x.

dy/dx ex sin x + ex cos x =y+ ex cos x,

diiy/dx2 dy/dx ex cos x ex sin x ;

and since ex cos x dy/dx y,

d ’y/dx2

2dy/dx 231.

2 . y= a sinx 6 sin 2x .

dy/dx 0 cos x 2b cos 2x,

a sin x 4b sin 2x,

d3y/dx3

a cos x Sb cos 2x,

d‘y/dx“

a sin x+ r6b sin 2x.

Hence, d 4y/dx4

5d’y/a

’x2

4y z o .

3. r“ 24x

3a’r/a

’x

4 . xy unq b o. .x’d fi

y/dx 2 xdy/dx 3y.

5 ( I + J") ( 1 n

ew fly/090 I +r’.

1 52 D I F FEREN T I AL CAL CUL US .

6 . y a sin x b cos x . dyQ

/dx2

y.

7. y ax a (x 1 )dy/dx = y.

8 . y (a —j bx )e0‘” d 2y/dx2

2 c dy/dx

9 . y sin x . (dy/dx )2 —j—y

21 .

10 . y excos x .

1 1 . y sin log x . xQ

d /dx’

x dy/dx +y 0.

12 . y log 5m x . dfly/dx2 —l (dy/dx )

21 .

1 3 J/2

2Rx “ i“ 2xy§§+y2 o,

14. y a cos (bx c) ; d 2y/dx2

15. y sin" 1

x . x2)d

2

y/dx2

x dy/dx o.

16 . y cx c”. y r: xdy/dx

17. y (x ay r“ ac

1 8 . y: ce"‘ tfm

- lw -tan“ 1 x—I . ( 1 x

2

)dy/dx y tan“ 'x.

19. y ( cx log x I )“ 1

. xdy/dx + y y’ log x .

20 . y2

2 cx y(dy/dx )2

2xdy/dx .=y.

2 C CO S ” bx—PC,

Sin mx c

77221} 0 .

y z: c cos (mx

2 2 . y c: x log [( c x3d 2y/dx

2

(y xdy/dx )2

23. y x sin ux/2u c cos nx c'

sin nx .

“WY/4x“ cos nx

y?Sin“ x 24} a” 0 2y cot xdy/dx =y

2

2 5° 1’ (5m {TU/(em y“

I dy/clx 0 .

26. e2y+2axey+ a

2=o. 1 .

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1 54 D I F FEREN T IAL CAL CUL US .

CHAPTER VI I I .

CHANGE O F THE INDEPENDENT VAR IABLE.

1 15 . Having any expression or equation containing differentials, or derivatives , of y regarded as a func tion of x

,

it is sometimes desirable to obtain a corresponding expression or equation,

in which y,or some other variable upon

which y and x depend, is the independent one. T h is operation is cal led c/tang ing tbc independent var iable.

T he principle deduced in 73 enab les us to make the

change in cases involving the fi rst derivative only.

When differential s of a higher order are involved it mustbe remembered that we have written

in which x is the independent variable, and dx is a con

s tant .

R egarding both dy and dx as variable, we have

dx d2

dyd

(dx da

y— dyd

s

x )dx+3 (dyd”x— dx d

2

y)d’x

dx5 7

CHAN GE O F THE INDEPEND EN T VAR I AB LE.1 55

wh ich should be subs tituted for d2

y/dx2

and dg

y/dxs

,re

spec tively, in order that the results may be general , that is ,in which neither x nor y is independent.

I f then y is made independent , we have dy constant,

d'

iy 0,da

y o,and ( 1 ) and ( 2 ) reduce to

d’i

y 3dy(d2x)

2dydx d 36

docs

which may be used when the independent variable is

changed from x toy.

EXAMPLES.

Change the independent variab le from x toy in the fol lowing :

diy dy2 dex dx 3 dx

3° yzfi + zzé+

I — O . ydy

2

d’x$

07

7 2I 0

5. (dy2

a dx d zy 0 . ( 1 dx9/dy9)3/2

a d zx/dy2z 0 .

I f x or y is g iven as a function of a th ird variab le , 6, which wew ish to make the independent one

,we first transform the given ex

pression,by means of ( 1 ) and into its general form in wh ich

neither x nor y is independent , and then substitute for x , dx , dflx

,

d3x , ory,dy,

d 2 d 3 their values in terms of 6 and its differential .

1 56 D I FFEREN T IAL CA LCUL US .

6. Having t z : x x 2, show that

Change the independent variab le from x to 9 in the fol low ingequations :

x dy h1 — x

9 dx I - x9

0 , w en x 0 0 5 9.

+ r = 0

dy a2

—éy

_ o,when x __

+y 0 , when x2

10 . x + by = o,when x :

d zy dy,

de2(a _ — O

1 1 . Having x r cos 6, andy z r sin 9, change the independent

variable from x to 6 in the equation

F rom ( 1 ) we havedx d 2y dy (fi x

;

and, since d’fi 0 ,

cos Bdr r sin 6 d6,

sin 6 dr r cos 6 d6,

cos Gd fir 2 s in 6 dr dfi r cos one“,

dzy sin 6 d2r 2 cos 9 dr dB r sin 6 dB”,

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PART I I .

ANAL YTI C APPL I CA TI ONS.

CHAPTER IX .

L IMIT S O F FUNCT I O NS WH I CH ASSUME INDETERMI

NATE FO RMS .

1 16 . T he symbol s

cc

are indeterminate forms .

When for a particular value of the variable a func tion of

the variable as sumes any one of the above forms its corresponding value is indeterminate.

T he limit of such a func tion, under the law that the

variab le approaches the particular value, is determinate.

39

In many cases this limit may be found by simple algebraie methods , otherwise a method of the Calculus is generally used.

*

1 1 7 . F orm

L et fx/q be a fraction such that both terms vanishlimit f x

when x a . I t is required to findx a gbx

Any operation by wh ich th is l imit is determined is genera l lycal led the evaluation of the corresponding indeterminate form

IND ETERM INA TE F O RMS . 1 59

F rom eq. 62,we write

f (x I.) f x s’

(x on) ,

¢ (x I.) gbx new an) .

Hence

f (a 5 ) f’

fa 512)¢ (d 72) - l fi

’b )

’

from which , as It 0, we have

I‘d/$4

limit f x-> a ¢x

Similarlywe may deduce

jux f

ua

¢”x (it

’d

lim

limit f x fmx »)—) d

f”x and ¢

”x representing , respec tively, the derivatives of

the numerator and denominator of the same and lowest

order,both of which do not vanish when x a .

I f f”a and ¢

”a are both finite

,the limit is finite.

I ff”a is z ero and qb

na is not z ero, the limit is z ero.

I f f”a is not z ero, and ¢

”a is, the limit is infinite.

I f f”a and ¢

”a are both infinite, the limit is undetere

mined.

T herefor e the required limit is the ratio, correspondingto the particular value of the variab le, of the derivatives of

the numerator and denominator, of the same and lowest

order,both of which do not vanish or become infinite.

160 D I FFEREN T IAL CAL CUL US .

Each ratio as obtained should be careful ly inspected and

fac tors common to both terms should be cancel led if possible before proceeding to the next ratio.

EXAMPLES.

2 limx a

( 8 111 x/x ) cos x]0 1 .

( I sin x )/cos xJfl/z cos x/sin xl r/z

10 . (ex e- x) /sin x]o (ex e

- x)/cos x1, 2 .

I I . s in x1, 2x/cos x] 0 o.

cos x — x sin x

CO S x 0 2 sin x 0 2 cos x

x cos x

Hereafter, for abbreviation,the forms f (x)x=a wil l fre

uentl be used to ex ressq Y px a

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162 D I F FEREN T I A L CAL CUL US .

1 18 . F orm 00 /ao

¢a , we maywrite

£2 gba j; and 1 1 7)

lim

lim

II ence

and since l imit s of equimultiples of two variables with

equal limits are equal , we have , multiplying by

¢xf'x/f x 915

Similarly, we may deduce

“ M alim

T he method is therefore the same as in the precedingartic le ; but by § 7 1 we have f”

a/gb”a oo / oo ,

when

f a/¢a for‘

a finite value of a . Hence for finitevalues of a th is method w il l fail to determine the limit unles s fac tors common to both terms of some derived ratio

become apparent or the l imit of f”x/gb

”x becomes otherwise

known.

Taylor’

s Ca lculus , § 79.

IND ETERM INA TE F O RMS . 163

EXAMPLES.

2 cot 2 s in x

cos x

1 sm x s in x2

sm 2x o 8 111 2x

cos x2 I .

2 CO S 2x 0

x] 0

sin x

I /singx 0

s in2 x

cot x cosec x 0 x cos x 0

2 sin x cos x

cos x

2 sin x cos x10

in wh ich n is a positive integer . I f n is fractional, the exponent of

x ul timately becomes negative , g iving the same result.7. Puttinn I /x ,

whence y o as x cc we have

e Jen/ex] 0 .

00

In some cases we may with advantage place or a it and sub

sequentlymake 12 0 . T hus 43/x f/x 2 a

2reduces to o/o . and

the ratios of all derivatives of both te rms become oo /oo when x a ;

but putting x a li , we obtain

0 4

x = a n= o

164 D I F FEREN T I AL CAL CUL US .

1 19 . Form

I f fa 0 and ¢a 00,we write

f x (bx

1 gbx 1/f x’

which takes the form or oo/ oo when x a and

limit may be determined by the methodwith the same limitations .

EXAMPLES.

I cl/xx

00

JO xQ/e

l/x]O

4 . e‘ x log x10

log x/ex

] 00 0 .

5. sin x log cot or} r: (sin x/x ) x log cot x1) log cot

6 . x tan"

tan

[ 1/(x24. I ll . 1 , when tan = 0 .

7. see x (x sin x (x sin x n/2 )/cos x ](x cos x sin x )/ s in xJF /Z 1 .

8 1 2 tan- lo (2 co t0 g

a1 g

a

log x

_

lx 10 x

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166 D I FFEREN T I AL CALCUL US .

1 2 1 . Forms 000,

I f z 00 ’

or 000or we write

10 8 61590 log fx ,whence evx log f x

,

(f’x lim enx log f x _ e

lim [d1x log f x ]

in which gba log f a 0 .oo in each of the above cases , and

the method of 1 1 9 wil l apply after the given expressionis placed in the form indicated.

EXAMPLES.

ex log x] 0 6 JO 2 I .

sin x log x

]

l . l . 8” I .

4, ( 1 x )1/x

]o el /( l l-x)

s (2 x/a)

6 . cot xsin x esin x log cot 431) z :

l /x 51135

n/x .

e “ 3 .

1 2 2 . Evaluation of Derivatives of Impl icit Functions.

Derivatives which for particular values of the variables assume indeterminate forms may be evaluated as in the pre

ceding cases.

INDETERMINA TE F O RMS . 167

EXAMPLES.

1 ya x

3

2

Whence zax 3x

Hence, (dy/dx)2

0 0 0 0z : :1: 0 0 .

2 x3

y3 0 or co .

3. V3

3ay"

2oxy ax“. I or - I /3.

4 . (x2+r

9are? r ) .

5 . u 2 : y4

311n—i

4a9xy a x z : 0 ,

Bu/dx 4a9y 2a

9x . au/dy z 4y

3 6aQy 4a9x .

za‘2y a x 2a

9 (dy/dx ) a

2y3

3a9y 2a or ( 6y

22a?

6. u z agyg — a x

7. u = x4 + an — ay

3 = 0 .

0 o

—o or zk l .

8 . u = ay3 Q O l

‘

i 1/b/a .

9 . u : a9xy+y"_—_ o.

0 o

—o or a

Q.

I O . u = x3+y

3+ a

3 — 3axy= 0 .

a

1 1 . u : : y3—3oxy+ x

4 = —o or oo .

1 2 . u = x4+ 2ax

2

y— aya z O , or : l:

dy/dxl ) . Ohas two limiting values for the same value o f the va ri

ab le , hence it is discontinuous a t the co rresponding s tate .

a — gx

and dy/dx]dy/dx ]dy/dx]

dy/dx]

2x

sin x

I .

6 . sec x tan

cos—1( I

O .

9 . 2 ” sin a. 10 . e e” .

tan x

1 1 . x‘ ” e

x] 00 .

CD

I log ( 1 —x )ex

—1sm x

x sm x

I 8 111 x70 S ln

fx

1 9. ( 1 ax ) 20 . log

x

2 1 ( 1 ax I .

O .

26 . 2 5° s in a .

cosec?cx co t x cosec x 1

2 7. (cos ax )

D I F FEREN T I AL CAL CUL US .

MI SCELLANEO US EXAMPLES.

= 6’

2 .

I .

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1 70 D I F FEREN T I AL CAL CUL US .

CHAPTER X.

D EV E L O PM EN T S .

1 2 3 . T he development of a function is the operation of

determining an equivalentfinite or convergent-infi nite series .

When this can be done , the func tion will be the sum of the

series,which

,in the c ase of a convergent-infinite series , is,

al so, the limit of tbc sum of n terms as n increases withoutlimit.

A convergent series having a given function as a limit

may be used to determine approximate values of the function, the degree of approximation depending upon the ra

pidity of convergence and the number of terms considered.

A divergent series should not be employed in finding ap

proximate values of a func tion, or in the deduction of a

general princ iple or formula.

Let S represent a func tion giving

Denote the sum of the first n terms by S” , and the sum

of the following terms by R ,called the remainder ; then

T he series is convergent if R is an infinitesimal as n in

creases without limit , in which case S is the limit of Sn.

When S is the limit of S ,. as n increases , I t is al so thelimit of Sn _ 1 , and we have

DE VEL O F MEN TS . 17 1

T hat is, in a convergent series the nthterm is an infini

tesimal as n increases , but the converse is not necessa

rily true unless Sn has a finite limit under the law ; forlim un 0 lim [Sn S n _ 1] may occur when S z 00 .

T herefore a series is not necessarily convergent whenthe nthl term is an infinitesimal as n inc reases.

1 24 . Taylor’s Formula bas f or its obj ect tbe development

of a function of tile sum of two var iables into a ser ies ar rang ed

according to tire ascending powers of one of tire var iables wit/z

coefi cients w/zic/z arefunctions of tire otber .

As suming an expans1on of the proposed form, we write

i (x ' l—k) 2 X1 + Xa

iz+ X sfi2

+ 0 R , (b)

in which X X2 ,etc . ,

are functions of x to be determined,

and R the remainder after n —l 1 terms.

A 0 gives f x X,.

Plac ing x 12 s,and differentiating, first with respect

to x and then with respec t to b , we have

df (x blMix

df (x Ic)/d/z (df s/ds)

But ds/dx ds/d/z, hence

df (x b)/dx df (x

172 D I F FEREN T I A L CAL CUL US .

Hence,differentiating the second member of (b) first

with respec t to x and then with respec t to it , we have

dX,

dX2

dX3

dX4 3 a W 1

dx dx/Z+

dx/Z

dx/z + etc .

—ldock + etc .

X2

2X3b 3X 4

/z2

4X 6lc3

etc . a H /z’z ‘ l—i etc .

,

wh ich is an identical equation,and by the principle of in

determinate coeffic ients we have

”X rn+l ’ etC0

Since X1— f x ,

dX,/dx = f

’x,

X2= f

’x .

T herefore x 2X8 ,

and X, %f

”x .

é 3X“ and X4

a

dx |n— 1

Substituting these expressions for X ,X

“X

, ,etc.

,

(b) , we have Taylor ’s formula

f (x + h) fx + f’x h i ” x h9/ 3

fflx h’t

/ln R,

in which f x represents what the given func tion becomes

when Ic z o, f

’

x, f"x , etc . , represent the first, second,

Formula pub l ished in 1 7 1 5 by Dr. B rook T aylor .

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174 D I F FEREN T IAL CAL CUL US .

As sume R F (X —l 1,in wh ich R 13 an un

known function of X and x,which wil l make ( 1 ) exac t for

all values of x and X , giving

fX — f x— f

'x (X — x )

— f"x (X xi

n

/lfi F (X r 0

Substitute z for x (except in R ) , and let F z representthe result which in general wil l not be equal to z ero,

F z = fX — f z — f’z (X z )

—f”Z(X NM w e z i

n t l/le z . (3)

F rom ( 2 ) and (3) we see that Ex 0,and from (3) we

have EX 0 . As z varies from x to X , F z increases andthen decreases or the reverse, and F

’z,if continuous, must

change its sign by vanishing for some value of z between x

and X .

D ifferentiating (3) with respect to z and reducing, wehave

,since the terms with the exception of the last two

cancel in pairs ,

F'z

L et x+ bin(X x ) , in wh ich (in is a positive number les sthan unity, represent the value of z for which F ’

z 2 : 0 .

T hen

P = fn+ l (x ‘ l‘ (TAX x»z f (x "i“ Hoff) ,

D EVEL OPMEN TS . 175

R n 1

in which 0 < 9n < 1 .

T h is expression for R enables us to determine the condi

tions of applicability of T aylor’s formula.

When as —> cc R is an infinitesimal , the formula givesa finite or convergent-infinite series for the function.

T h is condition is fulfi l led when as - > oo, f x and f

”x

remain continuous between states corresponding to all valuesof x from any as sumed value of x to x 12

,for then

is always real and finite,and 4 1 ) u+ 1

approaches z ero as a limit.With any assumed value of x wh ich fulfi l s the above

condition, it wil l frequently have limiting values . T hey are

the numerically least positive and negative values of it thatcause f x or any of its derivatives to become discontinuouswhen x it is substituted for x .

for any assumed value of x , f x or any of its deriva

tives becomes imaginaryor infinite, the corresponding limiting value of it must be z ero ,

and the formula is inapplicable .

I t may, however, develop the func tion for other values ofx .

T o il lustrate the use of R take the example

(x + y)

m(m (mxm‘ ”

r” R

.

in which f ”x

and Rn - m+1

’

Known as Lagrange’

s express ion for the remainder.

r76 D IFFEREN TIAL CALCULUS .

Wben m isfractional or negaz‘z’

ve,72 may increase without

limit the series will have an unlimited number of terms,72 will become negative when 72 > m numerical ly,

m(m r) (m 72+

I f x < o, the development fails when the exponent 72 m

is a frac tion with an even denominator.

I f x 2 f ”x ultimately becomes 00 and the formula is

inapplicable.

I f x 0, we have the ratio of R to the (72 2 ) th term

equal to

(x/ (x1

.

which vanishes as co ; and R,therefore, diminishes

indefinitely when the successive terms in order likewisedecrease.

The ratio of the (72 r) th term to the nth is

m— n+ r y m/x

72 x

the limit of which , as n oo is

Hence, when x is numerical ly greater than y the successive terms in order will ultimately decrease indefinitely.

I n which case R will be an infinitesimal ; and we conc ludethat when m is frac tional or negative, the binomial formuladevelops (x for all pos it ive values of x numerical lygreater thany.

EXAMPLES.

I . Deve lop (x y)1/2

.

f (x ) : V}: f (x ) I /2 Vx , f (x ) V973, etc.

(x —i—M 465+ y/2 407 y2/8v; ze,

wh ich fails for x or o .

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178 D I F FEF EIVI'

IA L CALCULUS .

o , the formula fails . I f x I , we have

10 sa( 1 + y) MaOf — y’/2 R ,

log ( I +y) y—y2/2 + y

3/3 y

“/4 R

7. ax” ax( 1 + 10 g ax lognay

n/ n) R .

R logn+l

a I

is an infinitesimal as —) oo, since log a is a constant, while

[y/(n I )] 0 . Making x 0 , we have

I log ay 4» log2ay

2

/2 log’i ay

n/ ln R 0 .

8 . Develop (xv2

f )"

The variab les are V; andyz .

When the variab les considered are not represented by the first

powers of letters , substitute the fi rst powers of other lette rs for the

var iab les , deve lop the result , and resubstitute the variables for the

auxiliary lettersT hus , placing x r

,andy

?.s'

, we have

(xi /2+xr f (r ) f ( ri srfl f

”

(r ) etc

Hence,

( “J/2+ J’

2) 5 (7' “ i“ 1 0 7

332

1 0 7233

57 34

55

x5/2+W I on/34+ o y

6+ sx

‘fzys+y

2

J’ J’

2Va + x

wh ich fails when x a or a .

1 0 ( x a ~l-y)5/2 (x a )

5/2s(x

x z : a gives 6 f>4/2 2 5 ;

but the development would fail w ith more terms .

i f x a,the formula fails .

I I . tan (x +y) tan x sec2xy+ 2 sec

2x tan xy

2

/z

2 sec‘2x ( I + 3 tan

"tx )y

3/ | 3 + R .

i napplicab le when x

DEVEL OPMEN TS . 179

2 2 2 11 i

f

" £1

1 2 . loga(x +J’ —1°g4 x2x

43x

64x

8

C C

ixn

—l-lyn—i- R '

- 1 - 1sec sec x +

r x2(x

2 2

cos2

(x -y) cos

2x +y sin 2x -

y2cos 2x 231

3sin 2x/3

+314cos 2x/3 fey

“s in 2x/ 1 5 R .

306 4-102

I x’(2x 3) I

60x”

xv-i 3( 2x 1h?

2y3

a a a 4a y? 2 8a y

3

—y W'

335 549 i/x"2 i3

(ax aly’

)s

03063

3x90

”

sat/x4 W“)

log sin (x +y) log sin x +y cot x -y2cosec

2x/2

+y3cos x/3 sina x -l R .

a 3

20 . (xx3

y x9

} 5x y

35—l /3

2 2 J’ x )“ 2

y“ ? my” sash/

"44x

3x

‘ 5 R .

23. sinh (x +y) sinh x ( I +y2/ [2 R )

cosh 960!+y3/I3

24 . cosh cosh x( I R )

sinh df (x+J’3/i3 ifs

/[5

1 26. Stirl ing’s Formula.

— In T aylor’s formula inter

change the symbol s x and II,and place /z 0

,giving

~

X—

2

(a )I I I

on2 i?

180 D I F FEREN T IAL CALCULUS .

moan is Siir ling’

s formula f or developing a f nnotion of a

sing le nar iaole into a ser ies a r ranged according to Me ascend

ing powers of Me zfar iaole,w iz

‘noonsz‘

anz‘

ooefi oienz‘

s.

f o , f 0, etc .

