/02/18

Transcription

1 3 09/0/8

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3 i III,,,, III,?,,,,,,,,,,,,,,,,,,,,?,?,,,,,,,,,,,,,,!!!,? 3,,,,

4 ii,,,!,,,, OK! :!,,,, :!,,,,,, 3:!,, 4:!,,,, 5:!,,! 7:!,,,,, 8:!,! 9:!,,,,,,,,, ( ),, :, ( ), ( ), 6:!,,, :... : 3 ( )...

5 iii,, ( ) : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ),, (ˆ ˆ;),,,,,, ˆˆ!,,,.,, ( ),,, OK,, : nasahara.kenlo.gw u.tsukuba.ac.jp, : shigen remedial/,, 0 (I, II) TA,, e-ptex p , emath (Version f0507c), GNUPLOT (Version 4.6), LibreOffice (Version 4..8.), Ubuntu Linux 6.04LTS 3 8

6

7 v i , ! (SI) ,, dimension check !

8 vi ,

9 vii ( ) :

10 viii : :

11 ix !

12 x 47

13 ,,?,??,?,,,. = a, b, a = b (.), 3, : a, a = a (.) a = b, b = a (.3) a = b b = c, a = c (.4)!,,? (.) (.4),,,,, (.) (.4),, A (.5), A = (.6), (.3), = A (.7), B, B = (.8) =B (.9), (.4) (.6) (.9), A = B (.0), A B!,?.,?,,,, (natural number) ( ),,, 3,., + ( ) + =,, ( ),, =, :=

14 (: ), :=,, OK, := + (.) ( + := ) 0?, 0 *, +3, + 3 = ( + ) + ( + + ) = ,, 3, 3, 3, a, b, a b a b,,,,!.3 a, b, a = b + x x (.) a b, a b, 5 (3), 5, ( ) (integer) ( ), 3,,,, = 0, 0, 3,,,,, *, 0,!?,, 3,,, 0,,, 3,.4 a, b, a = b x x (.3) a b, a b a b a/b, 0 (0 ),, 6 3 (), 5 4, (0 ), n, m ( m 0 ), n m (.4) (rational number quotient number) ( ), n, n/,.5,, ( ) π π, 3.459, n, m, n/m *,,,.44356,, n, m, n/m *3, π,, (irrational number), (real number) *4 * *3,, *4,,,,,,!,

15 .6 3.6!,,?, A, A B, B A,,,,,,,,,,,,,,, 3, 4,,, 3,,,,,,,, 3?, A 3.4 B, ,, B A,?,...,,,,,!,,,?...,,,,, 3...,,,,,, π,, ( π = ) 7 +,,, 4!, OK( P.5 ) π,, 4,,?...,,,,,? πr, π OK, r,, r,, r,?, 0,,, 3,, =, +,,,,,, π, e, cos,

16 4 sin,,,.7,, 0, 0, 0.000, /0.000 = 0000, 0.000, 0000, 0, 0, ( ) 0,, (infinity), ( ),,, 0 =, 0 = (.5),,, 0 5?... 6,,...,, ( ) ( ), 7?..., Q.E.D.,,,.8,,, 4, a, b, c,, a + b = b + a (.6) (a + b) + c = a + (b + c) (.7) a + 0 = a (.8) a 0, a + ( a) = 0 a (.9) a b = b a (.0) (a b) c = a (b c) (.) a = a (.) a 0, a, a (/a) = /a (.3) a (b + c) = a b + a c (.4) 0 (.5) (.6), (.0),,, (.6), (.0) (.7), (.),,, (.7), (.), (.4),,,,, a, b, a b a b,.3.8,, ( 3) ( 5),,!,,, (.6) (.5), (.6) (.5),, (.6) (.5),, (.6) (.5),,, (.6) (.5)?,, :. x, x 0 = 0?, (.8), a + 0 = a

17 .8 5 x, x (a + 0) = x a (.4), x a + x 0 = x a x a, x 0 = 0 ( ).3?, ( ) ( 3) 3?,, ( ) ( 3 + 3) (.4), ( ) ( 3 + 3) = ( ) ( 3) + ( ) 3 (.6), (.9) = 0,., 0, 0 = ( ) ( 3) + ( ) 3 (.) 3 = 3,, 3 = ( ) ( 3) + ( ) = ( ) ( 3) + ( + ) 3 = ( ) ( 3) = ( ) ( 3) (.7), ( ) ( 3) = 3 (.8) 4,,,,,,, a, b, a b a + ( b), a b (b 0 ), a (/b), (.9) b, (.3) /b (b 0 ).4 a, b, ab = 0, a = 0 b = 0 (.9) :, ab = 0 a 0, (.3), /a ab = 0, b = 0, b 0, /b, ab = 0 a = 0, a 0 b 0, a, b 0 ( ) 5 ( ):, ( ),,,, (.6) (.5) 8,,... (.6) (.5),,,,,,,,,, () x 0 0 () a, b, ab = 0, a b 0 ( ), 6, : () 35 5,, () A, 6 98 A,,, () 34, () 8,, (), 35 5

18 6, 35 36? (), A 99?,,,.9,,, a, b, a b ab, a b,, 3 3,, 3 ( 3 OK) a 3 a3, 3a, a 3, a 3 OK,, ( ), ( (.0)), ab ba OK,, 3?, 3=6, 3 =6,? 3=6 3 =6,,, ABC ( ) a 3 3a adcb, abcd,, ABC, ab + bc + ac (.30), a, b, c, a b, b c, a c,,,, ab + bc + ca ac ca (.30), ABC, a, b, c, a, b, c,...,, /, a, b, a b a/b a b, /, /ab /a / 3 /3 4, / (/ab b),, /(ab) /(a) /( 3) /(3 4),, (/a)b (/)a (/) 3 (/3) 4,, b/a a/ 3/ 4/3,,, ab b a a a ,, /,, /,, / /,,, ( ), { }, [ ] ( ), [{(a b)c + d}e + f]g (.3), ( ) { }, [ ], (((a b)c + d)e + f)g (.3) 4 3

19 .0 7,,,,,,, /, +,, + 3 ( + ) 3 + ( 3) + /3 ( + )/3 + (/3) 3 3,... ( 3),, ( ), ( ) 7,? () /a () 3 4 (3) ((x + ) 3)/4x : ( ), a, b, c,..., A, B, C,..., a, b, c,..., A, B, C,... x = 5 OK, x= 5,.0, 3 a, b, c, a + b + c (.33) abc (.34),, (a + b) + c, a + (b + c)? (ab)c, a(bc)?? (.7) (.),,,,,,,,, 8 4 (.35), (8 4) 8 (4 ),,,, 8 4, 8 4 (.36), 8 (4 ), (8 4),,,, 8 + ( 4) + ( ) (.37), +, 8 4,,. x n, x n, x n ( ) 3, = 8, ( ) ( ), x n n 3 3, (x, m n ): x m x n = x m+n (.38) (x m ) n = x mn (.39) (.38) x m, x n,, x m + n (.39) x m n,, x mn (.38) (.39),,

20 8, 3.3,,, m, n?,, (.38), m = 0, x 0 x n = x 0+n = x n (.40), x 0 (.40) x n (x 0 x n 0), x 0 = (.4), 0 0, (.38), n, m = n m, (.38), 3 3, 9 / = 9 = 3 ±3 (, ), : = ( ) 3 = ( ) 5 = ( ) :,,, : 5 4 4,,.5 : 4 3/ = (4 / ) 3 = 3 = 3 = 8 ( ) x n x n = x n+n = x 0 = (.4) 8 : x n, x 0, x n = x n (.43),, 3, /( 3 ), /8, (.39), n, m = /n m, (.39), (x /n ) n = x n/n = x = x (.44), x /n, n x x n, n n x /n n x, 8 /3 = 3 8 =, x /,, x, x n,, x /n n x, x n, 0, *5, 9 ±3, *5,, x, x n (!) () 5 () (3) (4) (5) 4 0 (6) : / = () / 3 () 3 5,, ( ), 9?... ( ; 3 ),.,,,, 3,,,

21 .3 9,,.,, ( 6 ), 4 (.45) 0, e,, π,, x, x,, e x, exp x, : exp x, e x e x,,.788 ˆ, : e 4,. (3) Shift, () *6, 4, =, ,, () (4),,,,, 4,, Shift, () ln,!,.3,. (), ˆ (, ˆ,, ˆ, x y ) 4.5, =, () 3.5 ().0 00 (3) 3 5 : 3, /3,, ,,, e, e =.788 (.45),. (5), EXP,, e x! 5 () exp () e ,,,,!.4 a, b, a b (a x = b x *6 () ln ( ), Shift, ln, e x,

22 0 ), log a b (.46) (, a ) (logarithm) ( ) *7 (.46) a (.46) b.6 log 8 = log = 0 0 log 0.5 = /, ( ): () log 4 () log 3 8 (3) log (4) log (5) log (6) log 0 0, e =.78, 4 : () () (3), log x,,, log 0 x,, ln x ln log natural,, *7 a, b,,, b = 8, a =, a b 3, b = 8, a =, b = 8, a =, a b,,, a b,, log x (log e x ) ln x I, II I,,! ln In n... l,, (l)!, log x OK, log x log x?... :, (,,,,, 0 e,,, ln (. ()), ln 3, ln, 3, =, 3, ln,,, log (. (4)), log, log 0, log,, =,, log 5 5 () log 0 () log (3) ln (4) ln 0.5, ( ),,, (, ),.0 m s, (

23 .5,!!,,, a, a 6., ( ), 3., :, :, 3: 3, 3,, C c, O o, P p, S s, V v, W w, X x, Z z, h n ),,, 3, (, ) a x, a, b, c,, x, y, z, A, B, C,, X, Y, Z,, a, b, c,, x, y, z, A, B, C,, X, Y, Z,. 3 ( ), a, b, c,...,,, A, B... A AB B A, B, AB,,, AB, A B ( )?... AB A, B, C D, CD AB ( ) AB = CD, CD AB ( ) a a = 0,, ( ) 3,,?..., α, a a α, a

24 α, αa,, a a a,, : : a, b, c,, x, y, z, A, B, C,, X, Y, Z : a, b, c,, x, y, z, A, B, C,, X, Y, Z,,.3 a, b, a = b + x x (.48) a b, a b, (.) : a a b a, b, 0 α, b αa = b (.47), a b, ( ), ( ) : a, b, a b, a b, a b, a + b b a a + b, : a, b, a, b ( ), a + b, a a + b b ( ),, ( ) :, a b (.48) b a?, a b a( ), a b b( ), 7 a, b, a + b, a b, a + 3b, b a,,.3, (x y ), O *8, ( ) x y,, (3, ), *8 O origin

25 .6 3, (z ), (3,, ) 3,! ( ),,,.7 a = (, ) b = ( 3, 4), 5a + b? 5a+b = 5(, )+( 3, 4) = (5 3, 5 +4) = (, 4) ( ).4 A B P 8 A, B (4, 5) (3, 6), () AB :. () AB :..6, ( ),,, O, P, OP,,,,,,,, :.8 A, B, a, b, O, a = OA, b = OB,, AB = b a, AB C c (c OC ), c = a + b (.49), AB m : n P (, AB, AP:PB= m : n P),? P p (.4), p = OA + AP = a + = na + mb m + n m AB = a + m + n m (b a) m + n (.50) m = n = (.49) ,,, 07 8,, 675 5, , ,,,,, 3 ( 30 ),,,,, 675 5,,,,, 6, 7, 5, 5,,,, ( 5),,, (, ),,,

26 4, 0, A,,?, A 00, ,, 0,, 0, A, 0, A 00, 90 0,,,, 0( 0 0),, 0, 0 0,, 675 5, 0,,,, , 00.0, 00,, ( ) 0, , 0 0,,,, 00.0,, 0, 0, 0 4, 4,, 00,, 0.00?,,, 3 0, 0, 0.00, 9, () 5.3 () 30.5 (3) 5300 (4) 0.030,,,, 0, 00,. 0 3 (.5), (.5) (.5) (00 ), (.5),, 0 (0 ), ,,.8, ( ), 4.56., :, , 6 6.,,,,,,,,.5 :.5, 5.76 ( )5, (0. ) 7,.,, 6, 0. 7, 6 8,

27 .8 5,, 5 7, 3 6,,, 6 5.8,, 5.6, 3 9,,, 6,,...,,,,, :, ,. 0.,,, , 0.,, 5.76, 0., 5 7, 0.0, 6, 5.8,,, 0,,,,,, 5 4, ,,, :, ,.,,, 3,,, ,, 4 (, ),,,,, () () (3) 00. 3, 4.56.,,.6 :, 5.47 ( )5, (0. ) 4, 7, 0. 4, 3 5,, () () (3) 00./3.9 6 cm 3,?, (6 cm)/3=5.333 cm, 6 cm, 3, 5.3 cm

28 6...! 3 3?,, 5 cm? 3, 3., 3,, 0,,, 3 4, 0, 3, , 0,, 6 cm,,, 0,,,,, , 3.47±0.0,, ( ),,, 3? 4?,,, 6 3 cm,,? 6 3 cm 5.3 cm,,,, ( ),, : = , 3, 4 (3), = 9.99,, : = , 3, 4 (83), = 0.0,?, ( ),, (9.99) 0.0, (0.0) 0., 0!? ( ),,, = 0.083, 4, 0.03? 6.4 cm, cm? () 6.4 cm cm =,, () 6.4 cm =,, (3)?, 4.,,?, ( 4. ) , , 4.!? , 4.? , 4 5, 4.!? 3, ±,? () 3.±0. () 3.±0. (3) 3.±0.6,

29 .9 7,,,?, 6...,,.9,,,.8 (λ, κ, µ, ν, ξ, π, ρ, σ), 4.7,.8,.9,.0, 3.9 (τ, ϕ, χ, ψ, ω),.0 (Γ,, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω),.7 (α, β, γ, δ, ϵ, ζ, η, θ), 4 : γ r, δ 8, η n, κ k, µ u, ν v, σ 6, χ x, ω w

30 8 7,?...,, 8 Ω?... Ω, Ohm A ( Ampere ) Ohm, O,, O Ω, 9, ι o?..., 0 ξ η,?..., x, x 0, (0 9 ) 3, x 3, ( )? () 0? () ( )? (3) (.46) a? 3 ABC, BC D, 3, AD :, : A, B, C a, b, c, 3 r, r = (a + b + c)/3 ( ),,,, A A, A (,,...), A 7 () a () 3 ( 4) (3) {}, x 8 () 3 () / = /4 (3) 0 = /00 (4) 9 / = 9 = 3 (5) (6) () 3 / () 5 /3 0 (?!) ().33?? ().704?? (3).709?? 3 5 () () () () 4 (3) (4) 3 (5) (6) () 0.30 (). (3) (4).30 7! 8 A, B a, b () a + b 3 = ( 3, 6 ) 3 () a + b 3 = ( 0 3, 7 ) 3 9 () 5, 3 (),, 3, 0, 5 5 (3) 5, 3, 0 (4), 3, ().9 () 3.5 (3) 87 () 4 ().8 (3) 7.7 ().8 cm () 3 cm (3) ( ) 3 () , 0., 3. () , 0., 3. (3).6 3.8,, 3

31 9., 3x = x + 4 x = x =,,, 00 m, *,, 60 cm.3, 60 cm, cm, 60, 60 cm 60 cm ( ), 60 cm, cm 60,.3.3,.3,, (unit) *, ,,,,,,, ,,,, *!!,,, *, ( credit ) 80 mg,, 80 ( 0 mg ),, 800 mg.3,, ( ) ml, 00 g,,, ( ),,,,,,, 0, (unit) (credit)?...,?,,,...?, ABC AB,?..., AB cm, m, km,, F, m...,?... F m m, m = 3 kg m = 3000 g

32 0, m, kg g t, m, 6 m t, 0 5 5, 0 cm 5? t =3? t =3?..., t t, t t = 3,,,,, 0 cm +.3 m=?, 0. m +.3 m =.43 m (.) 0 cm + 3 cm = 43 cm (.),, 0 cm.3 m?, cm!!, (.), m, =(0. +.3) m,,.43 m, ( (.4))!, 0. m.3 m,,,!., 0 cm 5,,,, 0 cm, 5 5 ( )?,,, 0 cm 5.3,, :.4 m 3 m, m 3 m = 6 m (.3), m,, m.5, 00 m 0 00 m 0 ( ) = 5 m/ (.4),, m/ /,, m/ m OK /,, m/ OK?..., / ( ) 5 m/ , m,......, 6 3,?, 6 (3 ), GDP, 4, 4, 4 /( ).7,

33 .4! 40 m, 40 m, 40 m/, 40 m.8 Sv ( ) Sv/h ( ), µsv/h µsv/s, /h /s,, 3600 ( ),!,,,.4!, (.3), : 3 = 6 m (.5),,? OK, cm 3 cm?, 3 = 6 cm (.6) (.5) (.6), ( (.4)), 6 m = 6 cm (.7)!, (.5) (.6),,,, (.5) (.6), *3,,, 3 = 6 [m ] (.8), [m ],,, (.3),.9 0 g 600, 900 g?, 0 g 600 g...,,,, : g g 900 g = = 4500 (.9) 0 g (.9),,,,,,,,, 0 g 900 g =... (.0) 600, g /,,.0,, = = =,, (.) (,, ) (,,, m/, m,, 3 m/ 5, 3 m 5 = 5 m (.) *3, m,

34 , m/ /m,,, ( ),,, 3,, 3,,...,,,, 6, () () m 3,? (3) 4900 ha,,000 t ( ),,? kg/m (4),, m /kg (5), 60 m,? (6) 60 kg 30,? (7), 0 ha?,,.5, 600 m, 4.4 /kg, ,,, d S, d S, (cm m ) d S S/d,, (, π )!,,, L, L = 5 cm cm ( cm), L/cm = 5 (.3), L cm,,, 5 (.3),,,,.6 (SI),,,,, Le Système International d Unités SI,, 7 : : m ( ) : kg ( ) : s ( ) : A ( ) : K ( ) ( ) : mol ( ) : cd ( ) 7 SI SI

35 .6 (SI) 3, SI m, m s, SI SI, 7? SI? m s m/s, /, / ( )!, kg/m s, s,,, kg/(m s) kg, m s, kg m s kg/m /s, SI,, kg m s m kg s OK, s kg m OK, ( ), (,, ),,, : min (minute,, ) min = 60 s h (hour,, ) h = 60 min = 3600 s a ( ) a = 00 m L l ( ) L = 0 3 m 3 cc (cubic centimeter) cc = cm 3 = 0 6 m 3 t ( ) t = 0 3 kg, (l),, (l) (L) 8, SI () min () h (3) a (4) L (5) cc (6) t (0 ), 0,, 000 m km, 0.0 m cm 0 3 k, 0 c,, (SI ) P p, M m, 0 5 P 0 5 f 0 T 0 p 0 9 G 0 9 n 0 6 M 0 6 µ 0 3 k 0 3 m 0 h 0 c 0 d. µm (, ) ps ( ), 50 kg 50 km 50, cm, 5 mm 5, : h (hour), m,, hpa mg,, 9 SI 0? () G () M (3) k (4) h (5) d (6) c (7) m (8) µ 4 SI g ( )? kg g k, kg g... kg SI (m s ),, kg k kg, mg ( ), (kg) (g) 3, 3,, km, (km), k(m )!

36 4. km = (km) = (0 3 m) = 0 6 m (.4) dm 3 = (dm) 3 = (0 m) 3 = 0 3 m 3 (.5) cm = (cm) = (0 m) = 0 4 m (.6) 0.03 km = 0.03 (km) = 0.03 (0 3 m) = m = m (.7). 5 km =000 m, cm 3 =0.0 m 3..., k c, m m 3, 5?, ab a (b ) km k (m )?..., km (km) 6,... km?,,,, km square kilometer km (square) kilo meter kilometer 7?... km km 000 m, 000 m = 3.6 m? ha, 00 a a, a=00 m, ha = 00 a=00 00 m =0000 m 8 ha, ka ( ) da ( )?..., ka, m = 0 5 m = 36. m,. (m km ) 30 : () m km () km m (3) cm m (4) m km (5) km m (6) cm m (7) m 3 km 3 (8) km 3 m 3 (9) cm 3 m 3 (0) dm 3 m 3 () dl m 3 () µm m (3) µm nm (4) mg kg (5) km ha : dl,, dl, (0.000 m 3 ) /3 = m 3 ( ): () km m () km 3 m 3 3, : () ml cm 3 () L dm 3 (3) kl m 3 (4) Gt Pg : L kg,,, L 0 3 m 3,, ( kg ), L, dm 3 ( ),, : : x = 5 OK, x= 5 : sin x OK, sin x : 5 m OK, 5 m : 5 m OK, 5m,

37 .7 5 : m s OK, ms : 3 kg OK, 3 k g, ms ( ) m s ( ) JIS *4, (, ),,,, [ ], 5.3 m/s, 5.3 [m/s] km h, m s 7.5 km h = 7.5 km h = m 3600 s =.08 m s. m s (.8) , s /s.4.0 g/s, kg/h.0 g/s =.0 g s = kg 3600 =.0 (/3600) h 000 kg/h = kg/h = 7. kg/h (.9) ( ), (h, s, km, m, kg, g ),,,,.3, 7.5 km h = 7.5 km h (.0) = 7.5 km 000 m h h km 3600 s (.) = km m h 3600 h km s (.) = 75 m 36 s =.08 m s (.3). m s (.4), (.0) (.), 000 m km h 3600 s (.5),, 000 m= km, km, (000 m)/km= h/(3600 s)=,, (.5),, km/h = m/s,..., 3600, 3600,,,, 7.5 km/h, ,,.4 34, OK () 340 m/s km/h ( ) () m/s km/h ( ) (3).0 g/cm 3 kg/l ( ) (4).0 g/cm 3 t/m 3 ( ) (5).3 g/l kg/m 3 ( ) (6) 0.05 kg/h g/s *4,,..., (),

38 6,, (),, 35,, : ha..., L... PET () ml () m 3 (3) mg (4) g (5) kg (6) t (7) µm (8) 0000 km (9) km,, (!), 3 t 3 m 3, 3 t = 3 m 3 (.6), 3 t = 3 m 3 (!) (.7) t, m 3, (.7), 3 t 3 m 3 ( 300 m 3 ) (.6),,,, 36, ( ),,, 0 kg 9 (.6) 3 m 3 =3 t, (.7)?... 3 m 3, 3 m 3 m 3 3,, 3 m 3.8, ( ),, m F,, a, F = ma (.8),,,, (.8),,,, (.8),, 30..., 3,?..., OK (.8), F force ( ), m mass ( ), a acceleration ( ),, 7 F = ma... F, m, a, 3 F a P.0,, ( )

39 .9,, 7, N ( ) (.8), F N, m kg, a m s, N = kg m s (.9) N, F = ma, 33 m s,...,, ( ) m s, (, ) m s ( s), m s, N,, /6, ( ) 0 N,,, kg, kg!?!?,,? (.8) (.8), F, m a, m a m,!, kg 0 kg,, kg, (.8),,,,,,,,,,, kg 9.8 N, kg, kgf (kg ), kgf := N (.30), kgf, kgf, kgf, kgf f kg,,, A 60 kg, A 60 kgf A 60 kg, kgf kg,, g = N (.7) 00 g, N,,, 00 g N ( 0.98 N), 00 g = N, kgf N?.9,,,, :,,, ( ), =, J ( *5 ) = *5

40 8, J, N, m, J = N m (.3) ( ) (.3), =, (SI J),,,, = /, Pa ( *6 ) = /, Pa, N, m, : Pa = N/m = N m (.3) Pa (.3), = /,,,,,,, = /, W ( *7 ) = /, W, J, s, W = J/s = J s (.33) W (.33), = / 8 W J... J W W=J/s, J=W s 35 W?, W=V A...,,, *8 W, W=V A, W V, V=J/C C, A,, A=C/s, V A=(J/C) (C/s)=J/s=W? 38 () () (3) (4) (5) 39 () kg 3 m s? () N 4 m? (3) m 0 N? (4) 00 W,? 40 : () N = J / m () Pa = J m 3 (3) J = Pa m 3 (4) J/Pa = m 3 (5) W = N m s (6) J = W s 4 J, Pa, W, SI (kg, m, s) 4 : () kw h J () hpa Pa, atm ( ) atm := 03.5 hpa (hpa h ) atm atmosphere atm *6 *7, 43 *8 = /

41 .0 dimension check 9 () atm Pa () atm kpa (3) Pa atm (4) 895 hpa atm, P, V, n, T, R, P V = nrt ( ), R = J mol K mol, 73.5 K, 03.5 hpa, L 45 R = J mol K, atm L mol K : J=Pa m 3 (cal), cal := 4.84 J ( ), cal g C,, J 46 () (, ) (), J kg K 36, g?..., g 37..., 70 cm.7 m, (3), 0 kg?.0 dimension check,,.5, m v, K : K := m v (.34) / m kg, v (m s ) =m s, kg m s J ( 4 ), K, J,, (!) ( ),, ( ), dimension check,, ( ), ( (.34) v ),,,,.9.0,, dimension check 48 (.5), dimension check 49,, m h, U *9 47 : U = mgh (.35) () 500 kcal J (), 0.5 m 3! K? g, g 9.8 m s dimension check *9,,,

42 30 50 r S V S = 4πr (.36) V = 4 3 πr3 (.37), A, (V = 4πr, S = 4πr 3 /3 ) dimension check, A,? * 0.!,,, 0 0, 0 m 0 ( ), 0 0,,,,!.6,, K ( ), 0 0 K,, 0 C 73 K, 0 K.7 ph, [H + ], : ph := log 0 [H + ] mol L (.38) : [H + ] = 0 ph mol L (.39) ph?, () mol L ph? () ph=5.6, (ph=7)? *0 S surface, V volume, r radius, ( ), (3) ph=5.0, ph=3.0,,, ph?,, C ph,,,,,,, (.3), dimension check,, 38,?..., C K, ph mol L 4 0 C 73 K, 0 C = 73 K (.40),, 0 C = 546 K (.4) (.4), 73 K = 546 K (.4)? 5,,,, SI : :,,,,, :,,,, :,,, : 6, ( ),,,,

43 .! 3,, 39...,? 5 6 () () (3) m3 = = = t 4900 ha = kg m =. 07 kg m = (/49) kg/m 0.43 kg/m (4) (3) /{(/49) kg/m }.3 m kg (5) 49 kg m m kg = 00 t (6) 60 kg (7) 0 ha.5 m kg 8000 m =.8 ha 4.4 kg = m.5 03 kg 600 m 4.4 kg = (= 500 ) 8 ( ), kg, m, s 3 () 60 s () 3600 s (3) 00 m, 30 ( ) () m=0 3 km m=0.00 km OK (4) m = (0 3 km) =0 6 km (8) km 3 = (0 3 m) 3 =0 9 m 3 () dl=0 L=0 0 3 m 3 =0 4 m 3 3 ( ) () m () m 3 3 ( ) () ml=0 3 L = m 3 =0 6 m 3 cm 3 =(0 m) 3 =0 6 m 3, ml = cm 3 (), (3) (4) Gt= kg=0 kg Pg=0 5 g=0 kg, Gt=Pg 33.0 g/s =.0 g s =.0 g s =.0 g s 34 ( ) kg 3600 s = 000 g h kg 3600 s 000 g h kg = 7. kg/h 000 h () 0 km/h ( km/h OK) () km/h (3).0 kg/l (4).0 t/m 3 (5).3 kg/m 3 (6).4 0 g/s OK! () ml... (0.5 ml) () m 3... (3) mg... (4) g... (5) kg... (0.5 kg) (6) t... (7) µm... (8) 0000 km... (3000 km) (9) km... (83500 km ) N 590 N 39 () kg 3 m s =6 kg m s = 6 N () N 4 m = 8 N m = 8 J (3) ( ) 5 Pa (4) ( ) 00 J 40( ) () J = N m m () Pa=N/m, N=J/m (3) 4 (.3) N (.9), J=kg m s m=kg m s (.43) (.3) N (.9), Pa=kg m s /m =kg m s (.44) (.33) J (.43), W=kg m s /s=kg m s 3 (.45)

44 3 4 () kw h = (0 3 W) (3600 s) = W s = J () hpa = 0 Pa = 00 Pa 43 () atm := 03.5 hpa = Pa = Pa () atm= Pa=0.35 kpa (3) (), Pa= /( ) atm = atm (4) 895 hpa = atm = atm 44 V = nrt P =.0000 mol J mol K 73.5 K Pa J = Pa =.44 0 J/Pa.44 0 m 3 = L =.44 L, J/Pa m 3, 40(4) 45 40(3), J=Pa m 3, m 3 =0 3 L, J=0 3 Pa L, 43, Pa = atm, J= atm L= atm L, R = J mol K = atm L mol K = atm L mol K 46 () m Q, T, Q/(m T ) (), g cal K, Q= cal, m= g, T = K, cal g K = 4.84 J 0 3 kg K = 484 J kg K 47 (, ): () 0 7 J () 5 K (3) 00 km 48 (.5),, 49 (SI ) J, kg m s, m kg, g m s, h m, mgh kg m s (kg m s ), 50 r, SI m A, V r m V, A, S 5 (.38) ( ), ph = log 0 [H + ] mol L (.46), 0 ph = ([H + ])/(mol L ) 5 (.39) ph=, 53 [H + ] = 0 ( ) mol L = 0 mol L () [H + ]=0.005 mol L, [H + ] ph = log 0 mol L = log mol L 0 mol L = log () (.39), ph=5.6 [H + ]=0 5.6 mol L ph=7.0 [H + ]=0 7.0 mol L, /0 7 = , 5 (3) x L H +, ph= 5.0, 0 5 x mol ph= 3.0, 0 3 x mol, H +, 0 5 x mol x mol x L + x L = mol L = mol L, ph= log 0 ( ) 3.3

45 33 3, ( ) 3., a, b, a b (a > b) a b (a < b) a b (a = b), (3.), a b (a > b a = b ), a b, a b, a b a b a 0 < a, a a a < 0, a :, a, b, c, a < b b < c a < c (3.) a < b a + c < b + c (3.3) 0 < a 0 < b 0 < ab (3.4), (3.5) (3.3),, a < b 0 < b a (3.5),, ( ) a < b 0 < c, ac < bc (3.6) a < b c < 0, ac > bc (3.7),, a 0 0 < a (3.8), 0, 0 < (3.9) 0 < a 0 < /a (3.0), 0 < ab (0 < a 0 < b) (a < 0 b < 0) (3.),, ab < 0 (0 < a b < 0) (a < 0 0 < b) (3.),, 0 a 0 b, a b a b (3.3), 0, 54 0 a, b, a = b a + b ab (3.4) : 0,, 0 :, <, >, 0 > a a < 0

46 a, a : 0 < a a := a a < 0 a := a 0 := 0,, ( ), : a, b, 0 a (3.5) a = a (3.6) ab = a b (3.7) a = a b b ( b 0 ) (3.8),, a, b, a b, a b 0,, 0 ( ) 3.3 n, n, n, n! ( ): n! := 3 (n ) n (3.9), n = 0 (3.9),, 0, 0! ( ) ( ( 3)! ) 40 0!=? 0!=0,... (3.9), n! = (n )! n, n =,,! = ( )!,! = 0!! = = 0!, 0! =, (3.9) ! 3, 6 ( ) : () 4! () 5! (3) 0! (4)! (5) ( 5)! a, b, c 3, ( )?, 6 : abc, acb, bac, bca, cab, cba, 6? a, b, c 3,,,, 3 = 3! 6 ( ) 3., n, n! ? a, b, c, d, e 5, 3,? 5 4 3, ( ) n m,, n P m 3.3, n P m, m : np m = n (n ) (n m + ) (3.0), n (n ) (n m + ) (n m) (n m) n!/(n m)!, : np m = n! (n m)! (3.) n n P n = n! 8,? 3.4, a, b, c 3,,, aa, cb?,,

47 3.5 35,, aa ab ac ba bb bc ca cb cc (9 ), 9? a, b, c (3 ), a, b, c (3 ), 3 3 = 3 9, n, OK m, n m 59 0,, 8,?, n m (, ), n C m n C m?, n P m, n m ( n C m ), m ( mp m = m! ),, np m = n C m m! (3.), nc m = n P m m! (3.), nc m = n! m!(n m)! (3.3) (3.4) (3.4), m = 0 (3.4) n C m, n C m, * ( ) n m 3.5 (3.5) 0 3,, : 0C 3 = ( ) 0! 3!(0 3)! = = 0 60 (n 3 ): () 4C () 5C (3) 5C 3 (4) nc 3 (5) nc 0 (6) nc n 6 : 6 n, m, n > m nc m = n C n m (3.6) 40, 3 ( 3 ) 3.5 ( ) ( ) 3.6 ( ) 3x xy abc, (polynomial) ( ) x + x, x + y + xy,, +x, + x + x, ( ) (term), ( ), n, n ( n ) 3.8 x 3 + x, x x 3, 3, 3, : x 3 y + x y + (3.7), x y x 3 y, 4, 4, 4 ( ) *, ( )!

