8

i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,,

ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,, ( ),,, OK, 3, nasahara.kenlo.gw u.tsukuba.ac.jp, http://www.agbi.tsukuba.ac.jp/ shigen remedial/,, 0 (I, II) TA,, LaTeX (Version 3.4596), emath (Version f0507c), GNUPLOT (Version 4.6), LibreOffice (Version 4..8.), Ubuntu Linux 4.04LTS 8 3 3

iii i..............................................................................................3...............................................4..............................................5....................................................6.............................................. 3.7................................................. 4.8................................................ 4.9.............................................. 6.0......................................... 7............................................ 8................................................ 9.3................................................... 9.4................................................ 0.5...............................................6................................................ 3.7............................................. 4.8.............................................. 6 9.,......................................... 9.................................................... 0.3......................................... 0.4!.......................................5.................................................6 SI.................................................7............................................... 5.8................................................ 6.9........................................ 7.0 dimension check............................................ 9.!.................................................. 9 3 33 3................................................. 33

iv 3.................................................. 34 3.3................................................... 34 3.4................................................ 34 3.5................................................. 35 3.6................................................ 36 3.7................................................ 36 3.8............................................... 37 3.9................................................. 38 3.0................................................... 39 3........................................... 39 3.................................... 40 3.3................................................ 4 3.4.............................................. 4 3.5.............................................. 43 3.6........................................ 44 4 49 4............................................... 49 4..................................... 50 4.3........................................ 5 4.4........................................... 53 4.5..................................... 53 4.6,...................................... 54 4.7..................................... 55 4.8....................................... 56 4.9............................................. 57 4.0................................................ 58 4.................................................. 59 4.................................................. 60 4.3........................................ 6 5 65 5................................................ 65 5................................................. 69 5.3............................................... 69 5.4................................................ 74 5.5............................................... 75 5.6.......................................... 75 5.7.............................................. 77 5.8............................................. 79 5.9......................................... 80 5.0......................................... 8 6 85 6................................................. 85

v 6.................................................... 87 6.3................................................ 89 6.4............................................... 90 6.5............................................... 90 6.6......................................... 9 6.7 ( )...................................... 93 6.8............................................. 94 6.9.......................................... 94 7 99 7............................................... 99 7.................................................. 99 7.3 :................................... 0 7.4................................................ 0 7.5............................................. 0 7.6................................................ 03 7.7........................................... 04 7.8........................................... 05 7.9.......................................... 06 7.0............................................... 07 7.............................................. 08 7.................................................. 09 7.3................................................. 0 7.4............................................. 0 7.5.......................................... 8 5 8................................................ 5 8............................................... 6 8.3................................................ 7 8.4............................................... 8 8.5.......................................... 8.6.............................................. 5 8.7................................................ 6 8.8................................................ 6 8.9.......................................... 8 8.0........................................... 9 9 33 9................................................. 33 9................................................. 34 9.3.............................................. 35 9.4............................................... 36 9.5......................................... 38 0 43

vi 0............................................... 43 0.................................................. 45 0.3............................................. 46 0.4................................................ 46 0.5............................................. 47 0.6................................................. 47 0.7................................................. 48 0.8................................................. 49 0.9............................................. 5 0.0............................................. 5 57.......................................... 57............................................ 58.3................................................... 58.4.................................... 60.5.................................... 6.6................................................... 6.7........................................... 63.8............................................. 65.9.......................................... 66 7................................................. 7................................................ 7.3................................................. 7.4................................................ 73.5.............................................. 73.6................................................. 75.7............................................. 76.8......................................... 77.9................................................. 78.0 :....................................... 79 3 83 3................................................ 83 3................................................ 83 3.3......................................... 84 3.4.............................................. 84 3.5............................................... 85 3.6.......................................... 87 3.7................................................. 88 3.8................................................... 88 3.9............................................... 9 3.0................................................ 9

vii 4 97 4.................................................... 97 4.................................................... 98 4.3................................................... 00 4.4................................................ 0 4.5................................................ 0 4.6................................................. 0 4.7....................................... 05 4.8.................................... 07 4.9............................................... 07 4.0........................................... 08 4.................................... 08 4...................................... 09 4.3.................... 4.4............................................. 5 9 5............................................... 9 5................................................. 0 5.3......................................... 5.4........................................ 5.5................................................. 3 5.6................................................ 3 5.7.............................................. 5 5.8........................................... 6 9

