Higher Mathematics 1410202674, 9781410202673

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I. SUVOROV

higher mathematics TEXTBOOK FOR TECHNICAL SCHOOLS

PEACE PUBLISHERS

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H a am.iuiicHOM Rjbinc

I. S U V O R O V

HIGHER MATHEMATICS TEXTBOOK FOR TECHNICAL SCHOOLS

Translated fr o m

the H a ssia n

f>V M.

V. O A K

T ranslation

E ditor

G EO lH .i; YANKOVSKY

PEA C E P U B L IS H E R S

M O SC O W

First Publish) Second Printin

CONTENTS A.

DA SIC A N A L Y T I C G E O M E T R Y IN T H E P L A N E

Chapter I. Method of Coordinates Sec. Sec. Sec. Sec. Sec. Sec.

1. The Coordinates of a Point . . . . 2. The Sum of Two Directed Segments 3. The Distanco Between Two Points . . . 4. Dividing a Segment in a Given Ratio 5. The Angle of a Straight Line with the Axis . . 6. Conditions for Parallelism and Perpendicularity

11 13 14 17 18 20

Chapter / / . The Straight Line Sec. Sec. Sec.

Sec.

7. A Straight Line as a L o c u s ......................................... 8. Equation of a Straight Line (Slope-InterceptForm) . . . . 0. General Form of the Equation of a StraightLine and Its Special C a s e s ....................................................................................... 10. Equation of a Straight Line (Intercept F o r m )........................ 11. Solved Examples 12. Construction of a Straight Line When ItsEquationIs Given 13. The Point of Intersection of Two Straight Lines . . .3 0 14. Equation of a Straight Lino Passing Through the Point (xi, y ]) in a Given Direction 15. Equation of a Straight Lino Passing Through Two Given Points (x |t y,) and (x2, y2) . . . 16. The Anglo Bolwccn Two Straight Lines

Sec. Sec. Sec. Sec. Sec. Sec. Sec.

17. 18. 16. 20. 21. 22. 23.

Sec. Sec. Sec. Sec. Sec. Sec.

22 23 25 27 27 29 33 34 35

Chapter / / / . Quadric Curves Equations of the C ir c le .................................................................. Solved E x a m p les............................................................................... The Circle as a Quadric Curvo . . . . . E l l i p s o .................................................................................................... The Equation of an E llip s o .............................................................. Investigating the Form of an Ellipso from Its Equation . . . Plotting an E llip s o ...........................................................................

39 40 42 44 45 46 48 5

Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec.

24. Relationship Between the Ellipse and the C ir c le ................... 25. Eccentricity of an E l l i p s e .............................................................. 20. H yperbola................................................................................................. 27. The Equation of the H y p er b o la ................................................... 28. Investigating the Forms of the Hyperbola from Its Equation 29. Plotting a H y p er b o la ....................................................................... 30. Asymptotes of the H y p er b o la ...................................................... 31. Eccentricity of a H y p er b o la .......................................................... 32. Equilateral H yp erb o la ....................................................................... 33. Solved Examples on the Ellipse and H y p e r b o la ..................... 34. Parabola ................................................................................................ 35. Equation of a P a r a b o la .................................................................. 36. Investigating the Forms of the Parabola from Its Equation 37. Plotting a P arab ola........................................................................... 38. Formulas for the Transformation of C oord in ates..................... 39. Equation of the Parabola in Parallel Translation of the Coordinate A x e s ................................................................................ Sec. 40. Equation of an Equilateral Hyperbola Referred to the Asymptotes . . . Sec. 41. Solved E x a m p le s............................................................................... Sec. 42. Quadric Curves as Conic S e c t io n s .................................................. B.

49 51 51 52 53 54 56 58 59 59 60 61 62 03 64 66 67 68 70

ELEM EN TS OF D IF F E R E N T IA L CALCULUS

diopter IV. Theory of Limits Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. 6

43. 44. 45. 46. 47. 48.

Absolulo Value and Its P ro p erties............................................. Infinitely Small Quantity (In fin itesim a l)..................................... Variable Quantities, Hounded and r n h o n n d e d ......................... Basic Properties of In fin itesim a ls.................................................. Infinitely Large Q u a n tity ............................................................... Relationship Between Infinitely Small and Infinitely Largo Q u a n tities................................................................................................ 49. The Limit of a Variable Q u a n tity .............................................. 50. t'lOumctrical Representation of a Number, Variable, and L i m it ......................................................................................................... 51. Relationship Between a Variable, Its Limit, and an Infini­ tesimal ..................................................................................................... 52. A Variable Can Have (inly One L i m i t .......................................... 53. The Limit of an Algebraic S u m .................................................. 54. The Limitofa P r o d u c t................................................................. 55. The Limitofa Q u o tie n t................................................................ :>li. Tho Limitofa RationalAlgebraicE x p ressio n ........................... 57. The Sign of aVariable andItsL i m i t ......................................... 58. Conditions for the Existence of a Limit of a Variable . . . . 59. On the Limit of a Quotient of In fin itesim a ls............................. OH. Examples in Finding Limits . . .