,represent what the given func tion and its

succes sive derivatives respectivelybecome when the variable vanishes .

Placing f x n, we write

awn

n :

0a’x

x —i

T o apply the formula,dif

ferentiate l ire f nnel ion and its

a’er ioa iioes in snooession nnz

‘

il one of Me nzgnest ora’er a

’esirea

’

is ooz‘

ainea’. I n Me f nnoz

‘ion ana’its der ivatives mane lne var i

aole equal to zero,ana

7saosoz

'

lnz‘

e lne resnlz‘

s,respeolively, f or

t/zeir cor responding symools inMeformula.

T hus, develop ( 1 x )

“

—n

f f (0 ) m(m etc ,

f u(0 ) m(m I ) (m 72+

1 2 7 . F rom 1 2 5 we have, by interchanging x and 11 and

making n o,

T h is formula is generally known as Maclaurin’s , but it was pub

lished by Stirl ing in 1 7 1 7 ; and not byMaclaurin til l 1 742 . I t is a

particular case of Taylor’s formula which was pub lished in 171 5.

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1 82 D I F FEREN T IAL CALCULUs .)

EXAMPLES.

I . Deve lop loga ( I x ) .

f ix ) Ioga ( 1 x)

f Ma

f (0 ) Ma

f (o) 2M“ .

IV 3

( I x )4 f (O ) l Ma

1 Ma

f n(x ) I )n—1

f n(o)- 1|n 1 Ma .

l i enee,

+ R ’

Inm Wt h R 1 )n I ( I

15 an 1nfin1te5 1mal under

the law that n increases w ithout limit, provided x or 1 nu

merical ly.

I x9/2 x 4/8 etc. R .

sin x x xV

/IZ-i R ,

in which R sin [(n 1 ) 7r/ 2 n' I is an infinitesi

cos x : I x6

/ 6 + R

in wh ich R cos ( (n I ) 7L’

/2 1,

D EVEL OPMEN TS .

S ince cos x a’ s in x/a’x

, the development of cos so may be

ob ta ined by diffe rentiating that of s in x

The radian measure of 1'

is for which value the developments of sin I

’and cos 1

’converge rapidly, g iving their values with

great accuracy.

2

+ log"a

in wh ich R _ . ae’w log

n l+ n+1/in -i I diminishes as 72 increases

e, we have

2 am n+1

in which x I g ives

e = 1 + 1 + —3i

and x x vi i

-

g ives

ex r-‘

x

cos x —i V: sin x .

Substituting x for x , we have

cos x 1 sin x .

Hence, cos x ex “4

e

' xV“ ) 2 ,

sin x = ( ex V:-

i

which are known as Euler’s expressions for the sine and cosine.

F rom (c) we obtain

V— I tan x

ex V—l

+ e

—x v—l(” CV

184 DIFFEREN TIAL CALCULUS .

In (a) and (5) put x mx , then

i m -1 0

exv

cos nix : l: V I s1n nix ,

ém —l s V—_

1)m

or, srnce e ( 6 (CO S x i V 1 5m

cos nzx : l: V— I sin V I sin x)m,

wh ich is De Mo ivre’

s formula.

Expanding the second member by the b inomial formula, and equat

ing separately the real and imag inary parts, we obtain,i n being a

positive integer , finite expansions for cos nix and s in nix in terms of

cos x and sin x .

T hus, nz 3 gives

cos 3x : i: V 1 sin 3x cos3x 3 cos ( i V I sin x )

-3 cos x sin2 x 1F V I sinsx.

Hence, cos 3x cos3x 3 cos x sin% ,

sin 3x 3 cos‘ix sin x sin3x .

cos x (exi sin x

F romwh ich , putting xi for x ,and multiplying both members of the

second by £2,we have

cos x i sin xi i(ex

I f f ( x ) f (x ) , the development of f (x ) will contain powers of x of an odd degree only. Such functions

are called odd functions .* Entire func tions all terms ofwh ich are of an odd degree with respec t to the variable are

examples , also 3m x, tan x

,cot x

,cosec x

,sin

“ 1x, and

tan x .

I f f ( x ) f (x ) , the development of f (x ) wil l contain

powers of x of an even degree only. Such functions are

R ice and Johnson’

s Calculus , 1 69 .

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186 DIFFERENTIAL CAL CUL US .

m s1n1x 1

9 . Develop o w ith respect to sin"

x .

- 1

cm 5m x —

1+ 771 sin" 1 x R .

1 0 . Develop em 8111 x w 1th reference to x .

The general rule may be applied , but the fol low ing method * is

perhaps simpler in th is case .

Place emShrl4”

y, then a’

y/a’x memSin

- 1x/ Vi i

-

x7

,

m2 em Sin"

mxem sin“ 1

x/( I x

Hence x a’

y/a’x

Assume y A O A l x A 296? A 3x

3 "i“ Anxn+ R .

The“ 6ZM496 A : 2 14 236’

nAnxn- l R

'

,

duly/doc

?r: 2 142 n(n

—2 + R

Substituting in ( I ) and equating the coeffi c ients of xn in the two

members , we find

An+2 Adm?

‘ i‘ “ i" 1 ) (n “ i“

from which A 2 , A 3 , A4 , e tc. , may be determined in order when A 0

and A 1 are known.

I . A ; 77281” Sin—l

x/vI x2

1) 2 772.

Hence A 2 mfl/lz . A 3 m(m”

etc. ,

m(m2 I )x 3 m2(nz"

4)x

3 [4emsin

“ 1x I + mx +

Comparing th is result w ith that of example 9, we have, byequatingthe coeffi cients of i n,

3252x7

ll .

3 2 5

I I . sin- l x

S

i:Similarly, by equating the coefficients of m2

,

Todhunter’s Ca lculus.

D EVELOPMEN T S . 187

2242 62x 82 9 . 4

'2x6

1 2 . S in- 1x

2 = x2

Equat ing the coeffi cients of 7723,we have

1 3 . ( s irr ‘ x )3

x3 + 3

2 I"

1x5+ 3

2521 +5

5

2

.

Dividing both members of example 1 2 by 2 and differentiating , we

s in- 1 x 2 . 4 . 6x"

2 x —.c

VI — x‘2 3

i

F rom which , multiplying both members by I —x 2, we obtain

x 2 x 2 xl

1 5. ( 1 S ln- l x z x

3 73 3 5

i .

s

in wh ich , putting x sin 6, we have

sin2 9 2 s in4 6 2 4 s in6 616 . 6 cot 6 = 1

3 3 5 3 5 7

When the determination of the successive derivatives of a higherorder is laborious , a s impler method may be employed provided the

deve lopment o ff’

(x ) is known.

T hus s ince sin- 1 x is an odd function, which vanishes with x , we

f (x ) sin‘ l x Ax 3 x3 -I Cx 5 Dx

"etc.

D ifferentiating , we have

f’

(x )4/

A 33 x“

5Cx4

7Dx6

etc.

I

Developing 1/ or”

, we have , provided x I ,

188 DIFFEREN T I AL CALCUL US .

( I ) and (2) are identica l . Hence ,

A I , B

3 5 7 .— I )

and coeffi cient of n I term

x

and s1n- 1 x z: x

2

3x5

. 3

wh ich is convergent for x 1 , since each term is less than the cor

responding term in the geometrical prog ression x x3

x°

etc .

I 3

2

From which n :

1 7 . Develop tan"1 x .

Since tan- 1

x is an odd function which vanishes w ith x , we assume

f (x ) tan- 1 x Ax + 8 x3 Car

6 Dx7

etc .

Differentiating , we have

f (x ) I /( I 962

) A 3s W e70 x“ ete

Developing I x 2 ) , we have , provided x I ,

—x 2+ x —x6+ etc.

(a) and (b) are identical . Hence,

A z l , B = C= 1/5, D etc.

7

tan—1x etc .+ I )”

2n+ 1

in wh ich R is an infinitesimal as —> cc provided x or < 1

numerica lly.

I f x I tan we have

7t/4 = I

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19o D IFFERENTIAL CALCULUS .

[Suggestion.— Put x/(x 1 ) z , and in development of ( I z )

put 2 I /( 1 xl-l

2 5 . ( I 2x I x 2x3

7x4/4 3x

5

/2 R .

2 6 . cos* 1 x : x — x

3/2 . 3

2 7. xQ/(ex x

4/2 R .

2 8 . ( I x9)"2 1 2x

23x

44x

6 R .

2 9 . ex/cos x I x x2

2x3/3 x

4

/ 2 3x5

/ 10 R .

30 . e“ in x I x2

x4

/3 R .

+ 4x + 1 2x21 :

1 I /x2

1/2x4

1/6x6 R .

I

( I xiii/3 I 5x

2/3 sa

d/9 R

tan—1 x I x x

2/2 x

3/6 7x

4

/24 R .

sin2 x

log sec x x /2 + x4

/ 1 2 xe/4s R .

(a2 6.x) 1/2 a boo/2a 62x 2/8a

3+ 36

3x3/48a5+R .

ex log ( I x ) x x2

/2 9x5

/l5_+ R .

tan x x x3/3 2x

5/ 1 5 1 7x

7/31 5 R .

log ( I sin x ) : x x2

/ 2 x3/6 x

4

/1 2 R .

log ( I e” ) log 2 x/ 2 x‘z

/z3

x4

/23

l4 R .

(ex 2”

( I nx 2/2 n(gn R .

cot x I /x x/3 x3/3

2. 5 2x

5

/33

. 5 . 7+ R . (Bymethodof Indeterminate Coeffi cients . )

tan4x z : x

44x

6/3 6x 3/5 R .

Bymeans of T aylor’

s and Stirl ing ’

s formulas deduce the fol

lowingsin (x ;t y) sin x cos y : l: cos x siny,

cos (x zic y) cos x cosy

:F siu x siny.

sinhx (ex e“ x )/2 x + x

3

/i

3‘ i‘ x

ii

/ii“ i" R .

DEVEL O RMENT 5 .19 1

cosh x z (e36 e‘ x )/2 I + x

2

/2 R .

cosh” x I nx2

/2 n(3n 2 )x4

/ 4 R .

log (s inhx/x ) x2

/6 x4/ 1 80 R .

tanh‘ l x :

sinh ‘

x/a x3/2 3a

3 -i 3x 5/2 4 . 55a5 R .

53 . Given ys

3y x z 0 ; deve lop y in terms of the ascending

powers of x .

y = f x , f o z o or i 4/3.

3y2a’

y/a’x 3a

’

y/a’x I z 0 g ives

619/d 0I / (3Jf

2I /3, fory 0 .

270134 ‘ i‘ J’2

(d’ 2

y/ (1902

) d /dx"

0 g ives

a’3y a

’ 3y

ydx , dx ,,

0 g ives

Hence, y x/3 x3/3

4 R .

54 . Given 2y3

yx 2 0 ; show that

y z I + x/ 2 . 3 x3/2

3

34 R .

55. Giveny2 8/y 6x ; show that

y z : 2 + x

5 6 . Giveny3x 8y 8x 0 ; show that

x x4

/8 3x"/2 6 R .

Given 4y3x -

y 4 z: 0 ; show that

4 44x s(4)7x

2 R .

58. Giveny3

a’

y ayx 0 ; show that

aca

/a2

x4

/a3

x ii/a"

R .

192 D I F FEREN T I AL CAL CUL US .

59 . Given sin a z : x sin (a a) ; show that

60 . Develop fex

2 ex

By Stirl ing’

s formula we write

x ex

I f (O ) + f (of-2

12

R .

Since f x f ( x ) , the deve lopment contains no power of x o f an

odd degree .

Writing (ex I ) 1 2/(ex I ) , we have

—+ xe

x

2l x I

1f (x ) l 2

— l

Differentiating , we have

(f (x ) f (x )

(f (x ) ar e ) H = f (x )

(flx) + sf (x>+ f (9 0) f

Making x z o in ( I ) , etc . , we find

f ( 0 ) — I /30 . fetc

Hence,

I +6 2 3o |4_

Edward’

s Differentia l Calculus.

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194 D I F FEREN T I AL CAL CUL US .

EMU) 8 QW) z)aw a

’x as a

’

y(Ex . 7, p.

Al so aux/er n,

we) aov),

a,

II ence,

g aw ) eon)a’i a

’

x

e/aoo) a2

¢ (z) aw 82W ) 8

2W),

or\ a’x d

’

x aw a’t e

’x as In a

’x2

a’

8 050 62

2250) 82

9750)Slm1larly, Z,

a’

y ofy/2

a’

y dxn

I I ence,

82

4W) 82

960)[ I 2

dyaé

(n

T herefore

(au/d z (aw/JIM,

are 8

224

2

dx afy/lé

a’

y2é

new)a W

e “(n one etc .

D EVEL O PMEN T S . r95

Substituting in Stirling ’s formula,and making I 1 , we

u + f,

In“ (A I ) etc .

I t should be notl ced thatis,in general , a func tion of x ii i andy RI

is the same func tion of x and y ;

is the same func tion of x nfinz‘ andy Rb’ni

'

is the same func tion of x an and y

T herefore the remainder term in (a) , wh ich is equivalent

is the same function of x + HM and

is of x and y. Hence, the ren + 1

mainder term, denoted byR ,may be written

eu-i—Iu a

n—H

F rom 1 2 5 and 1 2 7 we see that formula (a ) develops

the given function provided that,as n increases without

limit, is real and finite and R is an infinitesimal or,

196 D IFFERENTIAL CALCULUS .

what is equivalent, provided that n and all of its successive

partial derivatives of the nth

order are continuous betweenall states , corresponding to values of x and y from any

assumed values to x it and y k,under the same law.

Having a : f (x , y, we may,in a manner analogous

to above, deduce

f (x + o y+ e”o f,”

”

R S

the remainder term being indicated by the symbolic formdescribed in 1 0 9 .

In a similar manner,a formula for the development of a

func tion of any number of sums of two variables each may

be deduced.

1 2 9 . Extension of Stirling’s Formula — In (a) , {5 1 2 8 ,

put x z o and y 0 ; then write x and y for /z and R re

spec tively, giving

anf (o. 0 )

a’

xxx i

i

i—

5+

(0 0)

a n”J’

(My

(0, 0)

etc.

a n+ l

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198 D IFFERENTI AL CALCULUS .

Hence , substituting in fo rmula (a ) , 1 2 8 ,

(x 70201 + J’ W x ( ( z +303 2960l + r )

3/z 396202+30

93

(a +r>W ex <a ze ta

3<a +y>9fi i fe w +r>w xxs

3(a +y) /19R9 2xnk3 li 9k3.

3 . Develop u ex s in y.

f (O ; 0 ) z

9 3 3

Hence, €xS iIIy x) ;

33.

J

;x

6

},

1 30 . Theorems of Lagrange and Laplace.

Suppose y Z

in which x and z are independent, and let it be required todevelopf (y) according to the ascendmg powers of x .

Placmg n in which case a will al so be a func tionof x and 2

,then developlng by Stirling

’s formula,we have

x =0 1 0 2

in which the coeffic ients of the different powers of x are

functions of 2 In order to determine thei r values , we

transform etc . , before making x o.

DEVEL O fi MEN TS .

Differentiating ( 1 ) with respec t to x , we obtain

EDI / fix

whence ay/a’

x y)/[ I

D ifferentiating ( 1 ) with respec t to 2 , we have

I

ay/a’z I / [ I s eem/as].

Hence ay/a’

x ¢ (y)

an an By an an 83)SlnCe

dx ay dx,

dz dz?

we have

ail/47

x (au/47

3 )

O bserving that

or 8?8y a

’x (12 a

’x’

and that n f (y) gives

an of (ymy an aw ay

a’z By a

’z’

a’

x a) ! a’

x’

we have, differentiating

2 0 0 D I F FEREN T IAL CAL CUL US .

an 3Hence, by

a’

x" g,

as” 8

2

a’x3

a’x a

’

z

Hence,by

an” a

n- l

S1m11arly, frome’x"

an+1n 8 ”

a’

x” +1 a

’z”

which shows that the law of formation is general.

we deduce

0 gives y 2,Zl and S7?

Hence,a’

z

a z af tz )A. (W)

Substituting these expressions in we have

f (y) = f (z )2 a W) dz

which is cal led Lag range’

s Tneorein.

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20 2 D I F FEREN TIAL CAL CUL US .

2 . Given log y 2 xy, developy.

We may w rite y z exy, and putting xy z y

’

; we have y’z xe

wh ich may be developed by making and y =y’in example 1 ,

32xs

” 12" lx 7i

13 I”

Replacing y'

by xy and dividing by x , we have

In

3 . Developy z xy”

.

(2(y) = y”. no)

2

z as)”

seen ver-2 + R .

4 . Developy z e siny.

$0 0 S il ly, sin2 ,

a 2 0

a’z¢ (z ) 2 s1n 2 cos 2 2 sm 22 .

6 sin 2 cos2 2 3 sin3 2 sin 32 sin

Il ence ,

e“

e3

y z z —l—e sin 2 + Esin 22 + 32 sin z ) + R .

5 . Having y z e s iny, develop siny.

f (x) s iny. f ( z ) $04)“

2 cos 2 ,

¢ (z ) s in 2 cos 2 z: s in

s1ni 2 cos 2 ) (3 sin 32 sin z )/4.

D EVEL OPMEN T S . 2 0 3

Hence ,

2

s iny sin z + - sin 2 2 4—5

8s in 32 s in z ) + R .

6 . Having y z'—i a s iny, develop sin 2y.

Here f (y) s in 2y, f (2) s in 2 2 , ¢ (z )“

2 sin 2,

2 cos 2 2 , e.

I i enee ,

2 sm z 2 cos 22 s in 32 sin 2.

a’

s1n zy sm 2 2 (s rn 32 s 1n s)ea7z(sm

’2 2 cos R .

7. S imilarly, develop sin 3y .

In this case f (y) s rn $ 7, f (x) 2 s in 32 , (1503)

3 CO S 32 , x e .

Hence,

a’

sin 3y sin 32 e sin 2 3 cos 32 ZZZ-(s in

it2 3 cos R .

8 . S imilarly, deve lop cos y.

f (y) cosy, f (z ) cos 2 , ¢ (z ) srn z ,

sin 2 , x

Hence,

2

cosy : cos a e s in2 z 3 sin22 cos z

9 . Having a e s in n, deve lop a , s in n, s in 2a , sin 324, and

cos a in terms of z‘

and e. By comparison w ith examples 4, 5, 6 , 7 ,

and 8 we havee'2

e2

n nt ( sin nt)e+ s1n awj (3 sm 3nt srn nz

‘

)—éR .

e e"

S in n sm nt srn 2nt—2

(3 sm 3nz‘

srnm)?R .

sin 2a s in znl ( sin 3nz‘

sin nt)e R .

s in 3n sin 3n‘

t R .

cos n cos nt ( 1 cos 2 720-2 (3 cos 3nt 3 cos nt) -g R .

2 0 4 D I F FEREN T IAL CALCULUS .

1 0 . Kepler’s Problem.

*

Having fi t u 6 s in u

r z : 6 cos u) ,

find 9 and r in terms of t.

Firs t deve lop 6 in terms of a . In ( 2 ) put m, and since

from (d) , Ex . 5, § 1 2 7,

I tan x

we have

Vf lm I ) e

u

Hence , 66V 1

(m I ) (m

772

Placmg A, and V 2

, we have

ui+ log ( I Ar m) log ( I Rem)

A3

(Ac -m e- 2m

35- 32 :

M

(1 6162

m’

Me” r m

T’;(c3m

'

F 3” )

2 sin 2 22 +

* Price’

s Calculus , Vol .“

I I I . pp. 56 1- 564.

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20 6 D I FFEREN T I AL CAL CUL (7 3 .

CHAPTER XI .

MAXIMUM AND MINIMUM STATES .

FUNCT I O NS O F A S INGLE VAR IABLE.

13 1 . A Maximum state of a continuous function of a

single variable is a state greater than adj acent states which

precede or follow it. T hus, f ee has a maximum state

corresponding to x a, provided that as n vanishes we

have ultimately and continuouslyf a f (a : l: ii ) .

A maximum state is, therefore a state through wh ich,as

t/ze nar iaé/e increases conz‘

inaonsly, the function changesfrom an increasing to a decreasing func tion, and iz‘s firsta’ijj

’erential ceefi cienz

‘

cnanges its sign f rom pins in n nnas

63)AMinimum state is one less than adjacent states which

precede or fol low it. T hus, f a is a minimum provided

that, as n vanishes, we have ul timately and continuouslyf a < f (a iA m1n1mum state is

,therefore

,a state through wh ich ,

as z‘

x’ze var iable increases continuously, the func tion changes

from a decreasing to an increasing func tion, and iz‘s fi rsi‘

def erenz‘

iai caefi cienz‘

cnang es ifs signfrom ni inns z‘

apins.

A maximum is not necessarily the greates t, nor a mini

mum the least, state of a func tion.

A func tion may have several maximum and minimum

states.

MAX IM UM AND M IN IM Ufl/f S TA TES . 2 0 7

T o il lus trate, let AB CD -[ j be the graph of a function

T he ordinates PA ,R C

,TE

,and W] represent maxima

s tates of the func tion,and QB ,

SD ,ZP

,and VH

represent minima s tate'

s .

QB [ W illustrates the fac t that a minimum state maybe greater than a maximum.

The ordinate at U represents a z ero maximum,and the

ordinate at f a z ero minimum.

T he point I at which twobranches of a curve terminatewith separate tangents is called a salient point, and the

points C and E at wh ich two branches of a curve terminatewith a common tangent are cal led cusps.

1 32 . A continuous func tion mus t have at least one

maximum or minimum state between any two equal states ;for if, in passing through any state

,the func tion is inc reasing

it must change to decreasing, and if dec reasing it mustchange to inc reas ing

,at leas t once before it c an again

arrive at tnat state. T he maximum ordinate PA , between

the equal ordinates ML and M ’L

’il lustrates the principle.

Similarly, it may be shown that a continuous func tionhas at least one minimum state between any two maxima,

20 8 D I F FEREN T I AL CAL CUL US .

and one or more maxima between any two minima. T hatis

,as the variable increases , maxima and minima of a con

tinuous func tion occur alternately.