48 36 3,,, (3.7), x 3, y 3.0 ax + bx + c, x a, b, c, x ( ),,,, 3. x + x + 3 (x + )/(x + 3), ( ) x( ), χ, χ, x χ ( ) χ, x x 63 ( x χ ): (x + χ)(x + χ)(x χ) (3.8) 4 x χ... x 3.6 n (a + b) n, (a + b) = a + ab + b (3.9) (a + b) 3 = a 3 + 3a b + 3ab + b 3 (3.30) (a + b) 4 = a 4 + 4a 3 b + 6a b + 4ab 3 + b 4 (3.3),? (a + b) n, n (a + b), a b,, a b n,, a m b n m (m 0 n ) a m b n m, n (a + b) m a n m b,, a, n C m (a b, a ), a m b n m n C m, (a + b) n = n C n a n + n C n a n b + n C n a n b + + n C m a m b n m + + n C ab n + n C 0 b n (3.3), P.35 (3.6), : 64 (a + b) n = n C 0 a n + n C a n b + n C a n b + + n C m a n m b m + + n C n ab n + n C n b n (3.33) () (x + ) 7, x 3 () (x 3) 6, x x ax + bx + c ( a 0 ) (3.34), b, c a(x + b ) + c (3.35), 3. : x + x + 3 = (x + ) + (3.36) ( ), ax + bx + c, x x x : ax + bx + c = a (x + b ) a x + c (3.37)

49 (), x, x : ( x + b ) = x + b ( b ) a a x + (3.38) a, : x + b ( a x = x + b ) ( b ) (3.39) a a, (3.37) : ( ax + bx + c = a x + b ) ( b ) a + c (3.40) a a, b = b a, ( b ) c = a + c a, (3.40) (3.35) 65 : () x + 4x + 5 () x + x + (3) x x (4) x + 4x + 3 (5) 4x + x + 4?...,,,, ( ) x x = 0 (3.4) x y + xy + xy = x + (3.4) ( ) ( ), (3.4) ( ), x =, n n n (3.4),,, 0 ( ) 3.4 (3.4), x x = (x + )(x ) = 0 (3.43), x + = 0 x = 0, x =,... P.5 (.9)? ( ),, : 3.5 : x x + = 0 (3.44), (x ) = 0, x = ( ) ( ), ( x ),, ( ), 66 () x x 6 = 0 () x 3x + = 0 (3) x 4 5x + 4 = 0, : 67 x ax + bx + c = 0 (3.45) a, b, c, a 0 () a, : ( x + b ) b 4ac = a 4a (3.46) (), : x + b a = ± b 4ac a (3.47) (3), ( ) : x = b ± b 4ac a (3.48) (3.48),

50 38 3, (3.48), D,, D := b 4ac (3.49) a, b, c, D, D (3.48), x = b ± D a D=0, x = b a (3.50) (3.45) ( ), D 0, x = b + D a x = b D a (3.5),, D > 0 (3.5) OK, D < 0 D < 0 D!, D D ( 0 ),,,, i, i = (3.5), i =, i 3 i (i) = i = 4, ( 3 i) = ( 3) i = 3,, + i 3 i D < 0, D, (3.5), 3.6 x + 3x + 5 = 0 (3.53), D = = < 0, (3.48) ( = i ) x = 3 ± i 43 (3.54),,?...,, D, ( ), ± D, a, b i, a + bi 44?... a + bi, b 0 b = 0 a,,,, : n (n ),, n n, n, 68 () x + x + = 0 () x + 3x + = 0 3.9,,, 3.7 x = (3.55), x = x =, x = (x )(x + ) (3.56)

51 3.0 39, x,, ( ) 3.9,,, =, (6, 3, 3, 4,, 5, 6,, 3,, ) (3.6), 9 ( ) 3.0,,,, (a n ) = (, 3, 5,, (n ), ) (3.57),, ( ) *, a n, n ( n, n ) a =, a = 3, a 3 = 5 (3.58) a a n,,,, 0 OK , ( ), a n, n a n+ = a n + d (d ) (3.6), d ( ), (3.57), d a n, a n = a + (n )d (3.63) a n = a 0 + n d (3.64) (3.63) a, (3.64) a 0, a = a 0 + d (3.63) a (3.64) (3.57) 3.8 (b n ) = (,, 4, 8, 6, ) (3.59),, ( ) (3.57) n, a n = n (3.60), n 3 a = 3, a 3 = 5 n n, 45 a 0..., 0, 0,, ( ), a n, n a n+ = r a n (r ) (3.65), r ( ), (3.59), r a n, * ( ),, a : a n = a r n (3.66) a 0 : a n = a 0 r n (3.67)

52 40 3? 69, (), () 0, (3) 3, 3 (4), / (3.6) (3.65),,, () (3) (4) (5) (6) 0 (7) (a n = /n) n,,,, 3. 3.,,,, (,, 3, 4, 5, ) (3.68), (a n ),, n, a n < a n+ (3.69),,,,,, (,, 0,,, 3, 4, ) (3.70) (a n ),, n, a n > a n+ (3.7), *3 3.0 (,,,,,, ) (3.7) ( ) 70,,,, () *3 (3.69) (3.7),, < >,, ( ) n (3.73), (, 3 4, 7 8, 5 6, ) (3.74),,, ( ) 3.,, a, a 3. (,, 3, 4, ),,, ( ), ( ) 7,,?? () () (3) (4) (5) 0 (6) 0 (7) (a n = /n) n

53 ,, 3.3 (a n ) = (, 3, 5, 7, 9,, ), 3 9, = 4 (3.75) ( ), (a, a, a 3, ), m n a m + a m+ + + a n ( m, n, m n ), n a k (3.76) k=m ( ), 4 a k = a + a + a 3 + a 4 (3.77) k= Σ Σ, S S sum Σ k = m n,, k m n a k 46...,, a k, k,, (3.75) k k, (3.75) 5 (k ) (3.78) k= k 4, k, p, 5 (p ) (3.79) p=, 5 (k ) (3.80) p=, (p), ( a k ) k, p =, p = 3, p = 4, p = 5 k, (3.80) k k 4, 8k 4, a, n a = a + a + + a (a n ) k=, : n a = na (3.8) k= 7 () () (3) k= 3 p=0 p (4) 3 k= k 4 (n + ) n=, P.36 (3.3) : n (a + b) n = nc n k a n k b k (3.8) k= k=0 ( ) 73 (3.33), Σ,, (a k ), (b k ), ( ) α, ( ): n n n (a k + b k ) = a k + b k (3.83) k= k= k= k= n n αa k = α a k (3.84)

54 4 3 (3.83), (a + b ) + (a + b ) + + (a n + b n ) = (a + a + + a n ) + (b + b + + b n ) (3.84), αa + αa + + αa n = α(a + a + + a n ),,,,,, 3.4, (,, 3, ), ( ): n k = k= n(n + ) (n ) (3.85)?, n = =, = / = n = =+=3, = 3/ = 3 n = 3 =++3=6, =3 4/ = 6..., n n n = 00, n = 0 n = 0000 n,, (3.85) :, n =, (3.85),, n = N +, (3.85), N+ k= = k = N k + (N + ) (3.88) k= N(N + ) = N + N + N + = (N + )(N + ) + N + (3.89) = = N + 3N + (N + ){(N + ) + } (3.90) (3.85) n = N +, n = N + (3.85),, (3.86) n = n = + = (3.85), n = + = 3 (3.85), n = 3 + = 4 (3.85),,, n (3.85),..., n, () n = () N, n = N (3) n = N +, ( 3 ) =, = (3.86), =, N, n = N (3.85), N k = k= N(N + ) (3.87) 0 (), n = N..., n = N,! (), n = k...

55 3.4 43, (3.87), k k (3.9) k=, k, k, N, n k, (), N...,, ()! 74 A,, (3.87) N N +,...( )... N k = k= N(N + ) N N +, N+ k= k =...( )... (N + )(N + + ) = (N + )(N + )? A, 74 A, 3.5 n, (!?) (!?) : n k = n (3.9) k= (, ) n =, (3.9), n = N (3.9) (!), N k = N (3.93) k= N N +, N+ k= k = (N + ) (3.94), (3.9) n = N + (3.9) (!?), n =, (3.9) + = 3, = 4,! (3.9)! ( ) 3 (3.88), (3.89)...,, (3.87), (3.87), (3.88), N k= k, (3.88), (3.89) N(N+),, ( ) ( ),,,,,,? 47,...,,,,,,,,,,, 75 n, r, () () n k=0 r k = rn+ r n k = k= n(n + )(n + ) 6 (3.95) (3.96)

56 , (3.95) (3.96)...,,..., 48? () : 0 k=0 49 k () 0 k= k,?...,, 3.5, ( ),,!,,,,,,,,, *4,, Excel, *5 LibreOffice-calc, LibreOffice-calc *6, Excel *4,,, *5 *6 LibreOffice, ( Linux, Windows, Mac, 50?...,,,, ( ), B, 3 3,, 3, B3, A, =*3 : A B C =*3 3, A, 6, *3, 3,, * ( ),,,,, A 4, B, = (!) A, B =A, *5, B =A*5 A B C 6 4 =A*5 3, B, 0, A (4) , B, =A*5

57 3.6 45,,, A 7, B 35!,, 3.6, (,,, 3.6 : 0! +! +! + 3! + + n! + (3.97),,,,, (A, B, C, A, B) : A B C k /k! sum 0 3 (A, B, C, A (A, A3, A4,...), k, 0,,, 3, A (0) A3,,, 3,, 3... (, A3, =A+, A, A3, A4 A0 (,!),, 0, 00, A4 A,,, : A B C k /k! sum B, (/0!), B3 /!, B4 /!,...,, B3, =B/A3, B3, B ( ) A3 (, B3, B4 B *7 : A B C k /k! sum k (/k!) B 5 B!.75573E , ,, ,, 0, E, B, C ( C =B, C3 =C+B3 *7, (LibreOffice

58 46 3, C4 C, A B C k /k! sum C, (3.97) n = 0.788,, (.45)! ( ),,,,,,, C3 =C+B3, C4, =C3+B4?, (,!),,,, (,, 3, 77 n k= : k = n (3.98),, n π /6 = *8 3 n ( n = *8 II 987), (3.98), 7 : 3.6, A n k ( k ), B, /k B, =/(A*A), B3 ( A n ) (,!) ( 3.6 ) 5, ( ) =/A*A..., A, A,, 53 =/(A*A) =/Aˆ?...,, ˆ, ˆ,, ˆ,, = 3ˆ 9, 9?,, ( ), ( 3),,,,, *,, (, ) 4 5 *, * ( )

59 ,, ( ; ),,,,,,,,, #,, #,,, #,, #,, #,, 7 5, ( 3.) () r t :,,,,,,,... () 80, , (3.3), 3.,,,, (), 440 Hz? ()? (3), 88, 88,? (4), 0 Hz 0000 Hz? (5),? (6) 440 Hz,.5,,? (7),,,, 4:5:6, 8 r, t ( ) ( ) = a + ab + b ab 4 = a ab + b ( a b ) = 0 4, {(a b)/} = 0, (a b) = 0 (.9), a b = 0 a = b 55 () 4 3 = 4 () = 0 (3) (4) (5) 56 5! = 0 57 (3.), m = n, np n = n!/(n n)! = n!/0! = n! ,, 8 3, 8 P 3 = = () 4 () 0 (3) 0 (4) n(n )(n )/6 (5) n C 0 = n!/(0!n!) = n!/n! = 0! = (6) 6 (3.4), nc m = n! m! (n m)!

60 48 3, (3.4) m n m, nc n m = =, n C m = n C n m 6 40 C 3 = 9880 n! (n m)! {n (n m)}! n! (n m)!m! = n! m!(n m)! 63 x 3 + 3x χ 3xχ χ 3 7 () = 8 () = 4 (3) = 5 (4) = 4 73 n (a + b) n = nc k a n k b k (3.99) 75 k=0 64 () (3.3) a = x, b =, n = 7, (x + ) 7 = 7 C 7 x C 6 x C 5 x 5 x 3, 7 C 3 x 3 = 35x 3, 35 () (3.3) a = x, b = 3, n = 6, x 3 6C 3 (x) 3 ( 3) 3 = 430x 3, () (4) (x + ) + (x + ) + () (5) ( x + ) (3) ( 4 x + ) (x ) 66 ( ): () x =, 3 () x =, (3), (x 4)(x ) = (x + )(x )(x + )(x ), x = ±, ± 68 : () x = ( ± 7 i)/ () x = ( 3 ± 5)/ 69 () n () n (3) 3 n (4) (/) n 70 () : (,, 3, 4, ) () : (4, 3,,, ) (3) : (,, 4, 8, ) (4) : (,, 4, 8, ) (5) : (,, 4, 8, ) (6) : (, /, /4, /8, ) (7) 7 () ( ) () ( ) (3) ( ) (4) ( ) (5) 0 (6) 0 (7) 0 () n =,, = + r, = + r n = N, N k=0 r k = rn+ r, N+ k=0 r k = N k=0 r k + r N+ = rn+ r = rn+ + r N+ r N+ r (3.00) + r N+ = rn+ r, n = N + () n =,, =, = n = N, N k = k=, N+ k= k = N(N + )(N + ) 6 N k + (N + ) k= N(N + )(N + ) = + (N + ) 6 (N + ){N(N + ) + 6(N + )} = 6 = (N + )(N + 7N + 6) 6 (N + )(N + )(N + 3) = 6 (3.96) n N + n = N + 76 () (3.95), ( )/( ) = 047 () (3.96), (0 )/6 = 385

61 49 4 : 4. y (function),,, ( ) y = x, x = 3, y = 3 = 9,, x, y *, y = f(x) ( f(x) x ) f a(x) b(x), function f x, y = f(x) y, ( x) ( ), ( y) ( ),,,,, (, ),,, ( ) 4. y = (4.) x, x, y 4. x ( ) *,,,,, O y = O, x x, y y..., (O), x ( x), y ( y), y = x (4.) y = x, x = 0,,,, y = 0,, 4, 4.,,, y = x, 4.,, y= x y O x x y= -x y O x y = x y = x, a, y = ax (4.3), a, a, a, x y

62 50 4,, a, x y a,, y = ax x y, x y ( ) * x y, x y y x,, lim x ( 4.3 ), x y,, ( x y ) ( ), a, y = x (4.4) y = a x (4.8), x = 0,,, 3,, y = 0,, 4, 9, 4.3, y=x 4 3 O x y y=/x y 4 3 O x : y = x, : y = /x, y = x (4.5) 0, x = 0, x = /,,, y =,, / 4.3 (4.5), x ( ), 0 ( x ), x, /x 0, x, x 0 (4.6), lim x x = 0 (4.7) lim, * y = ax, y = ax +, x y, x y ( ),, ( (4.) (4.5), (4.8)), a, n, : y = ax n (4.9), ( (4.9)), n = 0 ( ) n = ( ) n = n = ( ), 4.,,, xy (, 3), (x ), (3, 3), (y ), (, 5) ( xy ) 78 (, 3),, xy () x 5 (7, 3) () y 4 (, 7) (3) x 3 (6, 3)

63 4. 5 (4) y / (, 3/) (5) x *3 (, 3) (6) y (, 3) (7) (, 3), xy (x 0, y 0 ), x a (x 0 + a, y 0 ) y a (x 0, y 0 + a) x a (ax 0, y 0 ) y a (x 0, ay 0 ) x (x 0, y 0 ) y ( x 0, y 0 ) ( x 0, y 0 ) 78, y = f(x) (4.0), x a, y b, y = f(x a) + b (4.)? y = f(x) P P (x 0, y 0 ) P (4.0), (x, y ), P (4.) 56 (4.) a b y = f(x a) b y = f(x + a) + b?... (4.), y b = f(x a) a b? (4.) ( (4.5)!), y = f(x), : ) x a, y b, y = f(x a) + b ) x a, y = f(x/a) (a 0 ) 3) y a, y = af(x) 4) x, y = f(x) 5) y, y = f( x) 6), y = f( x) y 0 = f(x 0 ) (4.) 79 4 P x a, y b, (x 0 + a, y 0 + b), P, (x, y ), x = x 0 + a (4.3) y = y 0 + b (4.4), x 0 = x a, y 0 = y b (4.), y b = f(x a) (4.5), y = f(x a) + b (4.6),,,,, 4. y = x (4.7) ( 4.4 ), y = x, y ( ) 4.3 y = x /4 ( 4.4 ) y = x y /4 ( 3), y = (x/), y = x x ( ) *3 x,, x,,, 4.4 y = x ( 4.4 ), y = x x ( 4)

64 y = x + 4x + (4.8), y = (x + ), ( 4.4 ), y = x y (y = x ), x, y ( 3, ) 4.6 y = /x ( 4.4 ), y = /x x ( 4) y ( 5) 4.7 y = x + x?, y = x + x = + x (4.9) (4.0) y=x- y O - - x y= -x y O - - x y= -/x y 3 y= x /4 y O - - x y= x +4x+ y O - - x y=/(x-)+ y 3 y = /x y, x, y ( 3, )! 4.4 ( ) 80 y = x, () x, y () x, y 3 (3) x, x (4) x, x (!) 8, () y = x + () y = x + x + 3 (3) y = + /x (4) y = (x)/( + x) 4.3 a, b ( a 0 ), y = ax + b (4.), y x (4.7) O x O x ,,,,, (4.), y = ax ( ) y b x = 0 y = b (0, b) y y (b ) y ( ), x y a, y,, (x 0, y 0 ), a?, O a y = ax x x 0, y y 0, (4.) : y = a(x x 0 ) + y 0 (4.)? x x 0 y = y 0 (x 0, y 0 ) x

65 a, a!,, y = ax (0, 0), (x 0, y 0 ), : (x 0, y 0 ) (x, y )?, (x 0, y 0 ) (x, y ), x x x 0 y y y 0 (y y 0 )/(x x 0 ) (4.) a, 8 y = y y 0 x x 0 (x x 0 ) + y 0 (4.3) () 3, y (), (, ) (3) (, ) (4, 3) 83, ( ) x, x F : 0 C 3 F 00 C F, x, y () y x () x y (3) 37 C F ( ) (4) 0 F C (3)(4) 4.4, (, ), (, ) x y = /x 0, y = x, x 0 y = /x, y = x 0,, 4.5 ( ) y 3 y=x+/ x y=x y= / x O 3 x 4.5 y = x + /x ( ) y = x y = /x! 84, y = x + /x 4.5, 4.6 x y, A, x y?,,, A x 0 30, y ), 4.8 : y = x + x (4.4), A, y = 00 y = 300, P, Q PQ, 0 < x y = x y = /x,, x, x = y = x y = /x y =,, PQ PA PA (mm cm ) A y, PA (300 00) + 00 (4.5) PQ!, A

66 54 4 y A B P.37, f(x, y,...) 57 (3.4) x y +xy +xy = x+, f(x) = 0... x y+xy +xy x = 0? f(x, y), f(x, y) = x, f(x) = 0 f(x) = 0, x = x 0, f(x 0 ) = 0 (x 0, 0) y = f(x), x ( ), x x, f(x) = 0,,, x = 0 x = 30,, x 85 A x y!, :, x = 4, y = 85 (x, y )! ,, y = ax + b (a, b ) : ( ),, y,, y x, a y y, y ( ), y = ax + b (x, y), 85 A, b, a = 0.6, b = 550! 4.6, x, y,... f(x, y,...), f(x, y,...) = 0 ( ) 4.9 y = x 3 x x 3 x = 0, x =, 0, 3 x x ( ), x y, x y,,,, 4.7 y - - O y = x 3 x ( ), y = x 3 y = x y = x 3 y = x, ( ) f(x) = 0, y = f(x) x 4.0 y = x + x + x +x+ = 0 (P.38), D = 4 = 3 < 0 x

67 ( ) y = x + x + x, 4.8 ( ) D<0: y=x +x+ y D=0: y=x -x+ y D>0: y=x -x- y - O x - O x - O x 4.8,, D, x D < 0 D = 0 ( ) D > 0 ( ) f(x), f(x) = 0, y = f(x), x x (, ) 4. y = x x + x x + = (x ) = 0 x = y = x x + x = x, 4.8 ( ) 4.9 y = x 4.7,, y = f(x), x, y, f(x), (x, y) P.45, 4. y = x x ( 4.9 A, x ( ) ( 0.) ( ), A, A3 =A+0., A3 A, x y B =A*A, B3 B A B (A 4.0!), ( ) (,, x ) 4.0, *4 ( ) 58 A? A?... B, B,, *4,

68 56 4 7,...,, :,, :,,,,, :,,,,, 4. y = x, y = x, y = x 3 87,, x () y = x () y = x (3) y = x 3 : x, (A B =A, C =A*A, D =A*A*A, ( B, C, D (B y=x ), A D, , (3.), (x ) (x+), < x < x+,, x < 0 0 < x +, < x <, 4.3, 4.4,, (4.6) y = x x, 4.8 (4.6), y < 0, x x < x <,, x + < x x? 3 < x, x 3 : ax + bx + c = 0 x = α x = β (0 < a α < β ), ax + bx + c < 0 (4.7) 4.4 α < x < β, x x < 0 (4.6) ( ) (x )(x+) < 0 ax + bx + c > 0 (4.8) x < α β < x? y = ax + bx + c, x = α x = β x, a > 0

69 x ( (4.7)), ( (4.8)), (4.7), (4.8), <, < 88 x () x + x 6 < 0 () x + 5x (3) x + 3x x () x + x + > 0 () x + x + < 0 (3) x + x > 0 (4) x 4x + 4 < 0 (5) x 4x (6) x 4x (7) x 4x + 4 > 0 x + x + > 0 (4.9) : (3) 0 x, (x ), (x )? y = x + x +, x, 4.8 (4.9), (P.33 (3.7) ) (x ) y > 0, x 0, (4) x, (7), (x ) x x, f(x),??... f( x) = f(x) (4.3),, f(x) ( ), x + x + < 0 (4.30) f(x)?,! 0 : 3 i (4.30) < x < + 3 i!..., (, + i 5i,?), (4.30),,, ( x + ) (4.3) (x + ) x 0, 3/4,, =0, (4.3) f( x) = f(x) (4.33), f(x) ( ),,,! : (4.33), f(x) = f( x),,, (4.33) 90??, y = f(x) y y = f( x) (4. 5), f( x) f(x),, y,, y, y = x, ( 4.3 ) y

70 58 4, y = f(x) y = f( x) (4. 6),, f( x) f(x),,,,, y = ax ( 4.), y = /x y = x 3 x, ( 4.3, 4.7) 9 f(x), ( ), y = f(x) () f(x) = 4x 4 5x + : x () f(x) = x + x (3) f(x) = x x 3 (4) f(x) = /( + x ) : x = 0 x ±? 59?..., OK,,, y,...,?,! 94 a 0, n f(x) = ax n, n, n 60,?...,, 9,,,,, 6, f( x) = f(x)?...! f( x) = f(x) f(x), 4.0 f(x), g(x) f(x), g(x),, f(x), g(x), f(x) g(x), g(f(x)) : 4.6 : 9 : f(x) = x +, g(x) = x () () (3) (4) (5) (6), 93 ( : ),,,, f(x) = { ( + x + x 4 ) 3 } 8 + x + x 4 (4.34),, x =, f() = + =, g() = = 4, g(f()) = 4, x, g(f(x)) = g(x + ) = (x + ) (4.35) x =, 4 (4.35) : g(f(x)) = {f(x)} = (x + ) (4.36),, f(g(x)) = f(x ) = x + (4.37)

71 4. 59 (4.35) (4.37)! ( ) g(f(x)),,,, 95 f(x) = x, g(x) = + x, g(f(x)) f(g(x)) 4.,, (4.5), y = /x, x = 0?,, y = /x,,, x, x, y = /x x = 0 x, y = /x 0?,,, y,, y = /x x, y, x y = 0?,, y = /x, y = x, 0 y = x + 3, 3 ( ),,,, x < 0 y = 0, 0 x y =, x = 3 x =, x = 0,,,,,, ( )!, : 4.8 x, x 0 f(x), 0 f(/3) =, f( ) = 0,, x x!! ( ) 4., y = f(x) x y y = g(x), f(x) ( ) 4.9 y = x, x y x = y y y = x/, y = x y = x/ ( ) 96, a, b a 0 () y = x + () y = /(x ) (3) y = a/x (4) y = x + b?, y = f(x), x y, x, y, x y,,, y = f(x), y = g(x), g(x) f(x),, f(x) g(x), 4.0 y = x x = 3, y = 6 x y = x/ ( y = x

72 ), y = 6/ = 3 x ( ),, : 4. y = x x y, x = y, y = ± x x = 4, y = ±,,, x = 4, y = x, x y,,,, y = x, 0 x, x y, y = x ( ), : f(x) g(x), 97 0 x f(x) = x, g(x) = x, (4.38) f(x) g(x), y = g(x) x = f(y), y = f(x) x y x = f(y), y = g(x) ( 4.) 4. y O y=f(x) y=x x=f(y) y=g(x) y = f(x) y = g(x) y = x x y, y = x y = x x y, y = x y = f(x) x = f(y), y = g(x), y = x (, ) x g(f(x)) = f(g(x)) = x (4.38) 98 y = x y = x? f(x), x, g(x) x, g(f(x)), x,, g(f(x)) x x,, g(f(x)) = x,, 4., y = x f(g(x)) = x y = x/, x /(x), 4. 4., y = x y = x/ f(x), g(x) f(x) = x, g(x) = x/ f(g(x)) = g(x) = x = x (4.39) g(f(x)) = f(x) = x = x (4.40) (4.38)! ( ) y = x, y... y = x x y = x, y! 96,, 99,, 0 x, () y = x () y = x (3) y = x 3 (4) y = x 4 (5) y = x / (6) y = x /3 (7) y = x /4

73 4.3 6 : ˆ A3 3 =A3ˆ3 A3 /3, =A3ˆ(/3)! (6), (7),, n, y = x n y = x /n, y = x,, y = x α,, α α, x y, *5 (4.) a b,? 6?...,,,,,, y = f(x),, (4.4), y = ± r x (4.4), ± x = 0 y = r y = r,, x y,,,, 4.3 F (x, y) = 0 x a, y b, F (x a, y b) = 0? F (x, y) = 0 P (x 0, y 0 ), F (x 0, y 0 ) = 0 P x a, y b P, (x, y ), x = x 0 + a y = y 0 + b x 0 = x a y 0 = y b F (x 0, y 0 ) = 0, F (x a, y b) = 0, P F (x a, y b) = 0 ( ) (x, y), (3, ), y = f(x), x y, x + y = 0, x y 0,, x y, x y F (x, y), F (x, y) = 0 x y, (implicit function) ( ) 00 r x + y r = 0 (4.4) ( r > 0 ), r 4.4, : x /x, 0, x ( x + y = x ) x = 0 y (y ) ( ) y y = 0 x (x ) ( ) x (x x y *5, y ),,

74 6 4, x y x y, 63 III,?... OK,, 0 x,,,,,,,,, 9,, :... 0 K (x), K (x), G (x), G (x),,,, ( ) () K (K (x)) () G (G (x)) (3) K (G (x)) (4) G (K (x)) (5) K (x) (6) G (x) F (x, y) = 0, () x, F (x, y) = 0 () F (x, y) = F (x, y), x (3) x a, y b, F (x/a, y/b) = 0 (4), : () x y = () (x /4)+(y /9) = v = V max[s] K m + [S] (4.43) V max K m v (, ), [S], v [S] () [S] = 0 v = 0 () [S], v V max (3) [S] = K m, v = V max / (4), (4.43) [S], v 0, 0 [S] (5) y = /v, x = /[S], (4.43) y x ( ), y? ( ):, V max K m [S] v,, V max K m, [S] v V max K m, 79 y = f(x) P (x 0, y 0 ) y 0 = f(x 0 ) : P x a, (ax 0, y 0 ), P, (x, y ), x = ax 0, y = y 0, x 0 = x /a, y 0 = y y 0 = f(x 0 ), y = f(x /a), P y = f(x/a) 4 : P x, (x 0, y 0 ), P 4, (x 4, y 4 ), x 4 = x 0, y 4 = y 0, x 0 = x 4, y 0 = y 4 y 0 = f(x 0 ), y 4 = f(x 4 ), y 4 = f(x 4 ), P 4 y = f(x) 80 () y = (x ) + () y = 3(x/) = 3x /4 (3) y = x ( 4.3 ), x, y = (x/) ( 4.3 ) x, y = {(x )/} ( 4.3 ) (4) y = x ( 4.3 ), x

75 4.4 63, y = (x ) ( 4.3 ) x, y = (x/ ) ( 4.3 ) y 0 = (x 3), y = 5 9 (x 3) = (5/9) x (60/9) () x = (9/5)y + 3 (3) y= x y=(x/) y=((x-)/) O 3 y= x x O 3 y=(x-) x O 3 y=((x/)-) x () =0 x = 3, 3 < x < () =0 x = 3, x 3, x (3) =0 x = ( 3 ± 5)/, x ( 3 5)/, ( 3 + 5)/ x O 3 x O 3 x O 3 x y = x y = x () y = x (y ) () y = (x + ) +, y = x (x ), (3) y = /x (y ) (4) y = /(x + ), y = /x,, (x ), x = 0 y = 0, y=x+ y 4 3 O y=+/x y 4 3 O x x y=x +x+3 y 4 3 O y=x/(+x) y 4 3 O () y = 3x () (4.), y = (x + ) +, y = x + 3 (3) (4.3), y = 3 4 (x ) +, y = x () (3, 0) (, 00) - x x 89 () =(x + /) + 3/4 x,, x (), ( ) (3) ( ) x <, < x (4) ( ) (5) ( ) x = (6) ( ) (7) ( ) 90,! () y = (x + )(x + )(x )(x ), x x =, /, /, x ±, y () y = ( + x)x, x x =, 0 y = (x + /) /4, y = x /, /4 (3) y = (x + )x(x ), x x =, 0, x, y x, y ( 4.9, y = x 3 x, 4.7, ) (4) x ±, y 0 x x = 0 y =! 9 f (x), f (x), g (x), g (x), f ( x) = f (x), f ( x) = f (x) g ( x) = g (x), g ( x) = g (x), () F (x) = f (x)f (x), F ( x) = f ( x)f ( x) = f (x)f (x) = F (x), F (x) () F (x) = g (x)g (x), F ( x) = g ( x)g ( x) = { g (x)}{ g (x)} = g (x)g (x) = F (x), F (x) (3) F (x) = f (x)g (x), F ( x) = f ( x)g ( x) = f (x){ g (x)} = f (x)g (x) = F (x), F (x) (4) F (x) = f (x) + f (x), F ( x) =

76 64 4 y=4x 4-5x + y - O - - x y=x+x y - O - - x 97 g(f(x)) = f(x) = x = x x 0, x, f(g(x)) = (g(x)) = ( x) = x y=x-x 3 y y= /(+x ) y y y=x y=sqrt(x) y=x - O x - O x O 4.6 x 98, sqrt(x) x y = x (, ) f ( x) + f ( x) = f (x) + f (x) = F (x), F (x) (5) F (x) = g (x) + g (x), F ( x) = g ( x) + g ( x) = g (x) g (x) = {g (x) + g (x)} = F (x), F (x) (6) y = y = x : y = + x, 93 : f( x) = f(x) : f( x) = f(x) ( ), 95 g(f(x)) = + x ( x 0), f(g(x)) = + x 96 () x y x = y +, y = (x )/ () x y x = /(y ), y = + /x (3) x y x = a/y, y = a/x (4) x y x = y + b, y = x + b : (3), (4) y 0.5 O 0.5 x y=x y=x, y=x / y=x 3, y=x /3 y=x 4, y=x / y = x, x /, x /3, x /4 (, ) 00 x + y, (x, y), r, r r, r 0,, (4.4) r =, x + y 4 = 0 x 3, y ( 4.3 ), (x 3) + (y + ) 4 = ,, 65,...,

77 65 5 :, 5., y y=f(x),,,, y = f(x), 5.,, y = f(x) P, y = f(x) ; 5.,, y f(x 0 ) P O x 0 y=f(x) 5. P a, x = x 0 f(x) P (x 0, f(x 0 )), a a f(x), x = x 0,, a,, x f(x )= f(x 0 + x) 5. f(x 0 ) O P x Q R x x 0 x =x 0 + x a, PQ, x 0 x 0 x, (x, f(x )) Q 5. P Q 5., QR/PR, (f(x ) f(x 0 ))/(x x 0 ), PQ P, Q P,, P a, P Q, a f(x ) f(x 0 ) x x 0 (5.) x x 0 x, x, x x = x 0 + x, (5.) a f(x 0 + x) f(x 0 ) x (5.) x 0 x x 0

78 66 5,, f(x 0 + x) f(x 0 ) a = lim x 0 x * 5. (5.3) f(x) x = x 0 a, a f (x 0 ), a P x 0,, g (x 3 ), g(x) x = x 3, () f(x) x = x 0 f (x 0 ), : f f(x 0 + x) f(x 0 ) (x 0 ) := lim x 0 x (5.5) x h, 5. P, (x 0, f(x 0 )), f (x 0 ), y = f(x 0 ) + f (x 0 )(x x 0 ) (5.6), P y = f(x), P mm,, P, (5.6) y = f(x), : f(x) f(x 0 ) + f (x 0 )(x x 0 ) (5.7), * x, (5.) f(x ) f(x 0 ) a = lim (5.4) x x 0 x x 0 (5.3), x x 0 = x x x x = x 0 + x, f(x 0 + x) f(x 0 ) + f (x 0 ) x (5.8), 5.3, S S f(x 0 + x) 5.3 y f(x 0 ) O P x 0 x y=f(x) S S x 0 + x y = f(x) S, S, x = x 0 + x x 0 x 0, x 0,, 0, 0 x, dx d x, dx, (5.8), : f(x 0 + dx) = f(x 0 ) + f (x 0 ) dx (5.9), = : x 0, S P, S S, P, P, S S P, S S, (5.8) S S, x 0 0,, 0, =, * 0, 0 *,, x dx, f df,, d, = x

79 5. 67 dx,, 0 0, x, dx x d, difference D x = x 0 x = x 0 + dx x = x 0 + x, (5.9), f (x 0 ), f(x) dx, dx, y = f(x), x y,, x y ;, 5.,, x dx,, (5.5),, (5.9), (5.9), () f(x) dx, f (x 0 ), x = x 0 f(x) : f(x 0 + dx) = f(x 0 ) + f (x 0 ) dx (5.0) 67 (5.5), (5.0)?..., (5.5),,, (5.0),,, (5.0) 68 (5.5)?..., (5.0), (5.5),, (5.0) 0 69 (5.0) (5.5) 5,,...,, (5.8) (5.9) : f(x 0 + x) f(x 0 ) f (x 0 ) x (5.) f(x 0 + dx) f(x 0 ) = f (x 0 ) dx (5.) f(x 0 + x) f(x 0 + dx), f(x 0 ), f(x 0 ), f df, f := f(x 0 + x) f(x 0 ) (5.3) df := f(x 0 + dx) f(x 0 ) (5.4), (5.) (5.) : f f (x 0 ) x (5.5) df = f (x 0 )dx (5.6) f x df dx f(x), x 0,,,, 70..., dx df,,,, (5.6) dx, f (x 0 ) = df dx (5.7)

80 f(x) = x + : d dx f (5.8) d, dx,,, (5.5) (5.0),,,,,,,, (5.5) (5.0),,, (5.5) (5.0),,,,,,, 7,...,, f(x), (derivative), f (x) (differentiate),, (5.0) x 0 x f(x + dx) = (x + dx) + = x + + dx = f(x) + dx (5.0), dx f (x), f (x) = (5.9) ( ) (5.9), : (x + ) = (5.0) (5.0), ( ), ( ), ( ), d (x + ) = (5.) dx, d/dx , f(x) = x f (x) = x x = 3 f (3) = 6, x = 3 03 p, q f(x) = px + q, f (x) = p, (px + q) = p (5.), y = q, 0, : (q) = 0 (5.3), x, 0, 0, px + q, q, p, p, px + q p 5. f(x) = x f(x + dx) = (x + dx) = x + x dx + dx = f(x) + x dx + dx

