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1 3 09/0/8

2

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

8 i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,, ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,,

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