1 33 . T he general definition given for maxima and

minima assumes that the function is continuous , and

that as the variable increases adjacent states precedeand fol low those considered . Some exceptional cases arisewhich are illus trated in the following figure.

As x increases, the func tion represented by the ordinate

of the curve AB C in passing through PA has no adjac ent

preceding states . f’

x does not change its sign,and is neither

z ero nor infinite ; yet as PA is smal ler than adjacent statesit is general ly considered a minimum.

At Q the positive ordinate is an asymptote to both

branches of the curve ; and although the unlimited value of

the ordinate does not represent a possible value of the

func tion, yet f’x changes its s ign ; therefore the ordinate

Q B is said to be an infinite maximum

At R the positive ordinate I S an asymptote to one branch

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Hence, f (a / 2 ) > f ( 7t/ 2 : l: b) as b vanishes ,

f (a / z ) 1 is maximum.

2 : 0 gives f o o, and f (o : l: k) sin ( : l:

Hence, f o f (o it) , and f o f (o n) , as nvanishes ;therefore f o 2 o is neither a maximum nor a minimum.

1 35 . In general , maxima and minima are determined by

finding those values of x corresponding to which , as thevariab le increases , f

’x changes its sign.

Assuming that 30 increases continuously,that f oo is continu

ous, and that every s tate considered has adjacent precedingand fol lowing states , a max imum state is cbaracter ized by a

elzang e of signfromplus to minus in tbe first differential co

efi cient, and a minimum state by a cor responding elzang efromminus toplus.

Conversely, if , inpassing tbrougb f’

a, f

'x c/zangesfromplus

to minus , f a is a max imum, and if f’

x cbangesfromminus to

plus, f a is a minimum.

f ee,if discontinuous, may change its sign by passing

through a double value, one pos itive and the o ther negative,

as il lustrated by a salient point or by pas sing through infinityas illustrated bya cusp with common tangent perpendicular to the axis of X .

f ee, if continuous, can change its sign only by passing

through z ero.

A maximum or minimum of f or: corresponding to a salient

point is an exceptional case distinguished bya double valuefor f ee.

Hence,in general

,values of x corresponding to which

f’

x changes its sign as 90 increases , are real roots of one or

the other of the two equations

and f’a z co .

MAX IM UM AND M I N IM UM S TA TES . 2 1 1

The figure, p. 2 0 7, illustrates the fact indicated by ( I ) andthat, in general , the tangent corresponding to a maxi

mum or minimum ordinate of a curve is parallel or per

pendicular to the axis of abscissas .

f or does not n ecessarily change its sign as 30 pas sesthrough roots of ( 1 ) and as may be seen in the casesrepresented byparticular ordinates of the fol lowing curves .

At the points A and B,where the tangents are parallel to

X , f’x 0 ; and at C and D , wh ere the tangents are per

pendicular to X , f’x 00

,but f

’oc does not change its sign

as f (x) passes through the corresponding s tates .The points A ,

B,C,and D are calledpoints of inflex ion.

T he real roots of ( I ) and ( 2 ) are, therefore, cal led cr itical

values,and the next step is to determine wh ich of them

correspond to s tates at which f’x cbanges its sign as x in

creases,and

, therefore, correspond to maxima or minima of

T he general method,for any critical value, as a

,is to

determine whether, as b vanishes , f’

(a b) and f’

(a b)ul timately have and retain different signs I f so

,a cor

responds to a maximum or a minimum,according as the

sign of f’

(a Ii) is plus or minus .

2 1 2 D I F FEREN T IAL CAL CUL US .

EXAMPLES.

I . Le t f x = b + (x

Thenf'

x 2/3(x a )1/3 00 g ives the critical value a .

As It vanishes, f'

(a li ) is negative and f’

(a -l 12) is positive .

Hence , f a r: b is a minimum.

2 . f x 2 6x 3x2

f’

x 6( I x 2x2

) 0 g ives the critical values I and

f’

( 1 2 ) 6143 wh ich is pos itive when 2

f’

( r li ) 6b( 3 wh ich is negative when ItHence , f r 5 is a maximum.

f’

( 12) is negative when lz o.

f’

( 2 ) is pos itive when 12Hence, f ( is a minimum.

3. f x a (x

f’

x I /3(x b)2/3 00 g ives x z 60

f’

(é :F li ) are both positive for all va lues of 12.

Hence, f (b) a is neither a maximum nor a minimum.

4 . f x (x I )4(x

f'

x (x us

e new 5) 0 g ives

x z I , x 2 , x

f t o is a minimum.

f ( _ 2 ) z : o is ne ither a maximum nor a minimum.

f ( is a maximum.

5. f r (x z )a

/ (x

f’x (x 2 )

'2

(x o and 00 gives

x = — 2 , x = 1 3, x = 3.

f 13 is a minimum.

f 3 2 do is a maximum.

6 . f x 2 (a x )3

/(a min.

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D I F FEREN T IAL CAL CUL US .

As It vani shes , the sign of the second member of ( 1 ) wil lultimatelybecome and remain the same as that ofand s ince lzz/ z is always positive,

f a a maximum requires f a 0,and

f a a minimum requires f a o .

I n the exceptional case when f a o,the sign of the

second member of ( I ) wil l, under the law, ultimatelydependupon that of ziz it?

” 'a/ |3, which changes with that of It . f a

c annot,therefore

,be a maximum or a minimum unless

ziz 0, wh ich requires f

’"0 .

I f also f a 0,the sign of the second member of ( I )

wil l , under the law,ultimately depend upon that of

n‘

f’va/ 4 ; and since is always positive,

f a a maximum requires f a o, and

f a a minimum requires f'

a 0 .

By continuing the same method of reasoning 1t may be

shown that, iff ”a is the first derivative in order wh ich does

not reduce to o, f a is neither a maximum nor a minimum if

n is odd, and that it is a maximum or a minimum if n is even,

according asf ”a is negative or positive. Hence, we have

the following rule :H aving f

’a 0

,substitute a f or x in tbc successive der iva

tives of f x in order,until a result otber tban o is obtained. I f

tbc corresponding der ivative is of an odd order, f a is neitber a

max imum nor a minimum but if it is of an even order, f a is

a maximum or a minimum according as tbc result is negative

or positive. If a result 00 is obtained,tbc met/cod of § 1 35

sbould be employed.

T he relations between the corresponding states of f x , f'

x,

MAX IM UM AND M IN IMUM S TA YES . 2 1 5

andf x,in a case where AB CDE is the graph of f x , are

shown graphically in the fol lowing figure.

*

EXAMPLES.

F ind the values of tbc var iable ionic/c correspond tomaxima

or minima of tbc f ollowing f unctions

x5

5x4

5x3+ I .

5x4

20x3 1 sa

20 g ives the critical values 0 , I , 3.

20x3 60x 2 30x .

1 0 . f 3 = 90 .

Hence ,

f r 2 , maximum. f 3 2 6 minimum.

f x = 60x2

I zox —l 30 , f and

f o 1 is neither a maximum nor a minimum.

2 . f x xi —

9x2+ 24x

— 7

f'x = 3(x

2g ives x = 2 or x

f fT herefore f 2 , maximum. f 4. minimum.

Calcul , par P. Haag , page 71 .

2 16 D I FFEREN T IAL CAL CUL as .

sin x cos x .

r: 3 s in2x cos2x s in4x 0 , g ives x etc .

”

x I O s iu3x cos x 6 sin x cossx .

Since sin and cos 60°

f 60°

3V372 , f 60°z : maximum.

4. x4 8x3+ 2 2x

2 2420+ 1 2 . x 3, min.

x 2 , max.

2 I, min.

x2 —

4x—l—9. x 2

, min.

6 . x3/3+ ax

2

3a2x . x z a

, min.

3a , max.

7. x3 6x 2 9x

-1'I O . x z 3, min.

x I, max .

8 . 3a2x3 b4x 6

5. x bi/3a , min.

x b2/3a , max.

x — x + I x = — I, max.

9 '

xQ —l—x -j— I

.

x = I , min.

x z: I max.

x

I I . xl /x . x e : max.

1 2 . s ec x cosec x x n/4 ,min.

x s7r/4, max.

I 3. f x e- x + 2 cos x ,

f’

x : ex — 2 sin x

,— 2 cos x ,

f x : e- x+ z sin x .

lv x = e + e- x+ z cos x.

f’0 = f 0 =f

Hence , f o 4 is a minimum.

xcos x , max.

I x tan x

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2 I 8 D I F FEREN T I AL CAL CUL Us .

spond to maximum and minimum states o f the power only

are rej ec ted.

T hus , having f (x ) 2 : l: 4/ 2 ax2

x we write

$00) 2o

T hen 4ax 3x2

0 gives x o and 4a/3 .

gb(o) is a minimum and gb(4a/3 ) is a maximum of

But f’

(x ) : l: (4a 3x )/ 2 4/ 2 a x z : 0 gives x 4a/ 3

only, and f (o) is neither a maximum nor a minimum of

f (x )Similarly, (x )

9

/ (x a?

) is a minimum when x= 2a,

z a )/4/x2

a2

18 not.

T o illustrate the use of the foregomg principles

f (x ) I / (S+ 10 gMi x”

2 5963

)

By 5°

we take 5 log V4b2

x”

2bx3

By 2°

and 4°

we take 4/4b2

By 6°

and 1°

we take 2bx2

4bx 3x2

0 gives x z : 0,x 4b/3.

(25 ( 0 ) 45, (45/3)

I I ence,

f (o) is a maximum and f (4b/3) is a minimum.

1 38 . I n certain cases it is not necessary to determine thesecond derivative.

In case of one critical value only, and it is knownthat the function has a maxrmum s tate or that it has a

minimum state.

MAX IM UM AND M IN IM UIVI S TA TES . 2 19

I f a is the only c ritical value, f (a ) is a maximum or a

minimum, provided f (a ) is greater than or less than both

f (a i ii ) for any assumed value of lz.

Whenf’

(x ) is composed of two or more fac tors , oneof which reduces to 0

,for x 2 : a

, f (a) may be determinedwithout using f (x) .

T hus,let f

’

(x ) ¢ (x) go(x ) .

T hen f”

(x )

Supposing that t/J (a) 0,we have

f (a )

Hence,to obtainf (a) , multiply t/ce difi erential coefi cient

of t/zatfactor of f (x ) wbic/z reduces to 0 by tlze otber factors,and substitute a .

T o il lustrate, let

f (x ) (x a)” 2

.

f’

(x) 2x (x a) ( 2x a) 0 gives

x 2 0 , x = a,

x a/ 2 .

f (o) 2 (x a ) ( 2x 2a“

,indicating aminimum.

f (a) 2x ( 2x 2a”

,indicating a minimum.

=‘

4x (x a ) dl/2 a2

, indicating a maximum.

EXAMPLES.

F ind t/ce values of tbc var iable zolziclc correspond to max ima

or minima of tbc f ollowing f unctions

(x 0 , min

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2 2 2 D I F FEREN T IAL CAL CUL US .

7 . O n the righ t line A B joiningthe two l igh ts A and B ,

find the po int between the l ights of leastil lumination.

Le t c number of miles from A to B .

Le t x number o f miles from A to required point .Le t a intensity of the ligh t A at I mile fromA .

Le t b intensity of the l ight B a t I mile from B .

T hen

x )iz : intensity of l ight at point required.

0 ives x

( c x )3 g

8 . F ind the relation between the radius of the base and the altitude of a cyl inder , open at the top, which shal l just hold a g iven

quantity of water and have i ts surface a minimum.

Let x radius of base , and y a ltitude . T hen

nyx”

volume r : y v/nx2

ax”

2 7rxy surface f (x ) .Hence,

7zx2 2 7txv/n

’

x2r : 7

'c 2v/x .

f’

x 2 7tx 2v/x ‘ 0 g ives x $77 72 and y

f $77 75) 67t , indicating a minimum.

9 . F ind the maximum rectang le that can be inscribed in a g iven

circle .

T ake the origin at the centre , and let 2x and 2y respective ly equalthe s ides of the required rectang le . Then xy area of rectang le .

x2+y

9 R 2,in which R represents the radius of the circ le , g ives

y : : t VR z — x”.

Hence, fx 2 i xVR ? xio

f’x : 1: (2R

2x 4x

3 ) r : 0 g ives x 0 and x R/ V2

f"(R/ V?) ind icating a maximum.

MAX IM UM AND M IN IM UM S TA TES . 2 23

I O . Determine the maximum rectang le wh ich can be inscribed in

a triang le o f g iven base and a ltitude .

I I . Show that the difference between the sine and the versed s ine

is a maximum when the ang le is

1 2 . The base and the vertical ang le of a triang le being g iven, showthat its area is a maximum when it is isosce les.

I 3 . O f all isoperimetrical plane triang les wh ich has the max imum

area

I 4 .

'

Divide a triang le whose sides are a , b, and c, respective ly, intotwo equal parts by the minimum right l ine .

Ans . 1/ (c a + b) (c + a b)/2 .

1 5. Through a g iven point (a, b) draw Y

the shortest straight line terminating in

X and Y.

Le t 6m: angle required l ine CB makesW i th

‘

X .

T henCB CP+ PB a/cos 6 b/sin 9,

which is a minimum when 6 tan—1 f/b/a , g iving CBSimilarly, show that

GB O C is a minimum when 6 tan‘ 1 b

—a ;

0B X O C is a minimum when 6

GB O C—l CB is a minimum when 6 tan"-1

OB X 0 C X CB is a minimum when

2a tan3 6 — b tan’ 6 + a tan 6 — 2b = o.

I 6. Determine the maximum righ t cone which can be inscribed in

a sphere whose radius is R .

2 24 D I F FEREN T IAL CAL CUL US .

Let x AP, and y PB .

Then

vol . v but y2

2Rx

T herefore v (2R 7t’

x2

nx3

)/3.

dv/dx 7tx (4R 3x )/3 0

x = 0,

x : 4R/3.

day/d ” (4k 6x )/3 4R 7t/3s 4R/3.

1 7. F ind the radius of a circle such tha t the segment corresponding

to an arc of a g iven leng th shal l be a maximum.

Let 2a leng th o f arc, and r radius .

Draw CD bisecting the arc , then

AD CA a/r , and 9 2a/r

Segment sector B CAD AB CA

r2sin

ra r2sin

which is a maximum when r 2 a/7t , and

the segment is a semi-circle .

1 8 . With a g iven perimeter find the radius wh ich makes the cor

responding circular sector am aximum. Ans . radius I /4 perimeter.

1 9 . F ind the maximum right cyl inder wh ich can be inscribed in a

g iven right cone.

Le t VA = a , B A = b, AC = x ,

CD : y, CV = a

Hence , vol . cylinder v

VA : AB : : VC : D C, y : b(a

Therefore v nb2(a x )2x/a

2.

O mitting we have

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D I F FEREN T I AL CAL CUL US .

Let r radius CA , AB AE x, BE 2 2y.

A lso, u

AB x AE BE x2

y

2 r

2

And i t

x ” A” x

x : r4/ 3, max.

24 . F ind the maximum cyl inder that can be inscribed in a g iven

pro late sphero id.

Let 2x axis , and y radius of

base , of required cylinder .

T hen, vol . of cylinder

v 2 7ty2x 2 71xb

2

(a2 x )/a

2,

which is a maximum for x a/4/3.

2 5 . F ind the minimum i sosce les triang le circumscribing a g iven

circle .

Let r radius CE, x 2 B F ,

T hen area A 7: xy.

S imi lar triang les , B CE, BF D , give

— r .

Hence, y 2 107767 2 r ) , and

area A x r Vx/ (x 2 r ) ,

c, wh ich is a minimum for x 2 3r .

26 . A boatman 3mi from shore goes to a point 5

mi . down the shore in the sho rtes t time . He rows 4 8

mi. and wa lks 5 mi an hour, Where d id he land ?

Let B be the boat 3 mi. from S ,which is 5 mi .

from the po int P. Let W be the land ing -place, and

T hen, number of hours

z ; —l- x2/ 4 ( 5 x)/5.

wh ich is a minimum for x 4 .

Todhunter'

s D lfi . Calc.. p. 2 13 .

MAX IM UM AND M IN IM UM S TA TES . 2 2 7

2 7 . T hrough a g iven point P with in an ang le

B A C draw a right line so that the triang lefo rmed shal l be a minimum.

Draw PD para l le l to A C, and let AD a .

- x . D s — a .

Then x — a zb zz s R .

AXR x AR s in A/ 2 , wh ich is a minimum for x 2a . XP : PR .

2 8 . The vo lume of a cylinder be ing constant, find its form when the

surface is a minimum. Ans . Al t itude diame ter of base .

2 9 . F ind the he ight of a ligh t A above the

straight l ine 0 B when its intensity at B is a

maximum.

intensity of ligh t at I foot from the

OA Z y, OE : b, Z O B A r - G.

‘2

The intens i ty var ies direct ly as S in 6, and inversely as B A

Intensity at B ay/(b2

y2

)3/2

,wh ich is a maximum when

y s z / z .

3o . Having y x tan a x2/4/z cos

2at , F ind the maximum

va lue o f y. Cons ider ing y o and a as varying , find maximum

va lue of x .

5 I st . yz : ns in2 a

,a max imum x z 1: s in 2 a .

l 2d . x 2 b , a maximum,a

3 1 . O n the righ t l ine CCI a jo ining the centre s o f two spheres( radi i [i r ) find po int from wh ich the maximum spherica l surface i s

2 2 8 D I F FEREN T I AL CA L CUL US .

CP 2 x PC1 a x .

Area zone A SEB 2 7rR X H S .

x : R R : CS . CS R2

/x and I I S z R R z

/x .

Hence, z one ASBH 2 7L'

R (R 162/x ) .

S imilarly,zone DEF 2 7zr ( r r

2

/ (a

Visib le surface 2 7c[R2

( R3/x 7

, 3

/(a which is

a maximum for x

32 . F ind the path of a ray of light froma po int A in one med ium to a po int B in

ano ther medium, such that a minimum

time w il l be required for ligh t to pass

from A to B ; the ve locity of l igh t in the

firs t medium be ing V, and in the second

V'

. [Ferma t’

s Problem. ]I t is assumed that the required path is

in a plane through A and B perpendicu

lar to the plane separating the media .

Le t A CB be the required path . T hrough A ,C, and B draw per

pendiculars to DE .

a z AE, s B ,

d Z DE.

Then AC a/cos (p, B C b/cos qb’

,

CE a tan Q) , CD b tan dfi'

.

a tan (1) b tan cb’

d. a cos2

cos2q) .

a bT ime I

, from A to BV cos m

l( I i d iV '

cos (ZED:d¢

0 g ives

(03 or

for t a minimum.

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2 30 D I F FEREN T I A2 CAL CUL US .

changes from increasing to decreasing , or vice versa , as PA passes

through AM and AN .

Let 6 ang le A OP. T hen x r cos 9, and

dPA/dfi : zar s in

i

9 2 0 ,

O therwise PA may be expressed in terms of y and the prohlenso lved .

36. F ind those conjugate diameters in an e l lipse whose sum is a

maximum or a minimum.

L et x’

and y’be any two semi-conjugate diameters and let

and a? b2 c

g. T hen [Ana l . Geom.] x

’2

+y

and s = x'

+ Vc2

ds/dx’2 : 1 x

’/ Vc2 z 0 g ives x z e2

/z =y'2.

T hat is , equal conjugate diameters are those whose sum is a maxi

mum.

Expressing x'

and y'

in terms of the inc lination of x'

to the trans

verse axis , denoted by 6, we have ds/d6

a and b are , respective ly, maximum and minimum states of x’

,

g iving dx’

/d9 o, and therefore ds/dd 0 . Hence the sum of the

axes is a minimum.

37. A rectangular hal l 80 feet long , 40 feet w ide, and 1 2 feet highhas a spider in one corner of the ceil ing . How long w il l it take the

spider to craw l to the opposite corner on the floor if he craw ls a foot

in one second on the wal l and two feet in a second on the floor?

Ans . seconds, minimum.

1 39 . To fi nd tlze max imum and minimum distancesfroma g ivenplane curve to a g ivenpoint in itsplane.

L et y = f x be the equation of any plane curve , (a ,b)

the coordinates of any point in its plane , and R the dis

tance from (a ,b) to the point (x

’

, y’

) on the curve.

Prob lem proposed by Pro fe ssor H . C. Whitaker in American

Ma tbema tica l Mont/zly, Vol . I . No . 8.

MAXIM UM AND M IN I M UM S TA TES .2 3f

I f move along the curve,R in general becomes a

varying distanc e measured on the radius vec tor j oining (a , b)with the moving point (x

’

, y’

) and R (x’

a )2

(y’— b)

2

.

We wish to find the maximum and minimum values of R .

Plac ing the first derivative of (x’

a )2

(y’

b)2

equal

to z ero,we have

a (r’

bled /M ) 0

T he equation of the normal to y f x at (x’

, y’

) is

y y’

H ence ( i ) expresses the

condition that (x’

, y’

) is on the normal through (a , b) .T he required value of R is therefore estimated along the

normal through (a ,b) , is a maximum or a minimum accord

ing as the second derivative,

1 (y

is negative or positive, and, in general , is neither a maxi

mum nor a minimum when the second derivative reduces

to z ero

As (a , b) may be any point upon any normal , we con

c lude that the radial distance of each point of a normalfrom

‘

the curve is , in general , a maximum or a minimum

when measured upon the normal . (See figure, page

T hus,let BAM be a normal to the curve N th/ O at M .

With A and B as centres , and with the radii AM and B M

respec tively, describe the c ircumferences e and RMR .

T he figure shows that the radial distance of A from NM O

is a minimum when measured upon the normal AM , and

that the corresponding distance of the point B is a maxi

mum. T his is evident from the fac t that the c ircumference

r /ld r,in the vic inity of and on both sides of M

,lies w ith in

the curve NM O ,wh ile the corresponding part of the c ir

cumference B MR lies without.

2 32 D I F FEREN T I AL CAL CUL US .