81 5.3 69, dx, d x, (dx),, dx d x dx ( ), dx 0, dx,, 0, dx!, f(x + dx) = f(x) + xdx (5.4) (5.0), dx x, f (x) = x ( ) dx, x x x 0, x 0, x dx,,,, f x, ( ) ( ) 73, dx?... dx, dx, 74,,... dx,, 75 dx 0,... dx dx, dx dx dx 0.0 dx 0.000, dx (dx 0, ) 5.3 f(x) = x n (n ) (P.36 (3.33)), f(x + dx) = (x + dx) n = n C 0 x n + n C x n dx + n C x n dx + = x n + nx n dx + n C x n dx + = f(x) + nx n dx + n C x n dx + (5.5) dx (5.0), dx nx n, : 5.4 f (x) = nx n (5.6) f(x) = /x, f(x + dx) = x + dx, x dx, f(x + dx) = x dx (x + dx)(x dx) = x dx x dx, dx, f(x + dx) = x dx x (5.7) = x x dx = f(x) x dx (5.0), dx /x, f (x) = x, (5.6) n = (5.6), n, n = 5.3,,,,, (5.5), : f f(x + x) f(x) (x) := lim x 0 x (5.8), x 0, 0,, : f (x) f(x + x) f(x) x (5.9),, f(x) (f(x + x) f(x), x ( x, f(x) = x, x 0.05 A,

82 70 5 f(x) B : A B C x f(x)=x f (x) (C ) f (x), C, =(B3 B)/(A3 A) B3 B f(x + x) f(x), A3 A x, ( ), x =, x = x = y = x (dy/dx), y = x x 0.05 y = x, 5.!, x ( x = 0.05),,, C, C3 C, C,, f (x), C,,, y = f (x) y 04, f(x) = x, y = f x ( ) x,, f(x), *3 f (x) = x, x 5.4! A B C D E 5.4 O x, 5.5, y = f(x) ( ), y = f (x) ( ) y y = f x ( ) y = f(x) A, B, C, D, E,, A C E, B, C, D, C A O B D E x,, A, E 0, B, D, *3 ( 5.5 y = f(x) y = f (x) y = f(x) C, y = f (x) ( C )

83 5.5 7,,,,, ,,, y () : {f(x) + g(x)} = f (x) + g (x) (5.30) : F (x) = f(x) + g(x), F (x + dx) = f(x + dx) + g(x + dx) = f(x) + f (x)dx + g(x) + g (x)dx = f(x) + g(x) + {f (x) + g (x)}dx = F (x) + {f (x) + g (x)}dx (5.3), dx,, F (x) = f (x) + g (x) 5.5 f(x) = x +x, - - O - - x f (x) = (x + x) = (x ) + (x) (5.3) 5., (x ) = x, (5.), (x) =, f (x) = x + (5.33) () y ( ) a, : O x {af(x)} = af (x) (5.34) : F (x) = af(x), () : (x = 0), 0 x < 0 0 < x 5.5, ( ) (5.0),,, ( ), f(x), g(x), ( ) F (x + dx) = af(x + dx) = a{f(x) + f (x)dx} = af(x) + af (x)dx = F (x) + af (x)dx (5.35), dx,, F (x) = af (x), Σ (P.4)? ( ) 5.6 f(x) = 3x, f (x) = (3x ) = 3(x ) (5.36) 5., (x ) = x, f (x) = 3(x) = 6x (5.37) ( )

84 7 5, 5.7 f(x) = x 3 + x + 3x +,, f (x) = (x 3 ) + (x ) + 3(x) + () (5.38), () ( x ), (5.3), 0 (x 3 ) (x ), (x) (5.6), = 3x + 4x + 3 (5.39) ( ) F (x) = (x + ) (x + x + 3) + (x + )(x + x + 3) = x(x + x + 3) + (x + )(x + ) = 4x 3 + 3x + 0x + (5.4),, F (x) = (x 4 + x 3 + 5x + x + 6) = (x 4 ) + (x 3 ) + 5(x ) + (x) + (6) = 4x 3 + 3x + 0x + (5.43),, ( ) 06 : 07 : () f(x) = x + x + () f(x) = 4x + 5x + 6 (3) f(x) = 3x + 3x + 4 (4) f(x) = 5/x ( : 5.4 ) (5) f(x) = x /x 3: {f(x)g(x)} = f (x)g(x) + f(x)g (x) (5.40) : F (x) = f(x)g(x), F (x + dx) = f(x + dx)g(x + dx) = {f(x) + f (x)dx}{g(x) + g (x)dx} = f(x)g(x) + f (x)g(x)dx +f(x)g (x)dx + f (x)g (x)dx, dx,, dx,, f(x)g(x) F (x), F (x + dx) = F (x) + {f (x)g(x) + f(x)g (x)}dx f(x) = (x + x + )(x x ) (5.44), () : x + x + x x, () ( ), f(x) (x),, 3, : (f + g) = f + g (5.45) (af) = af (5.46) (fg) = f g + fg (5.47),, 4: {g(f(x))} = g (f(x))f (x) (5.48) dx,, F (x) = f (x)g(x) + f(x)g (x) (5.4) : g (f(x)), g(x) g (x) x f(x), g(f(x)) : 5.8 F (x) = (x + )(x + x + 3) f(x) = x +, g(x) = x + x + 3, (x + ) 3, g(x) = x 3 x, f(x) = x +

85 g (x) = 3x, 4, {(x + ) 3 } = {(f(x)) 3 } = 3(f(x)) f (x) = 3(x + ) {(x + ) } = 3(x + ) (x) = 6x(x + ) (5.49), (5.49), 6x 5 + x 3 + 6x (5.50), (x +) 3,, dx 0, f (x), f (x)dx, 0, (5.0), f(x) g(x), x 0 f(x), dx f (x)dx, g ( f(x) + f (x)dx ) = g(f(x)) + g (f(x)){f (x)dx} (5.55) g(f(x)), F (x), (5.54) (5.55), F (x + dx) = F (x) + g (f(x))f (x)dx (5.56) {(x + ) 3 } = (x 6 + 3x 4 + 3x + ) = 6x 5 + x 3 + 6x (5.5) dx,, F (x) = g (f(x))f (x) (5.50), (5.5) ( ) ( ) 4,,, 4 : 5.0 x + (5.5) f(x) = x +, g(x) = /x, 4 (5.8) g (x) = /x, { } = x + (f(x)) f (x) ( ) = (x + ) (x + ) 4x = (x + ) (5.53), g(x) f(x), x +, x + x, (5.53), 4 : F (x) = g(f(x)), F (x + dx) = g(f(x + dx)) = g ( f(x) + f (x)dx ) (5.54),,, (5.7), dg dx = dg df df dx (5.57),, dg, df, dx, 08 : () F (x) = (3x + ) 3, () F (x) = (x + x + ) 3 (3) F (x) = (x 5 + x 4 + x 3 + x + x + ) (4) F (x) = ( + /x) 09 u(x) /u(x), *4 : ( u) = u u (5.58) : g(x) = /x, f(x) = u(x) 4 0 v(x) u(x) v(x)/u(x), : ( v u) = v u vu u (5.59) *4 (5.58), (5.59) u(x) (x)

86 74 5 :, () f(x) = + x () f(x) = x + x {f(g(x))} = f (g(x))g (x) (5.64), g(x) f(x), P.60 (4.38), f(g(x)) = x, : () u(x) = + x (5.58) () u(x) = + x, v(x) = x (5.59) 5. /x n (n ), g(x) = x n, f(x) = x ( ) n g(f(x)) = = x x n (5.60), (5.6) g (x) = nx n, 5.4 f (x) = /x, (5.48), : ( ) ( n ( ) = n x x) n x ( n = n ( x) ) x = n x n+ (5.6) (5.6) (x n ) = n x n (5.6), (5.6) n n (5.6), n,, n, 4 P.0 ( ) 5: g(x) f(x) g (x) = f (g(x)) (5.63) : f (g(x)), f(x) f (x) g(x) f(g(x)) : {f(g(x))} = (x) = (5.65) (5.64) (5.65), f (g(x))g (x) = (5.66) f (g(x)), g (x) = f (g(x)), (5.63) 5. (5.67) f(x) = x /n n g(x) = x n, g(f(x)) = x, g(x) f(x), g (x) = nx n (5.63)( ), f (x) = ( ) = g (f(x)) = n(f(x)) n = n(x /n ) n nx (n )/n = n x (n )/n = n x( n)/n = n x(/n) (5.68) (5.68), (5.6) n /n, (5.6), n (5.6), (5.6), (5.68), α, (x α ) = αx α (5.69), α, 0 ( ) *5,,, : *5,

87 : α 0, (x α ) = αx α (5.70) ( 3) (x x + ) (5.73) ( ) 5.3 f(x) = /x /x = x, (5.70) α =, f (x) = (x ) = x = x (5.7) (5.8) ( ) 5.4 x ( x) = (x / ) = x/ = x / = x ( ) () x 3 (5.7) 6, () x (3) x /3 5 (4) x 5,,,,, 5.5 x + g(x) = x f(x) = x + g(f(x)), ( x + ) = x + (x + ) x = x + = x x + ( ) 5.6 x x + (x x + ) = x x + + x( x + ) = x + + x = x + + x x + x x + (5.73) (5.74) 3,, (x + x) = x +, x + x = x + 4 : () (4) (7), x + 3 () + x x x (5) (8) x + x + x (3) (6) x + x + x (5.8), f(x), x = x 0 f (x 0 ), f(x 0 + x) f(x 0 ) + f (x 0 ) x (5.75) x 0, =, x f(x 0 + x),, ( ),,, 76,?...,,, OK, x 0 = 0, 0 x, x x, f(x) f(0) + f (0)x (5.76)

88 76 5,,, ( ) 5.7 f(x) = + x : f (x) = /( + x), f(0) =, f (0) = / (5.76), x = 0 + x + x (5.77),. = , (5.77),. = / =.05.0 = , (5.77),.0 = / =.005 ( ) 5 x = 0, ( a ) ( + x) a + ax (5.78) (5.78) 6, x = 0 : () /( + x) () /( x) (3) / + x (5.78), ( ) 5.8 (.0) 0 0,.0 = : ( + 0.0) =. (.0) 0 =.046, ?, 5 = 5, 6 = 5 + ( 6= 5 + = 5 + ) = ( 5 + ) = = , 3 (4 ), 5, + /5 /5 0, (5.78), ( 5) 7, (!): () (0.99) 0 ( ) () 0 /3 (.544 ) 5.7, f(x),, ( ), f (x) ( ) f (x), (5.8), d ( d ) dx dx f(x),, d dx f(x), d f dx (x), d f dx (5.79), n (n ),, f (n) (x) (5.80) d n f(x) dxn (5.8) d n f (x) dxn (5.8) d n f dx n (5.83) x,, (5.83), (x) (5.80), n, f (x), 3, (3) (n) 8, f (x), f (x), f (3) (x) () f(x) = x 5 + x 3 () f(x) = x 77 d x/dt dt d x?

89 , dt dt dt d x, t x, t t + dt dx, t t + dt t + dt dx ( ) d x, d(dx) dx, dt d x dt dx,, (0 ),,,,, dx/dt, dx dt,,, dx/dt dx dt,, 5.8,,, ( ), f(x) = { x x (x < 0 ) (0 x ), f (x) = { (x < 0 ) (0 x ) (5.86), x = 0 (5.0) ( (5.5) ) x 0 = 0, (5.0) f(0 + dx) = f(0) + f (0)dx (5.87) (5.85), f(0) = 0, (5.87) f(dx) = f (0)dx (5.88), (5.86), f (0) =, (5.88) f(dx) = dx (5.89) 5.0 dx 0, dx < 0, (5.89), f(dx) < 0 (5.85) f(x) = x (5.84),, (5.86) x = 0 *6, f (0), x = 0, f (x) = /x x = 0? (5.5) (5.0), f(x 0 ) x 0 = 0 f(0) f(0) (0 /0 =, ), x = 0 ( ), f (0), x = 0 ( ) 5. (5.85) 5.7 (x = 0),,,,, 0 *7 5.0,, 0 5. { 0 (x < 0 ) f(x) = (0 x ) (5.90) 5. f(x) = x (5.85) *6 *7 f(x) = x 3 x = 0 0, x = 0

90 78 5 y y - - O - x - - O - x 5.7 ( ) y = x x = (5.90) y = f(x) x = 0 ( ) 0, { f 0 (x < 0 ) (x) = 0 (0 x ) (5.9), f (x) = 0,, x = 0! (5.0) x 0 = 0, f(x 0 ) = f(0) =, (5.0) f(dx) = + f (0)dx (5.9), (5.9), f (0) = 0, (5.9) f(dx) = (5.93) dx 0, dx < 0, (5.90), f(dx) = 0, (5.93), f (0) = 0 x = 0, f (0), ( ) (5.90) 5.8 x = 0,, *8, 5.3 f(x) = x /3, f (0) ( ) 5.3, (x = 0), ( ) ( 5.9), x = 0 ( x ),, y - - O y = x /3 x = 0 ( ),?, (5.5) (5.0) f,, (5.5) (5.0),,, x, x, f (x) = 3x /3 (5.94), x = 0 *8,,, 5.9,, (, )

91 5.9 79,,, :, x P t P (x ) x(t) t 0 P x(t 0 ), t P x(t ), t t 0 *9 t > t 0 t t 0, t < t 0 t t 0 t t 0, t t 0 0 (displacement), x(t ) x(t 0 ) x(t ) > x(t 0 ), x(t ) < x(t 0 ) (distance), x(t ) x(t 0 ), 0, t... t t 0... ( ) t t 0... ( ) x(t)... x(t ) x(t 0 )... ( ) x(t ) x(t 0 )... ( ), t 0 t P v, : v := x(t ) x(t 0 ) t t 0 (5.95), (5.95),,, P x,, (5.95), t 0 t,, t 0 t, (5.95) t, t 0 ( ) *9,, : v(t 0 ) := lim t t 0 x(t ) x(t 0 ) t t 0 (5.96) t t 0 = t, x(t 0 + t) x(t 0 ) v(t 0 ) := lim t 0 t (5.97) (5.5), x(t), t = t 0 t 0 (velocity) ( ), t v(t),, x (t), x (t) ẋ ( ), v(t) = dx dt = x (t) = ẋ (5.98),,, ( ),, (speed) ( ) (5.95), v = x(t ) x(t 0 ) t t 0 (5.99) x(t ) x(t 0 ), t t 0, (5.99), = =,, (5.99) t t 0, ( ), ( ) ( ), 0, 4...,

92 80 5, ( ) 9 ()? ()? (3)? (4)? (5)?, (5.98), (5.04), a(t) = v (t) = x (t) (5.06), 78,...,,,,...,... (5.0), (5.98), : x(t + dt) = x(t) + v(t)dt (5.00) dt (5.00), dt ( x(t + dt)), ( x(t)), v(t)dt (,, t t + dt,, dt ),,, (5.00) x(t + dt) x(t) = v(t)dt (5.0),, (5.0), x(t + dt) x(t) = v(t) dt (5.0), (, v(t) ),, = (5.03) (acceleration) ( ), t a(t), a(t) = v (t) (5.04) (5.0) (5.04), v(t + dt) = v(t) + a(t)dt (5.05) 0 P t P ( ) x, (a, b, c t ): x = a t + b t + c (5.07) () t P? () t P?,,,,, /? (5.97) t,,, p q p / q,, / (5.07), a, b, c? 0, a = 4.9 m s, b = m s, c = 3 m, t = 0 s,,, 5.0, 3,, t, r(t)

93 5. 8, r(t) = (x(t), y(t), z(t)), t, (5.0), x(t + dt) = x(t) + x (t)dt y(t + dt) = y(t) + y (t)dt (5.08) z(t + dt) = z(t) + z (t)dt, r (t) = (x (t), y (t), z (t)) (5.09), 3, r(t + dt) = r(t) + r (t) dt (5.0) (5.0) r(t), (5.0), r (t) 79, r(t),?...,,!,, (5.0), r(t + dt) r(t) dr, dr = r (t) dt, dr dt? r (t) r(t), r(t) t, ( ), velocity, v(t), v(t) := r (t) = (x (t), y (t), z (t)) (5.), (5.0) : r(t + dt) = r(t) + v(t)dt (5.), ( ), a(t) := v (t) = r (t) = (x (t), y (t), z (t)) (5.3), P.6 (.8), F = ma (5.4) a,,,, m, F,, 3 xy P, t r(t) = (v 0 t, gt /) (v 0, g ) () P? () t P v(t)? (3) t P a(t)? 5. f(x) = x + f (x) = x, f (0) = 0 x = 0 0, x = 0 y = f(x),, 0?, f(x), x = x 0 (5.0), x, f(x 0 + x) f(x 0 ) + f (x 0 ) x (5.5) f (x 0 ), x f (x 0 ) x < 0, (5.5) f(x 0 + x) < f(x 0 ), f(x 0 ) * 0 f (x 0 ), x f (x 0 ) x < 0, (5.5) f(x 0 + x) < f(x 0 ), f(x 0 ), f (x 0 ), f (x 0 ) = 0, y = f(x), x = x 0 f (x 0 ), f (x 0 ) = 0, 0, ( ) *0

94 8 5, ( ) ( ),,, 5.4 f(x) = x 4 x f (x) = 8x 3 f (x) = 0, 8x 3 = 0, x = /, f(/) = 3/8 ( ),, f(x) x,,, *, f(x) = x 3 x, 4.7 (p.54),, f (x) = 3x, x = / 3 x = / 3 0 ( ), ( ),, ( x ),,, y? x = /, x f (x) < 0 ( f(x) ), x f (x) > 0 ( f(x) ), f( / ) = /4 x = /, x = 0, x f (x) > 0 ( f(x) ), x f (x) < 0 ( f(x) ), f(0) = 0 x f(x), 5.0 ( ),, 0,, 0,, 5.6 y = x 3 x = 0 0, P , y = x 3, x = 0 ( ) 5. f(x), f( x) = f(x) x, f ( x) = f (x) (!), f ( x) = f (x), f (x)! O - -/ y = x 4 x ( ) x 4 5 n 0,,! 5.5 y = x 4 x, x (y = 0), x = 0, ± f (x) = 4x 3 x, x = 0, ±/ f (x) = 0 () f(x) = x n () f(x) = x n+ (3) f(x) = /( + x ) (4) f(x) = x/( + x ) : (3), (4) *,

95 5. 83 y = /x n, x n, x,, (5.6), y = x /n, n, x,, (5.68) 3 ( (5.63)), :,, y = x 4 n 0, P n (x) * P n (x) := n n! d n dx n (x ) n (5.6) () n = 0,,, 3, 4, P n (x) () n P n (x), n P n (x) () (g(x) = x 3, f(x) = 3x + ) F (x) = 3(3x + ) (3x + ) = 3(3x + ) (6x) = 8x(3x + ) () (g(x) = x 3, f(x) = x + x + ) F (x) = 3(x + x + ) (x + x + ) = 3(x + x + ) (x + ) (3) (g(x) = x, f(x) = x 5 + x 4 + x 3 + x + x + ) F (x) = (x 5 + x 4 + x 3 + x + x + ) (x 5 + x 4 + x 3 + x + x + ) = (x 5 + x 4 + x 3 + x + x + ) (5x 4 + 4x 3 + 3x + x + ) (4) (g(x) = x, f(x) = + /x ) ( F (x) = + )( + ) x x ( = + ) ( x x = x + ) x 3 03 f(x + dx) = p(x + dx) + q = px + q + p dx = f(x) + p dx dx, f (x) = p, 08 + (x + x + )(x x ) = (x + )(x x ) + (x + x + )(x ) = 4x 3 4x 3 () f(x) = x 4 x 3x f (x) = 4x 3 4x 3 09 /u(x), /x u(x) (/x) = /x, ( u) = ( u )u = u f(x) p = 0, f (x) 0 0 v(x)/u(x), v(x) /u(x) 06 () f (x) = x + () f (x) = 8x + 5 ( (3) f (x) = 6x + 3 (4) f (x) = 5/x v ) ( = v ) ( + v u u u) (5) f (x) = + /x = v ( ) ) + v ( u u u 07 = v u vu u () f (x) = (x + x + ) (x x ) 4 () u x ( + x ) () x ( + x ) () x () x = 3/ x 3/ (3) 3 x /3 (4) 5x 6 () : g(x) = x, f(x) = x g (x) = /( x), ( ) = *, (K L ) f (x) f(x) = (x + 3) x + 3 = x + 3 = x + 3

96 84 5 () : g(x) = x, f(x) = x 4 g (x) = /( x), ( ) = f (x) f(x) = ( x ) x = x x = x x (3) : x + x ( ) = (x ) + x + x ( + x ) = x + x + x 3 + x (4) : g(x) = x /, f(x) = + x 4 ( ) = (f(x)) 3/ f (x) = ( + x ) 3/ x (x) = ( + x ) 3/ (5) : g(x) = /x, f(x) = + x 4 ( ) = x ( + x) ( + x) = ( + x) = x( + x) (6) : g(x) = x, f(x) = + /x 4 ( ) = + x ( + ) = x x + x (7) : x /(x + ) 3 ( ) = (x ) ( ) x + + (x ) x + = x + (x ) (x + ) = (x + ) : = /(x + ), (8) : g(x) = /x, f(x) = + /x 4 5, x, ( ) = x x + = x +,, ( ( ) = ) = x + (x + ) f(x) = ( + x) a, f(0) =, f (x) = a( + x) a, f (0) = a (5.76), 6 () (5.78) a =, /( + x) x () x x, /( x) + x (3) (5.78) a = /, / + x x/ 7 () (0.99) 0 = ( 0.0) = 0.9 () = 8, 0 /3 = (8 + ) /3 = { ( 8 + )} ( /3 = 8 /3 + /3 ( = + 8 8) ) /3 4 ( + ) = =.66 8 () f (x) = 5x 4 + 6x, f (x) = 0x 3 + x, f (3) (x) = 60x + () f (x) = /( x), f (x) = /( x) 3, f (3) (x) = 6/( x) 4 9 (), (), (3) (4) (5) 0 () : (at + bt + c) = at + b () : (at + b) = a 0(), a, a (SI m s ) 0(), at + b, b (SI m s ) x = at + bt + c, c x ( ; SI m) ( ) 503 m 99 m s 9.8 m s 3 r(t) = (v 0 t, gt /) = (x, y) () x, y t, y = gx /(v 0) xy,, () v(t) = r (t) = (v 0, gt) (3) a(t) = v (t) = (0, g),, v 0 5 () f( x) = ( x) n = ( ) n x n = x n = f(x) f(x) f (x) = nx n f ( x) = n( x) n = ( ) n nx n = nx n = f (x) f (x) ()

97 85 6 :,, 6. y=3 x y=(3 x ) y 5 4 a, a x a =, x, x = 0,,, 3, 4,,, 4, 8, 6,..., y = x, 6. y= x y=( x ) y O - x ( ) y = 3 x ( ) y = (3 x ) 6. ( ) - - O - y = x ( ) y = ( x ), y = x ( ), y = x,,, y = 3 x?, 6., y = 3 x, 6. 6., 6., ( ) ( ), 6. ( ) ( ), 6., a 3, y = a x x?, a.788, P.9 e, (e x ) = e x (6.) e ( e 0) e x, (exponential),, e x, exp x 6, y =.78 x, y =.78 x, x 0.0 x, e, : * e = lim h 0 ( + h) /h (6.) * dx e (e x ) = e x,, e x+dx = e x + e x dx, e x+dx = e x (+dx),, e x+dx = e x e dx, e dx = + dx /dx, e = ( + dx) /dx dx h (6.)

98 86 6, /h n, ( e = lim + n (6.3) n n) 7 (6.3) lim, n =, n = 0, n = 00, n = 000, n = 0000, 8 5 (6.), (6.3) 5 (6.3), n 0... n (6.3) lim.78, e x, * : ( e x = lim + x n (6.4) n n) 9 (6.4) lim, x =, n = 0000 e, x =, n = 0000 (6.4) (6.3), : 6., r, x rx, ( + r)x, ( + r), ( + r), ( + r) x, n, ( + r) n 30, 4 5 * : x = 0 0 < x N = nx, n = N/x (6.3) lim, ( + x/n) N/x n e, x, ( + x/n) N e x n N, e x = lim N ( + x/n) N N n, (6.4) x < 0 n + x = m (n = m x), ( + x/n) n = {(n+x)/n} n = {m/(m x)} m x = {/( x/m)} m x = {/( x/m)} m {/( x/m)} x = /( x/m) m ( x/m) x, n, m, 0 < x, /( x/m) m /e x = e x, ( x/m) x () %, r = 0.0, 00? () 0.0%, r = 0.000, 0000? (!),,.78..., e? n,, r = /n,, (6.3) lim n n (6.3) 80 30(),.0 00, P.76 ( + x) a + ax,.0 00 = ( + 0.0) = + =,.7?..., a 3 % 00,,? (6.4) 3, n (n ) (), (), (3), n (4) n,? (5) 000, 000? 4,, 8 e?... ( ),,

99 6. 87, y = e x 0, e 0 = (0, ), x = y = e =.78, (,.78 ) x y.78, x, x =, y = e = /e = /.78 x y /.78, x, x, y = e x , y = e x y = e x y ( P ) y 5 4, 6.4 f (x) = e x xe x = ( x)e x (6.7) f(x) = e a/x (a 0 ) e y y = a/x, f (x) = e a/x( a x 34 ) = e a/x ( a x ) = ae a/x x (6.8) () f(x) = e x () f(x) = e x (3) f(x) = exp( x ) (4) f(x) = x exp( x ) 6., a, x (a ), a x, log a x (6.9) O - x a, x, log a x = y (6.0), 6.3 y=exp x y=exp(-x) y = exp x y = exp( x) x = a y (6.) (6.0), (6.) y log a x, 33 () y = e x () y = ( e 3x ), 6. f(x) = e x e y y = x ( P.7!), (6.), f (x) = e x ( x) = e x (6.5) 6.3 f(x) = xe x x e x, (P.7), f (x) = e x + x(e x ) (6.6) (e x ), e x x = a log a x (6.) (6.), log a x, a x (6.), a x ( (6.) ) :,,,, log ( 8), 3 (( ) 3 = 8 ), log ( 8), log 8?,, log?

100 a, b, c a,, (8), (9) b, (0) c () a log a b = b (6.3) () log a a = (6.4) (3) log a = 0 (6.5) (4) log a b + log a c = log a bc (6.6) ( ) (5) log a = log b a b (6.7) ( b ) (6) log a = log c a b log a c (6.8) (7) log a b c = c log a b (6.9) (8) log a b log b c = log a c (6.0) (9) log a b = log b a (0) log a b = log c b log c a (6.) (6.) (6.),,,, 6.5 log 3 5,, 3 log 3 5 ( ),, log 3 5 = ln 5 ln 3 ln 5 =.609 ln 3 =.098,,, (6.5) (6.6), log 0 e (6.7) ln (6.8) 38 () (6.7) (6.8) 5 () (6.7) (6.8) (3), f(x) = e x, g(x) = ln x,, g(f(x)) = ln e x = log e e x = x log e e = x, f(g(x)) = exp(ln x) = e log e x = x ( (6.3) ) P.60 (4.38) 6.4, y = e x y = ln x y = x ( P.60 ) y exp x ln x log 3 5 = ln 5 ln 3 =.609 =.464 (6.3).098 : log 3 5 = log 0 5 log 0 3 = =.464 (6.4) O - - x y = e x y = ln x, y = x () log 7 () log x : log 0 x = (log 0 e) (ln x) ln x (6.5) ln x = (ln 0) (log 0 x).306 log 0 x (6.6),,,,, y = ln x 0 < x, ln x, x 0 ( ), ln( 5)?, e =.78,, 5, 39

101 () y = log x () y = ln x (3) y = ln( + x), f(x) = e x g(x) = ln x,, f (x) = f(x), ( 5; P.74), (ln x) = g (x) = f (g(x)) = f(g(x)) = x (6.9) ( 0 < x ), ln x /x 40, ln x (0 < x ), (ln x), (/x), : x = 0 x < 0 ln x, 0 x 6.5 : (3) x < 0 0 < x (6), x < < x (7),, (6.8) :,,, (7), 8 (3), ln x /x,..., 6.5 ln x, y, y (x < 0), x = 0, ln x, ( (ln x) ), x = 0,, (ln x) 0, x = 0, (ln x) ( ), /x, x < 0 ( ) 6.3 a, f(x) = exp( ax ) (6.30) (Gauss function),,, 4, : 6.5 y = ln x ( ) ( ) y = /x x 0.05, 0. 4 : () ln(x + ) < x () ln x 0 < x (3) ln x x 0 (4) ln(x ) x > (5) ln( x) x < (6) ln x x (7) ln x + x x (8) log 0 x 0 < x () () 0 (3) f (0) = 0 ( x = 0 ) (4) x, 0 (5) x = 0,, x = 0, 43, () a = () a? 6 y = exp( x ), y = e x...?

102 90 6 y 6.7, S (7, 30) S y = 40 T (4, 4) - - O x 44 U 6.6 y = exp( x ) y > 0, x, x, x, x 3 4 x OK( ),, 0.,, 0, 00 4, 00, 0. 7 y = exp( x ),... 4(3),! 6.4, 6.7,, : 0 0, 0 0,, (,, ),, y = Ae ax (6.3) ( (6.38), (6.39) ), A, a,,, log 0 y = log 0 A + ax(log 0 e) (6.3), log 0 y x ( ), (a log 0 e) a, (log 0 A) A 00,,,, ( ), y U T x S 0 log 0 y, ( ; ),,,,, ( y ), (log 0 y ) 6.7, T S, y,, 30 4 = 6, log 0 y,, T S,, ( ; 0 ),, = = a log 0 e (6.33)

103 6.5 9, a =.306 log 0 e (6.34) a, (6.3) y = A 0 ax, (6.34) / log 0 e , 3 S, T, U,, y = Ae ax A a,,, A a, 3 S, T, U,? 6.8, 00, y = Ax a (6.35) ( ), A, a,, (6.35), log 0 y = log 0 A + a log 0 x (6.36),,,, (,, ) a, (log 0 x = 0 y ) A * , 3 P, Q, R,, y = Ax a A a,,, A a, 3 P, Q, R,? 0 R Q P 6.5,, d f(x) = 3f(x) (6.37) dx , P (30, 0) 46 Q R,,,,,, *3,, *3, ( ), power law scaling power (6.37), f(x),, (6.37) 3, x,,, f(x), f(x) = x, (6.37)? (6.37), d dx x = x,, 3x x = 3x (x = x = ), f(x) = x, (6.37)! 48, (6.37) *4,, e 0,

104 9 6 : (5)(6) () x () e 3x (3) 3e x (4) 5e 3x (5) (6) 0,, 6.6 ( ),, 3 4, ( ), (isotope), 3 ( 3 C),, (6.37) e 3x, : 4 ( 4 C) *5, 4 ( 4 N) 4 C, t C(t) α 0, d f(x) = αf(x) (6.38) dx ( ), f(x) = βe αx (6.39) (β ) (6.39) (6.38) (6.38) (6.39)?..., (6.37), (6.38) α = 3, (6.39) β, β, (6.39) x = 0,, 0 C 0 4 C, C 0 := C(0), t C(t), t t + dt, ( ),, C(t),,,, dt,, dt, C(t),, dt, C(t + dt) = C(t) αc(t)dt (6.4) α 0 < α C(t),, C(t + dt) C(t) = αc(t)dt (6.43) dt, f(0) = βe 0 = β (6.40), β = f(0) (6.39), f(x) = f(0)e αx (6.4), (6.38) (6.4),,, C(t + dt) C(t) dt = αc(t) (6.44) dt, C(t) : dc dt = αc(t) (6.45), (6.45), C f, t x, α α, *5, :

105 (6.38), (6.45), (6.4), C(t) = C 0 e αt (6.46), C 0, t = 0( ) C, (6.46), C *6 5 4 C, /4,?, 50 (6.46) 6.7, ( OK),,, (6.46), : C(t) = C 0 ( ) t t h (6.47) t h, 4 C, t h = 5730 (6.48), A, B C : A B + C (6.49) A [A] [A] t, [A](t),, 5 (6.47), (t) dt dt A B+C :, A B C () (6.47), t, C(t) () C(t h ) C 0, t h (3) (6.47), (6.46) : / = exp( ln ) (4), (6.46) α? (5) (6.48), α,, A, [A] (,!), dt ( 0, 0!),, k, k[a] dt (6.50), [A], dt [A] d[a] *7,, C 0,,, C 0 C(t), t, :, CO 4 C 4 C,, 4 C, 4 C,, ( C 0 ), 4 C, d[a] = k[a]dt (6.5), [A], d[a] dt = k[a] (6.5) *6,,,,,,, *7 d[a] = [A](t + dt) [A](t)

106 94 6 *8, [A] (6.5), [A], (6.5), k,, (6.5), [A] f, t x, k α, (6.38), (6.5), (6.4), [A](t) = [A](0)e kt (6.53) [A](0) ( ) [A], (!), (6.53), A, 53 (6.49) (6.53) ( OK),, 6.8 (... ), : 6.9,, 3, (, ) *8 (6.5) (6.5) : [A](t + dt) [A](t) = k[a]dt [A](t + dt) [A](t) dt d[a] dt = k[a] = k[a] 6.9,,, (, : x, x I(x), dx, c, dx, I, x + dx, x, κ I c dx (κ, I(x + dx) = I(x) κ c I dx (6.54) (6.54), I(x + dx) I(x) = κ c I dx (6.55) dx, I(x + dx) I(x) dx = κ c I (6.56) dx, I(x), di = κ c I (6.57) dx (6.57), (6.57) P.9 : I(x) = I(0) exp ( κ c x) (6.58) (6.57) (6.58), (Lambert-Beer) Lambert, Beer,, Lambert Beer,,