,,?,??,?,,,. = a, b, a = b (.), 3, : a, a = a (.) a = b, b = a (.3) a = b b = c, a = c (.4)!,,? (.) (.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,., + ( ) + =,, ( )

,, =, := (: ), :=,, 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,,,,,, 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 *, π,, (irrational number), (real number) *3 * *,, *3,,,,,

.6 3.6!,,?, A, A B, B A,,,,,,,,,,,,,,, 3, 4,,, 3,,,,,,,,,,,!,,,?...,,,,, 3...,,,,,, π,, ( π = 4 3 + 5 ) 7 +,,, 4!, OK π,, 3?, A 3.4 B, 3.4596,, B A,?,...,,,!,! 4,?,,,?...,,,,,? πr, π OK, r,, r,, r

4 :.8,,,?, 0,,, 3,, =, +,,,,,, π, e, cos, sin,, 3,.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.,,,, 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)

.8 5,, 8, (.6) (.5),, (.6) (.5),,, (.6) (.5)?,, :.3 x, x 0 = 0?, (.8), 0+0=0 x, x (0 + 0) = x 0 (.6) x 0 + x 0 = x 0 (.7) x 0,?,... (.6) (.5),,??,...,,?...,,,,,, x 0 = 0 (.8) 4,,,.4?, ( ) ( 3) 3?,, ( ) ( 3 + 3) (.4), ( ) ( 3 + 3) = ( ) ( 3) + ( ) 3 = ( ) ( 3) 3 (.9), 3 + 3 = 0, ( ) ( 3 + 3) = ( ) 0 = 0 (.30) (.9) (.30), ( ) ( 3) 3 = 0 (.3) 3 3, ( ) ( 3) = 3 (.3)?,,, a, b, a b a + ( b), a b (b 0 ), a (/b), (.9) b, (.3) /b (b 0 ),.5 a, b, ab = 0, a = 0 b = 0 (.33) :, ab = 0 a 0, (.3), /a ab = 0, b = 0,, 3, 3 b 0, /b, ab = 0 a = 0, a 0 b 0, a, b,, 0 ( 5 ( ): ),,,, (.6) (.5) () x 0 0 () a, b, ab = 0, a b 0

6.9,,, 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 (.34), a, b, c 3, a b, b c, a c,,,, ab + bc + ca ac ca (.34), 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 3 3 3 4,, /,, /,, / /,,, ( ), { }, [ ] ( ), [{(a b)c + d}e + f]g (.35), ( ) { }, [ ] 4 3

.0 7, (((a b)c + d)e + f)g (.36),,,,, ( 3), 3 3...,! ( ), ( ) (.38) x m n,, x mn (.37) (.38),,, 3.3,,, m, n?,, (.37), m = 0, x 0 x n = x 0+n = x n (.39) 6,?, x 0 (.39) x n (x 0 x n 0), () /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 x n, x n, x n ( ) 3, = 8, ( ) ( ), x n n 3 3, (x, m n ) x m x n = x m+n (.37) (x m ) n = x mn (.38) (.37) x m, x n,, x m + n x 0 = (.40), 0 0, (.37), n, m = n m, (.37), x n x n = x n+n = x 0 = (.4) x n, x 0, x n = x n (.4),, 3, /( 3 ), /8, (.38), n, m = /n m, (.38), (x /n ) n = x n/n = x = x (.43), x /n, n x x n, n n x /n n x, n =, ( ), x, 8 /3 = 3 8 =