73 75 76 76 78 79 80 83 86 86 87 87 88 90 91 91 92 92

Chapter V. Function and Its Continuity Sec. 61. Argument and F u n c tio n ................................................................... Sec. 62. General Designation of a F u n c tio n .............................................. Sec. 63. Graphical and Analytical Representation of a Function . . . Sec. 64. Graph of a F u n c tio n ..................................................................... Sec. 65. Increment of the Argument and F u n c tio n ................................ Sec. 66. The Limit of a Function at a Finite P o i n t ................................ Sec. 67. The Limit of a Function When x —>- o o .................................... Sec. 68. Some O b servations......................................................................... Sec. 69. Continuity of a F u n c tio n ............................................................. Sec. 70. Another Expression for the Condition of Continuity of a F u n c t i o n ................................................................................................ Sec. 71. Testing a Function for C o n tin u ity ................................................ Sec. 72. The Properties of Functions Continuous at a Point . . . .

95 97 93 100 102 104 106 107 108 112 113 113

Chapter VI. Derivative Function Sec. Sec. Sec. Sec. Sec. Sec.

73. Linear Function, Its Rate of C h a n g e.......................................... 74. Examples in Finding Rates of C h a n g e ..................................... 75. Derivative F u n c tio n ...................................................................... 76. Tangent to a C u r v e ........................................................................... 77. Geometrical Meaning of a D e r iv a tiv e .......................................... 78. Relationship Between Differentiability and Continuity of ' a F u n c t io n .................................................................... . ' ..................

115 116 119 122 124 127

Chapter VII. Derivatives of Elementary Functions Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec.

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91.

Sec. Sec. Sec. Sec. Sec.

92. 93. 94. 95. 96.

Preliminary R em a rk s...................................................................... The Derivative of a C o n sta n t...................................................... The Derivative of a P o w e r .......................................................... The Derivative of tho Product of a Constant and a Function Tho Derivative of an Algebraic Sum of F u n ctio n s..................... The Derivative of a Product of F u n ctio n s................................. Tho Derivative of a F r a c tio n .......................................................... R e m a r k s ......................................................... Tho Function of a F u n c tio n .......................................................... The Derivative of a Function of a F u n c tio n ............................. Tho Limit of the Ratio of a Sino to an A r c ............................. Derivatives of Trigonometric F u n ctio n s..................................... Two Systems of Logarithms. Tho Number r. Changing from One System of Logarithms to thoO th e r ....................................... The Derivative of a L ogarithm ...................................................... Monotonic F u n ctio n s...................................................................... The Derivative of an Inverse F u n c tio n ..................................... The Derivative of on Exponential F u n ctio n ............................. The Derivative of Any P o w e r ......................................................

128 128 128 130 131 132 133 136 136 136 139 HO 143 145 148 149 |50 151 7

Sec. 97. Derivatives of Inverse Trigonometric Functions Sec. 98. Derivatives of Second and Higher Orders . . .

151 153

Chapter VIII. Studying Functions with the Aid of Their Derivatives Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec. Sec.

99. How to Determine Whether a Function Increases, Decreases or Is C o n s t a n t ..................................................................................... 100. Extreme Value P r o b le m s................................................................ 101. Maximum and Minimum of a F u n c tio n ...................................... 102. A Test for E x tr e m e s........................................................................ . . . 103. Procedure for Finding E x tr e m e s......................... 104. Examples in Finding E x tr e m e s................................................... 105. Second Derivative Test for Extreme V a lu e s .............................. 106. Extreme Value P r o b le m s.................................................................... 107. Maximum and Minimum of a Function at Points Where the Derivative Has No V a l u e ....................................................... 108. The Direction of Concavity of a Curve . . . . . . 109. Points of I n fle c tio n ........................................................................ 110. Constructing Graphs of F u n c tio n s............................................... 111. Mechanical Interpretation of the Second Derivative . .

154 157 159 160 162 162 163 167 170 171 172 173 174

Chapter IX. Differential Sec. 112. Comparing In G n ilo sim a ls.............................................................. Sec. 113. Differential of aF u n c tio n ............................................................. Sec. 114. The Differential of an Argument. The Derivative as a Ratio of D ifferen tia ls................................................................................ Sec. 115. Applying the Concept of Differential to Approximate C a lc u la t io n s ........................................................................................

C.

176 177 179 181

ELEM ENTS O F IN T EG R A L CALCULUS

Chapter X. Indefinite Integral Sec. 116. Integration as the In verso of D ifferen tiation ......................... Sec. 117. Tho Indefinite Integral as an Expression of the Aggregate of Antiderivatives of a Given F u n c tio n ...................................... Sec. 118. Properties of an Indefinite Integral . . . . . Sec. 119. Integration by F o rm u la s.............................................. . Sec. 120. Integration by Substitution . . . . . . . . Sec. 121. Standard Integrals and Their U s e s ............................................... Sec. 122. Integration of Powers ofsin *, cos x, tan x, cot r .............

187 189 190 191 195 201

Sec.

123. ^ V \i2— x‘ = A t lO.

I'i

Whence M xM2= M xO— Af20 , or M xM2= O M 2— OMx, because M xO = — O M x and M20 = — OM2. But OM2 = x2 and OMx= x x. Therefore, M xM2 = x2— xx.

‘2°. E xam ples. 1. T he m agnitude of a segm ent whose origin and term inus are th e points M x (3, 0) and M 2 (5, 0) is M XM 2 = 5 — 3 = 2. 2. If th e points are M x ( — 3, 0) and 7W2 ( — 0), then M XM 2 = = — 5 — ( - 3 ) = — 5 + 3 = - 2. 3. If th e points are M x (0, — 2) and A/2 (0, 3), then M XM 2 = = 3 — ( — 2) = 3 + 2 = 5. 3°. C o ro llary . TAc magnitude of segment AD (Fig. 6) parallel to the axis of abscissas {or ordinates) is equal to the difference iy

T"

i i i M, 0

! 1 ~ Ml X

xc =

—

yc-

= 1;

2.

l+ i XD :

/1+ 2- ( - 5 ) 1+2

— 2;

y D=

= — 1.