Consider the point (a ,b) to move upon the normal

,and

let ( x ,y ) be its variable coordinates. When the normal

distance of ( x ,y ) from the curve is neither amaximum nor

a minimum,we have

I (r -

y

whence by combination with ( I ) we obtain

x=x+ 1 +

for the coordinates of a point on the normal whose distancefrom the curve measured along the normal is

,in general

,

neither a maximum nor a minimum. R epresenting th is distance by p, we have

o“

(x 9 02

(y—y )2

»

which combined with ( 2 ) gives

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2 34 D I F FEREN T IAL CAL CUL 0 5 .

Having yr

"

f z , 2 $26,

(tr/fix (ctr/dz ) (fie/doc)

wil l give c ritical values of x .

EXAMPLES.

I . u = x3+ y3 — 3a

2x : :

au/dx z 3x—3a

'2_ x z zlz a .

Substituting in g iven equation, we have y : l: a

Oi

u/dx‘l 6x

, Ou/Oy 3y2

.

3

F , in, 3a V4

3a 15 a minimum.

y :

30 x) , + 3}

5 x z o, y z o, is a minimum.

3 3 o o

x z a 2, y a i s a maximum.

is a maximum.

b2’

4. 4xy—y4 — x

4z

5. 3 2x (yx x z y 2, a maximum.

6. y (b‘2

x2

)/x ,x b makesy a minimum

14 1 . Having v given as an implic it func tion of x,by two

equations v qb(x ,y) and u o,from whichy

is not readily eliminated, we may proceed as follows

dv an av dy

dx dx dx'

( 2 ) I n )

MAX IMUM AND MI N IMUM S TA TES . 235

dv 8 7) 8v 8 22 8 25Hence,

dx dx ay (ix"dy

’

do an 8 7) an

dx 8y dx

wh ich combined with u f (x , y) z 0 gives c ritical valuesof x . T he sign of the corresponding value of d

Q

v/dx2

wil l,

in general,determine whether v is a maximum or aminimum.

EXAMPLES.

I . v x9+y

2, (x a)

9

+ (y b)2

c2 o.

av/dx av/ay 2y,

au/dx 2 (x a) , au/ay 2 (y

Substituting in ( I ) , we have ay bx ; which combined with u = o

—l—b2

The positive sign g ives a maximum, and the negative a minimum,

for v.

2 . F ind the points in the circumference of a g iven circle wh ich are

a t a maximum o r minimum distance from a g iven point.

3 Given the four s ides of a quadrilateral , to find when its area is a

maximum.

Let a , b, c, d be the leng ths of the sides, to the ang le between a and

2. it that between c and d.

Then area 1 : v ab s in rib/ 2 cd sin

az -l- b

2zab cos cb : c

2

+ d2 2cd cos 1b,

each member being the square of the same diagonal .

v cdcos

2ab sin ab, 2cd sin 2b.

2 36 D I F FEREN T I AL CAL CUL US .

Substituting in ( I ) , we havetan qb z — tan ip.

That is , the quadrila teral is inscr ibab le in a c ircle.

dv ab cd dzbup 2

°°s ib l ? $27

45

ab s in qb cd sin

from wh ich dip/deb ab sin qfi/cd sin 1b.

Subs tituting in above , we havedv/dqb ab s in lb)/2 Sin tb'.

” W WW“z s in zb

indicating a maximum.

<5 +2

142 . H aving co given as an implic it function of x, by

th ree equations

w : F (x y, 2 ) , and u = f (x ,

and placing dz v/dx z 0 for a maximum or a minimum, we

writedzv 8w

+8w dy

+aw dz

dx dx ay dx 8 2 dx

dv 82) av dy av dz

dx dx ay dx+az dx

da an au dy au dz

dx dx+ay dx

+8 2 dx

Eliminating dy/dx and dz/dx ,we have a single equation

which,combined with to

“

2 F (x , y, v z 0,and u z 0

gives critical values of x and the corresponding values ofy,

2,and To.

By differentiating equations and eliminating

dy/dx ,(172/dx, d /dx

",d22/dx

2

,

an expres sion for d zv/dx2

may be determined .

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2 38 D I F FEREN T I A L CAL CUL US .

AMinimum state is one less than adj acent states . T hus,

f (a , b) is a minimum, provided as It and k vanish from any

values , we have ultimately and continuously

f (a ,2) f (a ziz 2 , 2 i

A minimum state is, therefore , one through wh ich,

as

ect/zer or bot/z var iables incr ease continuously, the func tionchanges from a decreasing to an increasing func tion, and its

par tial dzp‘erential coefiicients of tbc first order elzange t/zeir

signsfromminus toplus.

Any particular state of 2 2 : f (x , y) , as f (a , b) , maybe ex

amined direc tly by determining whether as It and b vanishfrom any as sumed values , we have ultimately

f (a , b) bzlz é ) , or f (a ,b) bi é ) .

144 . In general , however, maxima and minima are determined by tes ting those values of the variables corresponding to wh ich

,as the variab les inc rease

,both partial

derivatives of the firs t order change their signs from plusto minus

,or minus to plus .

Sets of roots of the equations

0, az/dy o

,

00, az/dy 00

,

o, az/dy 00

,

az/ex

are,therefore, c ritical

,and may b e tested as indicated in

1 43 .

A maximum or minimum state of a func tion of two va

riables is il lustrated by an ordinate of a surface wh ich iseither greater or less than all adjacent ordinates. T he

MAX IM UM AND M IN IM UM S TA TES . 2 39

conditions az/dx 0 and ae/dy 0 indicate , in general,that the corresponding tangent plane is parallel to X V.

145 . Lagrange’s Condit ion.

—When the successivepartial

differential coeffic ients to inc lude those of the n 1 orderare real and finite for a set of c ritical values

,as (a, b) , de

rived from equations § 1 44 , a condition for a maximum

or a minimum may be deduced from (a) , 1 2 8 .

Placing

we have

f (a : l: b, b i A) — f (a , : i: 2B bé Cb2)/2 R . ( I )

In general , as lz and b vanish , the sign of the second mem

ber will ultimatelydepend upon that of Ali 2 ziz 2B lié —l CP,

and when A 0 it may be written

[ (A2 ziz Ai r (AC

A maximum or a minimum state wil l then require that thesign of this expression shal l ultimately become fixed and

remain so,while 12 and A change their signs by passing

th rough z ero.

I f (AC B2

) o , the numerator of the above expressionw il l be positive when Az o

,and it wil l be negative for

values of It and 12 other than z ero wh ich makeAb : l: B b = 0 .

Hence, a condition for a maximum or a minimum state is

(AC B2

) > 0,

or AC B”

;

Xan

240 D I F FEREN T I AL CAL CUL US

T his condition being satisfied, f(a ,

b) is a maximum or a

minimum according as A and C are both negative or both

I f AC B”

,there is neither a maximum nor a minimum.

I f A 0 and B 0,we have the same result

,for the

sign o f 2B /zle —l CA”

,in the second member of eq varies

for a fixed value of k as . lz passes through the value

I f A : B = o, then

AC B2

o, and Ab2

ziz 2 B lzk+ CE Cé’

,

which vanishes when k o ,for all values of It leaving

the question in doubt. T he same result fol lows whenA z: B 2 : C 2 O .

EXAMPLES .

F ind sets of values of tbc var iables w/cicb correspond to

maxima or minima of tbefollowing f unctions

1 . 2 x2 -l xy+ J/

2“ i“ ( l

a

/x + 43

0 “

az/dx 2x + y a3

/x2 az/dy 2y x a

fl/y

20 .

F rom wh ich we obtain x : y a/

6 22/dx2

2 2a3

/x3,

2 2 2a9

/y3, ai z/dx dy I .

In which substituting values o f x andy,condition ( 2) is satisfied and

2 is a minimum.

2 . z z cos x cos a + s in x s in a cos (y - fi ) .

O ff/dx .: s in x cos cr+ cos x s in a cos (y

az/dy sin cr s in x sin (y — p’

) 2

g ive x z zz a, y z fi .

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24 2 D I F FEREN ?YAL CAL CUL US .

1 46 . Functions of Three Variables .—L et u f (x , y,

R easoning as in the preceding cases,it may be shown that

sets of roots of the equations

0, au/dy 2 : 0

, au/dz o,

any : one

are c ritical .Deno ting a set of critical values from ( 1 ) by a

, b, and c,

we have, (b) , 1 2 8

,

f (a ziz It, b ziz le, c : l: l ) —f (a , b, c) 2 [Ali2

.B /e2

Cl2

]/ 2

: l: D b/e : l: E/zl : l: F /el —l R ,

in which A,B

,C,D

,E

,F

,represent the values of

ai

u 82

22 ai

u an.

dx”

dy dz dx dy’

dx d2’

respec tively, when x a, y b

,2 c.

I n order thatf(a , b, c) maybe a maximum or a minimum,

4. 5 22

Cr) i D 22 : t E21 : l: FM,

if not z ero, should be either always negative or always positive , as ii , 12, and l varythrough z ero between certain positiveand negative limits .147 . Functions of n Variables — By extending the above

method of reasoning, it maybe shown that the sets of rootsof the equations formed by plac ing the partial derivativesof the first order separately equal to z ero are . c ritical .Each set of c ritical values when subs tituted in the corre

sponding expansion should render the quadratic function of

It , ls, l , e tc .,always negative for a maximum

,or always posi

tive for a minimum,as 12

,k,l,etc .

, vary through z ero

between certain limits.

PART I I I .

GEOMETR I C APPL I CA TI ONS .

CHAPTER X I I .

T ANGENT S A ND N O R MA L S .

RECTANGULAR CO O RD INATES .

148 . Equations of a Tangent and Normal . -T li e eGua_

tion of a straigh t line passing through (x’

, y’

) on the curve

y: f x is (Anal . Geom. ) y — y

’m(x

Placing m= f x we have for the tangent lineat 7 1 )

y —y’

and for the corresponding normal

(def /WWc

T hus , having y“

9x ,then f

’x g/zy, f

’

4

Hence, y 6 1 : (x 4) is the tangent, and

y 6 : -

4) is the normal at (4 .

I f the equation of a line is in the form u §b (x , y) z : 0,

we have 1 1 1 )

f fly/ (M

244 D I FFEREN T IAL CAL CUL US .

Subs tituting in above, we have

x’

) for the tangent ,

(y y’

) 1: (x x’

) for the normal.

T hus,having u y

2

9x o,

(an/D) 2x. 9.

and at the point (4 , 6)

(av/er ) 9

T herefore, we have

i z (y 6) 9 (x 4) for the tangent,

6) 1 2 (x 4 ) for the normal .

EXAMPLES.

Find the equations o f the tangent and normal at the point (x ’

, y)on each of the fo l lowing curves

I_ b?

I 2 I

I . 22y2+ 222 2 =

r r x /a r xx

2 . y2 = 2px .

2—r

'=

3. x2

+312 = R 2

.

4. y2

2Rx

I

5. y a log secy y

+y2/3 a

z/s

J/

29 49+

8 . x 7‘

VCI’

S“ ly

; Vz ry_ y2 .

r—r

’

(er/222 32

D'

e(x

x’

x x’

x R “.

r

r’

(1?

co t

y—y’

tan

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246 D I F FEREN T I A L CAL CUL US .

2 7 . F ind the ang le o f intersection made by the two curves

x2

+y2

( 2a )2

,and y

2x2

a2 Ans. tan

2 8 . F ind the ang le at wh ich the curve (y2

x?)2

z a2 (x2 —

y2 )cuts X .

29 . F ind that po int on an e l lipse at which the ang le be tween the

norma l and diame ter is a maximum. Ans . b/V2) .

30 . F ind the po int on a parabo la whe re the ang le be tween a s traightl ine to the ver tex and the curve is a maximum. Ans . x

’

z p.

T o fi nd tbe equation of tbe tang ent toy 2 f(x ) w/zic/i maleesa g iven ang le, qb, w itb X :

Let a 7: tan cb; then dy’

/dx’

a,wh ich combined with

y: f (x ) wil l determine for the point of tangency.

T hus,to find the equation of the tangent to y

“

6x

which makes an angle of 45°

with X :

dyl

/ [l 3/J),

y,

3 .

Subst i tuting iny2

6x,we have x

’and the required

equation is y 3 x

EXAMPLES.

I . Find the equation of the tangent to y a ( c x ) 2 wh ich isparal le l to X . Ans . y z a

,x 2

2 . F ind the equation of the tangent to x”

+y2

r2wh ich is

paral le l to y 2x 7. Ans . y : l: r /V5 2 (x ZF

3 . F ind the equa tion of the normal to y2/4 x2

/9 1 wh ich isparal lel to 2x —

y z 3 . Ans . y 2x 43 2 .

4 . F ind the equation of a tang ent to 1 8y2

8x 2

72 wh ich isperpendicular to the right l ine pass ing th rough the pos itive ends o f

the axes . Ans . 2y 2 3x2F

5 . F ind the equations of the tangents to 9y2

2 5x”

2 2 5

wh ich makes an ang le of 60°

w ith the transverse axis.

6 . F ind the po ints on y z x3

3x2

24x 85 where the tan

g ents are para l le l to X . Ans . (4 , 5 ) and 2 ,

7 . F ind the equation of the perpendicular through the focus o f a

parabo la to a tangent at (x’

, Ans . y z y’

(x

TAN GEN T S AND N O RMAL S . 247

8 . y9

4ax find the locus o f the points of intersection of

tangents , and perpendiculars upon them from the focus .

Ans . x a ; y o/o .

9 . y 2 (x I ) 4/x 2 ; find the ang le at which the curve

cuts X ,and the point where the tangent is perpendicular to X .

Ans. T an" 1 2 ; ( 2 ,

1 0 . Show that norma l to y2= 4ax is tangent toy2= 4(x

With oblique axes,inc lined at an anglefl,

dy’

/dx’repre

sents the ratio of the sines of the angles which the tangentand curve at (x

’

, y’

) make with the axes . T he equation of

the tangent remains unchanged in form,wh ile that of the

normal becomes

y—y' = + cosfl

149 . Let T ill,PM

,and NM ,

respectively, be the tan

gent,ordinate, and nor

mal at any pomt M

whose coordinates are

(x'

,

L et O s and O R r—‘g

be perpendiculars fromthe origin to the tan

gent and normal respec tively.

L et X TM 2 gb. T hen, since dy’

/dx’

tan (b, we have

Subtangent PT y’cot y

’dx’/dy

’

Subnormal PN y’tan (b y

’dy

’

/dx’

Normal = NM = y’sec (15 = y

’ V1 tan?

cb

V1

T odhunter ’s Calculus , page 283.

248 D I F FEREN T I AL CAL CUL US .

Al so,from figure, or equations ( 1 ) and ( 2 )

O T 7: I ntercept of tangent on X 2 x’

O S Interc ept of tangent on Y : y’

ON Intercept of normal on X x’

I I ence,

p CO Perpendicular to tangent O S CO S C15

— O S

9 O R Perpendicular to normal ON cos

x’

x’dx

’

+ y’dy

’

V1 —l d”

dy

T o apply these formulas , ob tain general expression for

dy’

/dx’from the equation of the curve

,and substitute

values of x ’andy

’.

EXAMPLES .

I . .y2

/b2

x2

/a2

1 .

Sub t (x a2

)/x’

Subn bzx’

/a2

.

I f a b,we have for the circle r a

, p

Subn

x2

/a2

I . p

(x'2 —l Subn bix

'

/a9.

I f a : b, we have for an equilateral hyperbola p aQ/Vx ’2

+ y

Subt Subn x'

.

y?

2px rfix

’.

Subt (2px’

r x ) ; Subn p rzx’

,

Tan

Nor Vs " 2 ’2

(p rix'

)2

.

I f r 2 z : 0 , we have , for the parabo la referred to axis and tangent at

vertex,

Subn p, Tan y'

t/pi

+ r /z>.

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2 50 D I F FEREN T IAL CA L CUL US .

8 . y3

ax2

x .

I ntercept of tan on Y

9 . y2 = 2x . x

’ = 8 , Tan

1 0 . x 2 sec 2y. p [(x2

sec"

4x9( x

2

Sub t x’

( 2a Subu 2 : x ( 3a 2a

Nor z y Subu

Tan =y'9/My c

“,

Sub t cy'

/4/y c2

I 3. p z c4/Vx

’6+y

14 . x3

3axy+y3

0

Eq. of tan r ar'

l/(y 2 x’

)

Subt (y x

1 5. x2/3 +y

2/3

Eq. of tan

I ntercept on X i/a x , on Y :

Leng th of tan between axes

Area between axes and tan

t i/ax'

r’

16 . y cc Subr a, Subn = y

’2a .

2a2 log x . Subn

1 8. xy2

2 a2

(a Subt 2 (ax’

19 3ay2

a 2x3 Subn

20 y2 = - 1 2 : 4 , Subt = 3

TAN GEN T S AND 1VO RMAL S . 2 8 1

PO LAR CO O RD INAT ES .

1 50 . Polar Tangent , Subtangent , Normal , and Sub

normal .

L et PM r,corresponding to the point of tangencyM

of any tangent as TM F rom the pole P draw PT per

pendicular to PM . Draw PO and MR perpendicular toTM .

PT,the part of the perpendicular to r

, from P to i ts

intersec tion with the corresponding tangent TM is the

polar subtangent corresponding to M . . PR is the polar

subnormal , and RM is the normal .

L et gb 2 : PM T,the angle made by r with TM

T hen 55. s 70 )

tan go r ) ] rdH/dr’

.

F rom T rigonometry,

cb tan qb/ tan2

(b, cos 9b VI / ( I —l tan“

Q5) .

T herefore, since t/dr r°dd

2

ds,

gb rdd/ds , cos 05 dr/ds , cot 9b dr/rdb‘.

2 52 D I F FEREN T IAL CAL CUL US .

Hence,

P T Subt r tan cb r’dd/dr ,

PR Subn r cot o

RM Nor i/r 27./do,

PQ p perpendicular to tangent

z 2

4719 a

’r sm to r s

VON/dd)”

r“

Since ds/dB is assumed to be pos itive , positive values only ofp are

considered .

I /p2

I /r2

Putting 1 r u,from which dr 2 2 : r

°

du2

, we have

I /p2

niz

PN g perpendicular to normal 2 r cos gb

i/ r2

(rdft/dr )2 p

Since ds rdr/r CO S 2 W7

77?»

r /g z r/ Vr fl m—pi

; ds = rdr/Vr 2 —p2.

T herefore ride prdr/ Vr

”

p”

.

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2 54 D IFFERENTIAL CALCUL US .

I S p z za r

a (ff2

I

I

Polar Equat ion

16 . r :

nates of any other point of

NM , as N . LetPMNn ; whence tan

T riangle PMN gives

sin MNP sin (9) Q5]sin PMN Cb

SH] ( 9’ 0) cot cos (9

’ 0)

sin ( 9’

(9) cos (6’

Putting r/r“

2 a and I /r'

a,whence

we have, dividing both members of ( I ) by r'

,

a’cos (0

’9) sin (0

’ H)

for the tangent .

O therwise it may be obtained from eq. ( 1 ) by

changing the reference to a system o f polar c oordinates .

L et r'and 6

”be the polar coordinates of the point of

tangency and r and 6 those of all other points ofthe tangent . T aking the pole at the origin and the refer

ence line to coinc ide with the axis of X, we have, as in

a Tangent— Let PM r

’and

XPM : H’,be the coordi

nates of M,the point of

tangency of any tangent,as

NM,and let PN : r

,and

XPN : (9 be the coordi

TANGENTS AND N O RM / 11, 5 .2 55

x z : r cos 6,

r'cos

y z : 7 sin 6, y’ z 7

'sin 6

'

r’cos 6

’a’6’

r'sin 6

'a’6"

Substituting in ( I ) we maywrite

sin cos 6’

o I o If sm 6— 7

‘

srn 6cos

-r’5 1116

’ (r cos 6— 7'cos

VVhence

s in 6 cos 6’

7 s in 6’cos 6

’r r

’

s in 6 s in 6’

r sin2 6’

I I

s in 6’

cos 6 r r'

cos 6 cos 6’

r s in 6’cos 6

’r cos

2

(s in 6 cos 6’

s in 6’

cos 6) r r’

(sin 6 sin 6’

cos 6 cos r

or,changing signs of terms ,

(sin 6’cos 6 sin 6 cos r r

’

(cos 6’cos 6 s in 6

’sin 6) r

sin ( 6’

cos ( 6’ 6) r

’

/r

Putting 2 a,

and r/r’ z a whence,

we have,dividing both members of ( I ) by r

’

,

a a’cos (6

’ 6) sin (6’

R epresenting the polar coordinates of all points of a normal exc ept the point of tangency by r and 6.

deduce in a s imilar manner from ( 2 ) 1 48)

a z a'

cos (6 —a sin (6

for the normal.

z 56 D I FFEREN T IAL CAL CUL US .

CHAPTER XI I I .

ASYMPT O TES .

RECTANGULAR CO O RD INATES .

1 52 . An Asymptote to a curve is a a’efim

’

z‘

e lz'

mzl z'

flg posi

tion of a tangent to the curve,under the law that the point

of tangency recedes from the origin without limit.Unless otherwise mentioned rec tilinear asymptotes only

wil l be considered.

Asymptotes Parallel to tire Coora’z’

aaz‘

e Axes.

153 . I f in the equation of a curve —> cc as jaw ed,then

z oo and x r z a is the equation of an asymp

tore parallel to Y. I f,otherwise

, y»—>a as x»—> oo then

y 2 : a is an asymptote parallel to X .

EXAMPLES.

I . ya

JCS/(2a

y»» 00 as x )»—) 2a .

Hence x 2a is an asymptote.

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D I F FEREN T I AL CAL CUL US .

1 5. x9y+ a

9y= a

3. y = 0 .

—y /y. y z o .

—x9) . x z i a , y :

1 8 . n‘”

a’(x 2—y9) . y i a .

19 . y(a9+ x

9) a

9(a x ) . y z o .

2 0 . y r ". y 0 .

2 1 . y =: a - c)2

y z a , x :

2 2 . a7y 96

931 a

‘3x . y O .

2 3 gas —

o

24 y(a9

y z o, x z zt a

2 5. 31Q

62496

3; if?“ 3a)

x = 2a , x o , y .L a .

ASYMPT O TES O BL IQUE T O T HE CO O RD INATE AXES .

1 54. L et TM be a tangent and SP an asymptote to any

plane curve, as NM .