8, x n,, x /n n x, x n, 0 *4, 9 ±3, 3 3, 9 / = 9 = 3 ±3 (, ), : =.44356 () 3 =.730508 ( ) 5 =.360679 ( ) :,,, : 5 4 4,,.,,,, 3,,,,,.,,.3 4.5.6 : 4 3/ = (4 / ) 3 = 3 = 3 = 8 () 7 () 5 () (3) 0 6 0 4 (4) 9 0.5 (5) 4 0 (6) 0 3 0 7 8 / = () / 3 () 3 5,, ( ), 9?... ( ; 3 ),.,, () (4),,.3,. (), ˆ (, ˆ,, ˆ, x y ) 4.5, =, 4.43998 9 5 *4,, x, x n (!) () 3.5 ().0 00 (3) 3 5 : 3, /3,, 0.3333333...

.3 9.,,, e, e =.788 (.44) ( 6 ), 4 0...,,,,!.3 a, b, a b 0 (.44) 0, (a x = b x ), log a b (.45) (, a ) (logarithm) ( ) *6 log a (base) e,, π,, x, x,, e x, exp x, : exp x, e x e x,,.788 ˆ, : e 4,. (3) Shift, () *5, 4, =, 54.598,,,, 4,, Shift, () ln,!,. (5), EXP,, e x!.7 log 8 = 3 3 8 log = 0 0 log 0.5 = /, 0.5 (): () log 4 () log 3 8 (3) log 0. 0.0 (4) log 0 000 (5) log 0 0.0 (6) log 0 0, e =.78, 3 : () () (3), 5 log x, () exp () e.5 *5 () ln ( ), Shift, ln, e x, *6 a, b,,, b = 8, a =, a b 3, b = 8, a =, b = 8, a =, a b,,, a b

0,, log 0 x,, ln x ln log natural,,,, log x (log e x ) ln x I, II I,,! ln In n... l,, (l)!, log x OK, log x log x?... :, (!),,,,, 0 e,? 0 e?,?,,,?, ln (. ()), ln 3, ln, 3, =, 3, ln,,, log (. (4)), log, log 0, log,, =,, log 4, 5 () log 0 () log 0 0.006 (3) ln (4) ln 0.4, ( ),,, (, ),.0 m s, ( ),,, 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,. 5., ( ), 3, A, B... A AB B A, B, AB,

.4,, : : a, b, c,, x, y, z, A, B, C,, X, Y, Z : a, b, c,, x, y, z, A, B, C,, X, Y, Z,,., C c, O o, P p, S s, V v, W w, X x, Z z, h n 3 ( ), a, b, c,...,,,!,, AB, A B ( )?... AB A, B, C D, CD AB ( ) AB = CD, CD AB ( ) a a 0 0 0 = 0,, ( 3.4 5 ) 3,,?..., α, a a α, a α, αa,, a a a,!!,,, a, a a, b, 0 α, αa = b (.46), a b, ( ), ( ) : a, b, a b, a b, a b, a + b b a a + b, a, b, a, b ( ), a + b,

a b a + b ( ), 6 a, b, a + b, a b, a + 3b, b a, ( ) : a, b, a = b + x x a b, a b, (.) : a b a b, a b a b a( ), a b b( ),,,.3, (x y ), O *7, ( ) x y,, (3, ),, (z ), (3,, ) 3 *7 O origin.3,,,,.8 a = (, ) b = ( 3, 4), 5a + b? 5a+b = 5(, )+( 3, 4) = (5 3, 5 +4) = (, 4) ().5, ( ),,, O, P, OP,,,,,,,, :.9 A, B, a, b, O, a = OA, b = OB,, AB = b a, AB C c (c OC ), c = a + b (.47), AB m : n P (, AB, AP:PB= m : n P),? P p

.6 3 (.4), p = OA + AP = a + = na + mb m + n m AB = a + m + n m (b a) m + n (.48) m = n = (.47)! () 7.4 (3, 6), A B P A, B (, ) () AB :. () AB :. 4....6,,, 7,, 697, 697 5678, 697,,,,, 3 ( 30 ),,,,, 697,,,,, 6, 9, 7,,,, ( 7),,, (, 7 6 8 ),,, 0, A,,?, A 00, 00 300 00,, 0,, 0, A, 0, A 00, 90 0,,,, 0( 0 0),, 0, 0 0,, 697, 0,,,, 00.0 00, 00.0, 00,, ( ) 0, 00.0 0, 0 0,,,, 00.0,, 0, 0, 0 4, 4,, 00,, 0.00?,,, 3 0, 0, 0.00,