The required points are C ( 1, — 2), Z)( — 2, — 1). 3°. If a point C divides a segm ent A B in to two equal parts, then AC = CB and = l a n d form ulas (IV) take the form _ xt + x2 . . . __ Vl + Vi —

2

*

y

------------------

2

(V)

S e c . 5. T h e A n g l e o f a S t r a i g h t L i n e w i t h th e A x i s 1°. D efinition. The angle formed by the axis Ox and the straight line A B (Fig. 10) is taken to be the angle of counterclockwise

rotation of the positive direction of the axis Ox so that the positive direction coincides with the straight line A B . 18

In Fig. 10a and b, th is angle is xCB. 2°. P roblem . Find the angle between the straight line passing through the points A ( x t, {/,) and B ( x 2, y 2), and the axis Ox. S o lu tio n . Let the s tra ig h t line A B form w ith the axis Ox an angle x D B equal to

because Z .C A B = / _ r D B (as corresponding angles). Introducing the values of Z. CAB and the sides BC and AC into (1), wo obtain (VI) tan

0).

M F K and M F are called the radii vectors of the point .1/. /■',/’ is called the focal length and is taken equal to 2c, 7J- =2c. Considering tho triangle F tFM , we seo th a t M F t -{-MF > F tF , i.e., 2n > 2c, or a > c, ( c > 0).

2°. Take a piece of thread 2a in length and fix its ends at points F and F t w ith two pins a t a separation of 2c. Then make th e thread ta u t w ith the point of a pencil as shown in Fig. 32

and describe a curve w ith i t keeping the thread ta u t all the time. In th is manner an ellipse can l>c drawn in two steps: first on one side of the s traig h t line F tF and then on the other. S e c . 21. T h e liq u a tio n o f a n E l l i p s e Let us take the middle of the focal length as the coordinate origin O, the s traig h t lino F ,F as the axis Ox and the perpendic­ ular to i t passing through the point O as the axis Oij (Fig. 31). The foci F , and F then have coordinates ( — c, 0) a n d (-)-c , 0 ). Inasmuch as the position of point M can vary with respect to the coordinate axes, its coordinates will be the current coordi­ nates (x, y). M F i = \ / r {x + c)2 + y2,

M F = V ( x — c)2-\-y2.

(1)

By the definition of an ellipse, M F\ + M F = 2a. Therefore \T(x + c)2 + y 2 + \ f ( x - c ) 2 + y 2 + 2a. Let us get rid of the radicals. To do this, first transpose one of the radicals to the right-hand side, V ( x + c)2 + y2 = 2 a — V ( x - c)2 + y 2, and then square both sides of the equation. Removing the brackets we havo x* + 2cx + c* + y2 = 4a2 — — 4a Y (x — c)* + p* + x 2 — 2cx + c2 -f- y 2.

Elim inating x 2, c2, y 2 and transposing the radical to the left side and 2 c .t to the right side of the e q u a lity sign, we obtain 4a y~(x — c)2 + y 2 = 4a* — 4cx, or, dividing the equation by 4a, V ( x — c)2 + y 2 = a — - ^ - x .

(2)

By squaring both sides of the equation and removing the brack­ ets we have x 2 — 2 cx + c2 + y 2= a 2 — 2 cx +

x 2.

E lim inating —2cx and transposing x 2 to the left side and c* to the right side of the equality sign, we get x 2- - £ - x 2 + y 2 = a2- c 2. Multiplying by a2 wo then have (a2 - °-c) x2 + a2y2 = a2 (a2 - c2). Since a > c we may put a2 — c2= b2

(XXX)

ami write the preceding equation in the form \r- + a*y- = a mb~. We divide this equation by a2b2 and obtain tin canonical equa­ tion of the oil ipse: + = i „2 ' J>2 S e c . SS.

I n v e s t ig a t i n g th e F o r m o f a n E l l i p s e I t s E q u a tio n

(X X X I) fro m

1°. Let us solve the equation -^*+ -£*-= 1 for V. >J = ± — V a 2— x 2, ami investigate y as a function of .r. For tho values of y to be real numbers, the radicand a 2 — x 2 must bo either positive or equal to zero. This will be the case if I.!•!< « .

Consequently x can take on values only in the interval — -f-a, where a > 0 (Fig. 33). If x = ± a , then ar — x- = 0 and t/ = 0. The points of intersection of the ellipse w ith -th e axis of abscis­ sas, A (a, 0) and A i ( — a, 0), arc called the vertices of the ellipse

and the chord A xA = 2 a is called the major axis of the ellipse. If x = 0, then y = ± - ^ - l / r a 2= ± b . The points B ( 0, b) and ®i(0 . — b) are the points of intersection of the ellipse with the axis of ordinates and are also called the vertices of the ellipse. The chord B xB = 2b is called the minor axis of the ellipse. To each value of x in the interval [ —a, + a ] there correspond two points of tho ellipse located on different sides of the axis Ox at a distance equal to the absolute value of //, since to each value of x there correspond two values of y equal in absolute value and opposite in sign. Thus, the ellipse is symmetric about tho axis Ox. When x increases from —a to zero, tho absolute value of y increases from zero to b; and when x increases from zero to a, tho absolute valuo of y decreases from b to zero. The equation — 1 can he solved for x and not y: x= ±