T he equation of TM is 1 48)

y—y

'

(x

and its intercepts are

0 1?3 : x’

O T : y'

A S YMPT O TES .2 59

As the point of tangencyM recedes from the origin

without limit , the tangent TM approaches the asymptote

SP,and the intercepts O R and GT approach 0 5 and O F

,

respec tively, as limits .Assuming that in the equation of the curvey

’00 as

x’

so and plac ing

tan O SP z K, OP Y

, ,and 0 5 2

- 250 ,

we have, omitting the dashes ,

any two of wh ich wil l serve to determine in regard to an

asymptote.

I f K = o or 00,or if V

, o oo,or if either lf

oor

X0is imaginary, th ere is no corresponding real asymptote

w ith in a finite dis tance from the origin otherwise there isand its equation is

y = Kx + Yo ,

or x z y/K -l—X O ,

W& +fln L

I f either Y,or X

0is z ero

,the corresponding asymptote

passes through the origin and its equation is

y rs - [ f x .

E X A M PL E S .

x y, dy/a’x p/y. I f 0 .

260 D I F FEREN T IAL CAL CUL US .

Hence , a parabo la has no asymptote .

x

x : l: w. y

=F w . dy/a’x (4dx sash/31

6.

Yo

Hence, y x 2a/3 is an asymptote .

3. y3

1 0

a’

y/a’x z — x

fi/yz .

[x yam-2

L [ I o/a l e

“2

/J’2

100 [( ro {I O/f I ] . I

Hence , y x is an asymptote .

2mx a

’

y/a’

x 2“

( p 7 2x)/ 1/ 2px+ 73m .

PX 0 2

p/x r

r2x2

: l: 4/ 21W r“

When 0 , Yo is imag ina ry. Hence , an e l l ipse has no asymp

0 , both resul ts are unlimited,showing that a parabola

has no asympto te .

When 72

0 , bo th resul ts are finite , and since Yo has two value san hype rbo la has two asympto tes

Putting p 62/a and 72

2 : a‘z

/a2

,we have

X o— ( l , Yo z i b.

cc y. fly/4x (w sofa/aw.

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262 D I F FEREN T I AL CAL CUL US .

1 5 . ys

z a x

16. y3: 6x9 + x . y z x

x

1 8 . F ind the pe rpend icular d istance from the focus of an hyperbolato an asympto te Ans . Semi-conjugate axis .

19 . F ind a tangent to a g iven curve wh ich forms with the coordi

nate axes a maximum or a minimum triang le .

Le t u/z area of triang le . Then

o and y'

0 are exceptiona l cases . Hence, in

a’u

a’

0 requi res 96 +y'

2: O , Whence y’ x'

x

T here fo re,in general , when the triang le is a maximum or a min

general ,

imum, the portion o f the tangent between the axes IS bisected at the

point of tangency.

Let the equation of the curve be x2+y

22 : R 2

Then

0 I

Hence , +y’z : 0 g ives y

'

x

and u/z [82 is a minimum.

155 . T he equation of any algebraic curve of the nth

degree may be arranged in sets of homogeneous terms and

written in the form

0

T hus,having

AS YMPT O TES . 263

(y3

xg

y) 2y2

+ x )

and x”

(y3

/x3 —

y/x ) x2

( 2y2

/x2

) x (4y/x 1 ) z o .

Let y mac c be the equation of a right line ; combineit with ( 1 ) by subs tituting mx c fory, giving

x"f .(m c/x ) oa

r-m(m c/x )

c/x)

the n roots of which are the absc issas of the a points common to the two lines .By c ausing m and c to vary, the righ t line may be made

to have any position in the plane X VDeveloping each term of ( 2 ) by T aylor

’

s formula,we

x

(m) etc . o. (3)

Any set of values of m and c wh ich satisfy the two equa

f .(m) 0. o

, (5)

cause the two terms in (3) containing the highest powers ofx to disappear, and the resulting equation to have two ihfinite roots . T hat is

,two of the points common to the

righ t line and curve are thus made to coincide at an infinitedistance from the origin

,and the corresponding righ t line

consequently is an asymptote.

Let m m, ,

m, be the 72 roots of the cor

responding values of c from (5) wil l beetc .

,and

y 2

D I F FEREN T I AL CA LCUL US .

.yz 772 x

etc .

y mflx

are the equations of the a asymptotes .H ence we have the fol lowing ruleI n Me egaaz

‘

z’

oa of Me came szMstz'

z‘az

‘

e mx e f or y,pZaee

Me eoefi ez’

enfs of Me two hzg/zesfpower s of x equal to z er o,

ana’

fi na’eor r espaaa

’mg sel‘s of va lues f or m ana

’e. Ear/z set

w ill determine an asyfi zpz‘

oz‘e r eal or z

'

mag z'

flary.

T hus,having x

3

2 x2

) » 2 x2

8} z 0,then

2 962

02296 e) 8 (mx e) z

and ( 1 2 770963

( 2 5 2 )ac2

877m 85 2 o .

1— 2m z o

,

— 2 e 2 0, give m z r/ z , e — 1 .

Hence, y x/ z 1 is an asymptote.

Having any2

x3

ay2

ax2

0,then

x (mx e)2

a (mx e)2

aoa2

and (m2

I )ac3

( 2 7215 am2

a ) .ac2

etc . o .

1 : 0,

2me

Hence, yz : l: (x a ) are asymptotes .

EXAMPLES.

CURVES. ASYMPTO TES .

I . y9(x - za ) z 2 2a , y : : l:

— I )2

x= x + 4 .

3 . x4 —

y‘2 a xy. y z x

, y

4 . x4 — x

2

y9 —l—a

9x9—l—a

4z o . x _ —

y,x z y,

x 2 0 .

5. x3

+y3+ 3a

9x + 36

2/+ c3 o . x

6 . x5

+y5

sax‘zyg. x +y a .

7. xy4 — ay

4 —l—x9ysz b5 z a

, y- x — a

, y z o.

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2 66 D I F FEREN T I AL CAL CUL US .

equal to z ero, determines two asymptotes real or imaginaryparal lel to X .

In a similar manner asymptotes paral lel to Y may be

determined. T hus , having x2

)?—x

°

y— xy

° x +y+ 1

in which x x”

, y“

, y”

are mis sing, y” —

y is the coeffic ientof x

2

and x as that of y“

.

Hence, y“

y o,x

'2

x 0 , give the asymptotes

x : and x :

EXAMPLES.

CURVES. ASYMPTO TES.I . y

’x —ay

gx9

axz 63 = 0 . x = a , y = x —l—a , y

- x—a.

2 . xy3+ x

9

y—a

4z o. y z o

,x 2 0 .

3 xi

y’

awe”+xi

) x ri: a . y : l: a .

4 . x2

y2 — a 2y

2 —x = o. x = zt a , y= o.

5. ay2—xy

2 —x9 = x z a .

Polar C0 0

157. I f for anyfinite value of 6,designated byat, r is infi

nite, and the corresponding subtangent PT’

designated by is finite,the curve has an asymptote paral

lel to the radius vec tor corresponding to a .

AS YMPT O TES .

Knowing a and S , the asymptote may be constructed as

fol lowsT hrough P draw PM ’

,making the angle XPM ’

a ; it

wil l be the direc tion of the corresponding infinite radius

vec tor. Draw PT ’

perpendicular to PM’

, and lay off

PT ’S

T’M ’

drawn paral lel to PM’is the corresponding asymp

tote.

Designate the angle XPT ’a by6 ,

and let r

and 6 be the polar coordinates of all points of Me asymptote. T hen from the triangle T ’PE we have

7 S/sin (a ( 1 )

Hence,knowing a and S

,the polar equation of the

asymptote is known.

I n some cases a and S maybe readily found. T hus, havingthe curve r a6/sin 6, 6 7r a gives r 00

, and

Hence,7 sin 6 2 mt is the polar equation of the asymptoz

‘

e.

Positive values of S are laid

off in the direc t ion

- 7t/z a

and negative values in the

direc tion

6 1 80°

T hus,havmg r

6= o a'

gives r : oo, and S a .

and direc tion of S is rf/ z . r sin 6 a is the equation of

the asymptote.

268 D I FFEREN T I AL CAL CUL (15 .

Having r z a a/6,a gives r z

S

Hence, r a/sin 6 is an

x asymptote .

As —> oo,

—>a, and the

c irc le I?z a is cal led an

asympz‘az‘z'

e circle

58. In general , let the polar equation of a curve be

r”

f n (0) W‘ I

f naw) ff ifi) 0 ( I )

Placing a 1/r , c learlng of fractions , and arranging withrespec t to a

, we have

zen

”fi t- 1M) 0

F or any point corresponding to r z 00 a must be z ero,and the roots of 0 are the values of 6 corresponding to 7

’00

Differentiating ( 2 ) with respect to 6, making a o and

6 a,we have

K“ ) + f

T herefore

d” a=o

Hence ( 1 ) the polar equation of the asymptote is

r sm (a

When a

7 sin (a 6)

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2 70 D I F FEREN T I AL CAL CUL US .

1 0 . r d ( SCC 6 : l: tan 7 cos 6 2a .

I I . 7 : 2a sin 6 tan 6. e cos 6= 2a .

1 2 . r : a sec 7716+ 6 tan m6.

1 3. 22cos 6 z : a

2sin 36.

1 4 . r I ) . a/[2 s in ( 1 80°

/ 7z

1 5 . r z a/cos 6+ 6. f z a/s in (90°

F ind the asymptotic circles of the fo l lowing curves

circle r 2a .

1 7 . r a69/(62

I ) . circle 7

7 s in [7z/2m —6] (a 6)/m.

r cos 6 = 0 .

D I RECT I ON O F CUR VA T URE. 2 7 1

CHAPTER XIV.

DIRECT I O N O F CURVATURE. S INGULAR PO INT

D I RECT I O N O F CURVAT URE.

1 59 . Let y = f (x ) be the equation of any plane curve,

and let 6 represent the anglemade by any tangent to the

curve with X .

T hen a’

y/a’

x z tan 6,

and (a’

tan

When aw

y/a’oc2is positive

tan 6 increases w ith x and,as may be seen from any

figure , the concave side is above the curve and Me a’

z’

r ee

z’

z'

oa of curvature is upwara.

When (fly/doc2

is nega tive, tan 6 is a dec reasing func tionof x ,

and Me of euroa ta re z

'

s a’owmoara

’

.

T he direc t ion of curva ture a t a po int is a long the normal towa rdsthe concave side .

2 72 D I F FEREN T IAL CAL CUL US .

I n a corresponding manner it maybe shown that positivevalues of a

mps/ofy

2indicate concavity towards the righ t

,and

negative values concavity towards the left.

T o illustrate, let y r : 2 (x

T hen amiy/a

’oc2

6 (x

(fix/fix”

9 6"

H ence,when x 2

,the direc tion

of curvature is downward and to

the righ t . When x 2,the curvature is upward and to

the left.Show that the direc tion of curvature ofy e

” is alwaysupward and to the left.160 . Polar System. R epresenting byp the perpendic

ular distance PQ from the po le P to the tangent at any

point M on a curve, it is apparent (when the tangent doesnot pas s through P) that the curve is concave towards

the pole when p is an inc reasing func tion of r,and that it

curves away from the pole whenp is a decreasing func tion

H ence, positive values o f a’

p/a’

r indicate concavity towards

the pole and nega tive values the reverse .

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2 74 D I FFEREN T I AL CAL CUL US .

SINGULAR PO INT S .

Points of any curve which,independent of coordinates ,

possess some unusual property are as a c lass called singular

points.

161 . APoint of Inflex ion is one at which , as the var iableincreases, a curve change s its concavity from one side of

the curve to the other . T he cor

responding tangent intersec ts thecurve

,and the direc tion of cur

vature is reversed.

L ety f oe be the equation of

a curve . I t fol lows from 1 59

that c is the abscis sa of a point of inflexion, if as x increases

f”x changes its sign in passing through f c.

T he real roots of the equations

x 2 0 and f

are therefore cr itical values , and may be tested bya methodsimilar to that described in 1 35 in the case of f x o

andf’x 00

Hence, Me g eaeral met/tori for any cr itical value as c is to

determine whether, as it vanishes from any definite value,

f”

(e 12) and f (c + fi ) ultimately have and retain dif

ferent signs .Whenf (x ) f (x ) , etc . , are continuous for values of x

adj acent to cr itical values , those derived from f (x ) 0

may be examined, as in the case of f’

2 o as

fo llows :H aving f (c) o

,saostitate cf or x inf (x ) , f (x ) , etc.

,

in order,until a result oMer Man o i s ootaiaea’. If Me cor

responding a’er i . a tive is of an odd oro

’er

, c isMe aéscissa of a

pot/i t of iaflexiozz, ot/zer zoise not.

S IN GULAR PO IN T S . 2 75

If a result 00 is ol tainea’

,Me general IneMoa’Moula

’

oe enz

ployea’

.

When f x is complex the general method is usual lypreferable.

I t should be observed that at a point of infiexion the

s lope of a curve is either a maxnnum or a minimum,and

that the determination of such pomts I S the same as findingthose at a maximum or a minimum.

EXAMPLES.

I . x3

2x9y 2 .x

2 8y .

a’

y x (x3

1 2x 16)

dx 2 (x2

a’ Qy 4(x

3 6x ”

( 2962 (x,

4)0 g ives

x = 2 . x V3) 4 2 + V3) = 7.s

Applying the me thod 1 38 , we have

0 . 54llx 0 -2

[x l x 0 -9 .

x = .54

7 5

[x 2 ] [x . 0 0 13 .

Hence, and are po ints

of inflexion.

2 . y 2 3(x

I

I

x

(6

2 5(x

0 has no real

m g00 g lVCS x 2 .

is negative, (a’ 2 is positive.

Hence, x 2 =y is a'

point of inflexion.

76 D I F FEREN T I A I ; CA L CUL US .

m m.

4 ai

r (x 6 9

5- r (x- I ) (x — 3) x :

6 x z o

y2

(x — a ) 2 x ‘ i' — 2a .

x :

y 2 x2log ( I x ) .

oc 2 4a2

( 2a y) .

y el /x

x3

+ y3 x 2 2 0

14 . y 2 x2tan x .

1 5. y 2 a sin 0,art

, etc.

1 6 . y 2 (a x )5/3

ax .

1 7. y 2 x2

x5/9.

I 8 . y 2 ex x 8.

19 o .r 2 i a

y :

2 I —a (3

2 ~ 1

x3

2 3 r—l—(flv

2 5. fla‘ 174 ) 2 x (x

26 . y 2 azx/(x

'za2

) . x 2 0, x

2 7 . y 2 xQ

/a xs

y/aQ

.

2 8 . y 2 x cos (oi/a)—36oc

2 — x x :

x r:

3 1 . axy 2 x 2

ai’

y 2 ova

/3 ax?

xy 2 a2 log 2 ae

3/9.

(ay am)2

2 ox s. x 2 (go/64.

a3

y 2 x2

(a2 — x ) . zh a/ya.

3/ (ao2 Jr a

?

) 2 a2(a

ai

r“

a 2 (r?

r?)

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2 78 D I F FEREN T I A I CAL CUL US .

163 . A Multiple Point is a point common to two or morebranches of a curve, and is double or tr iple, etc .

,according

to the number of branches . T hey are c lassed into Points

of I ntersection, S/i ooting Points, and Points of T ang ency.

A Multiple Point of Inter

section is one through whichthe branches pass and havedifferent tangents . a I S a doubleand o is a triple pomt of intersec tion. A double point of ih

tersec tion is also called a node.

A S/zooting Point is a multiple

point at wh ich the branches terminate with differenttangents.

A Salient Point a double shooting

AMultipleBhint of Tangency*

is one a t wh ich the branches havea common tangent.

A Cusp is a doub le mult iple point of tangency at whichboth branches terminate.

When in the vic inity of a cusp the branch es are on

oppos ite sides of the common tangent (F ig. c) , it is of the

Sometimes cal led points of osculation or double cusps.

S I N GULAR PO I N T S . 2 79

fir st species, or'

a éeratoia’

(horn) cusp; when on the same

side (F ig. it is of the second species, or a ramp/zoia’

(beak ) cusp.

A Conjugate Point is an isolated real point of a curve .

T hus,the origin x 2 is a real point of the curve

whose equation is y 2 : l: xMx 2,bUt J” is imaginary for

i x 2 . H ence the origin is a conjugate point .Let y 2 E(x ) be the equation of any curve.

F rom 1 24,

F (c I.) F (c) F (at / 2 etc .

When (c, a’

) is a conjugate point, F (c it) is imaginaryfor values of 12 near z ero, wh ile F (c) and ii are real . Hence,one or more of the expressions etc . ,

must beimaginary. T h is important charac teris tic of a conjugate

point is frequently used in tes ting c ritical points . T hus,

(c, a’

) is a conjugate point provided F (c) is real,and E ”

(c)is imaginary for any entire value of n.

In the example above we find E’

(0 ) 2 : l: V 2 .

Since the ordinates of points of a curve adj acent to a

conj ugate point are imaginary, the number of such ordinatesfor each pomt I S even. I t fol lows that a conjugate point isa mul tiple point in the immediate v10 1n1ty of which the

branches are imaginary. T he tangents corresponding to a

conjugate point may be real or imaginary, fcoincident Orseparate.

2 80 D I FFEREN T I AL CAL CUL US .

Having the equation of any curve with two or more

branches,if eiMer variable

,as y,

has but one real value,a’

corresponding to any real value of the other, as x 2 c,

(c, a’

) is a c ritical point.I f d, has two or more values

, (c, ii ) is a multiple

point of intersection or tangency according as the severalvalues of are unequal or equal .When the values ofy for points adj acent to and on both

sides of (c, cl ) are imaginary, (c, a’

) is a conjugate point.

EXAMPLES.

—4) Vx —

2 .

oc 2 , y is imaginary. oc 2, y 3,

2 x has two real values . oc z 4 , y 3.

x 4 , J! has two real values .

Hence, ( 2 , 3) and (4, 3) are critical .

a’

y/a’x 2 : l: Vac 2 i (se Vac

2 oo ( 2 , 3) is not a multiple point.

(fly/479 62 .3)

I i V;

H ence , (4, 3) is a double multiple point of intersection.

2 . y : V3.

x 2 0 2 y. y is imaginary when x < c,or 0 < x < 2 .

Hence, the origin is a conjugate point at which we have

(ofy/doc) 0 2 6.

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2 82 D I F FEREN T I AL CAL CUL US .

H ence the origin is a double point of tangency.

7 y - (x

a’y/a

’x 2 : l: 3 Voc

x < 2, y is imaginary.

cc> 2,y has double values .

2M” = 3, and 312 0 .

Hence, ( 2 , 3) is a cusp.

T he branches are on opposite sides of the common tan

genty 23, and the cusp is of the first Spec ies .

8 . ( 2x -

y>2

(x groc < 3 makes y imaginary, x 2

3 givesy2 6, and x > 3 g ives

real double values toy.

2 ziz 0 . Hence , (3 , 6) is a cusp.

Sincey 2 2x i (oc the branches are on oppositesides of the common tangenty 2 2x ,

and the cusp I S of the

first species.A charac teristic of a cusp of thefirst species is a change

in direct-ton of curvature from one side of the common tans

gent to the other ; while at one of the secoua7species the

direc tion of curvature remains upon the same side of the

common tangent . Hence,different signs for cl

”

y/a’x2

por

responding to the two real values of y in the immediatevic inity of a cusp indicate the fi rs t species , and like signsthe second Spec ies .

T hus,in example has val

ues with different signs .In some ca ses it is preferab le to considery as the inde

pendent variab le.

S IN GULAA PO IN T S . 2 83

9 . y 2 3 (x y has but one

value for each value of x . x 2 2 3.

cly/a’oc 2 2 /[3 (x is negative when

x > 2,is 00 when oc 2 2

,and is positive

when oc < 2 . ( 2 , 3) is evidentlya cusp of

the first species as in figure.

So lving with respec t to x, y

we have

Z i U’ ( It 3 ‘

U— S/Zhaa i 0

y < 3, x is imaginary. y 23, oc 2 y> 3 gives doub le

value for x . Hence, (3, 2 ) is a cusp.

has values with different signs ,

therefore the cusp is of the first species .Points corresponding to maximum or minimum ordinates

at which riy/a’

u 2 00 are cusps .1 0 . y 2 x

2

: l: x56

.

2x i 5.9c3/2/2 .

2 : l: 1 5x1/2/4.

0 2 y. x < 0, y 1S 1maginary.

oc> 0, y has two real values.

i 0 . Hence, the origin is a cusp at whichthe axis X is the common tangent .

Fer'

o < x < 1 both values of y are positive ; therefore the

cusp i s of the second spec ies . T his is al so indicated by thefac t that both values of are positive.

Examine the following curves for multiple points :

1 1 . y?2 x

3/ (2a (o, o) is a cusp of I st species.

(y x )?2 (o, 0 ) is a cusp.

16 . x2/3+y

2/3 2

1 7. 3x” 2 x4

1 8 . (y2

a9)3 2 x

‘

(2x

2 7. (y 2 (se a )5

2 8 . (xy+ 1 )9+(x 1 )3(x

29 . y?’

x3 2

D I F FEREN TIAL CAL CUL US .

2 o 2 y is a conjugate point.

(3, is a conjugate point.

(0 , o) is a doub le point of intersection.

(0 , :i: a ) and ( ziz a ,0 ) are cusps

of I st Species .

(o, o) is a double point of tan

geney.

3a/2 , t o ) are cusps of I st

species .

(0 , o) is a doub le point of tan

gency.

(o, o) is a doub le point of inter

section.

(a , a) is a cusp of I st species.

(0 , o) is a cusp of I st species.

(0 , a ) is a cusp of 2d species.

(0 , o) is a double point of inter

section.

(a , o ca”

) is a cusp of 2d species .

(0 , o) is a double point of inter

section.

(a , a) is a cusp of I st species .

( I , 1 ) is a cusp.

(0 , o) is a cusp of I st Species.

164 . L et u 2 f (x , y) 2 0 be the equat lon in a rationalintegral form of any algebraic curve, then ( 2 )

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2 86 D I F FEREN T IAL CAL CUL US .