107 : I(x) = I(0) 0 κ c x/ ln 0 (6.59) κ ( I(x) ) ln 0 c x = log 0 I(0) (6.60) (6.60) I(x)/I(0),,, x d ( d = cm ), (6.60) (absorbance),, A, A : ( I(d) ) A := log 0 I(0) 55 : (6.6) () I(d) I(0), A? () I(d) I(0) /0, A? 84 (6.6),?..., I(x) I(0), I(x)/I(0),, ln{i(x)/i(0)} 0 I(x)/I(0) 0, ln{i(x)/i(0)},,,, 0, x = d, (6.60), (6.6), κ ln 0 c d = A (6.6) A c = (κ/ ln 0)d (6.63) κ/ ln 0,,, d ( ), (6.63) A (ln 0/(κd)), K, c = K A (6.64), c A,,, cm cm, ( d = cm ) , a, nm, 76.8 mmol dm 3 cm, a 80 0, nm (d = cm),, A = 0. a : mmol dm 3 P.3! 6.9 a, b, c, f(x) = a + be cx (6.65),,, 57 (6.65), () f(0) = /(a + b) (), a + be cx, x (3) f(x), x (4) x, f(x)? (5) x, f(x)? (6) f (0) (7), y = f(x) 6.0, a = b =, (6.65),, ( )

108 96 6 y /a /( a + b) O x 0.0 m, p m n,, F, F = ( p) n (6.66) ( ) 6.0 (6.65)( ) x = 0 bc/(a + b) () m L m, np = L (6.67) 5, 0 3 5,,,,,? ( ) () 3 ( 3 I; 8.0 ) () 34 ( 34 Cs;.07 ) (3) 37 ( 37 Cs; 30. ) : (6.47) C(t)/C(0) 6 (Bq)?, *9, 37 Cs, 00 Bq ( 00 ), 37 Cs? ( ) : (6.43), 00 C(t) dt= α 7 a x x x ( < a) : a = e ln a, a x = (e ln a ) x, x x = (e ln x ) x, 8,,, () m, 0.0 m (00 cm ) F n L p (3) (6.67) L,, n, p 0, F = e L (6.68),, (4) m m 9,, E, M, : log 0 E = M (6.69) J () 995, M = 7.? (), (3) 0, M = 9.0?,? (4),, kw * 0, ( )? (5),, % *0,? *9,,, kw

109 (n = ) 9 (n = , e = ) ( + 0.0) 00 = ( + /00) 00 (6.4), e = 7.38,, ( + 0.0) 00 = () n, /n, /n (), ( /n) (3) 33 : x = 0 y, x, y 34 () e x () e x (3) x exp( x ) (4) ( x ) exp( x ) 35 ( ) () () a = a (3) a 0 = (4) a log a b+log a c = a log a b a log a c = bc = a log a bc, a, a log a c = c, = (9) (6.0), c a, (log a b)(log b a) = log a a = log b a, (0) (6.0), a c, b a, c b, 36 (log c a)(log a b) = log c b log c a, () log 7 = (ln 7)/(ln ).807 () log = (ln 5)/(ln 0.3) (6.0) 38 (3) (6.0) y - O () log x () ln x (3) ln (+x) x, log a b + log a c = log a bc (5) 0 = log a = log a {b (/b)} = log a b + log a (/b), log a (/b) = log a b (6) (4)(5) (7) a, (), a log a bc = b c, a, a c log a b = a (log a b)c = ( a log a b) c = b c, = (8) a, a log a b log b c = ( a log a b) log b c = b log b c = c () d dx ln(x + ) = x+ (x + ) = x+ () d dx ln(x) = x (x) = x d d (3) 0 < x, dx ln x = dx ln x = x (4) d dx x < 0, d dx ln x = d ( x) = x ( ) = x, 0 < x x < 0, d dx ln x = x ln(x ) = x (x ) = x dx ln( x) = x (5) d dx ln( x) = x ( x) = x ( ) = x (6) < x, d dx x <, d dx ln x = d dx ln x = d dx ln(x ) = x ln( x) = x

110 98 6 ( ), < x x < d, dx ln x = x d (7) dx ln x x+ = d dx (ln x ln x + ) = d d dx ln x dx ln x + = x x+ = x (8) ( ) /(x ln 0) 4 f(x) = exp( ax ) (a ), () f( x) = exp{ a( x) } = exp( ax ) = f(x) () e =.78 0, 0 (3) f (x) = ax exp( ax ) x = 0 (4) 0 lim x exp( ax ) = lim (6.70) x e ax 0 < a, x, ax, e ax, (6.70), (6.70) 0 (5) 0 x, e ax,, 43 x = 0 e ax, x 0 x = 0, e ax x = 0, () () exp( ax ) = exp{ ( a x) }, a, x / a, 44 (, 0.5) 45 (a = 0.69, A = 0.5 ( ) S, T, U y ) 46 Q: (, 7), R: (0., 3) 47 (a = 0.375, A = 5.5 ( ) P, Q, R y A,, x = ) 48 () f(x) = x, (6.37) C 0 C 0 / C C 0 /4 C 0 /8 O 6. t h t h 3t h t 4, =(x) =, = 3(x) = 6x,, (6.37) () (), (4), (6) 5 () 6. () (3) : C = C 0 exp[ (ln )t/t h ] ( ) (4) α = (ln )/t h (5) α = (ln )/(5730 ). 0 4 /...! 5 C = C 0 /4, t = t h = 460 (500 ) 54 : (6.58) e 0 log 0 e (6.59) log 0 e = /(ln 0) (6.59) I(0), (6.60) 55 (), I(d)/I(0) = /, (6.6), A = log 0 (/) = log 0 = 0.30 (6.7) (), I(d)/I(0) = /0, (6.6), A = log 0 (/0) = log 0 0 = (6.7) : 56 ( ).6 µmol/dm 3

111 99 7 :, 7., ( ) 58 ABC C C AB D ( 7.) () ABC ACD (), AB:AC=AC:AD, AB AD = AC (3) ABC CBD (4), AB:BC=BC:BD, AB BD = BC (5) AB = AD + BD : AB = AC + BC (7.) B D,,,, 360!,, ( ) 60 =, 60 =,,,, , , ( ) 60 =, =(/60), 4.5 = 4.5 (/60) , ( ) = , 60 =, =(/60), = (/60) 0.8, ( ) = , 0.07 =60, 0.07 = = 6.30, 6, 0.30, 0.30 = = 7.8, ( ) A 7. C, ( II B ), (radian) : 7.,,, ,, ( 7.) 7.3 4, 90,,

112 00 7,, π,, 7. π/4 = π/, 90 = π 7.4, 360,,, π, 360 = π ( ) ( ),,,,,, :, ( ) 7.4, 360 π, = π 360 = π (7.) 80 x, x = π x (7.3) 80, (7.3) 80/π, x = 80 π x (7.4) (7.3), (7.4),,,,, π π 85?...,? 86,..., r, l, l/r,, r l,, 59, 60, () 90 () 45 (3) 80 (4) 30 (5) 6, () π () π/3 (3) π (4) π/4 (5), 80/π π 3, 60 ( ), :,,,, 60 ( 7.3), 0 π,, 0 π, π, 0 π 3π π + π, π, π, π, 87, 3π = π?..., 3π/ = π/, 70, 90,,

113 7.3 : 0 C C A B 7.3 AB=BC=CA= ABC, ABC BC BC BAC π/3 (60 ), BAC, 60 6, 0 π () π () 3π (3) 6π 88, 80 π?..., 80, 90 0, 90 80, π π/...,?, P.30 4!, 360, 0.,!...,, 8,..., π,,,,, 7.3 : 63,,,,,,,,,,,, N,, θ a, l () θ a/l () R, d,, d = Rθ (7.5) (3) d R,, N R = d/θ (4) A, B, A = πr (7.6) B = Nπd /4 (7.7) (5),, B A = Nθ 4 (7.8) (6) a = cm, l = 60 cm, N = 0,? (7), (8), :,,, (Bitterlich),

114 (unit circle) 7.4, cos θ, sin θ xy,, (, 0) θ, x cos θ, y sin θ sin θ/ cos θ tan θ ( 7.4) cos θ, sin θ, tan θ, (trigonometry) cos, sin, tan,, 64? 65 θ, cos θ, sin θ, tan θ 89, x cos θ, y sin θ, sin θ/ cos θ tan θ,?... θ 66 cos θ sin θ, θ :? 67 θ, (sin θ ), (cos θ ), (tan θ ) ( ) 4, (4)(5)(6) () 0 () 30 (3) 45 (4) 60 (5) 90 (6) 0 (7) 80 (8) 70 (9) 30 (0) π/6 () π/4 () π/3 (3) (4) (5) 0. (6) 0.0 :,,, RAD R, DEG D,, 90 sin, (0.893 ), P.9.,, shift MODE,, (7.3) (7.4) (5), (6),, θ 0, sin θ θ, tan θ θ (7.9),, 90?...,,,,, cos sin, sec ( ) cosec ( ) : sec θ :=, cosec θ := cos θ sin θ 7.5 (7.0),,!,,, θ, : cos θ + sin θ = (7.)

115 7.6 03? (cos θ, sin θ),, cos θ + sin θ, (cos θ, sin θ),, cos θ + sin θ = (7.) π, θ θ + π, : cos(θ + π) = cos θ (7.) sin(θ + π) = sin θ (7.3), sin cos, π,, 7.5, θ, x ( ; ) θ, θ (x ), x, (cos( θ), sin( θ)) cos(π θ) = cos θ (7.6) sin(π θ) = sin θ (7.7) cos(π + θ) = cos θ (7.8) sin(π + θ) = sin θ (7.9) ( π ) cos θ = sin θ (7.0) ( π ) sin θ = cos θ (7.) ( π ) cos + θ = sin θ (7.) ( π ) sin + θ = cos θ (7.3) : π θ, θ y π + θ, θ π/ θ, θ x y ( y = x ), y ( ) θ π/ + θ, y ( ) θ 69 n, sin nπ = 0 (7.4) cos nπ = ( ) n (7.5) 70, + tan θ = cos θ (7.6) tan( θ) = tan θ (7.7) tan(θ + π) = tan θ (7.8) 7.5 (7.4), (7.5) (7.8), tan, θ π, sin cos π, tan, tan, (cos θ, sin θ), x ( 7.5), x y, : cos( θ) = cos θ (7.4) sin( θ) = sin θ (7.5),,, 7.5, ( ) 7 cos θ, sin θ, tan θ,, θ,??? 7.6,,,, 68 n, 7, x α

116 04 7 A, x β B, O A, B OAB, x (, 0) A, x α + β B, O A, B OA B () OAB OA B () OAB OA B (3) A, B, α, β (4) AB (AB ), α, β (5) A, B, α, β (6) A B (A B ), α, β (7) AB =A B, cos(α + β) = cos α cos β sin α sin β (7.9) (8) (7.0) sin(α + β) = cos(π/ α β), : sin(α + β) = sin α cos β + cos α sin β (7.30) : sin(α + β) cos((π/ α) + ( β)), (7.9) (7.9) (7.30), α, β, (7.9) (7.38),,, ( III!), 7.5 (7.9) (7.3), (±) sin cos, α = β = 0, α = 0, β = π/,, cos(α+β) = sin α sin β cos α cos β ( )? α = β = 0,,,!, cos(α+β) = sin α cos β cos α sin β ( )? α = β = 0,, 0,!, 7.6 (7.0), (7.9) : ( π ) cos θ = cos π cos θ + sin π sin θ, cos(π/) = 0, sin(π/) =,, sin θ ( ) 73 : 7.7 () cos(α β) = cos α cos β + sin α sin β (7.3) () sin(α β) = sin α cos β cos α sin β (7.3) (3) (cos ) cos α = cos α sin α (7.33), y = sin x y = cos x y = sin x 7.6 y = cos α (7.34) = sin α (7.35) (4) (sin ) sin α = sin α cos α (7.36) O -π -π π π y = sin x x 74 () cos α = () sin α = : + cos α cos α (7.37) (7.38), y = sin x, : sin 0 = 0 x π,

117 y x = π/ ( ), 7.6, ( 7.6 ) y = cos x (7.3), ( sin x + π ) = cos x (7.39), y = sin x (x )π/ y = cos x 7.7 cos x, y O -π -π - π π 7.8 y - y = tan x () y 0, x () y (3) (4) y ( ) (5) (7.38), x O -π -π π π - x y 7.7 y = cos x 7.6,!? y = tan x, tan x = sin x cos x (7.40), tan x, x = 0 0, x 0, sin x, cos x, tan x x = π/, cos x 0, tan x x = π/, cos x 0, sin x, tan x,, y = tan x 7.8, : 75 () y = sin x () y = cos x (3) y = tan x (4) y = sin x 7.9 O -π -π π π y = sin x x 7.0 y O -π -π π π y = sin x 7.6 y = sin x x, x y = sin x,,,,, x x 76 y = sin x OAB,

118 06 7 : () sin θ = AB OA () cos θ= OB OA (3) tan θ= AB OB (7.4) (7.4) (7.43) C 7. B P A 8 A O θ 7. B 77 P () BP = a sin C (7.47) II, (7.4), (7.4), (7.43), π/, () ABC S S = a b sin C (3) (7.48) 30 sin cos,...,, (7.4), (7.4), (7.43), AB = OA sin θ (7.44) OB = OA cos θ (7.45) AB = OB tan θ (7.46) 78 30, 00 m, m? S = b c sin A S = c a sin B (4) (7.48) (7.50) : a sin A = b sin B = c sin C (7.5) 8 (7.49) (7.50) (7.5) 79 3, 00 m, m? 3 0, (7.9) 7.9,,,, ABC BC a, CA b, AB c A, B, C A, B, C ( ACB C) B CA () CP=a cos C, AP= b a cos C () PAB, : c = (a sin C) + (b a cos C) (7.5) (3), : c = a + b ab cos C (7.53), ABC, : a = b + c bc cos A (7.54) b = c + a ca cos B (7.55)

119 7.0 07, (7.53) (7.55) : cos C = a + b c ab cos A = b + c a bc cos B = c + a b ca (7.53) (7.58) 8 (7.56) (7.57) (7.58) (7.53), C,,,, 7.0 cos, sin, tan, arccos ( ), arcsin ( ), arctan ( ),, cos θ = / θ arccos(/), cos, cos θ = / θ, ±π/3 ±7π/3 ±3π/3,,, arccos, 0 π, arccos x, cos θ = x 0 θ π θ, arccos = π (7.59) 3 9 (7.53) (7.58)?...,,,,,,,,,, a = 4 cm, b = 3 cm, c = cm θ, a b () (7.56) cos θ () cos θ + sin θ =, sin θ (3) (7.48), :, 83,,,, (7.48) 3 83(3)..., ( π/3 7π/3 ), arcsin x, sin θ = x π/ θ π/ θ arctan x, tan θ = x π/ < θ < π/ θ : arcsin arctan, arccos (θ ) : arcsin x Arcsin x sin x / sin x, arccos x Arccos x cos x / cos x, arctan x Arctan x tan x / tan x 3: θ cos θ sin θ, x, sin θ = x cos θ = x θ x, arccos x arcsin x, tan θ,, x arctan x 84 : () arctan () arctan 0 (3) arccos 0.5 (4) arcsin( 0.5) (5) arctan 5 ( ) (6) arcsin

120 (5), !? ?... (), (3) y = sin x x 0. OK (), (), (3) , ,?, 85 ( 880 m) ( 30 m), km, ( )? 7. 86, () y = sin x, 0 x 7 (), (3) y = cos x x 0. OK (), (), (3) y = cos x( ) ( ), y = sin x ( ) x 0., y = cos x y = sin x!, cos x, sin x!, (cos x) = sin x! tan x? tan x = sin x/ cos x P.73 (5.59), v = sin x, u = cos x, ( v (tan x) v = = u) u vu u = (sin x) cos x sin x (cos x) cos x = cos x cos x sin x ( sin x) cos x = cos x + sin x cos x = cos x (7.60) 7.3 y = sin x( ) ( ), y = cos x ( ) x 0., y = sin x y = cos x!, sin x, cos x!, (sin x) = cos x! 87, () y = cos x, 0 x 7, (sin x) = cos x (7.6) (cos x) = sin x (7.6) (tan x) = cos x 3 93 (7.63) cos x, cos x = sin x?... (cos x) = sin x

121 ,?... x = 0 = 0?, (7.6) x = 0, cos 0 =,, y = sin x,, P , x = 0 (7.6) x = 0, sin 0 = 0, y = cos x,, 7.7, x = 0 0, (, 7.7 ) (7.63) x = 0, / cos 0 =, y = tan x,, 7.8, x = 0 (7.6) (7.63),, 7.7 (cos 3x) = ( sin 3x)(3x) = 3 sin 3x (7.64) cos x 3x 88 () cos x () tan x (3) sin x (4) x cos x (5) cos x (6) cos x 89,,,!, xy P(x, y), r r = x + y P, r, θ x P, { x = r cos θ (7.67) y = r sin θ ( 7.5) (sin 3 x) = {(sin x) 3 } = 3{(sin x) }(sin x) = 3 sin x cos x (7.65) x 3 sin x (x sin 3x) = (x ) sin 3x + x (sin 3x) = x sin 3x + x (3 cos 3x) = x sin 3x + 3x cos 3x (7.66) x sin 3x ( ) 7.8 f(x) = x 3 cos 3x f(x) x 3 cos 3x,, f (x) = (x 3 ) cos 3x + x 3 {(cos 3x) } (x 3 ) 3x (cos 3x), cos 3x, cos 3x (cos 3x) (cos 3x), 3x, 3 sin 3x, f (x) = 3x cos 3x 6x 3 cos 3x sin 3x r,, r x θ, (polar coordinate),, x y (x, y), * 9, : () (,) * 3 ( )

122 0 7 () ( 3,) (3) (,3) : θ = arctan(3/) (4) r =, θ = π/3 (5) r = 3, θ =0 : 7.3,,, t x(t) : x(t) = x 0 cos(ωt + δ) (7.68), x 0, ω, δ t, x 0 > 0 x 0, ω 9,,, x x 0,, * : x(t) = x 0 cos ωt (7.69), t, x(t), t (), ω () t (3) t (4) ω,? 7.4,,, a, b, a sin x + b cos x (7.70), A, p ( 0 A, 0 p < π ) A sin(x + p) (7.7) :, (7.7), (P.04 (7.30)), : A sin(x + p) = A{sin x cos p + cos x sin p} = A cos p sin x + A sin p cos x (7.70), { a = A cos p b = A sin p (7.7) (7.67), xy (a, b) r A, θ p 7.9 sin x + cos x,, a = b = (, ), r =, θ = π/4 : sin x + cos x = sin ( x + π 4, sin(x + π/4), π sin(x + 4 ) = ( sin x cos π 4 + cos x sin π ) 4 = ( sin x + cos x) = sin x + cos x, 7.0 () sin x cos x = sin(x π/4) () 3 sin x + cos x = sin(x + π/6) (3) sin x 3 cos x = sin(x π/3) ) * ω,,,, x 0 0,,

123 , P.04 (7.9), (7.3), cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β, (7.74), (7.77), cos A cos B = sin A + B sin A + sin B = sin A + B sin A B cos A B (7.83) (7.84), cos(α + β) + cos(α β) = cos α cos β (7.73) cos(α + β) cos(α β) = sin α sin β (7.74), cos α cos β = sin α sin β = cos(α + β) + cos(α β) cos(α β) cos(α + β), P.04 (7.30), (7.3), sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β, (7.75) (7.76) sin(α + β) + sin(α β) = sin α cos β (7.77), : sin α cos β = sin(α + β) + sin(α β) (7.78) (7.75), (7.76), (7.78),,,, (7.73), α + β = A (7.79) α β = B (7.80), cos A + cos B = cos α cos β (7.8), (7.79)+ (7.80), α = A+B (7.79) (7.80), β = A B, α = A + B, β = A B, (7.8), cos A + cos B = cos A + B cos A B (7.8) (7.8), (7.83), (7.84), ( ),,,, 0 () y = arccos x, y = arcsin x, y = arctan x () y = arccos x, y = arcsin x, y = arctan x :, y () (7.9), sin θ lim = (7.85) θ 0 θ : (), (sin x) = cos x : (7.84) (3), (cos x) = sin x : (7.)?,,,, 3,,, ( ), 3 3, 80, 80!

124 7 (), 540 :,? (), 70 (3),? (?!) 58 () A,, A () ABC ACD, AB:AC=AC:AD, AC =AB AD (3) B,, B (4) ABC CBD, AB:BC=BC:BD, BC =AB BD (5) () AD=AC /AB (4) BD=BC /AB AB=AD+BD, AB=AC /AB+BC /AB AB, (7.) 59,, = π/80 60 () π/ () π/4 (3) π (4) π/6 (5) π/80 6 () 80 () 60 (3) 360 (4) 45 (5) 80 /π 6 () π () π (3) 0 63 (), l, a, θ a/l (), R, d, d/r,, θ = d/r, d = Rθ (3),, R,, R, N R = d/θ (4) A = πr, π(d/) N, B = Nπd /4 (5), B A = Nπd /4 πr, () d = Rθ,, = Nθ /4 (6) a = cm, l = 60 cm, θ a/l = /30 N = 0, B/A = (7) (7.8), d,, (8) d, d,..., d n,, d k N k ( k n) (7.8), d k C k, C k = N k θ /4, C, C +C +...+C n = (N +N +...+N n )θ /4 = Nθ /4,, N = N + N N n,,,, (cos θ, sin θ) P xy,, cos θ P x, sin θ P y,, P cos θ sin θ, 67 sin cos tan / 3/ / 3 45 / / 60 3/ / / / / 3/ / 3 π/6 / 3/ / 3 π/4 / / π/3 3/ / ?4 0.?40.?? ??9 0.9?5 0.??

125 nπ, n 0, n π,, x nπ P, n (, 0), n (, 0), y 0 sin nπ = 0 P x, n, n ( ) n, cos nπ = ( ) n 70 tan θ = sin θ/ cos θ, + tan θ = + sin θ cos θ = cos θ + sin θ cos = θ cos θ, cos( θ) = cos θ, sin( θ) = sin θ,, tan( θ) = sin( θ) cos( θ) = sin θ cos θ = tan θ tan(θ + π) = 7 (7.4), cos θ (7.5), sin θ (7.7), tan θ 7 ( ) 73 sin(θ + π) cos(θ + π) = sin θ cos θ = sin θ cos θ = tan θ () β β ( ) () β β ( ) (3) cos(α + β) = cos α cos β sin α sin β, β = α, cos α = cos α sin α, cos α+sin α =, sin α = cos α sin α, cos α = cos α, cos α = sin α cos α, cos α = sin α (4) sin(α + β) = sin α cos β + cos α sin β, 74 β = α, sin α = sin α cos α () (7.34), cos α = cos α () (7.35), cos α = sin α 75 ()()(3), 7.6, 7.7, 7.8 (4), 7.6 ( ) 76 () sin x, sin x 0 () sin x, sin x (3) x = 0 y = sin x = 0 (4) f(x) = sin x, f( x) = sin ( x) = ( sin x) = sin x = f(x) (5) (7.38),, y = cos x y / y /, , x θ P P x Q OPQ, OAB OP=, OPQ OA OAB () PQ= sin θ, AB=PQ OA=OA sin θ sin θ =AB/OA () OQ= cos θ, OB=OQ OA=OA cos θ cos θ =OB/OA (3), tan θ = sin θ cos θ = AB/OA OB/OA = AB OB 78 (7.44), AB, OA =00 m sin(30 )=50 m 79 =00 m sin(3 ), sin(3 )= , =00 m = 5.33 m, 3 = π/60, 0,, sin(3 ) = sin(π/60) π/60 = , 00 m = 5.35 m,, 3 mm

126 () CBP, BP=CB sin C = a sin C () ABC, CA= b, BP,, S = CA BP = ab sin C (3),, /, CA AB (7.49), AB BC (7.50) (4) (7.48) (7.50), 8 S = ab sin C = bc sin A = ca sin B /(abc) ( S ), sin C c = sin A a = sin B b () CBP CP=BCcos C = a cos C, AP= AC CP = b a cos C :, P CA () PAB AB c AP PB, AP= b a cos C, PB=a sin C (3), 83 c = a sin C + b + a cos C ab cos C = a (sin C + cos C) + b ab cos C = a + b ab cos C () cos θ = a + b c ab () sin θ = cos θ = (3) S = ab sin θ = = 4 = 7 8 ( 7 8 cm : (3),! 84 ) = 5 8 () arctan = π/4 () arctan 0 = 0 (3) arccos 0.5 = π/3 (4) arcsin( 0.5) = π/6 (5) arctan 5 =.373 (6) arcsin 85, arctan{(880 30)/000} 0.07 = ( ) 87 ( ) 88 () sin x () / cos x (3) cos x (4) cos x x sin x (5) cos x sin x, sin (7.36), sin x (6) x sin x 90, x θ P 0 P 0 OP,, OP= r, OP 0 =, P (x, y) P 0 (cos θ, sin θ) r, x = r cos θ, y = r sin θ 9 () r = + =, θ = π/4 () r =, θ = 5π/6 (3) r = + 3 = 3, θ = arctan(3/) = 0.98 (4) x = r cos θ = cos(π/3) = y = r sin θ = sin(π/3) = 3 (, 3) (5) (.95, 0.50 ) 9 () cos π, x 0 cos ωt, ωt = π, T, ωt = π, T = π/ω () (x 0 cos ωt) = x 0 ω sin ωt (3) (x 0 cos ωt) = x 0 ω cos ωt (4) x 0 ω, x 0 ω, ω,, 4

127 5 8 8.,,, 8., y = f(x) x, x = a x = b ABCD a < b,, x 8. n, x x 0, x,, x n x 0, a, x n, b x 0 < x < < x n 8. ABCD, k 8., 8. f(x) x, x = a, x = b ABCD,, ABCD x = a x = b f(x), b a f(x) dx (8.) CD, ABCD CD,, :, ABCD, y n x k, x k = x k x k,,,, 8.3 x ξ k ξ,, x k, f(ξ k ),, : k f(ξ k ) x k (8.), k =, k =,, k = n, ABCD, n ABCD f(ξ k ) x k (8.3) k=

128 6 8 8., f(x), x = a x = b, : 8.3, n,, (8.),, (8.3), n, x k k n, (8.3) (8.), n, x k, Σ ξ k x x k dx * 95 ξ k x k k... k, k = x k x k, k, Σ k = n, a b b, a x b a, 96..., sum S Σ S,,, *,, Σ, dx x k b a f(x) dx : = lim n x k 0 n f(ξ k ) x k k= (8.4), a = x 0 < x < x < < x n = b, x k = x k x k, ξ k x k x k,,,, ξ k, ξ k, (8.4), 97 ξ k (8.4)... f(x), b a f(x) x k, (8.4), b a f(x) dx k, b a f(x k ) dx..., k 34 b a f(x) (8.4),... dx

129 (8.4), a < x 0 < x < x < < x n < b... a x 0, x n b a b b a 99? P ! (8.4),, :,,, (8.4) x (8.4) f(x), (8.4),, ,, (8.4),, 98..., x,,,,,,,,,,,,, 8.3,,,,, b f(x)dx a k= n f(x k ) x k = f(x )(x x 0 ) + f(x )(x x ) + f(x 3 )(x 3 x ) + + f(x n )(x n x n ) (, x 0 = a, x n = b, x k = x k x k ) n, x k, =,, x k,, OK,, f(x) = x, 0 x 3 f(x) = x ( x 0. ): A B C x f(x) sekibun (C ), ( ) C 0, C3, =C+B3*(A3 A) B3 f(x ), A3 A x, C3, C4 C (!), C 0 x :

130 8 8 A B C x f(x) sekibun , C5, 0.3 x dx, 0 C3 3 x dx, 0, 0.0, 00 C 0?... ( (8.0) ) 94, f(x) = x, 0 x 3,, f(x), 8.4 y = x y = x...!!,, 95, f(x) = x : C, D =(C3-C)/(A3- A) D3 D3 D B ( x)!,! f(x),,, 36, OK?..., ( ),,,,,, 96, f(x) = cos x, 0 x 7, x y = x, 8.4, y = x ( 8.4 ), x = y, x = y 4, x = 3 y 9, y = x? ( ), 8.5 y = cos x ( ), ( ), ( ) x 0.,

131 , (8.4),,, ( ) 0? P !, a, b, c, f(x), g(x), ( ) *,, ( ) : b a b a = : {f(x) + g(x)} dx = {f(x) + g(x)} dx n f(x k ) x k + k= b a f(x) dx + b a b a f(x) dx + b a g(x) dx (8.5) n {f(x k ) + g(x k )} x k k= n g(x k ) x k k= g(x) dx P.4 (3.83) :, x k, n =, 37 b a f(x) + g(x) dx (8.5)...,, f(x) dx (dx g(x) ) *, ( : 0 x dx : b a b a : cf(x) dx = c cf(x) dx = c b a f(x) dx (8.6) n cf(x k ) x k k= n f(x k ) x k c k= b a f(x) dx P.4 (3.84) b a 3: f(x) dx + c b f(x) dx = c a f(x) dx (8.7) : a c m,, n (n m ) b x 0 = a, x n = b, x m = c m, n,, b a c b f(x) dx f(x) dx n f(x k ) x k (8.8) k= m k=n+ f(x k ) x k (8.9),, n f(x k ) x k + k= m k=n+ f(x k ) x k = m f(x k ) x k k= x 0 = a, x n = b, x m = c, n m x k,, b a f(x) dx + c b f(x) dx = c a f(x) dx,, b f(x) dx, a < b a ( (8.4) ), a b,, a b, a = b 3, b = c = a, a a f(x) dx + a a f(x) dx = a a f(x) dx

132 0 8, a a f(x) dx = 0, : 4: 0 a a f(x) dx = 0 (8.0), 7: f(x), a d dx x a : a, f(t) dt = f(x) (8.6), a > b 3, c = a, b f(x) dx + a f(x) dx = a a b a f(x) dx (8.) 4, 0, : 5: b f(x) dx = a a b f(x) dx (8.), 6: f(x) h, a x a + h f(x), a+h a f(x) dx f(a)h (8.3) : x 0 = a, x n = a + h,, a+h f(x) dx a k= n f(x k ) x k (8.4) x k a x k a + h, h, f(x k ) f(x 0 ) = f(a), n n n f(x k ) x k f(a) x k = f(a) x k k= k= k=, x k, a a + h n, h, n f(a) x k = f(a)h (8.5) k= F (x) :=, x a F (x + dx) = f(t) dt (8.7) x+dx a f(t) dt (8.8) (dx ) 3, F (x + dx) = x a = F (x) + f(t) dt + x+dx x+dx dx 6, x+dx x x x f(t) dt f(t) dt (8.9) f(t) dt = f(x)dx (8.0) (dx = ), (8.9), F (x) + x+dx x, (8.9), f(t) dt = F (x) + f(x)dx F (x + dx) = F (x) + f(x)dx (8.), (5.0)(P.67), d F (x) = f(x) dx 7, 95

133 8.5 8: f(x) x a d f(t) dt = f(x) f(a) (8.) dt :,, x a d n f(t) dt dt f (ξ k ) t k (8.3) k=,, f(x) F (x), 8. d dx x = x (8.5), x x ( ) 38 t 0 = a, t n = x, t k = t k t k, ξ k t k t k, 39 F (x) f(x) ξ k = t k t k,, f(t k ) f(t k ) + f (t k ) t k, f (ξ k ) t k = f (t k ) t k f(t k ) f(t k ) F (x)... F (x) f(x) f(x) F (x), :, (8.3), n {f(t k ) f(t k )} = f(t ) f(t 0 ) k= f(x) dx (8.6) + f(t ) f(t ) (8.6) (8.4) +, +f(t n ) f(t n ) + f(t n ) f(t n ),, (8.4) (8.4),,,, f(t 0 ) f(t n ),,,, f(t n ) f(t 0 ), f(x) f(a),, 7, 8 x x, ( 7),, ( 8),, ( ) 40 F (x) = f(x)dx F (x) f(x)..., f(x) F (x), F (x) f(x)? 8.5,, F (x) f(x), d F (x) = f(x) (8.4) dx, F (x) f(x) ( ), 0,, 8. d dx (x + 3) = x (8.7), x + 3 x ( ), f(x) F (x), G(x)

134 8, F (x) G(x), d dx {F (x) G(x)} = F (x) G (x) (8.8) = f(x) f(x) = 0 (8.9), F (x) G(x) ( 0 ), 9: f(x) F (x), C F (x) + C, f(x),, f(x),,,,, C, C 8.3 x,, x + C (C ), x dx = x + C (8.30) ( ) 4,,, : 0: x a a, x a dx = a + xa+ + C (8.3),,,, f(x), g(x) F (x), G(x), (F (x) + G(x)) = F (x) + G (x) = f(x) + g(x) (af (x)) = af (x) = af(x), F (x) + G(x) f(x) + g(x), af (x) af(x), : a, {f(x) + g(x) } dx = af(x) dx = a f(x) dx + g(x) dx (8.33) f(x) dx (8.34), ( 4 (8.33), f(x) + g(x) dx..., dx f(x) + g(x), f(x) + g(x) ( ) 8.4 : ( + 3x) dx (8.35),, dx + 3x dx = dx + 3 x dx (8.36) 0, (8.3),, :, a,, d ( dx a + xa+) = x a (8.3) 97 () x dx () x dx (3) dx ( x ) = (x + C ) C (8.37) C, C, dx x dx,, = x + 3x + C + 3C (8.38), C, C, C + 3C C + 3C

135 8.5 3 C, = x + 3x + C (8.39),,,,,, (8.35), (8.36), (8.37), (8.38) (8.39) ( ) :,,, (,, 98, ( + x + x ) dx (8.40), x a 0, a =, /x, a + = 0 0,, P.89 (6.9) : (ln x) = x (8.4), /x *3 : dx x = ln x + C (8.4), ln x, 0 < x, /x, x < 0, *3 x dx dx x x < 0 /x?, ln( x) + C (, x < 0, ln x,,, {ln( x)} = ( x) x = x, /x, (8.43) dx 0 < x, = ln x + C x (8.44) dx x < 0, = ln( x) + C x (8.45), ln, x, x, : dx x 0 : /x = ln x + C (8.46),..., 4(3),? 43 /x, ln..., ( ) 03 /x x 0?... (8.3), a =? a?, a =, x dx = + x + + C = 0 x0 + C = 0 + C ( ), 0,,,,,, 9,, :