V’b'1— y 2

and wo can analyse it in the same manner as before. We shall then come to tho following conclusions: x has real values only when y varies in tho interval — b ^ y ^ i b ] when y increases from — b to zero, the absolute valuo of x increases from zero to a; and when y increases from zero to b, the absolulo value of x decreases from a to zero. To each valuo of y in the interval ~ b ^ y ^ . b there correspond two values of r , equal in absolute

valuo and opposite in sign. The ellipse is symmetric about the axis Oy. 2°. The current coordinates x, y are contained in the equation of the ellipse only as squares. Therefore, if the point (x , y) belongs to the ellipse, then the point ( — x, — y) also belongs to i t because ( — x)* = x* and ( — y)i = y i . The chord connecting the points (x, y) and ( — x , — y) of the ellipse has the origin 0 as its middle point, since by formula V, Sec. 4. i ± ^ i ) = 0 and

? ' ± ^ = 0.

The point bisecting all the chords of the curve passing through it is called the centre of the curve. The origin O is the centre of the ellipse. S e c . 23. P l o t t i n g a n E l l i p s e 1°. Plotting by points (Fig. 34). Given 2a and 2c. On a straight line lay off a segment F tF = 2c and halve it. Then, from the centre 0 thus obtained lay off on the straight line F\F segments

OAx and OA equal to a to the left and right of O, respectively. Wo thus obtain the major axis A tA. Through the centre O draw a straight line 1>,I) perpendicular to A ,A . From tho focus F as the centre doscribo an arc with radius equal to a. The arc will intersect the perpendicular B tB a t points B i and B. Tho segment B tB = 2b is the minor axis of tho ellipse, since from the triangle OFB it follows th at OB* = FB* - OF* = a*— c* — b* (formula XXX). Thus wo have constructed four points which are the vertices of the ellipse: ^i| A, B t and B. 48

In order to construct other points of the ellipse, let us takeon the segment F,F to the left (or to the right) of the centre an arb itra ry point C t and from the foci F and /•’, as centres strike arcs (each time one arc above the s tra ig h t line A ,A and another below it) first w ith the radius equal to A tCt and then w ith the radius equal to C tA. The intersections of these arcs will give four points (in the figure all four arc labelled 1). The points 1 belong to the ellipse because the sum of the dis­ tances of each of them from the foci is 2a: I F , + 1F = AiC'i + CtA = A tA = 2a. T aking on the segment OFt (or OF) other points C2, C3, . . . and performing the same operations as in the case of the point 6',. we will each tim e obtain 4 points of the ellipse (in the figure they are labelled 2, 3, etc.). Having thus constructed a sufficient number of points, draw by hand or w ith the aid of a curved ruler a smooth continuous curve — the ellipse. Note th a t when constructing the ellipse, it is necessary to take more points on the segment F f i near the focus / ’, than near the centre O, and points Ct, C2, C3, . . . should be taken closer to each other as they approach 2‘. When the semiaxes a and b are given, it is necessary first to find c. To do this it is sufficient to construct a right-angled triangle w ith the hypotenuse equal to the sem im ajor axis a , one side equal to the semiminor axis b, and the other side equal t o e (formula XXX). S e e . 24. R e l a t i o n s h ip B e tw e e n th e E l l i p s e a n ti th e C ircle 1°. If in the equation b = a, wo got

of

the

ellipse —]- + -^- = 1 we

put

— + - § - = 1 or i* + y* = a 2, th a t is, the equation of a circlo of radius a. Thus a circle is an ellipse with equal semiaxes. 2°. Let us describe (Fig. 35) from the centre O of the ellipse (.2 - ^ - + - ^ = 1 a circlo with radiu9 equal to the semimajor axis a\ its equation is a:* + y* = a*. Let us solve the equations of the ellipse and the circlo for y and, in order to distinguish the ordi­ nates of the points of the ellipse from those of the circle having the same abscissa r , let us designate the ordinate of a point of 4 — 1320

'.9

the ellipse by ye, and th a t o f 1 the circle by y c. Then we have J’ e = ±

V a * — x *>

Vc= ± V a2 — x 2. Dividing the first equation by the second, we obtain Ve _

b

yc a ’ i.e., if a circle is drawn, with the major axis of an ellipse as its diameter, then for each value of the abscissa x the ratio between

the ordinates of the corresponding points of the ellipse and those of the circle is constant and equal to the ratio of the semiminor axis of the ellipse to its semimajor axis. Whence il follows th a t an ellipse can be obtained by ‘‘uniform compression" of a circle of radius a , i.e., by the reduction of all semichords (normal to its diameter) in the constant ra tio . This ratio is called tho coefficient of compression. 3°. Let the plane Q of a circle x 2 + y 2 = a2 form with piano P an angle

). By dropping a perpendicular on the plane P from each point AI of the given circle we obtain tho orthogonal projection of the circle on the plane P. From Fig. 36 it follows that if AIAI' ± P and AIN X Ox, \.\\on/_A1'NAI = q>. From tho A lf V right-angled triangle A I'N AI wo have — — cos 0).