Hence, the origin is critical . (3) gives :E1

“r has double values for + 1 > x 1

,henc e the origin 18

a double multiple point.

of intersec tion.

2 . u 2 x4

+ x2

y—y32 o

,

alt/ax 2 4x3

2 99 ) 2 0,give oc 2 0 2 y.

O il/OJ) 2 x2

2 p ,

aha/ax”2 1 2x a

Q

u/axay 2 2x,

a n/ax32 2 4x , a

su/ao

f

ay 2 2, a

a

u/ax ay22 0

,

F rom 2 0,and

'

i 1 . Hence, the origin isa triple point .

( 46 3x )”

(x give x 2

oe/ax 2 4y+ 2 4x 6 = 0

ate/ax 322 2 4x 0.

232u/ax

'2

3x 2 4 , ay. 2 . 2 4, afiu/8y

2'2 32 .

II ence,

I : [3x 2 V(x

x is imaginary. x 2 , 3) has double valuesSince 96 2 gives - one value o f y g reater and the otherless than 3x/4 , the x two"branches are on O pposite sides ofthe common tangent y 2

3x/4 . Hence ( 2 , is a cusp

of the first spec ies .

4 . u 2 y“

- x3 — a

2

x 2 0 .

“

3x2 —

4ax—

a22 o.

ay = o .

S IN GULA R PO IN T S . 2 87

Hence,

a,0 ) is c ritical . F rom ( 1

3962

+ 44 96 a

6x 4a

- a , 0) 2 a’

y/a’x

H enc e, a

,0 ) is a conjugate point .

5 . u 2 y2

-2 oc

2

y— oc

4

_y+ 2x42 0 .

alt/ax 4xy 4x3

) ; 8x32 0 .

alt/8) ; 2 2y 2oc2

x42 0 .

H ence x 2 0 2 y is a c ritic al point.F rom the equation of the curve,

y (or

x7 2 ) d: at v 4 4x

2

and is imaginary when x is near z ero.

a conjugate point .

Examine the fo llow ing curves for mul tiple p6 . x

33axy+y

32 o . (o, o) is a

section.

Hence the origin is

oints

doub le point of inter

7 . x4—zay

3~—3a

i’

y2— 2a2x

2+a

42 0 . (o ,

-a ) and ( :l: a , 0 ) are doub lepo ints o f intersection.

8 : x4

2ax2

y any2

+ a 2y2 0 . (o , o) is a cusp of 2d species .

9 . y‘

any2

x4

. ( 0 , o) is a cusp of I s t species .

10 . (ac2

2 a2

(oc2 -

y2

) . (o o) is a doub le po int of inter

1 1 . ay“ One (0 , o) is a conjugate point .

1 2 . ayfl—x

3-1—4ax2—5a x+2a

32 0 . (a , o) is a conjugate point .

1 3 . x 4 axi’

y axy2 my 2 0 . (o,o) is a conjugate po int .

14 . 312 2 x (x + a ) a , o) is a conjugate point.

2 88 D I F FEREN T I AL CAL CUL US .

1 5. o a)?(x rm

16. 36062

a?) 2

I 7. ac2

x9 9 6ax

‘2y a

iyQ

gency.

1 8. mix?2 x 6 (0 0 ) is a double po int of tan

19 . x4

axy‘i2 ay

3. (o , o) is a triple po int and a cusp.

165 . A Terminating Point (point d’

a r ré‘

t ) is one at

which a single branch of a curve terminates .

EXAMPLES.

1 . y 2 or log x .

2 0 2 y. y is real when x 0,and is imaginary

when x 0 . Hence,the origin is a terminating point.

2 . y 2

AS —> o, and as

—> o,

00 . Hence,

the origin is a terminating point for the right-hand branch .

3. xilog 2 +3» 2 ( 0 , 0 ) is a terminating point.

( I , 2 ) is a conjugate point.

(0 , o) is a conjugate point .

(0 , 0 ) is a double point of tan

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290 D I F FEREN T I AL CAL CUL US .

represent the angle which the tangent M T makes with X .

I t determines the direction,with respec t to X

,of the

motion of the generating point at M ,and, regarding 36 as a

function of the length of any varying arc of the curve,as

d 1. 0

0

AM 2 slb mm

1S the rate of curvature ofa’

s-> o

the curve s at M .

168. Rate of Curvature of a Circle at a Point — Let C

be the centre and r the radius of any c irc le. T hen

(i s m-9 0

Hence, in any circ le the rate of curvature is the same at

all points , and at any point is equal to , ,the rec iprocal of itsradius .169 . Circle and Radius of Curvature.

—When the radiusof a varying c irc le decreases continuously, the rate of cur

vature of the c irc le at any point increases continuously.

Hence,a c irc le may always be assumed having at all points

the same rate of curvature as that of anygiven curve at any

assumed point.

Such a c irc le tangent toe-the curve at the point assumed

,

and having the same direc tion of curvature, is cal led Me

CUR VA T URE O F O UR VES . 29 1

circle of curvature of Mepoint, and its radius and c entre are

cal led, respec tively, Me radius and centre of curvature. I t

follows that a radius of curvature is normal to the curve.

Any chord of a circ le of curvature wh ich pas ses throughthe point of tangency is c al led a Mora

7

of curvature.

1 70 . Curvature of a Curve at a Point — R epresentingthe radius of curvature at any point of any curve by p, we

have

dip/2’

s I /fl ( I )

T hat is,the rate of curvature of any curve at any point is

equal to the reciprocal of the corresponding radius of curvature

,and the rates of curvature at different points are in

versely as the corresponding radii of curvature.

corresponding to any point of a curve , mul tipliedby the unit of length of s is 68) the change that the cor

responding value of would undergo were it to retain its

rate at the point over the unit of length of s. I n otherwords , multiplied by the unit of s is the total curvatureof a unit of lengM of Me cor responding circle of curvature,and it is generally cal led the curvature of Me curve at Me

point or the curvature of Me cor responding circle of curvature.

I ts numerical value is the same as that of the correponding

rate of curvature, and for reasons similar to those given in

95 it is generally used instead of the rate.

I t is important not to confound this curvature of a circle,

wh ich measures the so-cal led curvature of Me curve at apoint,

with the total curvature of an are desc ribed in 1 66.

p and s must be expressed in terms of the same unit of

length,and at anypoint where p 2 unit of length the rate

of curvature is unity, and the corresponding curvature of

Me curve is a raa’ian, which is therefore the unit of cur

vature.

292 D I F FEREN T I AL CAL CUL US .

EXPRESS I O NS F O R RAD IUS O F CURVAT URE.

1 7 1 . F rom ( 1 ) (S R [is/deb.

which enables us to determine the rate of curvature at any

point from the equation of the curve in terms of s and 3b.

T hus , having s 2 c tan 5b; for a catenary,

w (CO S.

an as e“

)

172 . Let _y 2 f (x) be the equation of anyplane curve.

Having, as before, AM 2 s, let

PP’u’

oc. T hen

air 2 cos tba’

s,

a’

y 2 sin t/Jcls,

1/2 2 tan

I I ence

up a’xa

’

fy f (x ) 1

(2962

+ at

) ” [ I p’

I :

from which the radius , and therefore the rate of curvature smaybe found from the equation of the curve in rec tangula

coO rdinates.

Adopting the positive value of [ I pwill havethe same sign asf (x ) , which determines the direc tion of

curvature .

Ca l led the intr insic equation of the curve .

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294 D I F FEREN T I AL CAL CUL Us .

Hence, I /p 2 ao/ (a2 a x

2

CNN/ (0 36

+

At the vertices ( t a,0 ) I /p 2 a /o

i

, a maximum.

At the vertices (0 , : l: a) 1/p 2 O/uz

, a minimum.

Hence,at the vertices of the transverse axis the c irc le of

curvature is within the curve,at those of the conjugate axis

it is out side,and at all other points it cuts the el lipse.

Also x 2 x 2 (a’

y=y (ai

136W6‘

In (3) Plum/2: 5

2

(dz

and (4’

whence

52 a VI —e’. Then

I /P T 622

VI

EXAMPLES.

f (x) = 4: p 2 oo.

CUR VA T URE O F CUR VES . 29 5

p = 1 +

px=o2 p one ha lf the parameter.x 300 47 2 F

4. 4x92 64.

From (3) and (4) 1 72 ) we have at ( 2 , V3)

p x 9V3/4.

(agar2 69x? a

“)3/2 (a

‘iy2

(c’oc2

a

a464 ao

x 2 i a , y 2 o and p 2 og/a .

[ 22x (P M VP/2

p?

Hence (Example 3, g at anypoint of a conic, as g iven, p is

equal to the cube of the corresponding normal divided by the squareof half the parameter.

r verS“ 1 (y/r ) Vz rv y

g.

[ I ( fer/y 2 V2 77

Hence (Example 6, g at any point of a cycloid p is equal totwice the corresponding normal.

y 2 0 g ives p o, and y 2 2 r g ives 2 4r.

8. y 2 4 3(x

(x (x 2 )

x (x

(2 , 4) is a point of inflex ion at wh ich p 2 0 . Ex . 2 , S 1 6 1 .

296 D I F FEREN T I AL CAL CUL US .

9 r?

9m x y V425

1 0 . y? 2 8r . (3 2 2 (x +

—

2 .

1 1 .sci

/9 + y2/4 I .

1 2 . xy 2 m. p 2 (a:2

1 3. y a (Wa er r /“V2 . p r

i/a ,

14. 4.

yz/s a

2/3. p i/axy,

x x 3 65572:

1 5. rm? 2 23. p 2 (a

4

16 . y 2 x3

I . aw l/32 00 (point of inflexion) .

1 7. eyx’“ 2 sec p 2 a sec

1 8 . y3

2 6272 x [y4(4x

19 y 2 x 4 42 8 px =0 px =3

2 0 y a f t /a p (cl2

+yil3/2/ar

2 1 rs

aix (2 4 I SM/62 2 2 . 5/ (an

{9 (W4 MPM/64 42 .

gar m . p i (2a soar/2 Vic/a v3.

2 3. x 2 sec

24 . y 2 log .x . p 2 (M42

2 V2.

2 5. y 2 x3

x”

I . Py=min.

2 6 . We V} V5. p a(r V52 7 . ay

"2 2

3. p 2 (4a

2 8 . y?

2 x

.

/(2a p 2 a (8a

29 . a2

y 2 ox s cxz

y. p 0 2 00

30 . y 2 log sec at . p 2 sec at.

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298 [J I FFEREN T IA z: CALCUL [75 .

Putting r 2 I / l l , whence (Zr/cm2 ( r/tf

) and

(Pr/fie?2 and substitut

ing, we have

EXAMPLES.

p (warfa

re/(zan a,"

2 r 2 p 2 7 4/ i—

1 10 g2a .

a s in ” 9. na/z .

2a cos 9 —a . p 2 a (s 4 cos 6 cos

6 cos

2 a ( I 2 g cos e cos

r”cos 26 2 a

?

a( z cos 6 I ) .

7 cos2

2 p.

a sec2

7 2 ate—i

.

2 a9.

1 2 . 7 2 4a sin2

From which and g 1 50 ,

air/(me) 2 cot 2 cot q) .2

2p2 6 (p 2 and cit/J 2 3a’6/2 .

Therefore,

p (ls/dill 2 ( 2/3) r cosec qb 2 (8/3)a sin

p 2 7f (a2

72)3/2/a

3a( I

2 a?cos 29. p 2 a

2

/3r .

1 74 . From 1 72 we have2 cos z/m

’s,

and ti) » 2 sin zpa’s.

D ifferentiating w ithout assuming the independentable

,and writ ing a

’

s/p for cit/2 , we have

d’x 2 cos (paws sin

dfy 2 sin ¢d2

s cos

p

p 2 a (s 4 cos 6 cos

p 2 r 3/2/4/p

p 2 2a sec3

P 4 444

p 7/ (4cz4

CUR VA T URE O F CUR VES . 299

Squaring and adding,we obta1n

(fi x )?

(do)?

(M

wh ich is a general expression for p.

R egarding s as the independent variable d i s 2 o,and ( 1 )

becomes

from wh ich 1/p22

which is a convenient form when x and y are given as funct ions of s .

1 75 . In ( 1 ) x is the independent variable . Sub

stituting (a’x (n -a

’

_y for di

y/ay ,we have 1 1 5,

Exampl e 1 1 )p (aw de w/ (ax d

’

y a (fi x ) ,

in wh ich neither 90 nory is independent. T h is form IS con

venient when x and_y are given as functions of a th ird variable. T hus, having

1 . x d (¢ sin ¢ ) r J’ “ ( I CO S

we have, as in Exampl e 1 2 p 4a 811)

2 6 sin 2B( I cos2 6 cos 6.

y 6 cos Z6( I CO S[J 4 3

p W M am/(m .

2 a ( z cos 6 cos

y 2 a (z s in 6 sin8a sin

— b cos

sin - 9

— b sin

goo D I F FEREN T I AL CALCULUS .

1 76. p in Terms of r and p.— From 1 50 .

p 2 r2/

H ence,

— r”r

Comparing w ith ( 1 )we have

passes through P,and PM 2 7 . T hen

21) cos PMC 2 2p sin OMP 2 zpp/r 2

EXAMPLES.

2 . ar c 2

ynt-H z ampo

c z r/(m I ) .

Let C be the centre , and

CM 2 p, the radius of cur-o

vature corresponding t o 111 .

Let 6 2 MN represent the

chord of curvature wh ich

p 2 r/a , c

p

p 2 6

p a9620

3

p aQ/sr , 0

c 2

0 1M,

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30 2 D I F FEREN T I A L CAL CUL US .

T he m le of curvature of the curve at M is 1 67)

dgb l imitd; 0

Let a , y, and a + da, fl é

‘

fi , y 634 represent theangles wh ich the tangents M T and M ’

T’ make respec

tive ly with the coordinate axes.

From the triangl e EOE’ we have

EE’

/l = 2 sin hence,

2 l imit in I .

The coordinates of E and E are respectively1 cos a,

l cosfl, [ cos y,and 1 cos ( a da ) , Zcos (fl

1 cos (y dy) . Subst ituting in formula

0 mx (y (z

and div iding by l , we have

EE’

/l 2 V—

(Tcos (a da ) cos ad”

[cos (fl dfi ) cosfl]2

[cos (y dy) cos

D ividing by 63, tak ing the limit as 65 o

,whenc e

o‘

ep —> o, substi tuting 6¢ for we have ( I ) and

1 70 for the rate of curvature at M ,

[Viz/J Vd oos a d cosfl

From 88,

cos a: 2 (ix/ds, cosfl2 cos y”2 dz/a

’s.

CUF VA T URE O F CUF VES . 30 3

Differentiating and substitu ting in we have

in wh ich 3 is the independent variable .I n order to obtain a more general expression for I /p,

place 1 1 5)

47 296 ds a

’ 2x — a

’x 1z

’ 2

s dz

y a’s a

7 2

y— a

’

ya’ 2s

ace

dr’ etc"

(dr awx che f

’

s)“

(ds aw

y a’

yams)

2

wh ich may be written

2

a’

se

From 88, do“?2 dx

”dy

zdz

”

.

Whence , dr eams 2 dx d

2x + dyd § 7+ dz d

22 a

T herefore,

V(M ? (da

y)?

(a’

s

wh ich is a general expression for the rate of curvature at

any point of any curve .

I f the curve is of single curvature , its plane may be taken

as that of X V,2 w il l be z ero , and (5) reduces to ( 1 )

30 4 D IFFEREN TIAL CAL CUL US .

CHAPTER XVI .

INVO LUTES AND EVO LUTES .

1 78 . Each po int of any given curve as MM ’M ,has

,

in general,a centre of curvature .

T he locus of the c entre of

1 79 . Coordinates of the CentreC

of Curvature.—L et C be the

centre,and CM 2 [J the radius of curvature corresponding

to M,whose coordinates are T hen

curvature of any given curve iscalled its evolute. T hus

,CC

’

C"

is the evolute of MM 'IV

T he given curve M /W’Mp

is

cal led an involute of its evo

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30 6 DIFFERENTIAL CALCUL US .

I t fol lows that a radius of curvature wh ich is unl imitedin length is an asymptote to the evolute . H ence

,in general

,

the normal at a po int of infi exion is an asymptote to the

evolute .Also

,when in the vicini ty of anypo int a curve is sym

metrical w ith respect to the normal at the po int, the corresponding po int of the evolute is a cu p.

1 8 1 . Squaring both members of and addingeach to each , we have

( 07922

dig/ale”2 H ence, dp : t (3

7

372.

Let 5 represent the length of a varying portion of the

74) [J s ziz a , in wh ich a is a constant .

- s,

—a .

I I ence,

p—p’2 s

a 12 arc C

’C

”

, ( 2 )that is, in general , flze a

’zfi

'

erenee Oez‘

ween any two radii ofcurvature Of an z

’

naO /uz‘e is equal 10 Me are of z‘

fie evolute u

z‘

ween i/ze cor responding eenz‘

res of euraaiure.

Exceptions exist when the arc of the evolute includes a

singular po int or is discontinuous .

L et M

p be the radn of curvaturecorresponding to M ’

and Mrespect ively. Measuring the evolute from C, and denoting the

lengths of the arcs CC '

and C

by s‘and s

,respectively, we have

IN VO L UTES AND EVO L UTES.

Measuring the evolute from C,we have

arc CC’ —

p’ —

/J , or p’2 arc CC

’-l—p ;

arc CC 2 p—p, or /J - arc CC + p.

H ence, a 2 p.

Similarly it maybe shown that the constant a in equat ion

( 1 ) is, in general , equal to the radius of curvature wh ich

passes through the po int of the evolute from wh ich it ismeasured .

I t follows that as a right l ine ro lls tangential lyupon, or as

a string is unwound from, any curve , each po int describesan involute to the given curve as an evolute . H ence

,

wh i l e an involute has but one evolute,each evolute has an

unl imited number of involutes. Any two involutes correSponding to the same evolute are separated by a constantdistance measured along the normals

,and are cal led parallel

curves.

1 82 . In general,at cusps of the fi rst spec ies and at

po ints of inflexion p changes sign. H ence,at such po ints

we have p 2 0, or 00 and the evolutes pass through these

po ints or have infinite branches to wh ich the correspondingnormals are asymptotes.

30 8 DIFFERENTIAL CAL CUL US .

At a cusp of the 2d spec ies p does not change its sign,

and the corresponding point of the evolute w il l, in general,be a po int inflex ion.

Radius Curvature of an Evolute .—T he angl e

M (W) , between any two tangentsas those at M ’

andM is equalto that between the corresponding radii of curvature to the involute and since these radii of

vature of the evolute.

I

curvature are tangents to the

evolute,the angle wh ich they

make w ith each other is equalto C CC

'

,included between

the corresponding radu of cur

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3 10 DIFFERENT I AL CALCUL US ,

Comb ining w ith y?

2 apx , and el iminating x and y

we have

J’2

8 (x

for the evolute of the parabola.

C’CC is the evolute of the parabola C

”0 C

’.

2 0

gives x2 p 2 0 C.

x = 4p :-: OP

y : i pV8= i PCN

for po ints common to the pa=

rabola and its evolute .T he arc VE“ 1 )p.

T ransferring the origin to C,the axes remaining paral lel ,

we have,denoting the new coordinates by x andy,

x = F + n y = fl

H ence,we have 2 8x

3

/ (a7p) for the evolute .branch CC’ belongs to CC” and CC

”to O C

’.

L et r 2 FM ’represent the focal distance of any

as M ’

,and let 1 2 F Y represent the perpendicular

the focus to the tangent TM ’

T hen FM ’F 0

, or 12

2 pr/ z .

H ence,

2 1d! 2 pa’

r/ z , or a’

r/a’i 2

From 1 76, e 2 chord of curvature through F

H ence, e 2 812

/fi 2 Spr/zp 2 47’ 2 4FM

’

T hat is,in a common parabola the chord of curvature

through the focus is equal to four times the focal distanceof the po int of tangency.

I N VO L UTES AND EVO L UTES . 3 1 1

2 . 52x”2

a’

y/a’x 2 b2x/a

2

y, a’fy/a

’ec2 2 F /a

’

ya

.

Subst ituting in ( 2 ) and reduc ing,we have

x z (a? new; (ex/ [e err/3.

(e ewe ; (Ry/[e bur/3

Comb ining w ith a iy’ Fix”2 a

ié”

, we have

(were (we (a? We

for the evolute of the el lipse.

is the evolute of the ell ipse MM’X ; x

gives y 2 (a2

52

)/b 2 CC’"

When e22 (a

252

)/a2 we have a

2and

CC’”

2 b,in wh ich case the vertices C’” and C

’are on

the curve. T hey are w ithout (as in the figure) or w ith inthe ellipse according as e

“is greater or less than

Measuring the evolute from C,we have 1) 2 M C 2

and p’ M ’

C’

az

/i .

Hence , arc CC’ 2

a2

/b b“ a 2 (a3

Axis CC” 2a z 52/a 2 z (az 5

2

)/a .

Axis C’C’” 25 2 (a2

/O zé) 2 2 (a2

3 1 2 DIFFERENT I AL CAL CUL US .

H ence,Me axes of Me ovolaio are

,inverse!) as Me cor

responding axes of Me eZ/ifise.

3 . x 2 r Vary — y2.

a’

y/o’x 2 (ar/y r/y

”.

Substituting in reduc ing and combining w ith thegiven equat ion

,we have

x 2 r vers" 1 y/r ) V

for the evolute of the cyclo id .

Produce B A ,mak ing A O ’

2 B A ar .

OM ’O’O”is the form of an evolute of a cycloid . The

branch O O ’ belongs to O B and O’O ’to B O

T ransferring the origin to O’

, tak ing O’X ’

and O’A as

the new coordinate axes, deno ting the new coordinates byx andy, we have for any po int of the branch O O

’

,as M ’

,

x OA’

O’P’ m»

x, y A O

' P’M ’2 r+y,

wh ich substituted in (a) give

r vers“ 1

[ ( 2 r Vz ry -

y2.

r vers‘ 1

[( z r 2 r vers" 1

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3 14 D I F FEREN T IAL CAL CUL US .

r cos PM C 2 r sin PM s .