136 4 8 3: exp x dx = exp x + C (8.47) (5) exp(x + ) dx, f(x) F (x), F (ax + b) x (a, b ), af(ax + b), f(ax + b), F (ax + b)/a (8.48) 8.5 ( ) (8.48), (3x + ) 4 dx = 3 5 (3x + )4+ + C (8.49) = (3x + )5 5 + C (8.50) : (8.48), f( ) ax + b 44 : dx + x = x ln + x + C! x, 04,, dx = ln x + C (8.5) x dx (x + ) = (x + ) + C (8.5) exp x dx = exp x + C (8.53), () (3) (x + ) 3 dx dx x () (4) dx x + exp( x) dx 45 (3), ln x + C...,, /( x)!,, 4(6) P.08 (7.6), (7.6), (sin x) = cos x, (cos x) = sin x, 4: cos x dx = sin x + C (8.54) sin x dx = cos x + C (8.55) sin x cos x, 8.7 cos x sin x dx = sin x cos x dx = + C 4, P.04 (7.36), P.7 03, ( 8.8 cos x dx = + cos x dx = x + sin x 4, P.04 (7.37) ( ) 8.9 sin x dx = cos x dx = x sin x 4, P.04 (7.38) ( ) + C + C,,,,,,,

137 8.6 5, exp(x), P.89 exp( x ), (, ),,, (,,,,?!!,,,,,, x x + = x(x + ), : x(x + ) = x x + (8.6) (8.63),, 8. 5 (x + )(x + 3) (8.64) : dx x + x (8.56), :,, a x + + b x + 3 (8.65),,, : dx = x(x + ) ( = x x + dx = x (8.57) ) dx (8.58) dx x + (8.59),,, /x, /x, = ln x ln x + + C (8.60) ( C ),, ln, x = ln + C (8.6) x + ( ) 8.0 (8.57) (8.58), x, x, a, b, a x + + b a(x + 3) + b(x + ) = x + 3 (x + )(x + 3) = (a + b)x + 3a + b (x + )(x + 3) (8.64), 5 = (a + b)x + 3a + b (8.66), 0 = a + b (8.67) 5 = 3a + b (8.68),, a =, b =, 5 (x + )(x + 3) = x + x (8.69), (

138 6 8 () 0 ) dx x x () dx x (3) x dx x a, b (C dx (a x)(b x) = b a ln b x + C (8.70) a x 04,,?... 00(), 0, 8.7, P.7 (5.40): (fg) = f g + fg (8.7), P. (8.33), (fg) dx = f g dx + fg dx (8.7), (fg), fg, fg = f g dx + fg dx (8.73),, 5: f g dx = fg fg dx (8.74) f = sin x, g = x, x sin x (x) sin x dx = x sin x sin x dx = x sin x + cos x + C ( ),, cos x x, x, cos x?, cos x sin x, 0, (3) 0 < x, ( () xe x dx 8.8 () x sin x dx (3) ln x dx f(x) F (x), d F (x) = f(x) (8.77) dx, : F (x) = f(x) dx (8.78), ( f, ( g, 8. x cos x dx (8.75), cos x = (sin x),, x (sin x) dx (8.76), x t, F (x) t ( F (x(t)) t, d ( d ) dx dt F (x(t)) = dx F (x) dt (8.79), *4 *4 (5.48), g F, f x, x t (8.79)

139 8.8 7, df (x)/dx = f(x), (8.79), d F (x(t)) = f(x)dx dt dt, f(x) dx dt (8.80) (8.8), t, F (x(t)),, F (x(t)) = f(x) dx dt (8.8) dt (8.78), (8.8), 6: f(x) dx = f(x) dx dt (8.83) dt, x, t, 8.3 x cos x dx (8.84), x,, x = t (8.85),, dt = x (8.86) dx dx dt = x (8.87), x cos x dx = (x cos t) dt (8.88) x, cos t sin t = dt = + C (8.89) t x (!), = sin x + C (8.90),, (8.85) x t dx dt, x dx = dt (8.9) dx = dt x (8.9), (8.84) dx, 03, ( () (3) x + x dx cos x sin x dx () (4) xe x dx x x dx 04 : dx + x (8.93) () x = tan θ, : dθ (cos θ)( + tan θ) (8.94) : θ (8.83) t () cos θ( + tan θ) = (3) (8.94), dθ,, θ + C (8.95) (C (4), : dx = arctan x + C (8.96) + x 46,

140 8 8 (8.93) (8.95), (8.96)..., 05, x = tan θ,?... + x, + tan θ?,,, (8.93) 8.9,,,, : 8 (P.) : x d f(t) dt = f(x) f(a) (8.97) dt a, f F : x a d F (t) dt = F (x) F (a) (8.98) dt, d F (t) = f(t) (8.99) dt, F f ( (8.99) f (8.97) f ), (8.98), x a f(t) dt = F (x) F (a) (8.00), x b, t x, b a 7 f(x) dx = F (b) F (a) (8.0), : 8.4 x dx? x dx = x3 3 + C (8.0) (C ), (8.0), ( ) 06 x dx = = 7 3 (8.03) (8.03),, C?... C, x dx = ( 3 ) ( 3 ) 3 + C 3 + C? C (8.04),, C?..., (8.0) F, f x 3 /3 x, (8.0), F (x), [ ] b F (b) F (a), F (x) a (8.03), [ x x 3 ] dx = 3 = 7 3, (8.05) (8.03) x,,,, x t p,, x dx t dt p dp 05 (a, n () (3) 0 a dx () (4) 0 ax dx, F (x) f(x), ( ) (5) π π sin x dx (6) π π sin x dx,,, ( ) π π x sin x dx π π x cos nx dx

141 8.0 9,, 06 (R R 0 R x dx (8.06) 07 a, f(x) ( ), g(x) (, : () () a a a f(x) dx = 0 (8.07) g(x) dx = a a 0 g(x) dx (8.08) (8.) (8.), x dx, = (8.) (8.3),, =, (8.), (8.3), (8.), (8.) (8.3) f (x 0 ), (8.) f ( ), ( ) 3, x 0,,, (8.), (8.) 08, 8. () xe x dx () (x 5 + 3x 3 x) dx, 7 (8.0), : F (b) = F (a) + b a f(x) dx (8.09), F f, f(x) F (x), a = x 0, b = x 0 + x : x0 + x F (x 0 + x) = F (x 0 ) + F (x) dx (8.0) x 0 F (x) f(x), x0 + x f(x 0 + x) = f(x 0 ) + f (x) dx (8.) x 0, (8.) P.67 (5.0) (5.8): f(x 0 + dx) = f(x 0 ) + f (x 0 )dx (8.) f(x 0 + x) f(x 0 ) + f (x 0 ) x (8.3),,,,,,, ( ), P.5, x,, x 8.5, R, S, 0 x 0 y,, x y ( /4 (x, y), x + y = R (8.4),, y = R x (8.5), ( /4 X, : X = R 0 R x dx (8.6)

142 (8.6),, r k, : () : X = πr 4 () (8.7) S = πr (8.8) S = R 0 πr dr = [πr ] R 0 = πr (8.), (8.6),,, 8.,,, : 8.6 R, ( 8.6), n,, r, r,, r n r 0 = 0, r n = R k n, r k r k r k r k = r k r k r k, S k, S k πr k r k (8.9) ( *5, S, S k : 0 8.6,, x, y, z, R V R x R R = x 0 < x < < x n = R x 0, x,, x n, x, x = x k, R x k, x k x k, x = x k x = x k,, ( ) V k, 8.6 V k = π ( R x k ) xk = π(r x k) x k (8.) k,,, V = = ( ) lim n x k 0 R R n π(r x k) x k (8.3) k= π(r x ) dx = = 4πR3 3 (8.4) S n S k = k= n πr k r k (8.0) k=, P.?? (??), n *5,, r k, 47 (8.4), = R R π(r x k) dx =... x k k, x k k x k x k+ 0, k 8.7,

143 8.3 3,,,, : r,,, ( ) ( 8.7),,, d k,, (dx dv ) 8.8 ( 8.7 ) 3 x, y, z, R V, x dx R x, dv, : 8.7,,, dv = π( R x ) dx = π(r x )dx (8.5), V = ( ) R R π(r x ) dx = = 4πR3 3 (8.6), ( 4πR3 3 ),, 3( ) 4πr( 3 ( ), 8.7, ,, 8.6?!, r 4πr 8.9 R ( V ), r, dr dv, r, dv = 4πr dr (8.7),? (8.7) dr, dv/dr = 4πr!, (P.9 50 ), (πr), 8.9 (4πr ),,?... (πr),, ( ) (4πr ),, 8.7,, 8.9, 08, d, d =?... Σ, d, V = R 0 [ 4πr 4πr 3 dr = 3 ] R 0 = 4πR3 3 (8.8)..., , (P.78),

144 3 8 (8.) f(x), x0 + x f(x 0 + x) = f(x 0 ) + f (x) dx (8.9) x 0, x t, x 0 t 0, x 0 + x t, t f(t ) = f(t 0 ) + f (t) dt (8.30) t 0 OK, P x, t x(t), v(t), a(t), dx dt dv dt = v(t) (8.3) = a(t) (8.3) ( ), (8.3), x (t) = v(t), (8.30) x(t) f x ), t x(t ) = x(t 0 ) + v(t) dt (8.33) t 0 t, t 0, t 0 t, (8.3), v (t) = a(t), (8.30) v(t), : t v(t ) = v(t 0 ) + a(t) dt (8.34) t (8.33) P.80 (5.00) (8.34) P.80 (5.05),, a, (8.34) (,!), v(t ) = v(t 0 ) + a(t t 0 ) (8.35) t 0 = 0, t t, v(t) = v(0) + at (8.36) (8.33), x(t ) = x(0) + t 0 {v(0) + at} dt = x(0) + v(0)t + at (8.37) t t, x(t) = x(0) + v(0)t + at (8.38) (8.36) (8.38), ( ), a,,, (!, (8.34) (8.36), a, (8.37) a a, a, (8.33) (8.34),,, 09?...,,,,,, 8.4 P.9,, f (x) = af(x) (8.39) f(x) = f(0)e ax (8.40) (6.4) (8.40) (8.39),,,, f(x), f (x) x = 0 (8.4), f (x) = x +, f(x) x +, x +,

145 (8.4),, f(x) = (x + ) dx = x + x + C (8.4) C ( ), x + x +, x + x 5, x + x + 00, (8.4),,, (8.4) (8.4),,,, (8.39), f(x) = af(x) dx (8.43),, f(x) f(x), f(x) ( : (8.40), ), (8.39), :,, 0 x, f(x + x) f(x) + f (x) x (8.44), f(x + x) f(x) = f, f f (x) x (8.45), (8.39) f (x), f af(x) x (8.46) f(x) (, f(x) (x) ), f f a x (8.47), x x 0, x, x,, x n, f f 0, f, f,, f n, k 0 n, f k f(x k ) x k x k+ f k = f k f k (8.48) x k = x k x k (8.49), (8.47), f f 0 a x f f a x f 3 f a x 3 f n f n a x n,, n k= f k f k n a x k (8.50) k=, f k x k, (8.4) *6, f(x) f(x 0) df x f = a dx (8.5) x 0, x n x, d, = : ln f(x) ln f(x 0 ) = a(x x 0 ) (8.5), : ln f(x) = a(x x 0 ) (8.53) f(x 0 ) f(x) = exp{a(x x 0 )} (8.54) f(x 0 ) f(x) f(x 0 ) = ± exp{a(x x 0)} (8.55) f(x) = ±f(x 0 ) exp{a(x x 0 )} (8.56) *6 (8.50), f k (8.4) x k, f k (8.4) ξ k, /f k (8.4) f(ξ k ) f /f, f = f(x 0 ) f = f(x)

146 34 8 x = x 0,, f(x) = f(x 0 ) exp{a(x x 0 )} (8.57), (8.39), x 0 = 0, : f(x) = f(0)e ax (8.58) (8.40)! f(0), (8.58),, f(0),, x f(x) *7,, (initial condition),, (8.47), f = 0 0,?,,, x f(x) = 0, (8.39), f (x) = 0, x, dx, f(x + dx) = f(x) + f (x)dx = f(x) = 0 (8.59) x (x + dx) f = 0, x, f(x) = 0 f 0,, (8.58) f(0) = 0,, f = 0,,, f = 0,,, f (x) df/dx, (8.39) df = af(x) (8.60) dx, (8.46), df/dx, df dx *7, x = 0, (, ), (8.39) (8.60), f df, x dx, ( a ), df f = adx (8.6),, df f = adx (8.6),, df dx, ( f x) ( f, x ) *8,,,,, (,, ): ** ( ) ** df = af (8.63) dx, df f = adx (8.64), df f =, adx (8.65) ln f = ax + C (8.66) *8,,

147 (, C ), f = e ax+c = e C e ax (8.67) f = ±e C e ax (8.68) x = 0 ( ), f(0) = ±e C (8.69) 8.5, ( ) N(t),, N dt, α, αn dt (8.76) (8.68) ±e C f(0), f(x) = f(0)e ax (8.70), N, ( ),, 48 (8.69), f(0) = e C f(0) = e C...,, (8.70) f(x) = ±f(0)e ax (8.7),! x = 0 f(0) = ±f(0) (8.7),, (, f(0) = f(0), f(0) = 0, f(0) = 0, (8.7), f(x) 0 ),, C, e C, ±e C f(0), ±, : ±e C, +e C e C, f(0),, f(0), ±e C +e C, f(0), ±e C e C 5 : *9, dt, N, dt N, N, dt, N(N )dt N, N(N )dt, N dt, dt,, N dt β (β 0), dt : βn dt (8.77) dn,, (8.76), (8.77), dn = αndt βn dt (8.78) dt, dn dt = αn βn (8.79) (8.79), ( ) 8, f (x) = 3f(x) : f(0) = (8.73) 6 : f (x) 3f(x) = 0 : f(0) = (8.74) 7 : f (x) + xf(x) = 0 : f(0) = 3 (8.75) 9 (8.79) N(0) = N 0 (), N, t *9,,,

148 36 8 (), : β ( α βn + ) dn = dt (8.80) α βn (3), (C ): N ln = αt + C (8.8) α βn (4) : N α βn (5), : ±e C = (6), : N(t) = = ± exp(αt + C) (8.8) N 0 α βn 0 (8.83) N 0 e αt + N 0 β(e αt )/α (7) (8.84), : N(t) = ( β α + N 0 β α (8.84) )e αt (8.85) (8) (8.85), P.95 (6.65) (6.65) a, b, c (8.85)? (9) 6.0, (8.85) 8.8 N α/β, N, dn/dt = 0 (8.79) 0, αn βn = 0,, N = α/β, N α/β,, K,, α, r 0 : K r (8.79) dn ( dt = r N ) N (8.86) K 0 dn/dt = αn βn αn < βn,?... ( ), N,, dn/dt N,, βn = αn, dn/dt = 0, β = 0,?...,,,?...,?,,, 8.8 N 0 O (8.85) t 3 e x,?... ( ), y = e x + e x,,, N(t) 3 f(x) = exp( x ) (), y = f(x), 4 x 4

149 :, x -x, ( x), -x*x, -(x ) () f(x) x = 4 : F (X) = X 4 exp( x )dx (8.87), 4 X 4,, 0. (3) F (4) π,,, : exp( x )dx = π (8.88) (8.88), 4, P V γ, P, V, n, R, T, C v, U,,,,,,,, γ = (R + C v )/C v ( ) : U = nc v T dv, P dv, P V = nrt P, V T, T P V 5 dt * 0 w dw, ( w ),,, dw,,, *0,,, dw = α w dt (8.89) α dt,,,, α = w dw dt (8.90) α (relative growing rate), RGR, RGR := w dw dt (8.9) t = t t = t RGR, RGR = ln w ln w t t (8.9), (8.9) : (8.9) RGR,, (8.9) RGR, 4,...,,,,,?,, C 94 ( ) 97 () x 3 /3 + C () /x + C (3) x + C : dx, dx, ( ) 98 x + x / + x 3 /3 + C 99 () (x + ) 4 /8 + C () ln x + + C (3) ln x + C (4) exp( x) + C (5) exp(x + ) + C

150 () () (3) 0 x x = x(x ) = a x + b x, a(x ) + bx x(x ) = (a + b)x a x(x ) = x(x ) a, b x, a + b = 0, a =, a =, b =, x(x ) = x x, ( x(x ) dx = x ) dx x = ln x ln x + C = ln x + C x, dx x = dx (x )(x + ) ( = x ) dx x + = { } ln x ln x + + C = ln x + C x + = x dx x = x dx (x )(x + ) ( x + ) dx x + = { ln x + ln x + } + C = ln (x )(x + ) + C = ln x + C = ( b a a x ) dx b x = { ln a x + ln b x } + C = b a 0 () xe x dx = () (3) = xe x x(e x ) dx = xe x (x) e x dx e x dx = xe x e x + C = (x )e x + C x sin x dx = x( cos x) dx = x( cos x) (x) ( cos x) dx = x cos x + 03 ln x dx = (x) ln x dx = x ln x = x ln x cos x dx = x cos x + sin x + C x(ln x) dx = x ln x dx = x ln x x + C () + x = t, x dx = dt, ( x dx x) x dt + x dx = t = ln t + C = ln( + x ) + C, + x, () x = t, x dx = dt, e t xe x dx = dt = e t + C = e x (3) sin x = t, cos x dx = dt, sin x cos x dx = t dt = t + C = sin x + C (8.93) ( ) cos x = t, sin x dx = dt, sin x cos x dx = = t + C = cos x t dt + C (8.94) : (8.93) (8.94),,, (8.94), cos x + sin x = cos x, (8.93) + C,, 8.7,, (

151 (4) x = t, x dx = dt x x dx = t dt = 3 t3/ + C = 3 ( x ) 3/ + C 04 () x = tan θ, dx/dθ = / cos θ, dx = dθ/ cos θ (8.93) x tan θ, dx, () (cos θ)( + tan θ) = (cos θ)( + sin θ/ cos θ) = cos θ + sin θ = (3) (4) (θ = arctan x 05 () ] [ax 0 a dx = = a 0 () 0 ax dx = [ax / ] 0 = a/ (3) π [ π sin x dx = cos x = 0 π (sin x (4) π π sin x dx = = π (5) ( ) = π π ] π cos x dx π [ x ] π sin x 4 π π x( cos x) dx = [ x cos x] π π + π = π π = π + cos x dx = π π (x) cos x dx (6) ( (7.4), (7.5) ) = π x (sin nx) dx n π = n = n [ x sin nx ] π π n π π π π (x ) sin nx dx x sin nx dx = π n x(cos nx) dx π = [ ] π x cos nx n π π n cos nx dx = 4π cos nπ n = ( ) n 4π n π 06 x = R sin θ ( dx = R cos θ dθ = = = = = = π/ 0 π/ 0 π/ 0 π/ 0 R 0 R x dx R (R sin θ) R cos θ dθ R sin θ cos θ dθ R cos θ cos θ dθ R cos θ dθ = [ θ + (sin θ)/ R ] π/ 0 π/ 0 + cos θ R dθ = πr 4, x = 0 x = R θ = 0 θ = π/ 07 (), P.9 (8.7) a a f(x) dx = 0 a f(x) dx + a 0 f(x) dx (8.95), x = t dx = dt, t = a t = 0, 0 a f(x) dx = 0 a f( t)( dt) = a 0 f( t) dt ( 5 ) f(x) f( t) = f(t), a f(t) dt t x 0 a f(x) dx, (8.95), 0 a a f(x) dx = a 0 f(x) dx + a () : a a g(x) dx = 0 a g(x) dx + a 0 0 f(x) dx = 0 g(x) dx (8.96), x = t,, 0 a g(x) dx = 0 a g( t)( dt) = a 0 g( t) dt g(x) g( t) = g(t),

152 40 8, a g(t) dt t x, 0, a g(x) dx, (8.96) 0 a a g(x) dx = = a 0 a 0 g(x) dx + g(x) dx a 0 g(x) dx 08,, (), ( ) 0 09 () ( (8.6) 06 () X /4, S = 4X = πr 3 (8.33) t 0 t, t t + dt, x(t + dt) = x(t) + t+dt t v(t) dt (8.97) dt, t t + dt v(t), t+dt t v(t) dt = v(t){(t + dt) t} = v(t) dt 49 (8.99) (8.00) ±, (8.00) e C =... C, e C, ±, e C =,, ± 6 (8.74), df dx = 3f df, f = 3 dx, f = 3x + C, f = 3x + C df f = 3 dx,, f(0) = /C = C =, f(x) = 3x 7 (8.75), df dx = xf, df f = x dx, (8.0) df f = ( x) dx, ln f = x + C, f = ±(exp C) exp( x ), x(t + dt) = x(t) + v(t) dt, f(0) = ± exp C = 3, 4 (8.34) t 0 t, t t + dt, v(t + dt) = v(t) + t+dt t a(t) dt (8.98) dt, t t + dt a(t), t+dt t a(t) dt = a(t){(t + dt) t} = a(t) dt f(x) = 3 exp( x ) (8.0) 5?..., ( ),,, v(t + dt) = v(t) + a(t) dt 5 df dx = 3f, df f = 3dx ( C ), df f = 3dx, ln f = 3x + C, f = ±e 3x+C = ±e C e 3x (8.99) x = 0, f(0) = ±e C = (8.00), f(x) = e 3x

153 P.75,,,,, f(x) : f(x) = a 0 + a x + a x + + a n x n + (9.), x = 0 (x = 0 ), f(0) = a 0 f (0) = a f (0) = a f (0) = a 3 3 f (n) (0) = a n 3 n, f (n) (x), f(x) n (x n ) n 3 n = n!, a 0 = f(0) a = f (0)/ a = f (0)/( ) a 3 = f (0)/( 3) a n = f (n) (0)/n!, (9.), (0! = ): f(x) = f(0) 0! = n=0 + f (0)! x + f (0) x +! f (n) (0) x n (9.) n!, f(x) (9.) P.75 (5.76) f(x) f(0) + f (0)x x = 0, x = a, f(x) (x a) 3 a f(x), (x a) (b, b,, b n, ): f(x) = b 0 + b (x a) + b (x a) + +b n (x a) n + (9.3), b n = f (n) (a),, f(x) = f(a) 0! = n=0 + f (a)! (x a) + f (a) (x a) +! f (n) (a) (x a) n (9.4) n! : (9.) (9.)! (9.4) f(x) x = a, x = 0,,, (5.0)(P.67), x 0 a, f(a + dx) = f(a) + f (a)dx (9.5)

154 4 9, a + dx x, dx = x a,, f(x) = f(a) + f (a)(x a) (9.6) y y=x,,, x a, dx 0,, ( ), dx 0,, 4 e x, sin x, cos x (x = 0 ), : e x = 0! + x! + x! + x3 3! + x4 4! + (9.7) sin x = x! x3 3! + x5 5! x7 7! + (9.8) cos x = 0! x! + x4 4! x6 6! + (9.9) 5 e x, sin x, cos x (x = 0 ), (e x ) = e x (sin x) = cos x (cos x) = sin x -3π/ -π -π/ O x π/ π 3π/ - y= x-x 3 /3! y -3π/ -π -π/ O x π/ π 3π/ - y= x-x 3 /3!+x 5 /5! y -3π/ -π -π/ O x π/ π 3π/ - y= x-x 3 /3!+x 5 /5!-x 7 /7! y -3π/ -π -π/ O x π/ π 3π/ - 6 (9.7), : 9. y = sin x (, 3, 5, 7 ) y = sin x e = 0! +! +! + 3! + + n! + (9.0), P , 3,..., ( ), sin 3, sin ( ),, x (!) + x4 (!) x6 (3!) +,,,, 7..., ( ), 0,,,,,,,

155 9. 43,,, (9.), f(0) f (0),, 9. f(x) = /x, (x = 0 ), f(0) ( ) 9. f(x) = ln x, (x = 0 ), f(0) ( ), f(0), f (0), f (0),..., : 7 () : x = + x + x + x x n + (9.) () x =, (9.) (3) {, x, x, x 3, }, (9.), x,, x,, (9.) (x = 0, ), (9.7) (9.8), (9.9) ( x ) 9. P.38 (3.5), i = i, i a, b : z = a + bi (9.), a, b + 3i, 3 + 3i 3i 3 imaginary number i,,, ( ) real number, z z Re(z), z Im(z) (Re real, Im imaginary ), Re( + 3i)=, Im( + 3i)=3,,,,, z = a + bi w = c + di (a, b, c, d ), a = c b = d, z = w ( ) z = a + bi (a, b ), a bi, z, z ( ) 9.3 z = + i, z = i ( ) 8 z, :, ( ), OK Re(z) = z + z, Im(z) = z z i (9.3)

156 ,, : x, e ix = cos x + i sin x (9.4) 3 : cos x = eix + e ix sin x = eix e ix i (9.9) (9.0), 33 (9.9), (9.0), sin x cos x 9 () e z z = 0, z ix, (i, x ) (), cos x, sin x, cos x sin x, (3), e iπ + = 0 34, () (9.9) 3 : cos 3x = 4 cos 3 x 3 cos x (9.) () (9.0) 3 : sin 3x = 4 sin 3 x + 3 sin x (9.) 8 e ix,!!...,,? 9.4,, z, x, y, z = x + yi ( ),, z = x + yi 30, a, b, e a e b = e a+b (9.5), a = iα, b = iβ, x, Re(z), y, Im(z) ( 9.),,, e iα e iβ = e i(α+β) (9.6),,, (α, β ),, ( ) :, Re, Im cos(α + β) = cos α cos β sin α sin β (9.7) sin(α + β) = sin α cos β + cos α sin β (9.8) 35 z = + i 3 x e ix, x, cos x + i sin x x () z () z (3) + z (4) z i (5) iz (6) z (7) /z,,,

157 Im y z=x+yi 9.5 4, : 4 = 4 + 0i = ( 4) + 0 = 6 = 4 4, ( ) O x Re, z z,.8, , z = x + iy (x, y ) (9.5),,,,, < >, i + i? 9,??...,,,,,! 9.5 x + iy (x, y ), z x + iy : x + iy := x + y (9.3) i = + = 5 ( ) 50, + i = 5...! (9.3), y = 0, x = x (9.4) x,,,, r, r = z = x + y (9.6) (0 r), θ (0 θ < π), ( (7.67)), { x = Re(z) = r cos θ y = Im(z) = r sin θ (9.5), z = r cos θ + ir sin θ (9.7) = r(cos θ + i sin θ) (9.8) () ( (9.4)) e iθ,, z = re iθ (9.9),, 0 r 0 π θ,, z = x + iy,, re iθ, r, θ 9.6 z = i,, 9.3, (9.6),, r = z = 3 + ( 3) = = 3 (9.30)

158 z = 3 + 3i, (9.7), Re(z) = 3 = r cos θ = 3 cos θ,,,,,,,,,,,,, Im(z) = 3 = r sin θ = 3 sin θ 39 z, r, θ, α,, cos θ = =, sin θ = 3 = (9.3) () z e iα, z, α () z = re iθ, z = re iθ θ = π/6, z = 3 e iπ/6 (9.3) 9.7 e 4 i? 4 = e ln 4, ( ) :, 0 π, sin cos π,, π 4 i = (e ln 4 ) i = e (ln 4)i = cos(ln 4) + i sin(ln 4) cos(.386) + i sin(.386) i, π/6, π/6 3π/6,, 0 π, π π ( ) 40 i i :, (9.7) x y, (9.5) 36,, () z = + i () z = 3 + i (3) z = i (4) z = e iπ/4 (5) z = e iπ (6) z = e iπ/3 37, ( ) () e iπ/4 e iπ/3 () e iπ/4 /e iπ/3 (3) e iπ/4 i (4) (e iπ/4 ) (5) (e iπ/3 ) 3 38 () zw = z w (9.33) () z/w = z / w (9.34) () i = e πi/ () i i = e π/ (3) (), i i , re iθ, r(cos θ + i sin θ)... III,,,, re iθ,, 9.7,, (partial derivative), 9.8 f(x, y) = xy + 3x, x

159 9.8 47, y, 4 f(x, y) = exp(xy), (9.38) f x = y + 3 y, x, 43 f(x, y) = (e y + e y ) cos x, f y = xy ( ),, 6 OK 5, δ σ 6 4 x, y : () f(x, y) = x + y () f(x, y) = exp(xy) (3) f(x, y) = x + y (4) f(x, y) = x + y () f/ x () f/ y (3) : f x + f y = 0 (9.39) (9.39),,,, 0 (9.38),?..., f(x, y) = x + y, f x =, f y x = 0, y = 0 f/ y, 53 4(), f/ x = x + y f/ x y, (9.38) y = 0... x y, y 0, y, f/ x = x,, 9.9 f(x, y) = exp(xy), f = y exp(xy) x (9.35) f x = y exp(xy) (9.36) f y x = f = xy exp(xy) + exp(xy) y x (9.37) ( ),,, 9.8, P.67 (5.0) x, y f(x, y), (x, y), (x + dx, y + dy), f?, f(x, y) x, f/ x f x,, f(x + dx, y + dy) = f(x, y + dy) + f x (x, y + dy)dx, f(x, y + dy) y, f/ y f y, y, f(x, y + dy) = f(x, y) + f y (x, y)dy f y x = f x y (9.38), f(x + dx, y + dy) = f(x, y) + f x (x, y + dy)dx + f y (x, y)dy, y

160 48 9,, f x (x, y + dy)dx = f x (x, y)dx + f xy (x, y)dxdy (f xy f y x ), dx dy, dxdy, 0, f x (x, y + dy)dx f x (x, y)dx, f(x + dx, y + dy) = f(x, y) + f x (x, y)dx + f y (x, y)dy, f(x + dx, y + dy) = f(x, y) + f f dx + x y dy (9.40) 44 : P = ρrt ρ ( ), R (8.3 J mol K ) 0 (73 K), 000 hpa ( 0 5 Pa),, 00 hpa, () ρ, T, P (!) () ρ, T, P (!),, P.67 (5.0),, df = f(x + dx, y + dy) f(x, y) (9.4), (9.40) df = f f dx + dy (9.4) x y P.67 (5.6),, f(x, y, z, ), df = f(x + dx, y + dy, z + dz, ) f(x, y, z, ) (3) (4) : dρ = RT dp P dt (9.44) RT (5) T, P, dt = K, dp = hpa = 0 Pa (9.45), dρ,?... z = F (x, y), z = F (x + dx, y + dy) = F (x, y) + F F dx + x y dy, dx, dy, z,, ( ), 3 df = f f f dx + dy + dz + (9.43) x y z,, dx dy,,,?...,, ( ) ( ), ( ),,,,,,?... ( ),

161 9.9 49,,,,,,,,,, ( ) ( ) ,,, : xy, 0 x X 0 y Y S, S = XY,,,, ( / )R? 9.4 x n, y m, mn x x 0, x,, x n, y y 0, y,, y m (x 0 = 0, x n = X, y 0 = 0, y m = Y ) x i, y j R i,j, R,, m n R = R i,j (9.46) j= i=,, m n,,, (x, y), F (x, y), F (x, y) (x, y), R i,j F (ξ i, ζ j ) S i,j (9.47) S i,j, ξ i x i x i, ζ j y j y j (9.46), m n R F (ξ i, ζ j ) S i,j (9.48) j= i=, n m,, S i,j ds, = : R = F (x, y) ds (9.49),, ( ), (9.47) x x i = x i x i, y y j = y j y j, S i,j = x i y j (9.50), (9.48), m n R F (x i, y j ) x i y j (9.5) j= i=, x i y j 0, n m, R = Y X 0 0 ( ) F (x, y) dx dy (9.5)

162 50 9 (9.49) (9.5),,,,,,, 3,,,, x y( z), 45 : x (y ) x ( x y ), y 3 (x + xy)dxdy (9.53) 0 4, OK!,, ( ),, 5,?...,, (9.5),,, F (x, y),,, (9.5),, 6..., 46, (x, y, z) ( ) C(x, y, z) S, 0 x a, 0 y b, 0 z c 9.0, sin x, : sin x = x! x3 3! + x5 5! x7 7! + (9.54), x,, x, m, x 3 m 3, m m 3 ( ), (9.54), x ( ),, sin x,,,,,, 9. P.0, : x(t) = x 0 cos(ωt + δ) (9.55) (t, ω, δ ), cos θ θ, θ t,,, ω, ω /, s δ ( ) ( ), ω,,,,,

163 , x, : (7) P.37 (8.9) () x () /x (3) + x (4) /( x) (5) tan x (6) exp x (7) ln x 6 R, C, C C 0, 0, C r,, C 0? : r, dr? 7 ( (9.) ), : + x = x + x x 3 + x 4 x 5 + (9.56) ln( + x) = x x + x3 3 x4 4 + (9.57) + x = x + x 4 x 6 + x 8 (9.58) arctan x = x x3 3 + x5 5 x7 7 + x9 9 (9.59) (9.), x,, (9.),, 4 f(x) = e x, f(0) = e 0 =, f (0) = e 0 =, f (0) = e 0 =,, (9.4), e x = 0! +! x +! x + 3! x3 + 4! x4 + = 0! + x! + x! + x3 3! + x4 4! + f(x) = sin x, f(0) = sin 0 = 0, f (0) = cos 0 =, f (0) = sin 0 = 0, f (0) = cos 0 =, f (0) = sin 0 = 0, (9.4) sin x = 0 0! +! x + 0! x + ( ) 3! = x! x3 3! + x5 5! x7 7! x ! x4 + 5! x5 + f(x) = cos x, f(0) = cos 0 =, f (0) = sin 0 = 0, f (0) = cos 0 =, f (0) = sin 0 = 0, f (0) = cos 0 =, (9.4), cos x = 0! + 0! x + ( )! = 0! x! + x4 4! x + 0 3! x3 + 4! x ! x5 x6 6! + 5 (9.7) x, (e x ) = 0 +! + x! + 3x + 4x3 + 3! 4! = π 4 (9.60) = 0! + x! + x! + x3 3! = e x : (9.56), (9.) (9.57) (9.56) x = 0 (9.56) x x (9.58) (9.58) x = 0 (9.59) (9.59) x =, (9.60) (x =, (9.),, (9.59) ) 8, (9.56), (9.57), (9.58), (9.59) (9.8) x, (sin x) =! 3x 3! + 5x4 5! 7x6 7! + = 0! x! + x4 4! x6 6! + = cos x (9.9) x, (cos x) = 0 x! + 4x3 6x5 + 4! 6! = x! + x3 3! x5 5! + = sin x 6 (9.7) x =