Introducing the value of c into formula (XX XII) we obtain

(XXXIII) From this formula it follows that if b = a, i.e., if the ellipse is a circle, the eccentricity e = 0 ; if a remains unchanged and b decreases from a to zero, the eccentricity of the ellipse increases from zero to unity; the eccentricity is equal to unity when the ellipse turns into the segment A tA of a straight line. 2°. In the derivation of the equation of the ellipse (Sec. 21) we obtained equality (2): V ( x - c)2 -)- y 2 = a - - x. In it, y~(x — c)2 + y 2 = Mb' [Eq. (1), Sec. 21J — the radius vector of the point M . Let us denote it by r. The ratio — — r. Thereforo r = a — e-.r | Denoting the second radius vector of (he point M by r,. . l / / ' , = r , , we obtain

r + r, = 2«. S e c. 2 0 . H y p e r b o l a Definition. A hyperbola is the locus of points in a plane, /or each of which the difference of its distancesj/ron^'liro, giern points (fori) is constant. " L -. "v A

-

M

In accordance w ilh the definilion, if F, and F (Fig. 37) are the given points in the plane called the foci of the hyperbola, and M is an a rb itra ry point of the hyperbola, then the difference betwcen the distances M F X and M F (where M F t > M F , or, possibly, M F i < M F ) is constant and is taken equal to 2a: \W F l--M F \= 2 a . (a > 0 ) . M F i & n d M F are called the radii vectors of the point M . F tF is called the focal length and is taken equal to 2c: F tF = 2c. It is evident from the triangle F,M-F th a t I M F t — M F | < / ', / • ', i.e.,

2a a . If - a < r < + a, then to the values of x there correspond imagi­ nary values of y, and, therefore, if we draw straight lines x = — a and x = a (Fig. 38), there will be no points of the hyperbola in the infinite region enclosed between these lines. Thus the hyperbola docs not intersect the axis Oy. If £ = ± a , then xl - a J = 0 and y = 0. The points A (a, 0) and A t ( — a, 0) are the points of inter­ section of the hyperbola with tho axis of abscissas and arc called the vertices of the hyperbola. The chord A , A = 2 a is called tho real axis oT the hyperbola. To each value of x in the intervals — oo < x < — a and a < x < -{- co there correspond two points of the hyperbola located on both sides of the axis Ox and a t a distance from tho axis equal to the absolute value of y , since to each value of x in these intervals there correspond two values of y equal in S3

magnitude and opposite in sign. The hyperbola is a curve sym­ metric about the axis Ox. When .(• increases from a to + o o , the absolute value of y increases from zero to infinity. When x decreases from — a to — oo, the adsolule value of y also increases, taking successively the same values as in the case o f .th e increase of x from a to + oo, because x is contained in equation (1) only in the second power and, therefore, K ( — r) * - « * = / ( + i) * - a * . {\x\> a ). Thus, a hyperbola consists of two branches of the same form, symmetric about the axes Ox and Oy, one of which is located to the right of I ho straight line x = a and the other to the left of the straight line x = — a, both branches extending to infinity. On the axis Oy upwards and downwards from the origin O we lay off a segment of length b. From formula (XXXIV) it is evident that this segment is a leg of a right-angled triangle in which the hypotenuse is equal to c and the other leg is equal to a'. The segment B tB = 2b is called the imaginary axis of the hyperbola. The points Zi, and B are called the imaginary vertices of llie hyperbola. 2 \ Since in the equation of the hyperbola (XXXV) the current coordinates x and y arc contained only in the second power, if a point (x, y) belongs to the hyperbola, then the point ( — x, — y), symmetric about the origin O, must also belong to it. Thus, the origin O is the centre of the hyperbola. 3 . If we lake the segment B {B = 2b for the real axis and the segment . l , . l = 2« lor the imaginary axis, the equation of the hyperbola will lake the form

and the hyperbola itself will lake the form represented in Fig. 38 by the dashed curves. Two hyperbolas, the equations of which are . x2 1/2 , x: n- - £b- = i and -pr — -rs-= — 1, are called conjugate. S e c . VZ>. P l o t t i n y a H y p e r b o l a 1". By continuous motion (Fig. 39). Given 2a and 2c. Take a ruler and fix a thread to one end. Tho other end of the ruler and thread fix loosely (with pins, for example) in the foci F t ( o r rod)

and F. The length of Ihc thread must be such th at the difference between the length of the ruler (/■',Ar) and th a t of the thread (F M N ) is equal to 2a. W ith the point of a pencil make the thread ta u t so th at the ruler coincides with the straight line /•' then the point of the pencil will be in the vertex A of the hyper­ bola. Then draw a curve, keeping the thread tau t all the lime

f V r 2W /

F

C,CZC3 X

Fig . 39.

Fig. 40.

the pencil moves on the paper. In this manner one branch of the hyperbola may be drawn in two steps: first on one side of the straight line F tF and then on the other. To draw the second branch of the hyperbola, transfer the centre of rotation of the ruler from focus /•', to focus F and fix the free end of the thread in focus 2°. By points (Fig. 40). Given 2a and 2c. On a straight line lay off a segment F tF = 2c and then halve it. From the centre O thus obtained lay off on F,F sc•gtnenls O A t and OA, to the left and right of O, each equal to a. They will form Ihc real axis A ,A of the hyperbola. On the extension of the real axis take a number of points C2, Ca. . . . to the right of the focus F (or to the left of the focus F t). W ith the foci F and /■’, as centres, strike arcs (each lime one above the straight line A ,A and one below it) first with radius equal to A xClt then with radius equal to C\A. The intersections of the arcs will give four points of the hyperbola (labelled 1 in the figure). Points 1 belong to the hyperbola because the difference between the radii vectors of each of them is 2a: | I f \ — 1F | = | A ,6’, — C\A , = A ,A = 2 a . _ Performing similar operations for the points ( \ . C3. . . . each time wo get four new points of the hyperbola (labelled 2, 3, . . . in the figure).