H ence,

2 132

r zpp.

Also,

2 TM2

2 r2 —

p"°

and ( I ) , 1 76, 2

From the equation of the curve find p in terms of r,

F F 0”

)

By eliminat ing r, p ,

and 15, there resul ts an equation between r

land j ) , for the evolute . T hus

,having r : a

"thenA 2 er

,in wh ich c 2 1/ —l—log

’

a . (Ex. 3 , 1 50 )

p,”2 c

2r1

2and p,

2 or

for the evolute, wh ich is a logarithmi c spi ral simi lar to the

given curve .1 86. Having p in terms of equat ion 1 83, enables

us to express p,in terms of T hus

,

r 2 4a sin2

Exampl e 1 73 . When 2p2

p 2 8a sin

H ence, p 8e sin

1 83,

p, Wu 8a «us/9 8. sinW.

Let to, represent the angle wh ich p makes w ith X.

IN VO L U TES AND E VO L U TES . 3 1 5

r

rhen and

H ence, fl. 8“ Sin l( ffi.

187 . Equation of an Evolute — Comb ine the equations

y 2 E(x ) ,

(96 x ) (r o

el iminating x andy, and there w i ll result in general a differential equation involv ing x and y.

1 88 . Involute of a Circle.— L et a radius of c ircle ,

and let A be the ini t ial posi t ion of generating point. L et

TB 2 c ircum. A T be any posi tion of the tangent roll ingupon c ircum. A T T ake origin and pole at 0 . L et

2p angl e AO T ,and let 6’ 2 angl e A OE, wh ich radius

vector r makes w ith X . T hen tangent TB 2 ab, and we

have,for the rec tangular coordinates of any po int, as B ,

by

proj ect ing the two l ines O T and TB upon X and Y re~

spec tively,

x 2 OP 2 a cos ¢ + a¢ sin 1A

y 2 O Q 2

3 16 DIFFERENTIAL CAL CULUS .

polar coordinates we have , from the right triangle

0 angl e B O T 2

H ence, at!) 2 a9 a sec“ 1

(r/a)

abl a sec" 1

(r/a)"1

.

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318 D I FFEREN T I AL CAL CUL US .

the lines are tangent to each other, and are said to have acontac t of at least the fi rst order .

When,

= D B = [f (el l

the l ines have a contact of at least Me secoua’order .

When, also, f (a) 2

D B [f'

a (ole/ 4

the l ines have a contact of at least Me Mira’ oro’er and, in

general,the order of contact of any two l ines hav ing a po int

in common is denoted by the g reaiesz‘

number,beginning

w i th the fi rst, of successive derivatives of the ir ordinatescorresponding to the c ommon point, wh ich are, respectively,equal each to each .

D enoting the order of contact of any two l ines by n, wehave also f

”

(a) : making with f (a) : n 1

condit ions,and giv ing

D B (mar l

/ lo I (7)

From wh ich we see that when (n 1 ) is odd , the signof y

’-

y2 DE wil l change when the sign of A changes

,

that is, if D is above B when is is negative and vanish ing,

it w il l b e below B immediately after 12 becomes positive .T he l ines w i ll then intersect, as shown in F ig. 1 . When

(n 1 ) is even, the sign of y’ —

y 2 D B w i l l not changew ith that of 12, and the l ines w ill not intersect. (See F ig.

H ence , token Me order of conz‘

acc is even, l ines z'

nz‘er secl at

Me commonpo‘

ini , ana’

f

ie/Fen ii is odd, Mey do not.

T o i llustrate, take the two equat ions

4x , ( 1 ) and (x (‘y z

2232 . ( 2)

O S CULA T IN G L I NES . 319

Comb ining, we find that the point ( 1, 2 ) is common.

( 1 ) f ( I ) and f ( I )

( 2 ) gives 2 1, go ( 1 ) and 2

H ence,the c ircle ( 2 ) has a contact of the second order

w i th the parabola and intersec ts it at the po int ( 1 ,Determine the common points, and order of contac t at

each,in the following pairs of l ines

y“2 4x. ( 1 , 2 ) in common

Contact of I st order.

y = f

y 3x2

390 + 1Ans. ( 1 , 1 ) 2 d order.

422 - 96

2 —4

Ans. o,

1 d order.fi + x = n& s

M

Two l ines hav ing at a common point a contact o f the nthorder with a th ird l ine, have a contact w ith each o ther ofat least the nth order.

1 90 . O sculating Lines — T he l ine of any spec ies of l ine ,which at a given po int of a given l ine has the h ighest possible order of contact, is cal led an osculatr ix or osculating

line. T hus,the c ircle wh ich at the given po int has the

h ighest possible order of contact is called an oscil lating circle.

T he parabola of h ighest contact is cal led an osculaz‘ing pa

rabola .

T o determine the equation of an osculatrix at a given

po int of a given l ine , assume the general equation of the

spec ies of l ine in its reduced form.

T he problem then is to determine such values for thearb itrary constants conta1ned therein as w i ll cause the re

quired l ine to have the h ighest possible order of contact.

320 D I FFEREN T IAL CAL CUL US .

Since the osculatrix must pass through the given po intsubstitute its coordinates in the general equation, giv ingone equation between the required quanti ties, and diminishing the number of arb i trary cons tants by unity.

From the general equat ion of the spec ies and the equat ion of the given l ine determine expressions for the successive derivatives of the ordinates to inc lude those whoseorder is denoted by the number, l ess unity, of the constantsin the reduced form of the general equation of the species.

Substitute the coordinates of the given po int in each , and

place the resul ts corresponding to derivatives of the same

order equal to each other. T he resul ting equations w i ththe one before obtained wil l equal in number the requ iredquanti t ies, wh ich in general may be determined . T h eirvalues substitu ted in the general e quation w il l give the re

quired osculatr ix . T he order of contact w i ll in general bedenoted by the number, less unity, of arb itrary constantsentering the general equation, but in exceptional cases itmay be h igher.

EXAMPLES .

1 . F ind the equat ion of the osculating right l ine to the

parabolay22990 at the point ( 1 ,

Iny 2 ax O substi tute the coéirdinates ( 1 , giving

3 2 a O.

Fromy 2 ax b we find 2

From_y"2 926 we find 2 9/2y.

Substitut ing the coordinates ( 1 , 3) in each , and plac ingthe resul ts equal to each other

,we have a 2 2

which in ( 1 ) gives 5 2 H ence,y 2 3x/ 2 is the

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32 2 D I F FEREN T I AL CA L CUL US .

1 9 1 . O sculating Circle at any point of any planecurve whose equat ion isy 2 f (x ) .

Substituting in (x a ) —l (y O)2

2 R 2

,we have

a )2

(y,

2 R2

.

From (x a)2

-l (y b)22 I 3

” we obtain

x— a

and go (x ) 2

and from y 2 f (x ) we derive expressions for f’

(x ) and

no)Substi tut ing in each , and plac ing the resul ts cor

responding to the derivatives of the same order equal toeach o ther, we have

(x’

5)

[ 1 c)

wh ich w ith (0 ) give , omitting the primes,

R [ 1

a [ r

5 y [ I

Comparing these w ith ( 1 ) and 1 72 , we see

osculafozy circle at any point of a plane curve is Me

curoaz‘ur e.

EXAMPLES .

1 . F ind the equation and radius of the osculating c ircleto the curve 4 (y 1 ) 2 96

2

,at (o ,

Ans. y?

x22

3 ; radius 2 2 .

2 . F ind the radius of the osculating c irc le to the parabola

99s at (3.V2 7) Ans .

O S CULA T IN G L I NES . 32 3

3 . F ind the equation and radius of the osculating c ircleto the parabolay

22 1 6x at ( 1 ,

Ans . (x (y 2 1 2 5 ; radius 2 5 V5.

1 92 . In general,an osculating circle has a contact of the

second order wi th any plane curve , but 1 72 shows that at

a point where the curvature of a curve is a maximum or a

minimum,the c ircle of curvature

,and therefore the oson

lating c irc le,does not intersec t the curve . T he order of

contact is therefore odd , and of a degree h igher than the

second . T hat the contact in such cases is at least of the

th ird order may be shown as fol lows :From 1 72 , and 1 9 1 , we have

I?2 p 2 [ 1

and in order that [3 may be a maximum or aminimum,

zf'

rxff '

o) f (at:

whence f (x ) 23f

From 1 14 we have for a c ircle

Hence , rofien Me radius of curoaz‘ure is a maximum or a

minimum,Me cir cle of curvature nas a coniacz

‘coilli Me curve

of a t least Me Mird order .

I t fol lows that the order of contact of an osculating c ircleat a vertex of a conic is odd

,h igh er than the second, and

the c ircle does not intersect the conic .

324 D I F FER EN T IAL CAL CUL US .

CHAPTER XVI I I .

ENVELO PES .

1 93 . In

2 f (x.y.a )

let a be an arb itrary constant . Bygiving all possible valuesto a

, ( 1 ) wil l re present a series of l ines, all of the same k indor fami ly, unl imi ted in number

,and, in general , intersect ing

each other in order. a is then cal led a variable parameter .

T o i l lustrate , let

(90 9 0

By giving different values to a,the equation may repre

sent a series of c ircles having the same radius,the ir centres

on X,and intersecting each other in order in points a

m,m’

,etc .

I n general , any value of a in ( 1 ) corresponds to a determinate particular l ine , and a ii to ano ther l ine of thesame k ind having for its equation

a -l fl)

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326 D IFFEREN TIAL CAL CUL US .

the points of intersection, etc ., before obtained , and is

,

therefore , an envelope of the series.

H ence, an envelope of any series of l ines determined bygiving all possible values to a variable parameter in an equat ion involv ing two var i ables only may be defined as Me

l imit of Me locus,or the locus of Me l imiting positions of

points of intersection of the series of l ines,each w ith its

second state , under the law that the difference in positionbetween each second state and its primi t ive vanishes.

T o ob tain the equation of an envelope of a series of l inesgiven by an equat ion w ith a variable parameter, we havethe fol low ing rule :Combine Me g iven equa tion xvii/z its dif

ferentia l equation

taken w iM respect toMe var iableparameter , and eliminate Me

parameter .

F rom regarding a as constant,we obtain

du/dx 2 (an/ay) (dy/dx ) 2

for the differential equat ion of w eb of Me lines of Me ser ies.

From ( 1 ) and a 2 qb(x ) , wh ich substituted in ( 1 )gives the equation of an envelope . H ence

,differentiating

( 1 ) regarding a 2 we have

da an an dy au da

dx 6 x+ay dx

+aa dx

or, since au/au 2 o,

d’

u/a’x air/8x

for the differential equation of an envelope. Each l ine ofthe series, and an envelope , have , therefore , the same differential equation, and dy/dx at anypoint common to any l ineof the series, and an envelope w ill be the same for both.

An envelope is Meref ore tang ent to all of Me l ines of Me ser ies.

EN VEL O F ES . 32 7

EXAMPLES .

1 . F ind the equation of an envelope of the series of c irclesgiven by the equation

(x

when a is a variable parameter.

au/aa 2 — 2 (x —a) 2 ( 2 )

Comb ining ( 1 ) and ( 2 ) so as to e l iminate a , we have forthe required envelope

y= d: 3. = o/o o

H ence,the right lines MM 'M and MM

,M

, , ( see figure,

1 93) are the envelopes.

2 . F ind the envelope of the curves given by

_y 2 x tan (9 xi

/ (4li cos2

as 19 varies.

au/ad 2 x/cos219 no

”

sin cos39 2 0 .

H ence , tan 6 2 2 /z/x ,1 tan

” 0 2 (xi—l sec

26,

and cos

'2(9 2 x

Q

/ (xi—F4M) , 4/z cos

2(9 2 4/z

2

) .

Substitut ing in given equat ion,we findy 2 it x

z

/ (4b) for

the required envelope .

3 . F ind the enyelope of a righ t l ine of a given length cwh ich moves w ith its ends on the c oordinate axes.

L et (9 equal the angl e wh ich the right l ine makes w ithX. T hen intercepts are

,respectively, c cos (i and c sin 6,

u 2 x sec 0 +y cosec él - c 2 o

for the l ine , in wh ich 19 is the variable parameter.

au/afi 2 x sec 0 tan 9 — y cosec b’ cot 0 2 0.

Comb ining ( 1 ) and we find

x 2 c cosgb’, y 2 c sin

3

6’;

328 D I F FEREN T IAL CAL CUL US .

and el iminating (9,we have for the required envelope

x2/3+J/

2/3 2

O therw ise, let a and b represent the intercepts, respec

tively, giv ing

x/a +y/b 2 1 (3 ) and a2

b22 c

2

R egarding b as the variable parameter, we may bymeans

of (4) el iminate a from and proceed as before or, sincea is a func tion of b, we have

scan/a2

yet/R o , (5)“a“ "i“ Fab 2 0 . ( 6)

Comb ining el iminating aa/at , a and b,

we have

(a2_

,l_ D

2

1}, 53

; 5 and a (fryer/3

Subst itut ing these expressions in we have for the

envelope

In a simi lar manner,when there are n parameters and

n I equations of condition be tween them,we may differ

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330 z) I F FEREN TM L CALcoL US .

and a is the variable parameter .

Ans. x +322

z xy 2 5x 52

Vx V5» 12 Vb.

.8 . F ind the envelope of a righ t l ine mov ing so that the

product of its distances from two fixed po ints is constant.

T ake X to coinc ide w i th righ t l ine j oining the two

po ints, and the origin to be at its middle point. L et (a , o)and (2

,0 ) be the two fi xed po ints, and

x cos a°

+ y sin a = p

the equation of the l ine .T he distances from the fixed po ints to the line are, re

spec tively,

a cos af — p and - a cos a —p.

H ence ff (z?cos

2a z c constant .

From ( 1 ) and regarding a as the variable parameter,we find for the envelope

x2

/ (c2

02

) + J'2

/62

9. F ind the envelope of the right l ines whose equation is

2b(x b?)

wheny’is the variable parameter

,and we have

3) and b

El iminating b and x’

, ( 1 ) becomes

J’y 0

H ence, au/ay'

zyy'

+p2

apx

y’2 ( zax a

z

l/ zy

Comb ining (3) and el iminating y’

,we have

90 : : l: V

EN VEL OPES . 33 1

for the envelope , wh ich is a point at the focus of the

parabola, and is the Caustz'

b of rays of ligh t reflec ted fromthe concave side of a parabola, the inc ident rays be ing

parallel to the axis wh ich coinc ides w ith X .

1 0 . F ind the envelope of the right l ines whose equation is

y y’ V) ,

when x’is the variable parameter , and we have

y + x and

El iminating b, ( I ) becomes

x yl/Z

D ifferentiat ing and reduc ing,we ob tain

au/axlI

,

VI

a2/3

.y1/3 O

,or y

!

(3)

Subst ituting,in ( z ) , flb

w/a

2 for y’ we have , after

comb inat ion w ith (3) and reduc t ion,

x’

2 42/3x/ (a 2y

2/3) .

Squaring the expressions for y’and x

’

,and substitut ing

iny x z: 42

,we have for the envelope

da/sya/s —l 2y

2/3 )2

wh ich is the equation of the Causz‘

z'

c of rays of l igh t re

flec ted from a c ircle , the inc ident rays being paral lel to X .

1 1 . F ind the envelope of the polar l ine to the ellipse

9y2

+ 4x2 = 36 as the pole moves along the righ t l ine

L et x and y be the coordinates of the pole , giving

y 2 x I .

T hen gyy”

4xx 36 is the equation of the polar l ineSubstituting 2 x

”I for y w e have

2; 2 :9 11 4xx 36

z o,

332 D I F FEREN T I AL CALCULUS .

in wh ich x is the variable parame ter.

1 83) 4x o.

Comb ining the last two equat ions, we have ox gy 36,

or y 4 , x 1 8 for the required envelope .

Lines . Envelopes.

1 2 . y ax b/a . y?

4bx .

1 3 . ax -

y r : x90 a

9)/2p. x

2zpy 15

9.

I 4 . x2

/a +y2

/(a b) : I . (x +y’ = 0 .

1 5. x2

+y9

r

16 y'zz ax

I 7 . x cos 3¢ +y sin 3qb

a (cos an”?

m axi

fly mx a ( 72 d9)x

2.

”2

(m a’

)2

20’a a

‘.

19 . (y?2 fi x y

”4x

3/2 7c.

(x —b) 2 =

2 1 . y : 2ax+ a

2 2 , y2

+ (x — a )2 — 2pa = o. y

?

x :

y : l: cx/ ( I

2 5 . y2

: : ax 4Jfl= x

2

= 0 .

a2cos 6 b“ sin 9 c

”

x y a

2 8. y mx Mm . (fly? b2x

2

(375? x2/3 y2/3 c

2/3.

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334 D I F FEREN T I AL CAL CUL US .

from wh ich,el iminating x

’

, we obtain for the required en

veIO pe a]?

4 (x (See Example 1,

2 . Find the envelope of the normals to an ell ipse by theabove method .

O therwise , the equation of the normal to an ell ipse at a

po int whose eccentric angle is denoted by (7’ is

: ax sec fi — by cosec 0

R egarding 6 as the variable parameter,

an/a d ax sec (9 tan 9 J ; by cosec cot 9 o .

Comb ining and el iminat ing (9,we have for the requ ired

envelope (any/3

(by)2/3 ( 4

2 b2

)2/3

.

CUR VE T JCA CI A/G. 335

CHAPTER X IX .

C U R VE T R A C I NG.

195 . T he foregoing princ iples , w ith those from AnalyticGeometry, enab le us

,in general

,to trace curves from their

equations w ith great ac curacy.

RECT ANGULAR COORD INATES .

No fixed rule or direc tions apply in all cases,but

,in gen

eral,it is desirable to determine

I°

. Symmetry w ith respect to the coordinateaxes

L imiting coordinates and asymptotes paralel to them.

Points on the coordinate axes.

T erminat ing points .

D irec tion of curve at points on the coordi

nate axes.

O

Asymptotes obl ique to coordinate axes.

Multiple points .

Charac ter of cusps .

\D

OO

\I

O\

Maximum and minimum ordinates .

D irec tion of curvature and points of in

flex ion.

EXAMPLES .

I . y 2“

(fi x/ (612

Each value of x from 00 to 00 gives a real value

336 D IFFERENTIAL CALCULUS .

for _y. T he curve is therefore unl imited in both directionsw ith X

, and is l imited in both directions of Y.

As x j : : l: 0 . H ence , X is an asymptote in

both directions.

x = o givesy = o,

00

H ence , (o, o) is the only po int at wh ich the curve cutsthe coordinate axes.

f’

(x ) (f

(a?

x2

l/ (42

902

?

f'

(o) I . H ence , the direc tion of the curve at the

gin makes an angle of 45° w ith X .

f’

(x ) 2 o gives x": i a and : l: 00 .

f (x ) 2 429606

2x”

?

f (a) is negative ; y 2"d /a is a maximum.

f a) is positive ; y a/z is a minimum.

f”

(x ) 0 gives no r: o and i a V};

f (x ) is negative for values of x from co to

and the curve is concave downward .

F or values of x from a V3 to o f”

(x ) is posi tive , and

the concave si de is above .

As no varies from o to a V3 , f"(x ) is again negative

,and

the concav i ty is downward

Values of x from a to —'r 00 make f (x ) posi tive , and

the curve is concave upward .

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338 ELEEEEEN T LA L CALCUL US .

Values of x from 00 to 2 make f (x ) positive , andthe corresponding part of the curve is concave upward .

As no varies from 2 to f”

(x ) is negat ive , and the

concavity is downward . When x f”

(x ) is

positive, and the concave side is above the curve but

no makes f"(x ) negat ive , and the concavity is down

ward . I t fol lows that z, ( 0 -54, and

are pomts of inflexion.

3 . y azn/ (n a )

2.

(f (a a)s

.

f (x ) max an

T he curve is unl imited in both directions and

the positive direction of Y .

y = o gives x = o and zt 00 .

X is an asymptote in both directions, and since y 00

as x a , x a is an asymptote .

f’

(o) 1 . H ence , at the origin the curve makestan

“ 11 w ith X.

f’

(x ) 0 gives x a,

and f”

a) is positive.H ence, y cl/4 is a minimum.

T o the left of the point of inflexion

CUR VE TR A CJN G . 339

the concave side is below ,and to the right it is above , the

curve .

4 . y3

2 0 902

y”

2 x2/3( 2 a

f (x ) (4m 3x 3y?

T he curve is unlimited in bothdirections along X and Y. I t cutsX at (o , o ) and Y is

tangent to both branches at the

origin,wh ich is a cusp of the first spec ies ; and the tangent

at ( 2 a, 0 ) is perpendicular to X . J) x 2 cx/3 is the

equation of an asymptote in both directions. yz a V3 2/3,

corresponding to x 4a/3 , is a maximum. ( 2 a ,0 ) is a

po int of inflexion to the left of wh ich the curve is concavedownward , and to the righ t of wh ich it is concave upward .

5. y x (3

f (x ) (3 n)g

(s

f”

(x ) (3

As x »> n z o T he curve is

unl imited in the direc tions of X and Y . (o, o) and (3 , 0 )are po ints on X . f

’

(o ) 2 : o . f’

(x ) 0

gives x 2 : and 3 . f is negative ; henc e , yis a maximum. f

’

(x ) changes sign in passing th rough

340 D TEEEEEN T LAL CALCULUS .

f’

(3) 0 and‘

y 0 is a minimum. is a

po int of inflexion to the left of wh ich the curve is concavedownward, and to the right of wh ich the concav ity is up

6. 3722 x5 + x

f (x ) : i: W —l r .

f (x ) :i: ( 1 5902

2 490+ 8)/4 (x

T he curve is symmetrical with respec t to X.

Values of x 1 give imaginary expressions for y.

x 1 gives y : l: 0 . x 1 gives two values foryequal w ith opposite signs. As no cc

, j : 00 . T he

curve is,therefore , l imited in the direc tion of X negative

by the ordinate corresponding to x 1,and is unlimi ted

in the other directions along X and Y

I, : l: 0 ) and (0 , ziz 0 ) are po ints on X .

f’

( 1 ) : l: 00, and f

’

( 0 ) : l: o . H ence , at

i 0 ) the tangent is paral lel to Y,and X is tangent to both

branches at the origin, wh ich is amultiple po int of tangency.