164 5 9 7 f(x) = /( x) (), f(x) = ( x) f (x) = ( )( x) ( x) = ( x) f (x) = ( )( x) ( x) = ( x) 3 f (3) (x) = ( 3)( x) 3 ( x). f (n) (x) = = n!( x) (n+) = 3( x) 4 x = 0, f(0) =, f (0) =, f (0) =, f (3) (0) = 3!, f (4) (0) = 4!,, f (n) (0) = n! (9.), : f(x) = + x + x + 3! 3! x3 + 4! 4! x4 + + n! n! xn = + x + x + x 3 + x x n + () x =, =/( ) =, , (3) (3.95) r = x, + x + x + + x n = n k=0 x k = xn+ x (9.6) n x n+, x <, 0, /( x) x >, x n+ ( < x, x < ), x =, x n+, x = 0 (9.6), (9.6) ( ) x < 8 z = a + bi (a, b ) z = a + bi (9.6) z = a bi (9.63), z + z = a, a = (z + z)/ (9.3), (9.6) (9.63), z z = bi, b = (z z)/(i) (9.3) 9 ()() (3) e iπ = cos π + i sin π = + 0i =, e iπ + = 0 30 e iα e iβ = (cos α + i sin α)(cos β + i sin β) = cos α cos β + i(sin α cos β + cos α sin β) + i sin α sin β = cos α cos β sin α sin β + i(sin α cos β + cos α sin β), e i(α+β) = cos(α + β) + i sin(α + β), 3 f(x) = e ix, g(x) = cos x + i sin x f (x) = ie ix = i(cos x + i sin x) = i cos x sin x g (x) = sin x + i cos x, f (x) = g (x) 3 e ix = cos x + i sin x (9.64) e ix = cos x i sin x (9.65), e ix + e ix = cos x, e ix + e ix = cos x (9.66), (9.64), (9.65), e ix e ix = i sin x, e ix e ix = sin x (9.67) i

165 ( e (sin x) ix e ix ) ie ix + ie ix = = i i = eix + e ix cos ( ) 34 () = cos x ( e cos 3 ix + e ix x = ) 3 = e3ix + 3e ix + 3e ix + e 3ix 8 = e3ix + e 3ix 8 = 4 e3ix + e 3ix cos 3x + 3 cos x = eix + e ix 8, cos 3x = 4 cos 3 x 3 cos x () ( ) eix + e ix : () (5) () (4) z iz Im O 3 Re /z - exp ( π 4 i ) + i z 9.5 () (5) - -3 z z i 35 ( π ) exp 6 i (3) (6) +z ( 3π ) exp i + 3i 37 () e i7π/ () e iπ/ (3) e i3π/4 (4) e iπ/ (5) 8e iπ 38 () z = r e iθ, w = r e iθ (r, r, θ, θ r r 0 ) (9.6) (9.9),, z = r, w = r zw = r e iθ r e iθ = r r e iθ e iθ = r r e i(θ +θ ) (9.68), r r, zw = r r = z w, (9.33) () 39 z, w, () z = re iθ e iα z = re i(θ+α) z, α,, α () z = re iθ = r cos θ + ir sin θ, 40 z = r cos θ ir sin θ = r{cos( θ) + i sin( θ)} = re iθ () i,, π/, i = e πi/ () i, i i = (e πi/ ) i = e πi / = e π/ (3) i i = e π/ e () () (3) (4) f x = x, f f = y exp xy, x f y = y y f x = x x + y, = x exp xy f y = y x + y f x = x (x + y ), f 3/ y = y (x + y ) 3/

166 f/ x = y exp(xy), f = exp(xy) + xy exp(xy) y x, S = c b a C dx dy dz, f/ y = x exp(xy),, 43 f x y = exp(xy) + xy exp(xy) f y x = f = exp(xy) + xy exp(xy) x y () f/ x = (e y + e y ) cos x () f/ y = (e y + e y ) cos x (3) ()() 44 () ρ = P/RT ρ = mol m 3 () ρ = mol m 3 (3) dρ, (4) dρ = mol m mol m 3 = 0. mol m 3 ρ P = RT,,, ρ T = P RT dρ = ρ ρ dp + P T dt = RT dp P RT dt (5) dt = K, dp = hpa=0 Pa 45, dρ = 0. mol m 3 ( 3 0 = ) (x + xy)dx dy = ( 9 + 9y ) dy = [ x x y ] 3 dy 0 [9y + 9y 4 ] = ,,,,,,,,,,, 8...,, 9,?...,, 30,,,...,? 3 III... XXX,, III, 46 (x, y, z), dx dy dz ds, C dx dy dz

167 55 0 :.5 (P.0),,,, 0.,,, : 0. : a = (, ) (0.) : b = (x, y) (0.) : a = (, ) (0.3) : a(, ) (0.4) : b = (x, y) (0.5) : b = (x, y) (0.6) : b = (x, y) (0.7) (0.3) a (0.4) = (0.5) b (0.6) (, ) (0.7) ( ) 3?..., a, b, c, ab = c, b, a = c/b (b 0 ), a b, c, ab = c, a = c/b,,, 0.,,, ( ),,,, (3, ) (3,, ), ( ), *,, x y, (, ) x, y, z 3, (,, 4) 3, (,, 4), ( ),,, (0.8) 4,, *

168 56 0 : a 3a b, P., a, b, 0 α, αa = b (0.0), a b,, (, ) (, 4), (, ) = (, 4), : a = (,, 6), b = (x, y, ),, x y,,,, *???,,,, 0.,,,,,, ( ) OK a = (,, 3), a 5, 5a = 5(,, 3) = (5, 5 ( ), 5 3) = (0, 5, 5) (0.9) ( ), OK, a = (,, 4) b = (, 6, 0.), a + b = (,, 4) + (, 6, 0.) = ( +, + 6, ) = (3, 5, 4.) 48 : a = (, ), b = (, 3), () a () a b 0.3 a, a ( ) (x, y), (x, y) = x + y (0.), (x, y, z), (x, y, z) = x + y + z (0.) 5 : () (,, ) () (3, 4) (3) (0, 0) 5 a α, : αa = α a (0.3) 0, 0 (x, y), (x, y) = x + y = 0,, x + y = 0, x = y = 0, (0, 0), (0, 0, 0) 49 a = (,, ), b = (,, ), * 53 : (0.) (5) (0.3)

169 () (, 0) () ( /, / ) (3) (/ 3, / 3, / 3) (4) θ, (cos θ, sin θ) (5) a, a/ a 0.4 a, b, θ, a b cos θ (0.4), a b, a b ( ) *3, (, ) 54 a : a a = a (0.5) 55 0 a, b, a b = 0 (0.6) a b 56 0 a, b, θ, cos θ = a b a b (0.7), (0.4),, a = (a, a ), b = (b, b ) a b a, a, b, b, a, b A, B OAB,, AB = OA + OB OA OB cos θ (0.8) ), OA OB cos θ = OA + OB AB (0.9) OA= a, OB= b, a b,, OA = a = a + a (0.0) OB = b = b + b (0.) AB = AB = b a = (b a, b a ) = (b a ) + (b a ) = a + a + b + b (a b + a b ) (0.) (0.9), = (a b + a b ), = a b + a b (0.3) a b = a b + a b (0.4)!,! ( 3 ), a = (a, a, a 3 ), b = (b, b, b 3 ), a b = a b + a b + a 3 b 3 (0.5) 57 (0.5) : (0.4), (a, b, a, a, α ): () a a 0 () a a = 0 a = 0 (3) a b = b a (4) (αa) b = α(a b) (5) (a + a ) b = a b + a b (), (), (3), (4) (5), (0.4) ( θ AOB, a b 58 a = (, ), b = (, 3), a b *3

170 58 0 : 59 a = (,, ), b = (3, 4, 5),, (0.8) (0.9) a b, 60 a = (, ), b = (, 3), θ, (0.8) (0.7), cos θ 6 OAB O, A a, O, B b, b, ( ) e b, a e b (0.6), A OB P O ( AOB ), a b 33 (0.8) (0.7)?..., (0.7) ( 3 ), (0.8) ( ), (0.8) ( ), 54,... ( ),, ( ), a, b,, 6 a = (, ), b = (, 3), a b ab a b a b 3,, a, b, 63 OAB O, A ab a b a b a, O, B b, a b s 3,, ab *5 a b a b, () : 34? s = a b (a b) (0.7) () a = (a, b), b = (c, d)..., (, ) (, ), (3, 4) s = ad bc (0.8), (, )/(3, 4) (3) a = (, ), b = (3, 4) s, (0.7) (0.8)? (4) a b (a b ) S : 35, (, 4)/(, ) =?...!, (, 4) = (, ), S = ad bc (0.9) 63,, (0.7) (0.8) *4 (0.7), (0.8), 0.5,, xy (, y = (0.9), *4, *5,

171 ) ax + by + c = 0 (0.30) ( a, b, c, a, b 0 ),, x y (a, b),! : A(x 0, y 0 ), B(x, y ) : ax 0 + by 0 + c = 0 (0.3) ax + by + c = 0 (0.3) 36 0.,,?... (5, 3),, (5, 3), k(5, 3) (k 0 ), 5kx + 3ky + c = 0 (c c ), (, ), c = 3k, 5kx + 3ky 3k = 0 k,,, ( ),, a(x x 0 ) + b(y y 0 ) = 0 (0.33) : (a, b) (x x 0, y y 0 ) = 0 (0.34), (x x 0, y y 0 ) AB, (a, b) AB = 0, (a, b) AB AB ( ),, (a, b), (normal vector) (a, b), ax + by + c = 0 ( ) 64 (, ), (3, ) ( : (3, ), 3x y + c = 0 c OK) 0.6,, : ax + by + cz + d = 0 (0.35) ( a, b, c, d, a, b, c 0 ) (0.30), (0.35),, (a, b, c)!, (0.35) : A(x 0, y 0, z 0 ), B(x, y, z ) : ax 0 + by 0 + cz 0 + d = 0 (0.36) ax + by + cz + d = 0 (0.37) 0. (, ), (3, 5)? (3, 5) ( 0 ) (5, 3), 5x + 3y + c = 0 (c ) (, ), 3 + c = 0, c = 3,, 5x + 3y 3 = 0, (0.39), a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 (0.38) : (a, b, c) (x x 0, y y 0, z z 0 ) = 0 (0.39), (x x 0, y y 0, z z 0 ) = AB, (a, b, c) n 65 (, ), (, ) ( :!) n AB = 0 (0.40), n AB?, A B,

172 60 0 : (0.40), A B, n B, A, n ( ), (0.35), 3, x y z n = (a, b, c), 66 (0, 0, ), (,, ),,, 64, 0. 65,?, (0, 0, 0), (, 0, 0)?, (0, 0, 0), (, 0, 0),, x, x y, x z,,,,, (0, 0, 0), (, 0, 0) (0,, 0), x y ( z = 0), : 67 (0, 0, ), a = (,, 3), b = (,, ) n = (p, q, r) (p, q, r ) () : p + q 3r = 0 ( : n a ) () : p+q +r = 0 ( : n b ) (3) p q r (4) (p, q, r) = r(7, 4, ) (!) (5),,, a = a a a 3, b = b b b 3 (0.4), 3 a b, a b, a b = a a b b := a b 3 a 3 b a 3 b a b 3 (0.4) a 3 b 3 a b a b, a = (,, 3), b = (4, 5, 6), a b = 5 = = (0.43)!... (0.43), 55 (0.4), a b a b a 3 b a b 3 a b 3 a 3 b, a b 3 a 3 b (0.44) a 3 b a b 3 a b a b... x (a b 3 a 3b ), y (a b ) z (a 3 b 3), y z x, z x y, (a b ) 56,,...! 57...,

173 0.7 6,,,,,,, 68 () (0.4), : (a b) a = 0 (0.45) (a b) b = 0 (0.46) () (0.43), : ) a b, a b, 67,? : a = (,, 3), b = (,, ), a b :, (0.43), 0 : 3 6 = = = = , 0, 0,, : ) a b a b *6, (0.9), : 70, a b, a, b () a = (,, 0), b = (,, ) () a = (, 0, ), b = (,, ), : 3), b a = a b (0.47) 7 (0.47) : (0.4) a b 7 a, : a a = 0 (0.48) : (0.47) b a...?,,, ( 0.!): 4) a b, a b *7 38?...,?, ( ),,,, 73 e = (, 0, 0), e = (0,, 0), e 3 = (0, 0, ), : e e = e 3 (0.49) e e 3 = e (0.50) e 3 e = e (0.5) (e e ) e = 0 (0.5) e (e e ) = e (0.53) 74 a, b, c p,, *6 a b sin θ (θ a b ) *7, x y z

174 6 0 : (r, ω ): r(t) = (r cos ωt, r sin ωt) (0.56), r, x ωt (P.09!), ( ), ( x ), ω () P v(t), () P a(t), 0. ( ) a b b a, a, b S, a, b, a b b a a b a b (3) ( : ) (4) (5) (6),? : (pa) b = p(a b) (0.54) (a + b) c = a c + b c (0.55) 75 p, a, b, c, a b c a b c,, pb c,? 39...,,,, ab cos θ, ab sin θ ( ),, 40 76(), v(t) = ( rω sin ωt, rω cos ωt),?... v(t) v(t) 4 v(t) v(t)..., v(t) = r (t) (r!), 4?......,, 0.8, 76 xy, r, P t = 0 P (r, 0), t, P, r(t), OK,,,, ( ), r cos ωt rω sin ωt,

175 P.7 ( ),,,,, r 0, r, r := r r 0,, F r F θ,, r F,, r cos θ,, F r cos θ...!,, F r!, =,, : ( ): r F : r p : qv B, r, F, p (= ), q, B,,, 43 3,?...,,..., 77 : a(t) = a (t) a (t), b(t) = b (t) b (t) a 3 (t) b 3 (t), t : (a b) = a b + a b (0.57) (a b) = a b + a b (0.58) : (a b) (a b), 0.9,,,,, : n ( ), G := m r + m r + + m k r k + + m n r n m + m + + m k + + m n (0.59) ( ), m k k, r k k, 0.3 m 4 ( r r 4 ), 0.4 G := mr + mr + mr 3 + mr 4 m + m + m + m = r + r + r 3 + r 4 4 (0.60) m, r, m, r, G := m r + m r m + m (0.6) (.50),,, m : m ( ),,,,,,

176 64 0 :, ( ) ( ) , O G 4 A, B, C, D, a, b, c, d ( OA = a ) a = b = c = d = (0.6) (0.60), G, 0.3 ABCD 78 OG = a + b + c + d 4 (0.63), G O, OG = OO = 0 () : a + b + c + d = 0 (0.64) 0. ( ) (m/s) 00: : : : () θ : a b = a c = a d = cos θ (0.65) (3) (0.64) a : (4) : (5) : a a + a b + a c + a d = 0 (0.66) + 3 cos θ = 0 (0.67) cos θ = 3 (0.68) (6), θ, ( ) 79, ( ),,,? :, ( (7.67)),,,,,, 0, 45, , ( ),,, ( ), (, 3,, 6, 3, 5),,,,,,,, (,, 4, 3) 4,

177 0. 65 *8 44 4,...? , 3, 4, 4, 0.5 A,,,,,, 0, 30, 90, 80, a = (0, 30, 90, 80) A,, A ( ) 46??...,,, 4,, (0.4)! (0.4), 4!, 4, 0.6 A B C,,,, 0.5, B C,,, c = (,,, 0.5), A, B C, = 500, a c, a c!! 47 P39,, P65, ( )?...,,, 0.?,, 4,, *9 48?...?,?,,,,, ), ( ),,, a, b, α, β, αa + βb (0.69) *8, (,, 4, 3) (, 3, 4, ) *9, 4!!!

178 66 0 :, (,,, (vector)!, (0.69) (3 ) * 0 (linear combination) 80? 49,,,?...,, OK 50...!??..., 5,,?... (9.), f(x),, x, x,, x n (f (n) (0)/n! )?, f(x), x, x,,,,,,,,,,, ( ) (, ),?,,,,,,,, *0 II,,,,,,,, 9 ax + by + c = 0, P(x 0, y 0 ) (a, b, c, x 0, y 0, a b 0 ) P, ax 0 + by 0 + c a + b (0.70) : P Q(x, y ), (x, y ), PQ (a, b), Q Q, PQ OK 30 a = a a, b = b b (0.7) a 3 b 3, () a b (!): a b + a b 3 + a b + a b 3 + a 3 b + a 3 b (a a b b + a a 3 b b 3 + a 3 a b 3 b ) (0.7) () (0.7), a b (!), (0.7) : (0.7)

179 a, b, c 3 a (b c) = (a c)b (a b)c (0.73) :,!, (a c)b (a c), b, ( ) (,?!) 48 () a = (, ) = (, 4) () a b = (, ) (, 3) = (, ) 49 a = (, 4, ) 3a b = (, 8, ) 50 a = kb (k ), (,, 6) = k(x, y, ), = kx, = ky, 6 = k k = 3, = 3x, = 3y, x = /3, y = /3 5 () 3 () 5 (3) 0 5 a (a, a ), αa = α(a, a ) = (αa, αa ) = (αa ) + (αa ) = α (a + a ) = α (a + a ) = α a a ( ) 53 () (, 0) = + 0 = () ( /, / ) = ( / ) + ( / ) = / + / = = (3) (/ 3, / 3, / 3) = (/ 3)) + (/ 3)) + (/ 3)) = /3 + /3 + /3 = = (4) (cos θ, sin θ) = cos θ + sin θ = (5) a/ a = a / a = 54 a a = a a cos 0 = a 55 a, b θ a b = a b cos θ = 0, a = 0, b = 0, cos θ = 0, θ = a b = a b cos θ cos θ = a b a b 58 a b = + 3 = 8 59 (,, ) (3, 4, 5) = = 0 60 (0.7), cos θ = / 5 6 AOB θ OP=OAcos θ, a e b = a e b cos θ, a OA, e b, a e b = OA cos θ, OP 6 b e b, 53 (5), e b = b/ b = ( / 0, 3/ 0), a e b = / 0 = 0/5 63 () AOB θ (7.48), a = a, b = b, s = a b sin θ sin θ = cos θ, (0.7), sin θ =, s = a b ( a b ) a b ( a b ) a b = a b (a b) () a = a + b, b = c + d, a b = ac + bd, (a + b s = )(c + d ) (ac + bd) a d = + b c abcd (ad) (ad)(bc) + (bc) = (ad bc) = ad bc = (3), s = 4 3 = 4 6 = 64 (3, ), 3x y + c = 0 (c ) (, )

180 68 0 :, 3 + c = 0, c =, 3x y = 0 65 (, ), x + y + c = 0 (c ) (, ), ++c = 0, c = 3, x+y 3 = 0 66 (,, ), x + y z + d = 0 (d ) (0, 0, ), + d = 0, d =, 67 x + y z + = 0 () n a, n a = p + q 3r = 0 () n b, n b = p + q + r = 0 (3) q, p = 7r p, q = 4r (4), (p, q, r) = (7r, 4r, r) = r(7, 4, ) (5) (7, 4, ), 7x 4y + z + d = 0 (d ) (0, 0, ), + d = 0 d =, 7x 4y +z = 0 68 a b 3 a 3 b a (a b) a = a 3 b a b 3 a a b a b a 3 = (a b 3 a 3 b )a + (a 3 b a b 3 )a + (a b a b )a 3 = a a b 3 a a 3 b + a a 3 b a a b 3 + a a 3 b a a 3 b = 0 (a b) b 69 : (7, 4, ) 70 () a b = (,, ), (,, ) = 6 () a b = (, 3, ), (, 3, ) = 7 b a b a 3 b 3 a b a = b a = b 3 a b a 3 b 3 a 3 b a b a = a 3b a b 3 a b 3 a 3 b = a b 3 + a 3 b a 3 b + a b 3 a b a b a b + a b = a b 3 a 3 b a 3 b a b 3 = a b a b a b 7 (0.47) b a, a a = a a, a a = 0 73 ( ) 74 a = a a a 3, b = b b b 3 pa (pa) b = pa pa 3 = p(a b), c = b b b 3 (0.54) c c c 3 pa b 3 pa 3 b = pa 3 b pa b 3 pa b pa b (a + b) c a + b c (a + b )c 3 (a 3 + b 3 )c = a + b c = (a 3 + b 3 )c (a + b )c 3 a 3 + b 3 c 3 (a + b )c (a + b )c = a c 3 a 3 c a 3 c a c 3 + b c 3 b 3 c b 3 c b c 3 = a c + b c a c a c b c b c (0.55) 76 () v(t) = r (t) = ( (r cos ωt), (r sin ωt) ) = ( rω sin ωt, rω cos ωt) v(t) = ( rω sin ωt, rω cos ωt) = rω ( sin ωt, cos ωt) = rω

181 0. 69 () (3) (4) a(t) = v (t) = ( ( rω sin ωt), (rω cos ωt) ) = ( rω cos ωt, rω sin ωt) a(t) = ( rω cos ωt, rω sin ωt) = rω ( cos ωt, sin ωt) = rω r(t) v(t) = (r cos ωt, r sin ωt) ( rω sin ωt, rω cos ωt) = r ω cos ωt sin ωt + r ω sin ωt cos ωt = 0 0 v(t) a(t) 77 (a b) = ( a (t)b (t) + a (t)b (t) + a 3 (t)b 3 (t) ) = ( a (t)b (t) ) + ( a (t)b (t) ) + ( a3 (t)b 3 (t) ) = a (t)b (t) + a (t)b (t) + a (t)b (t) + a (t)b (t) +a 3(t)b 3 (t) + a 3 (t)b 3(t) = ( a (t)b (t) + a (t)b (t) + a 3(t)b 3 (t) ) + ( a (t)b (t) + a (t)b (t) + a 3 (t)b 3(t) ) = ( a (t), a (t), a 3(t) ) ( b (t), b (t), b 3 (t) ) + ( a (t), a (t), a 3 (t) ) ( b (t), b (t), b 3(t) ) = a b + a b (a b) = { a a a 3 b b b 3 } = a b 3 a 3 b a 3 b a b 3 = a b 3 + a b 3 a 3b a 3 b a 3b + a 3 b a b 3 a b 3 a b a b a b + a b a b a b = a b 3 a 3b a 3b a b 3 + a b 3 a 3 b = a b + a b a b a b 78 a 3 b a b 3 a b a b () (0.63) OG = 0 () OG OG ( a b ) θ (0.6) a = b =, a b = a b cos θ = cos θ a c, a d (3), a ( ) = a (a + b + c + d) = a a + a b + a c + a d = ( rω sin ωt, rω cos ωt) ( rω cos ωt, rω sin ωt) = r ω 3 sin ωt cos ωt r ω 3 cos ωt sin ωt = 0 0 (5) a(t) = ω (r cos ωt, r sin ωt) = ω r(t), a(t) r(t) ω, ω, a(t) r(t), (6) (), v ω, ω, v (), a ω, ω, a 4,, a ( ) = a 0 = 0 (4) a a = a, (0.6), (0.65) (0.66) (5) (0.67) (6), arccos( /3).?06 09.?? 09??6 79 x, y, (U x, U y ) *, * m/s, (, 0) m/s,, x

182 70 0 : 0. ( ) (m/s) U x (m/s) U y (m/s) 00: : ????? 3.34 : ????? 8: , 00:00 θ = 5, U=3.0 m/s, U x = U cos θ = 3.0 (m/s) cos(5 )=.897 m/s,.9 m/s, 0.30 m/s, θ, tan θ = 0.30/.9 = 0.56, θ = arctan 0.56 = 0.? 8.?, 8.?, 5...,,,, 53, ,,,,

183 7 :,,,,,,,,,. (matrix),. A = A, B [ ] 3, B = 0 [ ] 3 4 (.) (.) A, (,, 3) (,, 0),,,, (.) A, [ ], [ ], [ ] 3 0 (.4), 3,, m, n, m n (.) A 3, (.3), (.) 3,, ( ) 55 [ ]? ( ), n n n n,,..., (row), (.) A, [ 3 ] (.),,, (column), (.) A, [ ],, (.3),,,,,, (.) B, (component),, i j, (i, j), a ij, (a ij ) (.) A, A = (a ij ), A (, ) a =, A (, 3) a 3 = 0 *, (i, j) ( ),, * a,,

184 7 :,. ( ), ( ). A = [ ], B =, A 3, 3A = ( ) [ ] 0 3 [ ] 3 3 = 3 ( ) 3 [ 3 ] (.5) (.6) ( ) ( ), (, ) ( ), (i, j) ( ).3 (.) A, B,, ( ).4 (.5) A, B,, : A + B = ( ) [ ] = [ ] 3 3 (.7),, A, B, A, B, A B, AB, A i ( ) B j ( ) (i, j), ( )!.5 A = [ ] 3, B = 0 3 (.8) 0 4, AB (,), (,, 3) = 8, AB (,), = 3, [ ] 3 AB = [ ] ( ) = ( ) [ ] 8 = (.9) 3 ( ) 56! (.7), ( (.9))?...,, 57?...,.5,,,, ( ),,,! A, B, AB OK, A B, A B 8 A = A, B, C [ ], B = [ ] [ ] 0 3, C = (.0)

185 .3 73 () A A B () AB BA, AB BA (3) A + B = B + A (4) (A + B) + C = A + (B + C) (5) (AB)C = A(BC) (6) A(B + C) = AB + AC 8 (.8) A, B, BA 3!, A, B, C, : A + B = B + A (.) : (A + B) + C = A + (B + C) (.) : (AB)C = A(BC) (.3) : A(B + C) = AB + AC (.4) : (A + B)C = AC + BC (.5) ( (.), (.) 5 ), 8() 8, : AB = BA (.6) (, ) 83 λ, λ, [ ] λ 0 B = 0 λ (.7) A, AB, A λ, A λ :, [ ] a b A = c d (.8), AB,.3,,.6,,,, 5, 8,,, 7, 4, 3, F (Food ): F = /( ) (.9) 3,, 3,, 5 7 [ 8 4 / = 3] 3 8 / (.0) 3 3, 3, 8, 3, 3,, 3, 4,, 3 5, 3 M (M Monkey M): M = [ ] 4 5 (.) 3 3 3,, F M : 5 7 [ ] F M = / = / , (i, j), j, i(i =, i =, i = 3 ),,,,,,,, ( ) :,,

186 74 : ( ), , (5, 7) /( ) [ 3],,, 84,, 0, 5, 3 7,,,?,,, 80, 60, 50,? :, F M,, F M, 58,...,,, P.48 f (x, y, z) 3, (9.43), f, df = f f f dx + dy + dz (.) x y z? ( f/ x, f/ y, f/ z), (dx, dy, dz)?, f g, g (x, y, z) g, dg = g g g dx + dy + dz (.3) x y z (.), (.3), [ ] df = dg [ f x g x f y g y f z g z ] dx dy (.4) dz,, P.67 (5.6) (5.6),, (.4), (x, y, z) 3 (f, g),,,.4 0,, [ ] 0 0, 0 0 [ ] O (.5), O, A, AO = OA = O (.6),, 0,, 0 0,, [ ] [ ] = [ ] 0 = 0 0 [ [ ] ] [ ] 0 = 0 0 [ ] , O O.5, (.0) B, (, ), (, ),,,,, (.0) B, (, ), (, ) 0,,, 0, (unit matrix)

187 .6 75, [ ] 0 E = (.7) 0 E 3 = (.8) 0 0, n n (.7), (.8) 3, n, E n, n, E * E, *3 A : AE = EA = A (.9), (.9), ( ),, x x = x = x (.30),, (.0) A, (.9) n E, n A AE = EA = A n E?... n A,, E n AE = EA = A 60, n A AE = EA = A n E OK?..., A = E = [ ] 0 0 0, AE = EA = A, E,, 0 n OK,,, 6, 0?... [ 0 ] ,, (determinant) : [ ] a b A = c d, det A *4 : (.3) det A := ad bc (.3) 59...,, (.3) (.3) 3, :, : det A, det(a), A 3: det A A,, A, 86 (.0) A, B, det A det B 6?... n A AE = EA = A n E, (.7) * I n I *3 AE EA! *4,

188 76 :,, [ ] a b A = c d det A, a = [ ] a, b = c [ ] b d (.33) (.34) S, S = ad bc = det A (.35) ( : ) P.58 (0.9),,, a, b, a + b A, B, C OACB,, : O: (0, 0), A: (a, c), B: (b, d), C: (a + b, c + d),, S = (a + b)(c + d) (ac/) (bd/) bc = ad bc A, B, C,, ad bc *5, ad bc S *6 89 a = (, 8), b = (9, 9) :,, a b (a b) 63 S = ad bc ( )? A, A *7 det A = det A (.36) 9 A, B, y V S C [ ] [ ] a b p q A =, B = c d r s (.37) R B det A, det B, det(ab), : A Q det(ab) = (det A)(det B) (.38) O P U x., A, B, C. C x, y U, V, A x, B y P, R A CU, B CV Q, S, OAP CBS, ac/, OBR CAQ, bd/, PAQU SBRV, bc OACB, OUCV,,, ( 3 ),, ( ), determinant, (matrix), ( ), *5 A B, S = bc ad *6 ad bc, a b, ( ) *7, det A

189 .7 77,, *8.7,,?,, A, B, AB = BA = E (.39), B A (inverse matrix), A ( ) *9 9 det A 0 [ ] a b A = (.40) c d, * 0 det A [ d ] b c a (.4) (.4), det A 0 0,, det A = 0,,,, 0,, O,, E ( ) [ ] (.4), 9,, 0 * : (.4), (.4) (.39), 64 AB = BA = E?..., 65, A, B, E, AB = BA = E?... A, B, E, AB, BA 5, 66, A, B, E, AB = BA = E B?...,,, B A A = 94, [ ], B = [ ] 0 A, B () AB () : (AB) = B A (.43) *8 ( ) *9 (.39), AB = E BA = E,,, *0 3, * 3, 3,! 95 : A, det(a ) = det A (.44)

190 78 :.8, { x + y = x + y = (.45), : [ [ ] [ x = (.46) ] y ] A, (.4), A =, [ ] [ ] x A = y [ ] =, A, [ ] [ ] A x A = A y, A A = E, [ [ ] x = A y] [ ] [ ] [ ] = = 3 (!) [ ] (.47) (.48) (.49),, a, b, c, d, p, q : { ax + by = p cx + dy = q, [ ] a b A = c d, : [ ] [ ] x p A = y q (.50) (.5) (.5) ( A),, *,, A det A 0, A? : { x + y = x + 4y = : [ [ ] [ x = 4] y ], A, det A = 4 = 0 (.54) (.55), A (.54), (x + y = ), (x + 4y = ),, : x + y = (.56),,, (x, y) = (, 0) (0, /),,,? { x + y = x + 4y = 3 (.57), 0,, 0 = (.58) x, y,,, A A,, [ [ ] x = A y] p q (.53),,,, *,,,,

191 A, λ, (0 ) x, Ax = λx (.59), x A (eigenvector), λ A (eigenvalue) ( ),, : (.59) xa = xλ ( ),, (.59),, 68, x 0?... x = 0, A,, λ, Ax = λx,.7 : A = [ ] (.60) λ, x,, Ax = λx, Ax λx = 0 (.6), λx = λex,, Ax λex = 0 (.6), (A λe)x = 0 (.63), (A λe)x = [ 5 λ 3 ] [ ] x 4 λ y = [ ] 0 0 (.64), x = (A λe) [ 0 0 ] (.65) (A λe),, [ 0 x = = 0 (.66) 0],, A λe * 3, 0, det(a λe) = 0 (.67) (.67) A ( ), (5 λ)( λ) 3 4 = λ 6λ 7 = (λ + )(λ 7) = 0, λ =, λ = 7, λ =, (.64), (A λe)x = [ [ ] [ 6 3 x 0 = 4 ] y 0] (.68),,,, [ ] [ ] x = y,, λ = 7, (.63), [ ] [ ] 3 x (A λe)x = = 4 6 y [ ] 0 0 (.69) (.70),,, [ [ x 3 = y] ] (.7),, A λe, (A λe) *3

192 80 :, A,, 7, ( ) [ [ ] 3 ] (.7),,,,, ( ) :,, ( 96),,, 7, 96 [ [ ] 6 4 ] ( 0 ), :! 97 [ ] 3.0,.7(P.79), A = [ ] (.73) (.74) A, (.7), p =, [ [ 3, p ] = ] (.75) Ap = p, Ap = 7p (.76), p p, P, p, p P = [p p ], P = [p p ] = [ ] 3 (.77), A : [ ] [ ] AP = A[p p ] = (.78) 4,,, Ap Ap,, AP = A[p p ] = [Ap Ap ] (.79) ( Ap, Ap ) (.76), [Ap Ap ] = [ p 7p ] (.80) ( p, 7p ), 83, [ ] 0 [ p 7p ] = [p p ] 0 7 (.8) (.79), (.80), (.8), AP = P [ ] (.8), P P, P AP = P P [ ] (.83) P P,, E (.9), (.83), P AP =, [ ] [ ] [ ] [ ] = 4 [ ] (.84) (.85)

193 . 3 8, ( A A, ), P, P AP, 0 ( ),, A, : () A (), A (3) (4) A P,, (5), P AP ( ), A :,, (.78) (.83), A P, : 98 (.74), P, P AP [ ] [ ] 3 3 () () 99 : [ ] 6 6 (3) [ ] 6 4 (.86) 69?...,, P AP,,, [ ] 0 (.87) (!),,, : ), 8, (! ) ),, 9 n (bare) (forest) b n km, f n km, [ bn+ f n+ ] = [ ] [ ] bn f n (.88) t (b n, f n ) c n, A, (.88) : c n+ = A c n (.89) 300 () A () c n = A n c 0 (c 0 ) (3), A n ( : (P AP ) n = P A n P ) (4) 80 km, 0 km 3 (5) (n ),?,, ( ),,,, c n 4, A ( ) 4 70? ,, ( ),,, ( ),,, :.8 00 km. 3,,

194 8 :,,, [ ] a b A = a b (.90) det(a), a b a b,., (. ; a b ), (. ; a b ), * 4. ( ), 3 a b c B = a b c (.9) a 3 b 3 c 3 det(b), :.3, 3, a b c 3, a b 3 c, a 3 b c,,.3, 3, a 3 b c, a b c 3, a b 3 c,, det(b) := a b c 3 + a b 3 c + a 3 b c a 3 b c a b c 3 a b 3 c (.9) 30 () : () (3) (5) (7) ( ) (4) (6) ,,,,, A =, Ax = [ ] [, x = 0 3] [ [ ] = 0 ] 3 [ ] [ = ] (.93),,,, (linear transformation) * 5,,,, *4,,, *5,