Having constructed a sufficient number of points of the hyper­ bola in the mauncr outlined above, we draw a smooth curve through them cither by hand or w ith the help of a curved ruler. 3°. When the semiaxes a and b are given, i t is first necessary to find c. To do this it is sufficient to construct a right-angled triangle, the legs of which are equal to the semiaxes a and b. The hypotenuse of this trianglo will be c, since from formula (XXXIV) it follows t h a t a* + b* = c*. S e c . 30. A s y m p t o t e s o f th e H y p e r b o la Let us examine tho relative positions of the s tra ig h t line (Fig. 41)

and the right branch of tho hyperbola in the first quadrant. We represent llic equation of tho hyperbola (XXXV) in the form: j/= +

(2)

We will regard x ns an arb itra ry abscissa in the interval « < . ' ■ < ~ oo. Substituting this value of x into the equation of the s traig h t line ( 1) and the equation of a branch of the hyperbola (2), we find the ordinate of the point whose abscissa is x, which point lies on the straight line (1) and the hyperbola (2). Let us denote their values by ya and t/i„ i.o., b

l/a = 7 J'.

b m/" «i

•»'

>Jh = - V x 2 — a--

Since a ' - > 0 , x > | ' r — a ! and ya > V h. The difference between the ordinates is a3

Ua — y>, = or ya — Uh

b ( x — Y x t — a g)

a Rationalising the numerator of the fraction, wo obtain >ya — >Ju =

(x —

x - — qg) ( r - t - V ^ j 2 — a*) a (x + \ / ~x l + a?)

b (xZ — z t + gZ) _ a ( x - j - l ^ x - — a*)

ab x ^ - \ x - — a-

The numerator ab of this fraction is a constant, while the magni­ tude of the denominator x - r \ rx - — a3 depends on x and increases with x. Consequently, the difference ya — y^ decreases with increase in ./. When x increases indefinitely, y„—yt, tends to zero. r,r,

Let point P on a s tra ig h t line and point M on the hyperbola correspond to some value of x. Now ya — yi, = M P . Drop a perpen­ dicular M Q on the s tra ig h t line OP from point M . In the right-angled triangle M Q P Z. P M Q = Z :(OP = a, e > 1. From formula (XXXIV) it follows th a t c = l ^ a 2 + 6*. Substituting this value of c into the eccentricity for­ mula, we get f a*- 6*

(XXXVII)

It follows from this formula that with a constant real axis 2a and Fig. 43. decreasing imaginary axis 26, the eccentricity of the hyperbola ap­ proaches unity. At the same lime, the smaller the value of 6, the smaller the angle formed by the asymptotes ^ the ratio -£■= tan | a | —|b | or |a — — |a |. Indeed, let a — b = c. Hence, a = b-j-c. From tho preceding arguments, Solving this inequality for |c |, we gel \c\>\a\-\b\ or

|« — 6 | > ! a | —16|. The absolute value of a number doos not change with change of sign. Hence it is always true that |n — i | = I 6 — a |. Hut from the foregoing |i> — a \ > \ b \ — | a [. Hence the following inequality is also true: \ a - b \ > \ b \ — \a\. Note that the difference |« | —' fcj or |6J —|a ; can be uegative. The absolute value of a product is equal to the product of the iihsolute. values of the factors, i.e., \ a b ' = \a\-\b]. 3.

4 . The absolute value of a quotient is equal to the quotient ob­ tained by the division ol the absolute value of the dividend by the absolute value of the divisor, i.e.,

5. The absolute value of a power with positive integral exponent is equal to the same power of the absolute value of the base, i.e., | 0 "i = i « r Properties 3 to 5 follow directly from the properties of multipli­ cation and division. Sec.

44. I n f i n i t e l y S m a ll Q u a n tity (In fin ite s im a l)

In engineering and in natu ra l sciences, one particular kind of variable plays an especially important role, namely, the infi­ nitely small variable (infinitesimal). 1°. D efinition. A variable is said to be infinitely small, if, from a certain moment onwards in its process of variation, the absolute magnitudes of all subsequent values of the variable ultim ately be­ come and remain smaller than any positive number e. 2°. Examples. 1. The side of a regular inscribed polygon is an infinitely small q uantity when the number of sides of the polygon is doubled indefinitely, since in this process the side can be made as small as desired. 2. The fraction is an infinitely small quantity when the abso­ lute value of x increases indefinitely ^ e . g., o r T = r rio . - j o o

’ = r!d d d -

In d ccd -

no

= — , —^ m a lle r

. how

sm aH

the given positive number e, a time will come, as | x | increases indefinitely, when, from Ibis instant onwards, | a-! will be greater than — : e 1i x |1> -e and |I --I i < e. 3. The fraction — , where p is constant and x increases indefiX 1 nilely, is an infinitely small quantity, since it is only necessary to take | x '| > —i e to obtain |I x^ |I < c. 3°. Wo shall denote infinitely small quantities, or infinitesimals, by the Greek letters u, (J, y. • • • By definition, u is an infinitesimal if |u |< e where e is any given small positive number. 4°. An infinitesimal should not bo confused with a small num­ ber. Any small number 0 is non-variable, and soino other

positive number e can always be found such th at | c | will not be less than e. Hence, no small number c, not equal to zero, can be called an infinitesim al.