(Exampl e 6, page

f’

(x ) 0 gives x 0 or f (0 ) is positive forthe upper and negative for the lower branch . H ence, thez ero ordinates at the origin are

,respectively, a minimum

and a maximum. f"( I S negative for the upper and

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342 ELEEEEEN T JAL CALCUL US .

point for the right-hand branch . As x 0 , f’

(af) 00 ;

and as x 0, f

’

(x ) H ence,X is a tangent at

the origin. Y is an asymptote to the l eft-hand branch , and

y r: 1 is an asymptote to both branch es. As no varies con

tinuously, f’

(x ) does not change sign, and there are nomax

imum or minimum ordinates . Corresponding to x 2

there is a point of inflex ion, to the left of wh ich the curveis concave upward , and to the right of wh ich the curvatureis downward .

9 . The Logarithmic Curve.

f (n) is negative for all points of the curve, and the con

As y» x 1,

yz 0 . As -> oo . T he

curve is l imited in the direc tionof X negative by Y,

wh ich is an

asymptote, and is unl imited in

the other directions along X and

Y . f'

( 1 ) 1 . H ence,at ( I , 0 )

the curve makes tan“ 11 wi th X .

CURVE TRA CING'

. 343

cave side is below . R S,the subtangent

xa’

y/a’x 1 2 : 0 A.

1 0 . y"

an2

bxs

.

f (x) 2 : l: (Ll 3bx/ 2 )/ Va + bx .

f”

(x ) : l: (4ab 3b2

x )/4 (a bx )3/2

.

T he curve is symmetrical with re= B

spect to X .

As x»»—> oo y» ziz 00

x — a/b, y : : i: 0 .

y is imaginary.

T he curve is l imited in the direction of Xthe ordinate corresponding to x z : a/b, and

in the other directions along X and Y

21: 00 f'

(0 ) ziz Vii. T he origin is,there

fore , a double multiple po int.

f (x ) 2 0 gives x za/3b, for whichy : l: 2 4 V3a/9b

are maximum and minimum ordinates.

F or x a/b the first value of f”

(x ) is positive , and

the second negative . T he branch EEOC is,therefore ,

c oncave upward, and the other is concave downward .

POLAR CO O RD INAT ES Z EXAMPLES.

I . r z a sin 2 9.

a’r/dfi 2 4 cos 2 9.

As H varies from 0 to n/4 , r

x changes from 0 to a , and as 0 variesfrom 7r/4 to 7

’ changes from a

to 0 ; compl eting a loop in the firstangle . As 9 varies from 7r/2 to

negative, changes from 0,through a , to Q

,form~

344 E LEEEEEN TLAL CALCUL US .

ing a loop in the fourth angle . As 9 varies from 7: to

T is posi tive and changes from 0,th rough a

,to 0

,forming

a loop in the th ird angle . As (9 varies from 3 7r/ 2 to 2 7r,T

is negat ive and changes from 0,through a

,to 0

,forming

a loop in the second angl e .As (9 passes through 7t/4 , changes from —l to

H ence , T t : a is a maximum. As 0 passes throughziT/a

’d changes from to and T a is a minimum.

As 0 passes through s7r/4 , dT/a’d changes from to

and T a is a maximum. As (9 passes th rough a’T/ zz

’fi

changes from to and T a is a minimum.

2 . T cz tan 0.

dT/cz’fi cl/eos

2 H.

T is always an increasing funcT

/

tion of (9.Values of 6’ from 0 to 7r/ 2 give

the branch PM ’As H varies

from 7r/2 to 71,T is negative and

increases from co to 0,giv ing the

branch in the fourth angle . Values of (9 from 7r to 3 7t/ 2

determine the b ranch in the th ird angle,and the branch in

the second angle is due to negative values of T corresponding to values of 0 from to 2 71 .

T he sub tangent T2a’f9/a

’T 2 : a sin2 (9.

(9 or gives T cc,and the sub tangent 4 .

H ence T cos 0 1: : l: a are asymptotes.

3 . The Spiral of Archimedes , T ab.

Estimating from the pole P,where T : 0 z 0

,T ih

c reases direc tly with d. Denoting the value of T after one

revolution by

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346 E LEEEEEN T LAL CAL CUL US .

5. The Hyperbol ic Spiral , T

2 (x/ 2 71 ; a z z n’

T,

Z 7I 7’

1/0.

At the pole T 0 and 6 on . T varies inversely w ith(9 H ence

,PO

’2 4T , , P0 2 T PO etc .

dT/ (ZH subtangent PT

(9 z 0 gives T cc and the subtangent PBH ence B C

, paral l el to PA ,is an asymptote .

From 1 50 , tan PM T : T zz’H/ (ZT 0

,

- wh ich leads to

a construction of the curve by po ints . T hus,w i th P as a

centre and radius cl,describe a c ircle . D raw any radius

vec tor as PM,and the corresponding subtangent PT.

Lay off PE arc H N,and draw EN . TM drawn par

al lel to EN w ill determine a point 111 of the curve ; for

tan PM T tan PLVE PE/cz H IV/ (z 6.

CUEVE TEA CLN C . 34?

6 . The Logarithmic Spiral , T a

M a/ f’

,wh ence 1 50 ) tanPM T : tan qb M

a ,

and gb is constant .

Also, gb TAG/E

’s.

H ence , j )z T sin gb TWH/a

’

s 2 CT,

in wh ich C represents the sine of the constant angle madeby the radius vector w ith the curve .

I f 9 0 be increased by equal angles, the corresponding values of T w i l l be in geometrical progression. Withany convenient radius

,as PA ,

describe a c irc le,and lay off

equal arcs Ab,bnz

,Tnc

,etc . D raw the r igh t l ines PA

,Pb

,

Pnz,etc .

,and let PA be the ini tial side of (9. PA w ill then

represent unity. Make 0 : Ab/PA ,and de termine the

corresponding value of T PP. PE/PA w i ll be the ratioof the progression,

and the distances PM,PC

, etc .,from

P to corresponding points of the curve are readi ly determined . Since T : 0 requires (9 co

, the number ofspires from A to P 18 unlimited .

CHAPTER XX.

APPL ICAT I ONS T O SUR FACES.

1 96. n F (x ,y,z ) 0 ( I ) or 2: f (x ,y) ( 2 )

is the general equat ion of any surface .

Assuming the co-ordinate axes perpendicular to eachother, and x and _y as the independent variables, we have

for the general differential equation of the surface.

197 . T 0 find f/zo donation of tile tangonz‘

plane to W T

faoe at a g ivenpoint.

L et 2: ¢ (x , y) be the equation of the surfacez’

) be the given point.

T hen

in wh ich are constant for a fixed point

T he equations of the tangent l ine PQ in the vertical

plane PMN are, from the figure,

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350 D JEEEEEN T JAL CALCUL US .

(F or tangent plane to surfac e of 2 d order only. Compare

8 14 an 8 1;C. Smith ’

s Sol id Note . I fa’

z'

the plane is indeterminate .

EXAMPLES .

1 . F ind the equation of the tangent plane at the po int

( 2 , 3 ,V2 3) on the surfac e x

222

z 36.

2 : x'

/z'

2/ V272,3 V2 3.

Subst itut ing in we have

(x z ) ( z/ 3/M ) ,

or 2 90 V2 32 36 , for the required plane,

an/a’x'r: 2x

’

4 , au/a’y'

6,an/a

’z'

2 : 2 V2 3

Substituting in we obtain same result .2 . F ind the equat ion of the tangent plane at (x

’

, y’

,z'

)on the ellipso id —

y’

/b2

1

Ans. ococ’

/a2

az’

/oz

1 .

3 . F ind the equation of the tangent plane at z’

)on the surface whose equation is

2 —ny2

+pz + 772 x —i—Zz o .

Ans. 2 7nx’x —l- 2ny

’

y-l—2pz

'z —l—na

4 . F ind the equat ion of the tangent plane to the ell ipsoidwhose equation 15

4302

+ 2y2

+ z2 = 1 0

,at ( 1 ,

Ans. 2x —y

AEEL / CA TTO N S T o S UEEA CES . 35r

5. F ind the equation of the tangent plane to the ell ipso id

org

/ 1 6 +yz

/9 3 1,at (3 , I

,V4

AnS sac/ 1 6+ y/9 V47/36 ) Z/4 I .

6 . F ind the equation of the tangent plane at anypoint of

the surface 90 +3} + 2 2 : and show that the sum of the

squares of the intercepts on the axes made by a tangent

plane is constant .

1 98 . Timnormal at anypoint of a surface.

Let equation of surface be z and po int beT he normal passes th rough z

’

) and is

perpendicular to the plane given by 1 97 ; hence itsequations are

x— x’

y—y’ z

I f equation of surface be u F (x , y, z ) z: 0,equat ions of

normal arex x y

—y’ z z

’

au/a’x' an/a

’

z"

lEXAMPLES .

1 . Deduce formulas for the direction cosines of the nor

mal given by by And find the cosines of the anglesthe tangent planes corresponding to ( 1 ) and ( 2 ) make w ithXX

,X2

,and YZ.

2 . F ind the tangent of the angl e that the tangent plane

to x’

+ 2."j

36 at ( 2 , 3 , V5?) makes w ith X Y.

1 99 . To fi na’ t/ze eguaz

‘

z’

ons of tbe tangent Zine fo a g iven

eur oe af a g ivenpoint.

352 D I F FEREN T IAL CAL CUL US .

L et equations of curve be

F (x , y,2 ) 0

,

¢ (x aJ ” Z) O;

and let z’

) be the point of tangency.

T he required tangent l ine l ies in the tangent plane to

each of the surfaces and (b) at point henc eits equations are

I f the curve be given by two of its proj ecting cyl inders as

the equat ions of the tangent become

3f(z — z ) =

(y —y’

) 33; z Z’

) 0,

wh ich 1 48) are the equat ions of l ines in X2 and YZ

respectively tangent to the curves (a'

) and (b'

) at the

proj ec tions of (x'

, y’

,T he prob lem is thus reduced to

finding the equations of righ t l ines tangent to two of the

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354 D I F FEREN T I AL CALCUL US .

2 0 1 . T he numerical value of the expression

measures tbe slope of the surface 2 f at the po intM

,along any sec tion,

as M B,made by a vertical plane

th rough M,and whose trace on YX makes w ith X

Writing tan QM T tan s

we have,

BZ/a’x

VI

tan s

Plac ing

ziy/an tan gb 772,az/o

’

x 2 p,and az/a’y 9 ,

we have

tan s (p Tno)/ 1/ 1 7722

.

APPL I CA TTON S T o S UEEA CES . 355

Applieaz‘

ion.

—At the po int ( 2 , 3 , V2 3) on the surface

x2

find the slope of the curve cut out by the plane

y 2x

From (a) , az/o’

x and az/ ( ly

H ence , 2/Vg , and o 3/ V2—

3.

From (b) , a’

y/a’x 2 2 on. T herefore

,

tan s : 2/V2 3 — 6/ V53VV1 + 4

H ence,

is the requ ired slope.2 0 2 . At any po int, as M , tan s varies w ith Tn. T o deter

mine nz,in order that the slope shall be a maximum, we

whence o mp 0 , or m o/p.

When tan s is positive , maximum values of tan s and the

slope are the same ; but when tan s is negative, the sl ope isa maximum when tan s is a minimum.

Application— F ind the equat ion of the vertical plane

wh ich passes through the point ( 2 , 3, V2 3 ) on the surfacex2

+ y z”

36, and wh ich cuts from the surface thel ine with the maximum sl ope .

y mx b,A is the general form of the requ ired

equation.

At ( 2 . 3.M ) T

e fie/fly

356 ELEEEEEN T JAL CALCULUS .

H ence, nz 3/2 . T he trace on XY must pass through

(2 , H ence,we have

3 = 3+ b, or b z o,

and y = 3x/2 ,

is the required plane . T he maximum slope is approxin

mately .75 1 .

When p 7729 0,

or on —p/g, tan s 0

,and

the slope is a minimum, since numerical values only of

tan s are considered .

In the above appl ication 772 p/g H ence,

5.giving 5 and y fax/3

is the plane wh ich cuts out the curve whosetangent at M is paral lel to X Y.

T he intersect ion of the surface by the horiz ontal planeth rough the given point is a horiz ontal line, anda e

,are its equat ions.

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Thurston’s Text-book of the Materials of Construction 8vo,T il lson’s Street Pavements and Paving Materials 8vo,Waddell’s De Pontibus. (A Pocket-book for B ridge Engineers . )

16mo, morocco,

RAILWAY ENGINEER ING.

indrews

’s

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Handbook for Street,RailwayEngineers . 3x5 in . mor .,

1B erg’s Bu1ld1ngs and Structures of American Ra ilroads . . 4to, 5

Brooke’s Handbook of Street Railroad Locat ion 16mo, morocco,

Butts’s Civil Eng ineer ’s F ield-book 16mo, morocco,

Cranda ll’s Transition Curve 16mo, morocco,Railway and O ther Earthwork Tables 8vo,

Dawson’s Electric Railways and Tramways . Sma11 4to, half mor .,

Engineering and Electr ic Traction Pocket-book.

16mo, morocco,Dredge’s H istory of the Pennsylvania Railroad : ( 187 Paper,Drinker

’s Tunnel ing, Explosive Compounds, and Rock Drills.

4to, half morocco,Fisher

’s Table of Cubic Yards Cardboard,

Godwin’s Railroad Engineers’ F ield-book and Explorers’ Guide.

16mo, morocco,Howard’s Transition Curve F ield-book 16mo, morocco,Hudson

’s Tables for Calculating the Cubic Contents of Exca

vations and Embankments 8vo,Nagle’s F ield Manual for Railroad Engineers . 16mo, morocco,Philbrick’s F ield Manual for Engineers 16mo

,morocco,

Pratt and Alden’s Street-railway Road-bed 8vc ,Scarles’s F ield Engineer ing 16mo, morocco,

Railroad Spiral 16mo, morocco,Taylor’s Prismoidal F ormulm and Earthwork 8vo,Trautwine

’s Method of Cal culating the Cubic Contents of Ex

cavations and Embankments by the Aid of Dia

grams . 8vo,The F ield Practice of Laying O ut Circular Curvesfor Railroads 12mo, morocco,

Cross-section Sheet . Paper ,Webb’s Railroad Construction 8vo,Wellington’s Economic Theory of the Locat ion of Railways .

Small 8vc,

DRAWING.

Barr’s Kinemat ics of Machinery 8vo,

Bartlett’s Mechanical Drawing 8vo,

Coolidge’s Manual of Drawing 8vo, paper ,B urley’

s Elementary Text-book of the Kinematics of Machines .

( I n preparation. )Hil l’s Text-book on Shades and Shadows, and Perspective. . 8vo,Jonm’

s Machine Design :Part I .

—Kinematics of Machinery.

Part I I .— Form

,Strength and Proportions of Parts

MacCord’s Elements of Descr iptive GeometryKinematics ; or , Practical MechanismMechanical DrawingVelocity D iagrams

Mahan’s Descr iptive Geometry and Stone-cutting

Mahan’s Industrial Drawing . (Thompson )

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Sheep,Manual for Courts-martial 16mo, morocco,

Wait’s Engineering and Ar chitectural Jur isprudence 8vo,Sheep

Law of O perations Preliminary to Construction in En:g ineering and Architecture 8vo,

Sheep,Law of Contracts 8vo,

Winthrop’

s Abr idgment of Military Law . 12mo,

MANUFACTURES.

B eaumont’s Woollen and Worsted Cloth Manufacture . . 12mo,B ernadou

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s Smokeless Powder— Nitro cellulose and Theory of

the Cellulose Molecule 12mo,Bolland’s I ron F ounder . 12mo, cloth,

The I ron F ounder Supplement 12mo,Encyclopedia of F ounding and Dictionary of FoundryTerms Used in the Practice of Moulding . . l 2mo,

Eissler ’s Modern H igh Explosives 8vo,Efi

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ront’s Enzymes and their Applications . . 8vo,

F itzgerald’s Boston Machinist 18mo,Ford’s B oiler Making for Boiler Makers 18mo,

Hopkins’s O il-chemists’ Handbook . 8vc ,Keep’s Cast I ron . 8vo

Leach’

s The I nspect ion and Analysis of F ood with SpecialR eference to State Control . (I n preparation. )

Metcalf’s Steel . A Manual for Steel-users 12mo,Metcalf’s Cost of Manufactures— And the A dministrat ion of

Workshops, Public and Private 8vo,Meyer

’s Modern Locomotive Construction 4to,

Reisig’s Guide to Piece-dyeing 8vo,Smith’s Press-working of Metals 8vc,

Wire : I ts Use and Manufacture Small 4to,Spalding

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s Hydraul ic Cement 12mo,Spencer

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s Handbook for Chemists of B eet-sugar Houses .

16mo, morocco,Handbook for Sugar Manufacturers and their Chemists 16mo

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Thurston’s Manual of Steam-boilers, their Designs, Construction and O peration 8vo,

Walke’s Lectures on Explosives 8vo,West

’s American F oundry Practice 12mo,Moulder ’s Text-book 12mo,

Wiechmann’s Sugar Analysis . Small 8vo,

Wolff’s Windmill as a Prime Mover 8vo,Woodbury’

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MATHEMATICS.

Baker’s Elliptic Funct ions 8vo,

B ass’s Elements of D ifferent ial Calculus 12mo,

Br iggs’s Elements of Plane Analytic Geometry 12mo,Chapman

’s Elementary Course in Theory of Equations . . 12mo,

Compton’s Manual of Logar ithmic Computations 12mo,

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Mounted on heavy cardboard, 8 x 10 inches,10 c-Opies for

Elementary Treatise on the Integral Calculus .

Small 8vo,Curve Tracmg m Cartesian Coo rdinates 12mo,Treatise on O rdinary and Partial DifferentialEquations Small 8vo,

Theory of Errors and the Method of Least

Squares 12mo,

Theoretical Mechanics 12mo,

Laplace’s Philosophical Essay on Probabilities . ( Truscott andEmory. ) . 12mo

,* Ludlow and B ass. Elements of Tr igonometry and Logarith

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Merr iman and Woodward . H igher Mathematics 8vo,Merr iman’s Method of L east Squares . 8vo,Rice and Johnson’s Elementary Treatise on the Differential

Calculus Small 8vo,Diff erential and I ntegral Calculus. 2 vols .

in one Small 8vo,Wood’s Elements of Co-ordinate Geometry 8vo,

Trigometry : Analyt ical, Plane, and Spher ical . . 12mo,

MECHANICAL ENGINEERING .

MATERIALS O F ENGINEERING, STEAM ENGINESAND BO ILERS .

Baldwin’s Steam Heat ing for Buildings 12mo,B arr

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B artlett’s Mechanical Drawing 8vo,B enjamin’s Wr inkles and R ecipes 12mo,

Carpenter’s Exper imental Engineer ing 8vo,Heating and Ventilating B uildings 8vo

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Clerk’s Gas and O il Engine Small 8vo,Coolidge’s Manual of Drawing 8vo, paper,Cromwell’s Treatise on Toothed Gearing ” . . 12mo,

Treatise on B elts and Pulleys 12mo,Durley

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( I n preparation. )F lather’s Dynamometers, and theMeasurement of Power 12mo,

Rope Dr iving . 12mo,Gill’s Gas an Fuel Analysis for Engineers 12mo,Hall’s Car Lubrication 12mo,Jones’s Machine DesignPart I .

— Kinematics of MachineryPart 11.

— F orm, Strength and Proportions of Parts 8vo,Kent’s Mechanical Engineers’ Pocket-book . . 16mo, morocco,

Kerr ’s Power and Power Transm1ss1on 8vo,

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Velocity Diagrams . 8vo,Mahan’s Industrial Drawing. (Thompson ) 8vo,Poole’s Calorific Power of Fuels . 8vo,Reid’s Course in Mechanical Draw ing 8vc ,

Text-book of Mechanical Drawing and ElementaryMachine Design 8vo,

Richards’s Compressed Air 12mo,

Robinson’s Principles of Mechanism.

Smith’s Press-working of Metals .

Thurston’s Treatise on Fr iction and Lost Work in Machin

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Animal as a Machine and Prime Motor and theLaws of Energetics 12mo,

Warren’s Elements of Machine Construction and Drawing . 8vo,Weisbach’s Kinematics and the Power of Transmission. (Herr

mann—Klein . ) 8vo,Machinery of Transmission and Governors. (Herr

mann—Klein . ) 8vo,Hydraul ics and Hydraul ic Motors . (Du B ois . ) . 8vo,

Wolff’s Windmill as a Prime Mover 8vo,

Wood’s Turbines

MATER IALS OF ENGINEERING.

Bovey’s Strength of Materials and Theory of Structures . . 8vo,

Burr’s Elasticity and R esistance of the Materials of Engineermg

Church’s Mechanics of Engineering

Johnson’s Mater ials of ConstructionKeep’s CastLanz a

’s Applied Mechanics

Mertens’s Handbook on Testing Materials. (Henning ) . 8vo,Merriman’

s Text-book on the Mechanics of Mater ials . . 8vo,Strength of Mater ials 12mo,

Metcalf’s Steel. A Manual for Steel-users 12mo,Smith’s Wire : I ts Use and Manufacture Small 4to,

Materials of Machines 12mo,

Thurston’s Mater ials of Engineer ing 3 vols., 8vo,

Part 11 .— I ron and Steel 8vo,

Part I I I .—A Treatise on B rasses, B ronz es and O ther Alloys

and their Const ituents 8vo,Thurston’s Text-book of the Mater ials of Construction . . 8vo,Wood’s Treatise on the Resistance of Mater ials and an Ap

pendix on the Preservat ion of T imber .

Elements of Analytical Mechanics

STEAM ENGINES AND BO ILERS.

Carnot’s Reflections on the Mot ive Power of Heat . (Thurston )

12moDawson’

s“ Engineering and Electric Traction Pocket-book .

16mo, morocco,Ford’s B oiler Making for Boiler Makers 18mo.

Goss’s Locomotive Sparks 8vo,

Hemenway’s I ndicator Practice and Steam-engine Economy.

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