195 . 83, A = [ ] 0 0 (.94), : [ cos α ] sin α sin α cos α (.98), A, (x, y), [ ] [ ] 0 x = 0 y [ ] x y (.95), ( x, y), x,, y 30,? () (4) [ ] 0 0 [ ] 0 0 () (5) (3) [ ] 0 0 [ ] 0 0 (6) [ ] 0 0 [ ] 0 0 0, 7 cos sin... (.98), α = 0 0,, (,?) sin cos, 304,, : () x () (3), π/4 305 (.98) A : () det A () A (3) A 303, (), (3), (4),,?, (P.09), P ( ), [ ] x = y [ ] r cos θ r sin θ (.96) r O P, θ x OP ( ),, α,,, [ ] r cos(θ + α) r sin(θ + α) (.97) (P.04), [ ] [ ] r cos θ cos α r sin θ sin α x cos α y sin α = r sin θ cos α + r cos θ sin α y cos α + x sin α [ ] [ cos α sin α x = sin α cos α y] 3 A, λ, λ, det(a) = λ λ : λ, λ, P, P AP (!), det(p AP ), (.38) (.44) 33 (.9) a = a a a 3, b = b b b 3, c = c c c 3 () (a b) c (.99), det(b) (), det(b), a, b, c ( 3

196 84 : ) (3) a = b a = c b = c, det(b) = 0 :!), 3,?,,,,,, ( ) 34 3 B, () B det B = det B (.00) () 3, 3, (3), B, ( ),, 3,, ( ),,,,,,,!, [ ] [ ] 4 () A =, A B = 0 [ ] [ ] 4 4 () AB =, BA = 0 3, AB BA [ ] 3 (3) : = = 0 [ ] 6 3 (4) : = = 0 [ ] [ ] [ ] (5) (AB)C = =, 3 [ ] [ ] [ ] A(BC) = = 4 3, (AB)C = A(BC) [ ] [ ] [ ] 5 9 (6) A(B + C) = =, 3 [ ] [ ] [ ] AB + AC = + = 3 3, A(B + C) = AB + AC 86 det A = ( ) = 3 det B = 0 = 87 det E = 88,, 89 a b, det [ ] 9 = = a b,, 37/ 90 A [ ] a b A = c d, det A = ad bc, A A, [ ] A c d = a b

197 . 85, det A = cb da = (ad bc) det A 9 [ ] ap + br aq + bs det(ab) = det cp + dr cq + ds = (ap + br)(cq + ds) (aq + bs)(cp + dr) = acpq + adps + bcqr + bdrs (acpq + adqr + bcps + bdrs) = adps + bcqr adqr bcps (det A)(det B) = (ad bc)(ps qr) = adps + bcqr adqr bcps, det(ab) = (det A)(det B) 9 (.4) B [ ] [ ] a b d b AB = c d deta c a = [ ] [ ] a b d b deta c d c a = [ ] ad bc ab ab deta cd cd ad bc = [ ] [ ] deta 0 0 = deta 0 deta 0 = E (BA = E, ),, B A A = [ ] /3 /3, B = /3 /3 [ / ] 0 / () (.38), det(ab) = (det A)(det B), A, B, det A 0 det B 0, (det A)(det B) 0, det(ab) 0, AB () (AB)(B A ) = A(BB )A = AEA = AA = E, B A AB 95 AA = E, det AA = det E =, det AA = (det A)(det A ), (det A)(det A ) = det A, det A = / det A 97 A,, (.67), det(a λe) = ( λ)(3 λ) = λ 5λ + 4 = (λ )(λ 4) = 0, λ =, λ = 4, λ =, (A λe)x = [ [ ] [ x 0 = ] y 0],, [ ] [ ] x = y,, λ = 4, (A λe)x = [ [ ] [ x 0 = ] y 0],, [ ] x = y [ ] (.0) (.0) (.03) (.04),, A,, 4, * 6 [ ] [ ] 99 ( : ) [ ] [ ] [ ] = , : 300 [ ] [ ] [ ] = [ ] [ ] (.05) (.06) *6, (0 )

198 86 : () ( ) λ, A λ.7λ = (λ )(λ 0.7) = 0, λ =, 0.7,, [ ], [ ] P, P =, () (3) [ ] P AP = [ ] (.07) (.08) (P AP ) n = (P AP )(P AP ) (P AP ) = P AP P AP P AP = P AA AP = P A n P,, (P AP ) n =, P A n P = [ ] n [ ] 0 0 = n [ 0 ] n, [ ] A n 0 = P n P = = [ ] n 0.7 n n n (4) n = 3, [ b3 f 3 ] [ ] = A 3 b0 = f 0 [ ] [ ] 80 = 0 (.09) (.0) [ ] , (50 km ) (5) (3) n, A n [ /3 ] /3 /3 /3 (.) 30, [ bn f n ] [ /3 /3 /3 /3 ] [ ] 80 = 0 [ ] 00/3 00/3 [ ] 33 67, 33 km, 67 km () 0 () (3) (4) (5) 3 (6) 3 (7) 30 () (,, ) () x (3) (4) y = x (5) x (6) x ( ) 303 (), (3), (4),,,, 304 () 305 [ ] 0 0 () [ ] 0 0 () det A = cos α + sin α = () [ cos α ] sin α sin α cos α (3) [ ] (.) :, ( α) (3) ( ) [ cos α ] sin α sin α cos α (.3) :, α

199 87,,,,,,,,, p q,., n 4 n ( )., (condition) : n 4 n *,,,,,?,, (proposition),, p, q, p q. n 4 n, p: n 4 q: n, p q, ( ), p q,,. 4 *,.3,,,,,, ( ),, =.4.,.3, : n 4 = n = ( ).,,, n, n n, p p.5 x x 0 ( )

200 88.6 ( ),,,,.7,,,, ( ),,,.3,,.8,,,,,, 3 3,,,,,, 3 3,, OK, OK ( ),, p, q, p q = p q (.).9,, 3,, 3 ( ),, p, q, p q = p q (.).0, 3,,, 3 ( ). n, m, n,, m,, n,, m,, n, m ( ) 306 : () x () x,y, (3) x,y, p, q, r p, q 3, r? () p q r () p q r (3) p q r (4) p q r.4 p = q,,, : q = p p = q

201 .5 89 q = p. n 4 = n,,,, : : n, n 4 : n 4, n : n, n 4 ( ),,, n = 4 ( ) n = 4, ( ),,,,,,,.3,,,, : : : : ( ) 308,, ( ) () x = 0 x = 0 () x > 0 x > 0 (3) 309 (, (3) ) () () (3).5,?, p q, p q ( ).4?, ( ),, : ( A) ( B) ( C), A B( ),,, A B,,, C,, ( ),,,!

202 90...,...,....5, ( ),...,....6 :,, ( ),.7,,,,,,!...,, ( ).8 4, 4 4,, 4,, 8 4, 4,, 4,, ( ) 30, (), (), (3), 0 (4) 3,,? (),, (), (3), (4) (5) (6) (7) (8) (9),,,.9,,,, ( ),.0,,, ( )

203 .6 9,,. 3,,,, = 3 3+, 3 3 ( ), OK,,,,,,,,,, : p q (.4) p q q p, p q, p q, 33 p, q, p q,,,, p: x, x = q: x 34 p, q, p q,,,, p: x, x = q: x 3, (0 ),.6,, p = q (.3) ( ), q p, p q, p q, q p p q,, q, p, q p, =, p q,, q p,, p q,, p q,, A B, A B., ( ), ( ), A B,, A B B A, A B A B ( ),,,,,,,?,,,,,,

204 9.3? ( ).4?, ( ),,, ( ).5?,.6?,, ( ) ( ),,?,, 35?,?,, ( ),,,, ( ).8 n, n, n n, n, n, n, n, n, n 36 n, n, n 37 n, m, nm, n, m 60, n, m..., P.87.7,,,.7 (?) :,,, (? ) 6, nm... nm 6, n, m, nm, n, m...,,, n, m, nm, n, m ( )?

205 ,... 38?? nm, n, m,,,, nm n, m,, x, X, : x X (.7) ( (.7) ): X x (.8).9, S.8, (set) ( ),, ( ) x, X, : x / X (.9).9, / S *6.9 S = {,,,, } (.5) S, * ( ) ( ), *3 (element) ( ).9, *4 S, (.5), { }, { },.30 B = {, 4, 6, 8, } 63 (.) (,),..3 {.,.3}, {...3} :, {},, 0 / {} (.0) ( ( ) ), 0 / (.) ( ( ) ( ) ) 0 < (.), B = {n n } (.6) ( ), *5 ( ) 0 < {} (.3) ( * *3 ( ) ) *4,!!,,!! *5 n, n B = {m m },, OK *6,,,

206 94 : X, Y, X Y, X Y (subset),, ( ): X Y (.4) ( (.4) ): Y X (.5) X Y (.8).3 {,, 3} {, 3, 4} = {, 3} (.9) ( ) X, Y,,.3 S, : S S, S S S ( ).33 X Y (.0) {,, 3} {, 3, 4} = {,, 3, 4} (.) 39 X Y ( X ( ) Y ),, ,, X, X X (.6) X Y, X Y X Y,, X Y, Y X, X Y, : X = Y (.7) {,, 3} {4, 5, 6} = (.) ( ),,, N *7 Z Q R C, N Z Q R C (.3) 30,, : () {,, 3} () {,, 3} *8,,, x R (3) {} {,, 3} (4) {} {,, 3} (5) {, } {,, 3} (6) {, } {,, 3} (7) {,, 3} {,, 3} *7, 0 X, Y,, *8, x, y x + iy, y = 0,, x,, x = y = 0, 0,

207 .9 95 x, x {, 3, 4} A U, A U, A = {, 5, 6} ( ), x x R 3, () {n n N} () {n n N} (3) {, 0,,, 3} N.38 U = N, A, A U, A U, A ( ),. (4) {n n Z} {x 5 < x < 5, x R},,.35,, 7 ( ), ( ) *9 ( ), U * 0. ( ) A, ( ) A ( A) U, A, B, : A B = A B (.5) A B = A B (.6) (.5),., U A, A U A, A *, A := {x x U x / A} (.4) ( ).36 U. (.5) A,, A U, A, ( ) 3, (.6).37 U = {,, 3, 4, 5, 6}, A = *9, *0 universe * c A c complementary.9 X, Y,

208 96, X Y, X Y, X Y := {(x, y) x X, y Y } (.7).39 A = {, }, B = {5, 6, 7}, A B = {, } {5, 6, 7} = {(, 5), (, 6), (, 7), (, 5), (, 6), (, 7)} (.8) ( ) ),,?,,,,,,,,, (,, ), 33, ( ),,,,,,? ( ),,, :.40 M = {, } (.9) D = {, } (.30), M D ={(, ), (, ), (, ), (, )} (.3), ( ), ( ) ( ),,.39, A, B 3, A B 6,.4 A = {, }, A A = {(, ), (, ), (, ), (, )} (.3) (, ) (, ), (, ) {, } (, ) {, }, (, ) ( ), {, } * ( ), X, Y, Z, X Y Z := {(x, y, z) x X, y Y, z Z} ( ),, 65.39, A B = {5, 6, 7, 0,, 4}...!, ( R R, R R R,,, R R = {(x, y) x, y R} (.33) *, {, } = {, }, (, ) (, )

209 .0 97?, ( ), R R R = {(x, y, z) x, y, z R} (.34), 3 R R, R R R, R R 3 3, 3 = 8, 34 {,, 3} {, 3, 4} (.35) 66..., 35 (.36) (.37), (,, 3) (4, 5, 6) (.36) {,, 3} {4, 5, 6} (.37).0 ( ),,,,?,... for All A... exist E! = (, ) (, ) =, ( ) := ( ) ( ) / ( ) ( ) N Z Q R C (a, b) {x a < x < b}, a, b, a, b [a, b] {x a x b} [a, b) {x a x < b} (a, b] {x a < x b} ẋ x(t) t z z, z z z * 3 f : A B f, A B ( ) f : a b f, a b ( ) g f *3, = ( / x, / y, / z),

210 98 g f, g f : x g(f(x)) 37 : i.e. e.g. s.t. such that cf. compare 74 ϕ (a, b),?..., 4 (n ) (.43) n= 38, (, = < OK) * 4 () x, y R, x y = a R, y = ax () x R, n Z s.t. x < n (3) n Z s.t. x R, x < n 76 (a, b), 39, :?...,.43 x, x 0 x R, x 0 (.38) A, 0 A = {x Z 0 x} (.39) N Z (.40) () z, z z, () x, x =, x,, :.,, : ;, x 3 ( ) x [, 3] (.4) 35 < 3, = 3, 3?? 36 3 : {, } (, ) [, ] 36, 0,, Σ,, Π, {a, a, } p, q p q, q a k := a p a p+ a q (.4) k=p 306 *4 () (3), (), (3)

211 .0 99 () x 0 () x, y, 0 (3) x, y 0, 5, 7, 9, :,? 307 : (), () (3), (4) (3), (p q) r p (q r),,! 308 () : x = 0, x = 0 : x 0, x 0 : x 0, x 0 () : x > 0, x > 0 : x 0, x 0 : x 0, x 0 (3) :, :, * 5 :, 309 () ( ) () ( ) (3) 3 0, (x =, p = q, q = p ) 34 (p x = x = q x = 3, q = p ) 35, :,,,,,,,, 36 n, n, n, n, n, n 37 n, m 30 () ( ) (), (3) (4) 3 6,, 8, 3, 4 *5,,,,, n, n = k (k ) nm = km km, km nm, m nm, nm,, n, m 38 ( ) 39 ( ) 30, : (): ( ) (3): {} {,, 3}, (5): {, } {,, 3},

212 00 3 () {, 4, 6, 8, } () {, 3, 5, 7, } (3) {,, 3} (4) {0,, 4} 3.3 (3) n, x, x < n (, ) 39 () z C, z + z R () x R, x = = x = ±.3 (.6) 33 X, Y, X Y, X Y 34 {(, ), (, 3), (, 4), (, ), (, 3), (, 4), (3, ), (3, 3), (3, 4)} 35 (.36), (.37),, (,, 3) (4, 5, 6) = ( 3, 6, 3) {,, 3} {4, 5, 6} = {(, 4), (, 5), (, 6), (, 4), (, 5), (, 6), (3, 4), (3, 5), (3, 6)} = = () x y, a, y = ax () x, n, x < n (,, )

213 0 3,,,,,, (,, ) 3.,, 3( ) 4,,, ( ) (trial), 3 (event) ( ) 77?,,?...,, 3( ),, A B A B, A B (3.) ( ) 3., A, 4 B A B, ( ) A B A B, A B (3.) ( ) 3. A, B A B, 4 4 6, A B, ( ; ), A B = (3.3), A B (exclusive),,, 3,, ,, A, A = 4 6, 3 ( A ) *, U *, U = 6, 6 * * U

214 0 3,, A, A, Ā A Ā (, A Ā = ), A Ā,, 3, 5,, (3.) (3.),,,, 330 () () (3) (4) (5) (6) (7) (8) 3. N A n(a), A (probability) P (A), : n(a) P (A) := lim N N (3.4) P, probability P (A) P r(a), 3., A,,, N, N, n(a) N/, (3.4), P (A) = n(a) N = N/ N = (3.5) / 3.3,,,,?,, N, n, n/n, ( ) 67 ( ),,, n/n 0, A, : 0 P (A) (3.6), (, ) N, U N, (3.4), P (U) = n(u) N = N N = (3.7) 6 33 : P ( ) = 0 (3.8) A B, (3.3) A B =, (3.8), P (A B) = P ( ) = 0 (3.9), A B, A B A B, A B n(a B), n(a) + n(b) *3, N, (3.4) *3 n(a B) = n(a) + n(b) n(a B), A B, n(a B) = n( ) = 0

215 3. 03, P (A B) = = n(a) N n(a B) N + n(b) N = n(a) + n(b) N = P (A) + P (B) (3.0),,, 3, ( /6), 3 ( /6) /6 = /3 A Ā, (3.0), P (A Ā) = P (A) + P (Ā) (3.), A Ā, A,, U, (3.7), P (A Ā) = P (U) = (3.) (3.), (3.), P (A Ā) = P (A) + P (Ā) = (3.3),, P (Ā) = P (A) (3.4),,, C, C,, 3, 4, 5, 6, P (C) = /6, : P ( C) = 6 = 5 6 (3.5),, 6 6,,,, 3, 6,, /6 = /3,, ( 3 ), ( 6),, A ( ) m(a), U m(u), P (A) = m(a) m(u) (3.6) (3.4) ( ),,?,,,, (3.6) (3.4) (3.6), (3.4), (3.6), ( ) *4 33 0? 333,,,,, 50? *4,, 0, 3, ( ), /6

216 04 3 A, B, A B, P (B A) *5,, A,, B, P (B A),,,, ( ),,,,, P (B A) > P (B) (3.7),, C,, P (B C) < P (B) (3.8),,,, A B, A B A B,, A ( P (A)),, A, B ( P (B A)), : P (A B) = P (A)P (B A) (3.9) A B, P (A B) = P (B)P (A B) (3.0) (3.9), (3.0), P (A)P (B A) = P (B)P (A B) (3.) P (A), P (B A) = P (B)P (A B) P (A) (3.) (3.),,, 334, A 3, B 5 P (A), P (B), P (A B), P (A B), P (B A), A, B 3.3, (independent) 3.4, 3 ( ) ( ) A, B, A B, P (B A) = P (B), P (A B) = P (A), (3.9) (3.0), P (A B) = P (A)P (B) (3.3) (3.3) 335 A, B,? (), A: B: 3 (), A: B: 3 (3), A: B: , 3.4 (, ) (stochastic variable)?, 3.5, 6, D *6, D ( ) 3.6,,,,, *5 P A (B) *6 D dice

217 , 50, 0,... *7, ( ), 3.5 D, 6 (, ), 3.7 :, 0 ( ) ( ) 0, ( ) 00, ( ) 337 ( 3.7,, ), 3, 3.8 3,, D = 3,,, /6, P (D = 3) = /6 3 D 3, P (D 3) = / ( ) *7 ( ) X x, x,, x n, n P (X = x k ) = (3.4) k=,, :,,, : (3.4),,, X, Y, X x, Y y,, (x, y), X Y, ( ) 3.0 A, B A D A, B D B 3.5, D A D B, A B A 3, B 5,, x, y x, y 6, D A = x D B = y, D A D B 3. A, B A, B H A, H B H harvest H A, H B, H A H B,, A, B, H A H B,

218 A, B A D A, B D B 3.5, D A D B, A B A 3, B 5,, x, y x, y 6, D A = x D B = y, D A D B ( ) X Y P.04 (3.3), P (X = x Y = y) = P (X = x)p (Y = y) (3.5), (x, y) (3.5) 3.5 D, D + 0, D + 0,,, 3, 4, 5, D, 0D, 0D, 0, 0, 30, 40, 50, 60,, ( ) 3.0, D A + D B S, S A B, S,,, S ( ), 3., H A + H B H, H A B,,, 3.5, /6 3.5 D, ( ), A, B, A, B D A, D B, D A, D B ( 3.3 ), D A = D B A B,,?, PK,,,,, OK,,, 0 X p (p 0 ), p, { P (X = ) = p P (X = 0) = p (3.6) (3.6),,,,, p = 0.5,, 5,,

219 3.6 07, 0 5,, X := (3.7), ( X (3.6) X ) X,, B(n, p) (3.8) (n, p ),, k ( 0) X k, X k (3.6) X X n n, (3.7) X, X = X + X + + X n (3.9),,, : X B(n, p) n, k,, p k ( p) n k (3.30) ( 5,, 3, p( p)( p)( p)p = p ( p) 3 ),, k P (X = k) ( ), n k, k, n C k (3.30), : P (X = k) = n C k p k ( p) n k (3.3) 338 5, 3? 339 5, 3,? 3.6 X,,,, X (expectation), E[X], x, x,, x n X, : n E[X] := x k P (X = x k ) (3.3) k= 340 (3.3) 5, x, x,, x n 34 A,, (3.3), x, x,, x n, n?, A..., A,? D, E[D] = = 3.5 (3.33) 6 ( ) :,,,,, 34, 343 (3.6) X, : E[X] = p (3.34)

220 08 3,?, X x, x,, x n, p, p,, p n (n ) a, b, Y = ax + b Y ax + b, ax + b,, ax n + b, p, p,, p n, n n n E[Y ] = (ax k + b)p k = a x k p k + b k= k= k= = ae[x] + b (3.35) ( P.4 (3.83), (3.84), (3.3), (3.4) ), : X, a, b, E[aX + b] = ae[x] + b (3.36) X, Y X + Y, ( ), E[X + Y ] = E[X] + E[Y ] (3.37) ( ), X, Y,,,,,,,,, (3.37) :, X x, x,, x n, Y y, y,, y m (n, m, n m ) X + Y, x + y, x + y,, x + y m x + y, x + y,, x + y m x n + y, x n + y,, x n + y m nm (, ),, X + Y p k ( ),, p i,j := P (X = x i Y = y j ) (i, j n, m ), E[X + Y ] = = = n i= j= n i= j= n i= j= m (x i + y j )p i,j (3.38) m (x i p i,j + y j p i,j ) (3.39) m x i p i,j + n i= j= m y j p i,j (3.40), (3.38) (3.3) (3.38) p i,j (3.39) (3.39) (3.40) P.4 (3.83) (3.40) : x i j, j x i, n i= j= m x i p i,j = n m x i ( p i,j ) (3.4) i= j=,, m j= p i,j, X x i, Y, P (X = x i ), (3.4), = n x i P (X = x i ) = E[X] (3.4) i= ( (3.3) ), (3.40), n m m n m n y j p i,j = y j p i,j = y j ( p i,j ) i= j= = j= i= j= i= m y j P (Y = y j ) = E[Y ] (3.43) j= (3.38), (3.40), (3.4), (3.43), (3.37) (3.37), 3, X, X,, X n, ( ): E[X + X + + X n ] = E[X ] + E[X ] + + E[X n ] (3.44) (3.37), E[X + X ] = E[X ] + E[X ] (3.45)

221 3.6 09, X + X Y, (3.37), E[Y + X 3 ] = E[Y ] + E[X 3 ] (3.46) Y = X +X, E[Y ] = E[X +X ] = E[X ] + E[X ], E[X + X + X 3 ] = E[X ] + E[X ] + E[X 3 ] X + X + X 3 Y, E[Y + X 4 ],..., n (3.44) 344 (3.37) , 00, 346 B(n, p) X, : E[X] = np (3.47) : (3.9) (3.44), : X, Y, E[XY ] = E[X]E[Y ] (3.48) : (3.48) (3.37), (3.37) (3.48), (3.38), : n m E[XY ] = x i y j p i,j (3.49) i= j=, p i,j = P (X = x i Y = y j ), X, Y, (3.5), P (X = x i Y = y j ) = P (X = x i )P (Y = y j ), p i,j = P (X = x i )P (Y = y j ) (3.49) : E[XY ] = n i= j= m x i y j P (X = x i ) P (Y = y j ) (3.5) j j Σ, E[XY ] = = n x i P (X = x i ) i= m y j P (Y = y j ) j= n x i P (X = x i ) E[Y ] (3.5) i= = E[X] E[Y ] (3.53), (3.48) 347 X, Y, (3.48),, (3.48) , 00,,, ( ) X, Y X Y, 0 00 E[X] = 50, Y = 0,, /4, E[Y ] = 0 ( ) = 75 (3.54) 4, : E[X]E[Y ] = = 3750 (3.55), XY XY = 0000 X = 00 Y = 00, X = 00, Y = 00, XY = 0000, P (XY = 0000) = (3.56) (3.50) P (XY = 0) = = (3.57)

222 0 3, : E[XY ] = = 5000 (3.58) (3.55) (3.58),, E[XY ] E[X]E[Y ] ( ) V [D] = E[(D µ) ] = ( 3.5) 6 + ( ) ( 3.5) (6 3.5) 6 = , () E[X + Y ] = E[X] + E[Y ] (), ( ), E[XY ] = E[X]E[Y ], (3.48), X Y,, Y X, E[X ] = (E[X]), 34 (E[D ] 5., (E[D]) = 3.5 =.5), (3.37) Y = X, E[X] = E[X] (3.36) a =, b = 0,,?, (3.36), (3.44), (3.48), (3.48),, 3.7 X E[X] µ µ, X µ ( ),, (X µ) ( ) (X µ) X (variance), X V [X], ( µ X ): V [X] := E[(X µ) ] (3.59) 3.6 D D µ, P.07 (3.33), µ = E[D] = 3.5, (3.59),, X ( ) X (standard deviation) ( ), X σ[x], σ[x] := V [X] (3.60) 3.7 D V [D] =.96, σ[d] = V [D] =.96 =.707,,,?,, X,,, (3.59), X µ,,,,, (X µ), X (V [X]), (σ[x]),,,,, E[ X µ ] (3.6)?,, E[ D µ ] = =.5

223 3.8 σ[d] =.707,,,, (3.6), (3.60),, (3.60) (3.6),,,,,,,,,, ( ),,,,, D 3.5,.7,,, 3.5 ±.7,.8 5. D,.8 5.,, 5,,, 6, 3,, 6 ( ), 5 (4 ),, 6, 5,, 3.8,,,,, X, 0 cm 30 cm 0.5 cm, () X 5.0 cm P (X = 5.0 cm)? () E[X]? (3) V [X]? (4) σ[x]?, ( ),,,, ( ),,,,,,,, 3.9, X, a, b ax +b (3.59) X ax + b, V [ax + b] = E[(aX + b E[aX + b]) ] = E[ ( ax + b (ae[x] + b) ) ] = E[(aX + b ae[x] b) ] = E[(aX ae[x]) ] = E[a (X E[X]) ] = a E[(X E[X]) ] = a V [X] (3.6) P.08 (3.36) (3.60) (3.6), σ[ax + b] = V [ax + b] = a V [X] = a V [X] = a σ[x] (3.63), (a), ( ), (b), 349,, X, Y

224 3 X + Y? V [X + Y ] = E[(X + Y E[X + Y ]) ] = E[ ( X + Y (E[X] + E[Y ]) ) ] = E[ ( (X E[X]) + (Y E[Y ]) ) ] = E[(X E[X]) + (Y E[Y ]) +(X E[X])(Y E[Y ])] = E[(X E[X]) ] + E[(Y E[Y ]) ] +E[(X E[X])(Y E[Y ])] = V [X] + V [Y ] + E[(X E[X])(Y E[Y ])] (3.64) P.08 (3.44), P.08 (3.36) (3.64) E[(X E[X])(Y E[Y ])], : E[(X E[X])(Y E[Y ])] = E[XY E[X]Y E[Y ]X + E[X]E[Y ]] = E[XY ] E[E[X]Y ] E[E[Y ]X] + E[E[X]E[Y ]] = E[XY ] E[X]E[Y ] E[Y ]E[X] + E[X]E[Y ] = E[XY ] E[X]E[Y ] (3.65), X Y, P.09 (3.48), E[XY ] = E[X]E[Y ], (3.65) 0, (3.64) : X, Y, V [X + Y ] = V [X] + V [Y ] (3.66) (3.66), (3.66), 3, X, X,, X n, V [X + X + + X n ] = V [X ] + V [X ] + + V [X n ] (3.67)? X + X X, X 3 Y (3.66), 3 (3.67), (3.67) 35 (3.6) X, : V [X] = p( p) (3.68) 35, : B(n, p) X V [X] = np( p) (3.69) : (3.9) (3.67) 78 ( ) [ ]?..., [ ] ( ), 79...,......,,,,,,, 3.0, (3.64), E[(X E[X])(Y E[Y ])], X Y (covariance), Cov(X, Y ) ( ): Cov(X, Y ) := E[(X E[X])(Y E[Y ])] (3.70) (3.65), E[XY ] E[X]E[Y ], X Y, E[XY ] = E[X]E[Y ] Cov(X, Y ) = 0 X Y, Cov(X, Y ) 0 : ρ XY := Cov(X, Y ) σ[x]σ[y ] (3.7), X, Y ( ) , 00,,, (X E[X])(Y E[Y ]),

225 3. 3 X Y,, X Y, X Y,, X Y,, X X 0 60,, X ( ) (),, X 0,,,..., 59 60, P (X = 30) (), 0., X 0, 0., 0.,..., , P (X = 30) (3), 0.0, P (X = 30)? (4) 0 n (n ), P (X = 30)? (5), P (X = 30)?,, 30,,,, (5), 0, 0,,, 0 0, 0 80, , 30, 30,, 30.0, , , a, ( ) a ( (3.7)) 8 0, 0...,, 83,!...,,,,,, )! (), (), (3), (4),, X,,,, (, ) (5),, 0 60, *8 3.,, 0 ( ),, 0,, 3.8, X 0, ( ), 0,, *8

226 4 3,, X, dx 0, x < X x+dx, a P (x < X x + dx) = a dx (3.7) a x, f(x), P (x < X x + dx) = f(x) dx (3.73) f(x) X (probability density function),,,, 354 X f(x) a < b, *9 : P (a < X b) = b a f(x) f(x) dx (3.74) f(x) 0 x a b 3., (3.74), P (a < X b) : (3.73) (3.74) P (), < (3.74), P (a X b), P (a X < b), P (a < X < b), <,,, 0 (, ),,,, f(x) f(x) dx = (3.75),, X, X 0 x a b 3.,, 3., (, ),, 3. a b, x ( ), a b ( (3.74) ),,, x, X, * , () X f(x), x () X f(x) (3), P (30 < X 30.) x X, x, P (X x) x ( ) *0 (3.75), (3.4), *9,

227 3. 5, X F (x), F (x) := P (X x) (3.76) ( ) 3.9 D x D F (x), : 0 x < /6 x < /6 x < 3 F (x) = 3/6 3 x < 4 4/6 4 x < 5 5/6 5 x < 6 6 x ( ) (3.77),,,, *,, X,, (3.74) a =, b = x, F (x) = x f(x), F(x) f(x) dx (3.78) F (x), (3.78) 356 X f(x) F (x), d F (x) = f(x) (3.79) dx (3.79),, ( ), * 357 ( )X F (x) x < x, : P (x < X x ) = F (x ) F (x ) (3.80) (3.80),,,, 358 ( )X F (x), 3 : () (x ) () (x ) 0 (3) x F (x) ( ) , 0 x a b f(x) ( ), F (x) ( ) f(x) 360?? :,,, *,,,, *,,

228 , OK,, 0,, X E[X], E[X] := xf(x) dx (3.8) f(x), X X,, 355, 0 x < 60 f(x) = /60, f(x) = 0, E[X] = ( ) xf(x) dx = 60 0 x [ x ] dx = 0 = 30 0 X, P.0 (3.59), V [X] := E[(X µ) ] (3.8) µ, X E[X], P.0 (3.60),,, σ[x] := V [X] (3.83) X, 3.0, X µ = E[X] = 30, V [X] = E[(X µ) ] = = 60 0 (x µ) f(x) dx (x 30) dx = , 300 = 7.3 ( ) 36 c, a X, c a c + a, * 3 () X, c a x c + a /(a), 0 () E[X] = c (3) V [X] = a /3 (4) σ[x] = a/ 3,, (3.37), (3.44), (3.48), (3.6), (3.63), (3.64), (3.66), (3.67), * 4 ),,, ( (3.70), (3.7)) 3.4,,, ( A B),, AB C, AC CB AC L A, CB L B AB,, L A + L B,,,,,,, AC X A, CB X B, ( m m * 5 ), X A L A, *3, P.9 3, 4 *4 (3.37),, x i, y j x, y, p i,j f(x, y) dxdy, f(x, y), f(x, y) dxdy = P (x < X x + dx y < Y y + dy) (dx, dy ), X Y *5,

229 3.4 7 X B L B : E[X A ] = L A (3.84) E[X B ] = L B (3.85), AB, X A + X B ( ),, X A + X B,, (3.44) E[X A + X B ] = E[X A ] + E[X B ] = L A + L B (3.86), AB,?,,,, * 6, X A (X A ) X B (X B ), X A +X B (X A +X B )? (3.66), V [X A + X B ] = V [X A ] + V [X B ] (3.87), (σ[x A + X B ]) = (σ[x A ]) + (σ[x B ]) (3.88), AB, AC CB, * 7! 3. X A ( ) 3 cm, X B ( ) 4 cm, X A + X B ( ) 3 cm + 4 cm = 7 cm, (3 cm) + (4 cm) = 5 cm ( ),, (3.84) (3.85),, (3.66), X A X B AC CB, * 8,, AB *6,, *7,,,, *8,, 3., AB 3 cm+4 cm=7 cm, AB n, X, X,, X n, : Y = X + X + + X n, (3.67), V [Y ] = V [X ] + V [X ] + + V [X n ] (3.89), (σ[y ]) = (σ[x ]) + (σ[x ]) + + (σ[x n ]) (3.90), * 9 : ( ), ( ),, 36,, ( ), ( ),,?? 363 km, m 000, mm, ( m ) mm,?,,,,, 3.3 V, r ( ), h, V = πr h V ( ), *9,

230 8 3?, y, y = f(x, x ), x, x ( r x, h x, V y, πr h f ) x X = µ + dx µ x ( ), X dx (dx ), x X = µ +dx µ x ( ), X dx (dx ), y f(µ, µ ), y Y f(x, X ) (Y ),, dx dx,, P.48 (9.40), Y = f(x, X ) = f(µ + dx, µ + dx ) = f(µ, µ ) + f x dx + f x dx (3.9), [ V [Y ] = V f(µ, µ ) + f dx + f ] dx x x (3.9) f(µ, µ ), (3.6), : [ f = V dx + f ] dx x x (3.93) f, x dx f x dx (, dx dx ),, (3.66), [ f ] [ f ] = V dx + V dx x x (3.94) (3.6), (3.94) : = ( f ) V ( f ) V [dx ] + [dx ] (3.95) x x, (3.6), V [X ] = V [µ + dx ] = V [dx ] (3.96) V [X ] = V [µ + dx ] = V [dx ] (3.97) (3.95) V [dx ] V [dx ] V [X ] V [X ],, (3.9),, V [Y ] = ( f ) V ( f ) V [X ] + [X ] (3.98) x x, σ y = ( f ) σ ( f ) σ x + x (3.99), σ[y ] σ y, σ[x ] σ, σ[x ] σ, (3.88), y (x, x ),, x, x,, x n ( ), : σ y = ( f ) σ ( f ) σ ( f ) σ x + x + + x n n (3.00) (3.00), (3.90), y = x + x + + x n, f/ x = f/ x = = f/ x n =, (3.00), (3.90), (3.00) 3.3,,,, f/ x, f/ x,, µ, µ, (, ),, ( ) P (9),,, :,, dx dx, (, ),,,,,