S e c . 45. V a r i a b le Q u a n ti t ie s , B o u n d e d a n d U n b o un d e d 1°. Definition. The variable x is called a bounded quantity if> from and after a certain moment, |* |< m (1) where m is some positive number. Otherwise, it is called an un­ bounded quantity. For an unbounded quantity it is impossible to choose a number m, lor which (at a certain time) inequality (1) would bo fulfilled; on the contrary, an unbounded variable can have values for which the inequality j . r | > m holds for any m. 2°. Any given number may be regarded as a bounded quantity. 3°. An infinitesimal a is necessarily a bounded quantity since, from and after a certain moment, its absolute value not only becomes less than some definite positive number m but less than any assigned small positive number e, | a | < e.

S e c . 40. B a s i c P r o p e r t i e s o f I n f i n i t e s i m a l s 1°. An infinitesimal (infinitely small quantity) remains an infinitesimal on a change of its sign. 2°. Theorem. I f a and fi are infinitesimals, their sum or differ­ ence is also an infinitesimal. Proof. Assume that to every moment of approach of one of the infinitesimals to zero there corresponds a definite value of each of the infinitesimals being considered. (For example, when a = 0.1, 0.01, 0 .0 0 1 ,... P is, at the same moments, equal respectively to —0.01. —0.0001, —0.000001, . . . ) . In any case, no matter how differently the values of a and P may change in I heir approach to zero, there will bo a timo start­ ing with which the absolute value of each of the infinitesimals will remain less than — : 1« ! < | - and | P i < \ , and, consequently, their sum will remain less than e, | a | + | p | < e. 70

But | a + p | < | a | + | p | (Sec. 43, 2*). Hence

ia+

p|
J = / U) which is read “ y is equal to / of x ” or “y is a function of j " . the letter / hero signifying the dependence of ij on j , i.e., the rule under which, for every permissible value of the argument r, a corresponding value of the function y can be definitely deter­ mined. In accordance w ith the n otation y = /(.r), the fact that the length I of a rod is a function of the temperature i is written / = /(0 . and th at the volume V of a gas is a function of the pressure p (temperature being constant): V = 0. * Hcrc>

fy rnusl he less th an the radius of th e neighbourhood, 1. 105

In this case we say that at the point x = c the function has an infinite limit. '. Definition. + oo (or —oo) is the infinite limit of a function f (x) at the point c, when 1) x tends to c, and 0 < | x —c|, 2) for any large positive number N there exists a positive number d such that f ( x ) > N (or / (x) < —N) if | x _ c | < f l . In this example:

xa

>N

if

x*
0 , for any definite value of x 10' —0 < e

if

.r< lo g e.

Example 3. lim 10' = -‘-oo. Indeed, we may lako any positive •T-( t- '•) number .V, however large, and we will find that 10T > too

.X

if

x

> log A\

2°. Definition. The number A is Ihe limit of the function f (x) o> x — oc. If for a given small positive number e there exists a positive numbir A' such that l f ( x) — A I < e if r > A ' (or x < — A').

Sec.

68. S o m e O b s e r v a tio n s

1°. By definition (Sec. 06) Ihe number A is the limit of the function f (x) a t the point c when x —*c by any taw whatsoever. Sometimes i t happens t h a t the lim it of the function / ( .r ) at c

differs when x tends to c from the loft (i.c., remains less than c) and when x tends to c from the rig h t (i.e., is greater than c). For example, the function y = f ( x ) = ------ -—j— , where c > 0 , 1 + to*—c has the lim it 1 “ on the left” and the lim it 0 “ on the r i g h t ” (Fig. 80). Indeed, let us denote the value of the fraction —

by =,

If x approaches c from the left (x < c ), then the difference * — c approaches zero, remaining negative all the time. And z in this case will lend to — oo. Since lim 10: = 0, l i m / ( , ) = r i-0 = l . X c ) , then the difference x — c will approach zero, remaining positive all the time. And z 107

in this case will tend to -|-oo. Since lim 10: = + a o , the fraction l 1 + 10*

has 0 for its limit: lim f (x) = 0 . 1-K *>C

When the limits of a function at the point c “ from the left” and “ from the right” differ, the function cannot, in the ordinary sense of the term, be said to have a limit at c. 2°. The limil of a function f ( x) at c —number A — should not be confused with the number /(c), which is the value of the

function / (r) at c. The following cases may occur: 1) the number /(c) docs not exist, whereas the function has a limit lim / (a) = .4. In Fig. 81, the point x = c is taken out of the I-*C

graph of the function y = f (x). Thus the function f ( x) has no value at x — c, but tho limit lim /(x ) exists, and on the graph it is equal to OA\ 2) the function has a value at the point x = c (Fig. 82), /(c) = — CM, but the function has no limit a t x —>c\ 3) the value of the function at tho point c, /(c) and the limit of the function at that point, A, both exist, but A ^ f ( c ) . (Fig. 83). S r r. (}{). C o n t i n u i t y o f a F u n c t i o n

I3. We shall make cloar by a concrcto example the meaning of the concept “ continuity of a function at a point” . Let it be required to coinpulo tho valuo of tho function y = xa when .i - - l 2. Now [ 2 is the limiting valuo of a sorios of its 108

approximate values: x = 1.4, 1.41, 1.414, . . . or i = 1.5, 1.42, 1 . 4 1 5 , . . . as the degree of accuracy of calculation increases: .X First approximation Second approximation Third approximation

1.4 and 1.41 and 1.414 and

I I 1.5 1.42 1.415

JC'» 2.7 and 3.4 2.80 and 2.87 2.827 and 2.833

In Fig. 84, the values of the function y = x 3 are represented to a first approximation by points A/, and A/j, to a second approx­ imation, by points M 2 and A/j, to a third approximation by M 3 and A/j, . . . . and th ro ­ ugh each point are drawn y straight lines parallel to Ox and Oy. The straight lines drawn through M 2 and A/, a* parallel to Ox are more closely spaced than those drawn through A/, and A/j; and, similarly, those drawn U(V2' iP) S through M 3 and A/j are still w more closely spaced than those drawn through M 2 and A/', and so on. The distance between the straight lines parallel to Ox can be made as small as wo please by sufficiently brin- 7 gmg together the straight F ig. 84. lines parallel to the axis t'Ki and the point M will always lie in the rectangle thus formed. Consequently, the given function y = a* has the value ( \ ^ 2) 3 at the point x ■ — \ 2 and has u limit also equal to ( | / 2 ) s. This properly of the function is decisive in defining the continuity of a function a t a point. 2 . Definition. 1. The function f (x) is said to be continuous at the point x = c if the lim it of the function at c is equal to its value at this point, i.e., if lim f ( x ) = f (c)

(

111 ) too

Thus the condition for continuity of a function / (x ) a t a point x = c is that: a) the value of the function at x = c be some definite num­ ber / (c); b) the limit of the function f (x), when x tends to c on the left as well as on the right, be one and the same definite number, l i m /(*)*; x-*c c) the numbers lim /(x ) and /(c) be equal. .T-*C

2. Every point where the function is continuous is called a point of continuity of the function.

Fig. SS.

Fig. 86.

3. A function is said to be continuous over an interval if it is continuous at every point of the interval, including the end points. 4. A point where the condition of continuity of the function is not fulfilled is called a point of discontinuity of the function and the function itself is said to be discontinuous at the point. 3°. Non-fulfilment of the condition of continuity (III) may consist, for example, in Ihe following. 1. The limit / (.r) at Ihe point c is not the same on the left as on the right. For example: a) the function f ( r ) = -----j - , where 1 + i0*"c c > 0 (Fig. 80), at .r = c has a limit 1 on tho left and a limit 0 on tho right (Sec. 68), hence r = c is the point of discontinuity; b) the function / (r) — — (Fig. 74) at .r = 0 has a limit — 1 on the left and a limil -| 1 on the right, .r = 0 is the point of dis­ continuity of the function. • Tlir function / (j) is one-sided if tlic point x = c servos as the boundary point of Ilie domain of the function. lit)

2. The lim it f (x) a t c is not equal to the value of the func­ tion when x = c. For example, the function / ( j ) = | a | if r is any real but non-integral number, and 0 if x is an integer: it has one and the same lim it 3, 2, 1, 1, 2, 3 on the left as well as on the rig h t a t points — 3, — 2, — 1, 1, 2, 3, respectively, but none of these lim its is equal to the value of the function a t these points, which value is zero. The graph of this function (Fig. 85) is a broken line consisting of the bisectors of the coordinate angles in the second and the first quad­ rant, the points having integral values of the abscissa “ taken o u t” of the bisectors. The values of the function a t these points are situated on the i-axis. When x = — 3, — 2 , — 1, I, 2, 3_, etc., the function is discon­ tinuous. 3. The lim it of the function / ( x ) at c is infinilc. For example. the function y = — (Fig. 86) a t the point r = 0 has an infinite. not a finite, lim it (Sec. 66, 5C); .r = 0 is the point of disconti­ nuity. The function y limit — oo on the and no point on x = 0 the function

— 2l 1 = 2 (Sec. 67) at .r = 0 has theleft and the limit + cd on the right (Fig. 87). the graph represents the point i = 0. When is discontinuous.

4°. The foregoing instances of discontinuities in functions occur in engineering practice. For example, girders used in construction work often carry a load like this: to the left of a given cross-section I lie load is d is­ tributed cvonly lengthwise and has one value, to the right it is also distribu­ ted evenly but has quite a different value. Thus at the given cross-section of the girder there is a jump in the linear distribution of load along I lie girdor. The law of this distribution of load corresponds to a discontinuous function, and tho jump in linear loud distribution corresponds to the point of discontinuity of tho function. I.oads concentrated at individual points of a girder may correspond to isolated points of the graph of distribution °* l°ai* al°nK the length of the girder. 5 . Equality (III) may be replaced by two inequalities: l/( * ) — / W ! < e if | i —c | < f t and tho continuity of the function of the point is represented geometrically as follows (Fig. 88). Construct on tho axis Oy a neighbourhood of the point /(e) of any small radius 2*. If the function y ~ H-r) is continuous at c. then on the axis Ox ono can find around the point c a neighbourhood of radiua ft such that all its points arc mapped into the neighbourhood 2e of

111

the point /(c). In other words, if wo draw two strips: one of width 2e bounded by two straight lines parallol to the axis Ox, y = f (c) + e and y = f (c) —e, and the other of width 26 bounded by two straight lines paral­ lel to the axis Oy ( x = e + A and x = c —6), then all the points of the graph of the function lying in the second strip belong also to the first strip. This condition is not satisfied in tho case of a discontinuity in the func­ tion; namely, a strip of some particular width 2e bounded by straight lines parallel to the axis Ox w ill be found such that a point of the graph lying

Ui A

* 1 1h 4 B II H ** i