231 r h, r = 35. mm, h = 73.4 mm r σ r =.0 mm, h σ h = 0.3 mm, V σ V 37 (3.48), E[XY ] = E[X]E[Y ], X, Y (!), X, Y,?, 330 () () (3) (4) (5) (6) ( ) (7) (8), 33, n( ) 0, (3.4), P ( ) = n( )/N = , (6, 6)(6, 5)(6, 4)(5, 6)(5, 5)(4, 6) 6, 36, 6/36 = /6 333 ( ), 334 : P (A B) < P (A B) < P (B A) 335 (), 3,,, 3,,, * 0 (), 3,,,, (3) 4 A, B,,,, 4, 6, 4, 336 (/) 0 = / B(5, /) X, (3.3), P (X = 3) = 5 C 3 (/) 3 ( /) 5 3 = 5C 3 (/) 5 = 0/3 = 5/ , 5/6 34 /6, = E[X] = P (X = ) + 0 P (X = 0) = P (X = ) = p 345 k P (k), P (0) = /8, P () = 3/8, P () = 3/8, P (3) = /8, 0 P (0)+00 P ()+00 P ()+300 P (3) =00 /8 = k (0 ) X k (k n ), X = X + X + + X n, E[X] = E[X + X + + X n ] = E[X ] + E[X ] + + E[X n ] (3.0) ( (3.44) ), X, X,, X n, P.07 (3.34) ( : X, X,, X n!, (3.34) X E[X ] = E[X ] = = E[X n ] = p (3.0), (3.0) np 348 *0,,

232 0 3 () P (X = 0, Y = 0) = /4, P (X = 00, Y = 0) = 0, P (X = 0, Y = 00) = /4, P (X = 00, Y = 00) = /, E[X + Y ] = 0 (/4) + 00 (0 + /4) + 00 (/) = 5, E[X] = 50, E[Y ] = 75, E[X] + E[Y ] = = 5 (), P (X = 0, Y = 0) = P (X = 00, Y = 0) = P (X = 0, Y = 00) = P (X = 00, Y = 00) = /4, P (XY = 0000) = P (X = 00, Y = 00) = /4 P (XY = 0) = P (XY = 0000) = /4 = 4.3, E[XY ] = 0 (3/4) (/4) = 500, E[X] = E[Y ] = 50, E[X]E[Y ] = () ( ) x, x,, x n M, x k m k, x k, (3.6), m k /M, P (X = x k ) = m k /M /,, k, P (X = x k ), () (3.3), (3.3), x k, (3) µ (3.59), n V [X] = E[(X µ) ] = (x k µ) P (X = x k ) k= (µ = E[X]), (), (x k µ),,,, (4), /,,,, ,, (0 50 ) P (0) + (00 50 ) P () + (00 50 ) P ()+( ) P (3) = 7500,, : 00, (0 50 ) (/)+(00 50 ) (/) = ,, 3, = 7500 :, 35 (3.34) E[X] = p µ, V [X] = E[(X µ) ] = E[(X p) ] = ( p) p + (0 p) ( p) = p( p) + p ( p) = p( p)( p + p) = p( p) 35 (3.9), V [X] = V [X + X + + X n ] (3.03), X, X,, X n, (3.67), V [X] = V [X ] + V [X ] + + V [X n ] (3.04), X, X,, X n, (3.68), V [X ] = V [X ] = = V [X n ] = p( p) (3.05) (3.05) (3.04), V [X] = p( p) + p( p) + + p( p) = np( p), 353 () X 60, 30,, /60 () X , 30,, /600 (3), /6000 (4), /(60 0 n ) (5),, (4) n, /(60 0 n ) 0, a b, n, a = x 0 < x < x < < x n = b, 0 k < n, x k = x k+ x k (3.73), P (x k < X x k+ ) f(x k ) x k (3.06)

233 3.4, n P (a < X b) = P (x k < X x k+ ) k=0 n f(x k ) x k (3.07) k=0 x k 0, n,, 355 ( ) (), x, dx, x x + dx, (3.73), dx x, f(x) x () f(x) = /60...! (3) P (30 < x 30.) = f(x)dx = (/60) ( ) = / (P.66 (5.5)), d F (x + x) F (x) F (x) = lim dx x 0 x (3.78),, lim x 0 x+ x f(x)dx x f(x)dx x = lim x 0 x+ x f(x)dx x x, x, x x + x f(x),, 6, = lim x 0 f(x) x+ x dx x f(x) x = lim = f(x) x x 0 x 357 X x X x, X x x < X x,, (3.0), P (X x ) = P (X x )+P (x < X x ),, P (X x ) = F (x ), P (X x ) = F (x ), F (x ) = F (x ) + P (x < X x ) x ( ), X x X,, X, P (X x) () F (x) = P (X x), X x X, x x ( ), X,, P (X x) 0 (3), x < x P (x < X x ) = F (x ) F (x ) (3.6) 0, 0, 0 F (x ) F (x ), F (x ) F (x ), x F (x) 359 : x 0 0, 0 < x 60 x/60, 60 < x 360 X, dx, P (x < X x + dx) = f(x)dx f(x) X, P (X x) x, X 36 () X c a x c + a, X f(x), c a x c + a, A, c a x c + a X, c a x c + a f(x) = 0, f(x)dx = c a f(x)dx + + f(x)dx = c+a c+a c a c+a c a f(x)dx A dx = Aa (3.75), Aa =, A = /(a) () E[X] = (3) V [X] = c+a c a c+a c a x dx [ x ] c+a a = 4a = c c a (x c) dx a (4) σ[x] = V [X] = a 3 [ (x c) 3 = 6a ] c+a c a = a () F (x) = P (X x), X x X, x 36, ( ) ( )

234 3,,,,, 363 X, X,, X 000 Y = X + X + + X 000 mm, V [X ] = V [X ] = = V [X 000 ] = ( mm),, V [Y ] = V [X ] + V [X ] + + V [X 000 ] = 000 ( mm) = 000 mm, σ[y ] = V [Y ] = 000 mm 3 mm,, 3 cm 364 (3.00) n =, x = r, x = h, σ = σ r, σ = σ h, σ y = σ V, σ V = ( f ) σ r r + ( f ) σ h h f = πr h, f r, = πrh, f h = πr σv = (πrh) σr + (πr ) σh, σ V = (πrh) σr + (πr ) σh, V = mm 3, σ V = mm 3 σ V, σ V = 6000 mm 3 V ( 6; ), 0 (7 ), V = mm 3, V = (.86 ± 0.6) 0 5 mm 3 :,,,,!,,,,,, ( ),,,,,?,,,,,,,,,,,,?,,,,,,,,,,,,,,,,

235 3 4 ( ) 4., (population),,,, ( ) (sample), ( ) ( ) (, ), 00, 00,, 00,, 3, 00 3 ( 300 ), 3 (300 ) ( ) :! (= = ) (= )! 4., 00,, 00, 00,, ( ) 365? 366?? 367?,,,,,,,,,,,,,,,

236 4 4,, 3 /6 ( ),.74 m,,,?,,,, ? P.07 (3.3),,,,,,,,, {X, X,, X n } (n ), X := X + X + + X n n (4.),,,, 4. 8,, 3, 5, 3, 6,, 4, 5, X = ( ) = 3.65?,,,,,,,, (n) (, ),,,,,,,,,, n,, :, X, x, x,, x N, N 4. N = 6, x =, x =,, x 6 = 6 n * {X, X,, X n },, x k m k (k N ),, n, m + m + + m N = n (4.), n, 8, m =, m =, m 3 =, m 4 =, m 5 =, m 6 =, m + m + + m 6 = 8,,,, X + X + + X n = x m + x m + + x N m N (4.3), , , 9 * n N!

237 4.3 5, (4.) (4.3), X = x m + x m + + x N m N n m = x n + x m n + + x m N N n (4.4) (4.5), m k /n, n, x k p k ( ), (4.5), x p + x p + + x N p N (4.6),, n, E[X] X,, n,, (, ) 369,, :,, (3.67), V [X + X + + X n ] = V [X ] + V [X ] + + V [X n ] = σ + σ + + σ = nσ (4.) (3.6), [ X + X + + X ] n V [X] = V n = n V [X + X + + X n ] = n nσ = σ n (4.),, σ[x] (standard error) ( ), (4.), σ[x] = σ n (4.3) 4.3,, X,, X,, :,, n X, X,, X n (4.7), (4.7), µ σ, E[X ] = E[X ] = = E[X n ] = µ (4.8) V [X ] = V [X ] = = V [X n ] = σ (4.9), (4.), P.08 (3.36), P.08 (3.44), [ X + X + + X ] n E[X]= E n = E[X + X + + X n ] n = E[X ] + E[X ] + + E[X n ] n = µ + µ + + µ = nµ n n = µ (4.0) n, (4.3) 0,, ( ) / n,, ( ),, /0, 00, :, X, n σ, ( ), σ/ n 69,...,,,

238 ,,, ( ),? P.0 (3.59),,,,, (3.59)? {X, X,, X n } s := n n (X k X) (4.4) k= ( ) ( ) s ( ) *, P.0 (3.59),,, ( ),,, (4.4), u, : u := n n (X k X) (4.5) k= u ( ) n n, 370, 0, x, : 3, 4, 3, 5, 6, 4, 5, 6, 4, 7 (4.6), : () X 4.7 () s.6 (3) s.7 *, ( ),,, A, (4.6) : A B C A, =sum(a:a0)/0, A sum(),,, X k, (X k X), B, =(A-A$)*(A-A$) A,, B, B B0 B, n, s, B, =sum(b:b0)/0 s, B, =sqrt(b), : A B C () () A, =average(a:a0) (3) A3, =stdevp(a:a0),,

239 4.5 7 ( ) 4.5 X, µ, σ, X µ σ (4.7) ( ) ( ) 37 0, ( ),, µ, σ, X, s( u) 373,, 74 cm,.5 cm 7 cm, P.,,,,, (4.7), (4.7), (4.7),,,,,,,,,,,,, 0 50 (!): := X µ σ 374 () 373? (4.8) (), :,!,, 4.6, (normal distribution) (Gaussian distribution) : f(x) := { exp (x } µ) πσ σ (4.9) µ, σ (4.9)! * 3,,,,, : σ, f(x) = (x } µ) exp { π σ σ (4.) (4.9) µ, σ, N(µ, σ ) (4.9), x = µ (f(x) ) ( 4.), P.89 (6.30), a = /(σ ), (6.30) (P ) x µ, y / πσ, (4.9) *3,,, exp, (4.7) : x µ σ (4.0) ( ) ( (4.7) X x, X, x...!) P , exp,, (4.0),

240 8 4 f(x) 4. O µ-σ µ µ+σ N(µ, σ ) (4.9) 375 (4.9) 0, 376, (4.9) (3.75), { exp (x } µ) πσ σ dx = (4.) () t = (x µ)/( σ), (8.83), (4.) : π e t dt (4.3) x 84 N(0, )? N(0, 4)...,, σ, ( σ ),, 0,, N(0, ), (standard normal distribution) ? N(0, ) f(z), : f(z) = π exp ( z ) (4.4) :, x,, z f(z) () (8.88), (4.3) - - O z 377, () N(0, ) () N(0, ) (3) N(, ) :, x,,, 4. N(0, ) (4.4) 380 (4.4) : 38 N(µ, σ ) X Z, * (), σ = = 4,... σ =, σ = () *4, Z, z,

241 Z, (P.7) : P.4 (3.74) () P ( Z ) (4.5) () P (.64 Z.64) 0.90 (4.6) (3) P (.96 Z.96) 0.95 (4.7),,, (4.5) (4.7), 4.3 : () P (Z.64) () P (Z.96) (3) P (0 Z.96) 38, 383, 4.3,,,,, P(-<Z<)=0.68 f(z) 0.4 P(-.64<Z<.64)=0.90 f(z) 0.4 P(-.96<Z<.96)=0.95 f(z) *5, O - z O z O z, P(< Z )=0.3 f(z) 0.4 P(.64< Z )=0.0 f(z) 0.4 P(.96< Z )=0.05 f(z) 0.4 O z O z O z P(<Z)=0.6 P(.64<Z)=0.05 P(.96<Z)=0.05 f(z) f(z) f(z) O z O z O z N(0, ), 4.3,,, Z ( 0.3), Z ( ; 0.68), ( )! 7...,,,, ,?, X X 0, µ = 0.5, σ = 0.5 n, n {X, X,, X n }, X n, X µ = 0.5, X σ/ n = 0.5/ n ( ) X, N(0.5, 0.5 /n) ( ), n X 383 Z 4.3, : *5,,

242 30 4,, X / n E[X],, X ? 0000, 500? : 4.3 n = 0000, P (0.5 X) 4.8, X, µ, n, X µ ( ), :,, n, X, µ, σ/ n (, σ ), X, Z : Z = X µ σ/ n (4.8) 38, Z, (4.6), P (.64 Z.64) 0.90 (4.9), ( P.64 X µ ) σ/ n (4.30) (4.30) (), P (X.64 σ µ X +.64 σ ) 0.90 n n (4.3), µ, 90, X.64 σ n, X +.64 σ n (4.3),, ( µ), ( 90 ),, ( 90 ), ( (4.3)) : µ 90,,? µ, µ,,, (n ), (4.3), µ,, ( 0000 ), 90 ( 9000 ), µ,,? / n, (P.5 (4.3)), (P. (3.66))!, (4.9), (4.7), 95 X.96 σ, X +.96 σ (4.33) n n,,,, 386,,,?,, /a, a,

243 4.8 3 /0, 00! : (4.3) (4.33), σ, (,, ), 387, σ s u (s u ),, X, t,,,, 387 t, , ( ), 400,,.30 cm, 0.0 cm ( ), 90 95,, 85?...,,,,,,,,! 86?! , 366 ( ) 367 ( ),, 368, 369 X, X,, X,, 370 () () (3) P.08 (3.36) (a = /σ, b = µ/σ ), [ X µ ] E σ [ X = E σ µ ] = E[X] µ σ σ σ = µ σ µ σ = 0, P. (3.6) (a = /σ, b = µ/σ ), [ X µ ] V σ [ X = V σ µ ] = V [X] σ σ = σ σ = 373 (7 cm 74 cm)/(.5 cm) =. 374 () (.) = 38 () 375! 376 () t = (x µ)/( σ), x dx = σdt, t = (x µ) /(σ ) (4.) *6 (), π e t dt = π π = 378 0,, N(0, ) 379 (4.9), µ = 0, σ = (, x z ) 380,, *6, x t, x t,

244 3 4 µ=0, σ= µ=0, σ= µ=, σ= f(x) 0.4 Z N(0, ) O X, Z = X µ σ, P (x < X x + dx) = x (4.34) { exp (x } µ) πσ σ dx (4.35), X = x Z = z, X = x + dx Z = z + dz, (4.34), z = x µ σ z + dz = x + dx µ σ, = x µ σ + dx σ = z + dx σ (4.36) dz = dx/σ (4.37), x X < x + dx z Z < z + dz, P (x X < x + dx) = P (z Z < z + dz) (4.35), (4.38), (4.36), (4.38) { P (z Z < z + dz) = exp (x } µ) πσ σ dx = ( exp πσ z ) dx (4.39) dx (4.37) σdz, P (z Z < z + dz) = = π exp ( z ( exp πσ z ) σdz ) dz (4.40) () P (Z.64), , 0.05 = 0.95 () P (Z.96), , 0.05 = (3) 4.3,, , = 0.475, 4.3 ( 0.05 = 0.95) X,, 500/0000 = 0.5, X,, 0.5, 0.5/ 0000 = X = 0.5, Z = =.0 P (.0 Z) (Z ) 38,, P (.0 Z) 0.0??, 0.0?? (? ) 386 n, / n ( (4.3) (4.33) ), /, / n /, n, n X =.30 cm, σ 0. cm, n = 400 (4.3), (4.33) σ/ n = 0. cm/ 400 = 0.0 cm,.64σ/ n = cm,.96σ/ n = cm, µ, 90,.4 cm,.46 cm 95,.0 cm,.50 cm (, 4, 5 ) 7,.4,.46, dz, N(0, )

245 33 5, ( ),,,,, i, j, k, l, m, n, N 5.,,,, i, i, j, j i j, (i, j),, A (i, j), [A] ij ( ij i j i j ), A = [ ] 3 4 (5.), [A] =, [A] =, [A] = 3, [A] = 4,, A, B, A B, AB, A B, A i a i, B j b j, a b a b a b l a b a b a b l AB = (5.) a n b a n b a n b l, [B] j [B] j [AB] ij := a i b j = ([A] i, [A] i,, [A] im ). [B] mj = [A] i [B] j + [A] i [B] j + + [A] im [B] mj (5.3) = m [A] ik [B] kj (5.4) k=, m, n, l, n A, m A B, l B i, k, j m, n, l 5., (Σ), (Σ ) (5.4), [AB] ij := k [A] ik [B] kj, k A ( B ),,,, OK, a ij,, AB (i, j) [AB] ij 3 i= j= a ij 3 j= i= a ij

246 34 5? 3 a ij = a + a + a + a + a 3 + a 3 i= j= 3 a ij = a + a + a 3 + a + a + a 3 j= i=,,, i, j,, 3 A, B, C A =B, B =C, (AB)C = A(BC) (5.5) ( ) P.73 (.3) : i, j (5.4), 5.3, A, B, AB = BA = I (5.9) (I ), B A,, B = A, AA = A A = I (5.0) 390 n A, B, : (AB) = B A (5.), (5.) [(AB)C] ij = l = l = l = k = k [AB] il [C] lj ( [A] ik [B] kl )[C] lj k [A] ik [B] kl [C] lj k [A] ik [B] kl [C] lj l [A] ik [B] kl [C] lj l 5.4 A ( ), A, A, t A A T *, ( ) i, j, [ t A] ij := [A] ji (5.) = [A] ik [BC] kj k = [A(BC)] ij, (AB)C = A(BC) 5. A = , t A = (5.3) (5.5), A, B, C a ( A B, C ), : A(B + C) = AB + AC (5.6) (A + B)C = AC + BC (5.7) (aa)b = a(ab) = A(aB) (5.8) ( (.4) (.5)) : ! ( ),,, A, : t ( t A) = A (5.4) : ( ) i, j, (5.), [ t ( t A)] ij = [ t A] ji = [A] ij,, [ t ( t A)] = A, A, B, t (A + B) = t A + t B (5.5) * t T, transpose ( )

247 : ( ) i, j, (5.), [ t (A + B)] ij = [A + B] ji, [A+B] ji = [A] ji +[B] ji ( ), [ t (A + B)] ij = [A] ji + [B] ji = [ t A] ij + [ t B] ij = [ t A + t B] ij,,, : 5. x =, t x = (,, ) (5.6) :, (,, 3), ( 3),, () [] ( ), (5.) : A =B, i, j, (5.),, [ t (AB)] ij = [AB] ji (5.3) 5.3, (5.4), a = (4, 5, ) (5.7) a, t a, 4 t a = 5 (5.8) ( A B ), t (AB) = t B t A (5.),, : n n A, n ( ) x, t (Ax) = t x t A (5.) (5.), (5.), : 393 (5.3) A (5.6) x, () Ax, t (Ax) () t x t A, t x t A (3), t (Ax) = t x t A [AB] ji = k [A] jk [B] ki (5.4) (5.3), (5.4), [ t (AB)] ij = k [A] jk [B] ki (5.5) ( ), (5.), 39 a = 3 (3 ): a b, x = c x y z, : (5.9) () t ax () t xa (3) a t x (4) x t a 39, n a, b, : t ab = a b (5.0), : A, B, AB [A] jk = [ t A] kj, [B] ki = [ t B] ik, (5.5), [ t (AB)] ij = [ t A] kj [ t B] ik (5.6) k ( ), = [ t B] ik [ t A] kj = [ t B t A] ij (5.7) k, t (AB) (i, j), t B t A (i, j), t (AB) = t B t A (5.) (5.),

248 (5.), 5.5,, n A tr(a), tr(a) := n [A] ii (5.8) i=, A = 4 5 (5.9) 7 6 3, tr(a) = = 6,,,, 395 n A, B, : () tr(a + B) = tr(a) + tr(b) (5.30) () tr(ab) = tr(ba) (5.3) (3) tr(a) = tr( t A) (5.3) () : (5.4) 73 tr(ab) = tr(a)tr(b)... ( ) A B, AB, tr(ab) =, tr(a)tr(b) = = 4, 5.6 A, t A = A, A (symmetric matrix) ( ), (5.33) ( ) (5.34), 396 A ( ) : () t AA () A t A : (5.) t ( t A) = A 397 A, A + t A (5.35) 5.7,, 5.4, k, X k Y k,, N N, : { } (X, Y ), (X, Y ),, (X N, Y N ) (5.36), X, Y, s X, s Y, X := N s X := N s Y := N N X k, k= Y := N N k= Y k N (X k X) (5.37) k= N (Y k Y ) (5.38) k=

249 : s XY := N N (X k X)(Y k Y ) (5.39) k=,, [ s S := X s XY s XY s Y ] (5.40) S,, S (, ) (, ) s XY t S = S, S ( ),,,, r := s XY s X s Y (5.4) r,, r, r 74 (5.4) s Xs Y... s XY X Y s XY s X X ( ) X 75 (5.40), [ sxy s X S := s Y s XY ] (!) (5.4)......, N : X X X X D c =. X N X Y Y Y Y. Y N Y (5.43) D c,,, ( ) (mean centering) (D c c centering ) (5.43),, 398, D c S : S = t D c D c (5.44) N (5.44), 399 (5.43) D c, a, b r : r = a b a b (5.45) 87 (0.7)?... (0.7) 3, (5.45),,,, 400 (, 5 ), 7 ( N = 7) (k = k = 7 ), : k (X k ) (Y k )

250 38 5 () X k Y k, X, Y () 398, D c, (, 3 4 ): D c = (5.46) (3) (5.44) S (4) s X, s Y, s XY (5) r, 3,, Z, X X Y Y Z Z X X Y Y Z Z D c =... X N X Y N Y Z N Z, S, (5.47) S = t D c D c = s X s XY s XZ s XY s Y s Y Z (5.48) N s XZ s Y Z s Z 3,, Q (5.49), (5.49) (5.50),, (5.50) Q, (5.49) (5.50) 40, n, θ () n I () [ ] 0 Q = 0 (5.5) (3) [ ] cos θ sin θ Q = sin θ cos θ (5.5) cos θ sin θ 0 (4) Q 3 = sin θ cos θ 0 (5.53) 0 0 cos θ 0 sin θ (5) Q 4 = 0 0 (5.54) sin θ 0 cos θ 40 Q Q,, : t QQ = I, 5., 403 n n,, n P ( : t P P, ) 5.8 Q, t Q = Q (5.49), Q (orthogonal matrix) ( ) (5.49) Q (I ): t QQ = I (5.50) 5.9! 404 n n A λ, λ A,, λ λ x λ, x λ x x :, ψ := t x (A x ) (5.55)

251 5.9! 39 () ψ = λ t x x () ψ = t x ( t Ax ) : (3) ψ = ( t x t A)x : ( (5.5)) (4) ψ = t (Ax )x : (5.) (5) ψ = λ t x x (6) λ t x x = λ t x x : (),(5) (7) (λ λ ) t x x = 0 (8) t x x = 0 : λ λ (9) x x 404 :,! : : A, λ = λ, p, p,,, p a, p b ( ), λ = λ,, ( ) 405, () [ ] () : 3 A,, det(a λi) = 0,,,,,, A P, P AP, P A, A,, Q, 403, Q A,,,! n A, Q, λ λ t QAQ = λ n λ, λ,, λ n A A ( ) Q 406,, () 407 A = [ ] () A = n A, P, : λ λ P AP = λ n

252 40 5, tr(a) = λ + λ + + λ n (5.56) : (5.3),, 5.0 (principal component analysis: PCA),, (5.40), (5.48), n ( n ), S n, n Q, : λ λ t QSQ = λ n (5.57), λ, λ,, λ n, S,, λ λ λ n λ, λ,, λ n,, q, q,, q n, Q S, Q i n q i,,, i,, i ( ), (loading vector), i, j n, i j q i q j = 0 (5.58), k,, d k, d k := (X k X, Y k Y, ) (5.59), k , d k,, d k = c q + c q + + c n q n (5.60) (,, P.65 ) q i, d k q i = c q q i + c q q i + + c n q n q i (5.6),,, q i q i 0, d k q i = c i q i q i (5.6),, q i =, q i q i =, d k q i = c i (5.63), (5.60) c i, d k i d k i, i,, 3 ( ) q, 3, i,,, i

253 5.0 4, PC, PC 5. d k ( (5.60) c, c ), d k (q, q ), d k ( ) (, d k q = d k q cos θ θ, q =, ), ( 5.) ,,, (, ) ,, :,,, : ( ),,,,,, : PC, PC,...,, S, (5.44) (5.48), D c, = ( 3 ) (5.44) (5.57) t QSQ : ( ) t Q t D c D c Q = N [ ] λ 0 0 λ, : (5.64) ( ) t Q t D c D c Q = N N ( t Q t D c )(D c Q) (5.65) = t (D c Q)(D c Q) (5.66) N (5.65) (5.66) (5.), D c Q D c, D c := D c Q (5.67), (5.66) : t D N cd c (5.68), (5.44), D c D c, D c Q, D c Q (5.), D c Q D c, Q ( S ), d q d q D c d q d q = D c Q =.. d N q d N q (5.69), d i D c i ( i ), q j Q j ( j ), d i q j, i j, D c, D c,, D c

254 4 5 D c ( ), D c, 408 Q, D c, D c = D c Q : D c = D c Q = (5.70) , : (5.64), (5.68), [ ] t D N cd c λ 0 = 0 λ (5.7),,, 0 (, (5.40) ) 0, (5.4), 0,,,, (,,, 0,, 5., D c ( ), 5.3, D c (, ),,,,,,, 5., 5.3,,, Q 5. English - - O math (5.46) ( ) k O 7 PC score PC score (5.70), 5.,,, (5.7), 407, ( ),,, i,, i i ,, 6

255 5. 43,,,,, 88?...,,,,,, 5.,,,,, 3,,,,, A, : 0, : 0, : 5, (0, 0, 5 ), A,,?, 3,,,, ( 5.4 ),, 5.4,,,, 4?, ( ), 3 ( ),, ( 5.4 ), 5.4 A ( ), 5.4,,? 44,? 89..., 389 [A(B + C)] ij = l [A] il[b + C] lj = l [A] il([b] lj + [C] lj ) = l ([A] il[b] lj + [A] il [C] lj ) = l [A] il[b] lj + l [A] il[c] lj = [AB] ij + [AC] ij = [AB + AC] ij A(B + C) = AB + AC (A + B)C = AC + BC [(aa)b] ij = l [aa] il[b] lj = l a[a] il[b] lj = a l [A] il[b] lj = a[ab] lj (aa)b = a(ab) [A(aB)] ij = l [A] il[ab] lj = l [A] ila[b] lj = a l [A] il[b] lj = a[ab] lj A(aB) = a(ab)

256 ( ) AB B A, (AB)(B A ) = A{B(B A )} (5.7) (5.5) ( (5.5) C B A ) {} (5.5) ( (5.5) (5.5) A B, (5.5) B B, C A ), (5.7), A{B(B A )} = A{(BB )A } (5.73), BB (5.73), = I, A{(BB )A } = A(IA ) = AA = I (5.74) (5.7), (5.73), (5.74), (AB)(B A ) = I (5.75), AB B A (......), (B A )(AB) = I (5.76) (5.75), (5.76),, B A (AB) (5.) 39 () ax + by + cz () ax + by + cz (3) 395 ax ay az bx by bz cx cy cz () tr(a + B) = i (4) [A + B] ii = i ax bx cx ay by cy az bz cz ([A] ii + [B] ii ) = [A] ii + [B] ii = tr(a) + tr(b) i i () tr(ab) = [AB] ii = [A] ik [B] ki i i k = [B] ki [A] ik = [B] ki [A] ik i k k i = k (3) tr( t A) = i [BA] kk = tr(ba) [ t A] ii = i [A] ii = tr(a) 396 () (5.), t ( t AA) = t A t ( t A) (5.77), t ( t A) = A, (5.77), t AA, t ( t AA) = t AA, t AA () 398 t D cd c = X X Y Y [ ] X X X X X N X X X Y Y Y Y Y Y Y N Y.. X N X Y N Y [ ] (X X) + (X X)(Y Y ) + = (X X)(Y Y ) + (Y Y ) + [ N = k= (X k X) N k= (X ] k X)(Y k Y ) N k= (X k X)(Y k Y ) N k= (Y k Y ) (5.78) N, (5.37), (5.38), (5.39), [ s X s XY s XY s Y ] = S (5.79) 400,, 3 (!) () X = 3.4, Y = () (3) S = t D c D c = 7 (4), s X =.55, [ ] s Y =.38, s XY =. (5) s XY /(s X s Y ) =. / = 0.83,, R : mth <-c(4,3,,3,4,5,) ( mth ) eng <-c(3,4,,3,5,5,) ( eng )

257 5. 45 mean(mth) ( ) mean(eng) ( ) Dc=cbind(mth-mean(mth),eng-mean(eng)) ( D c ) Dc ( D c ) S=t(Dc)%*%Dc/7.0 ( (5.44) ) S ( S ) S[,]/(sqrt(S[,]*S[,])) ( (5.4) ) cor(mth,eng) (, R ) 40 ( (5.50) ) 40 q q q n q q q n Q = q n q n q nn i, q i, q i q i q i =. q ni, q q q n q q q n t q q q n q q q n Q Q = q n q n q nn q n q n q nn q q q q q q n q q q q q q n = (5.80) q n q q n q q n q n (i, j) t Q i Q j q j,, t Q i Q i q i, (i, j) q i q j, Q,, 0 0 t 0 0 Q Q = I = (5.8) (5.80) (5.8), q i q i = q i = q i =, Q,, q i q j = 0 (i j ), Q 403, n n, p, p,, p n n P, P = p p p n (5.8), t P t P = t p t p. t p n (5.83) ( t p n ), t P P = t p t p. t p n p p p n p p p p p p n p p p p p p n = p n p p n p p n p n (5.84), i p i p i = (!), i, j (i j ) p i p j = 0 (!), (5.84),, 0,, t P P = I

258 46 5, P 406, : [ ] [ ] / 5 / 5 () Q = / 5 /, 3 0 QAQ = 5 0 /3 /3 / () Q = /3 /3 /3, t QAQ = /3 /3 / :, ± OK () Q = OK [ ] / 5 / 5 / 5 / , (.4,.4) ( (5.46) 3 ) 4 40, 3 (.4,.4) d (5.85) q, q = d q = d q = 0.4, =.54, 4,,,...? 407 (5.3), tr(p AP ) = tr(p (AP )) = tr((ap )P ) = tr(a(p P )) = tr(ai) = tr(a), tr(a), P AP P AP λ, λ,, λ n, tr(a) = λ + λ + + λ n 408 ( ),, 4 ( ), ( ), λ =.69..., λ = 0.4, [ ] 0.73 q =, q 0.68 = [ ] (5.85), [ ] Q =, (5.86) [ ] t QSQ = (5.87) :, ( ),, t QSQ, 43 λ =.69, λ = 0.4, tr(s) =.94, =.69/.94=0.9 =0.4/.94=0.08,, ( ), ( ), ( ) cos θ cos θ..., (5.85) q, q

259 47, 07 arccos, 07 arcsin, 07 arctan, 07 Beer, 94 cf., 98 cos, 0 cosec, 0 det, 75 dimension check, 9 e.g., 98 exp, 85 i.e., 98 Lambert, 94 ln, 0 log, 0 s.t., 98 sec, 0 SI, sin, 0 SI, tan, 0 t, 3, 07, 07, 07, 45, 5, 75, 66, 8, 3, 39, 6, 88, 63, 85, 7,, 44, 34, 49,, 70, 5, 83, 63, 0, 6, 06, 4, 04, 4, 4, 66, 4, 80, 8, 90, 50, 04, 36, 49, 60, 89, 5, 37, 7, 44, 55, 57, 07, 9, 95, 39, 94,, 37, 43, 45, 50, 66, 09, 8, 8, 8, 38, 38, 43, 34, 79, 4, 88, 59, 77, 34, 07, 7, 55, 7, 75, 8, 7, 94, 0, 97, 30

260 48, 3, 57, 0, 78, 4, 93, 4,, 76, 4, 39, 38, 39,,, 36, 99, 79, 79, 79, 03, 37, 0, 58, 7, 8, 8, 0, 0, 3, 3, 3, 99, 95, 7, 7, 85, 7,, 0, 0, 4, 93, 40, 40, 34, 39, 0, 87, 30, 30, 0, 55, 0, 35, 7, 63,, 43, 05, 37, 37, 50, 49, 9, 7, 38, 34, 87, 04, 4, 0, 4, 63, 30, 03, 4, 7, 70, 55, 64, 39, 4,, 57, 7, 7, 0, 06, 05, 64, 58,, 0, 77, 34, 55, 7, 7, 94, 77, 5,, 7,, 3, 5, 8, 75, 66, 40, 4, 8, 34, 45, 40, 50, 95, 48, 4, 33,, 37, 50, 46, 79, 8, 8, 8, 74, 88, 5, 36, 0, 90, 50, 35, 50, 37, 9, 0, 74,, 56

261 49, 35, 0, 40, 40, 38, 5, 37, 59, 63, 6, 37, 9, 96, 38, 0, 87, 3, 59, 49, 5,, 88, 3, 56, 34, 34, 4, 09, 8, 8, 0, 39, 6, 39, 79, 63, 36, 68, 45, 9, 03, 04, 05, 49, 95, 57, 36, 3, 35, 36, 07, 9, 85, 0, 9, 40, 93, 94, 50, 38, 90, 04, 49, 74, 9, 9, 87, 89, 7, 5, 0, 3, 37, 3, 3, 37, 3, 3, 6, 6, 50, 0, 66, 67, 9, 99, 43, 38, 43, 44, 8, 78,, 56, 78, 95, 6, 9, 94, 6, 5, 0, 37, 4, 07, 4,, 5, 76, 36, 79, 45, 7, 34, 46, 46, 04, 7, 50, 9, 0, 66, 60, 06, 06, 9, 59, 50, 95, 3, 3, 3, 3, 4, 96, 94, 8, 5, 76, 67, 4,, 94,

262 50,, 87, 50, 95, 67, 3,, 77, 93, 0, 07, 0, 99, 94, 3, 9, 03, 7, 4, 83, 7, 74, 7, 55, 59, 3, 78, 35, 40, 94,, 6, 95