4 8, () 5.3 () 30.5 (3) 5300 (4) 0.030,,,, 0, 00,. 0 3 (.49),.0 0 3 (.50) (.49) (00 ), (.50),, 0 (0 ),. 0 3.0 0 3,,.7, (), 4.56., :, 4.56 3, 6 6.,,,,,,,,.5 :.5, 5.76 ( )5, (0. ) 7,.,, 6, 0. 7, 6 8,,, 5 7, 3 6,,, 6 5.8,, 5,...,,,,, :, 4.56 0.0,. 0.,,, 0. 0.0, 0.,, 5.76, 0., 5 7, 0.0, 6, 5.8,,, 9,,, () 5.3 + 6.6 () 0.03 + 3.5 (3) 00. 3, 4.56.,

.7 5,.6 :.6, 3 9,,, 6,, 5.47 ( )5, (0. ) 4, 7, 0. 4, 3 5,,, 5 4, 3 7 7 5.5,,, :, 4.56 3,.,,, 3,,, 5.47 5 4,, 4 (, 7 4 5 ),,,, 0,,, () 5.3.6 () 0.03 3.5 (3) 00./3.0 6 cm 3,?, (6 cm)/3=5.333 cm, 6 cm, 3, 5.3 cm...! 3 3?,, 5 cm? 3, 3., 3,, 0,,, 3 4, 0, 3, 3.0000, 0,, 6 cm,, 6 3 cm,? 6 3 cm 5.3 cm,,,, (),,., 3.47.88 3.47.88 = 9.9936 3.47.88 3, 3, 4 (3), 3.47.88 = 9.99,, 3.47.89, 3.47.89 = 0.083 3.47.89 3, 3,, 4 (83), 3.47.89 = 0.0,?, (.88.89 ),, (9.99) 0.0, (0.0) 0.,

6 0!? (),,,, 0,,,,, 3.47 0.0, 3.47±0.0,, (),,, 3? 4?,,,,, 3.47.89 = 0.083, 4, 0.03?, 4 5, 4.!?, ±,? () 3.±0. () 3.±0. (3) 3.±0.6,,?, 6,...,,.8,,,,, 3.7,.8,.9,.0, 3 6.4 cm, cm? () 6.4 cm + 6.4 cm =,, () 6.4 cm =,, (3),?, 4.,,?, 4.5 4.5 ( 4. ) 4. 4.3, 4.0 4.4, 4.!? 4. 4.3, 4.? 4. 4.3.7 (α, β, γ, δ, ϵ, ζ, η, θ),

.8 7 4 : γ r, δ 8, η n, κ k, µ u, ν v, σ 6, χ x, ω w 7,?...,, 8 Ω?... Ω, Ohm.8 (λ, κ, µ, ν, ξ, π, ρ, σ), A ( Ampere ) Ohm, O,, O Ω, 9, ι o?..., 0 ξ η,?...,.9 (τ, ϕ, χ, ψ, ω), 0,? (.45) a a =,? : a =, log?.0 (Γ,, Θ, Λ, Ξ, Π, Σ, Φ, Ψ, Ω), 3 ABC, BC D, 3, AB :, : A, B, C a, b, c, 3 r, r = (a + b + c)/3 ()

8, (,, ),,,, A, A = A, A (,,...), A,,, (0 ) 3 3.4596,,, 3.4596..., 3.4596...,, 4 (!) 5 ( OK) 6 () a () 3 ( 4) (3) {}, x 7 () 3 () / = /4 (3) 0 = /00 (4) 9 / = 9 = 3 (5) (6) 0 4 8 () 3 / () 5 /3 9 3 5 ().33 ().704 (3).709 0 (!) 3 5 () 7.389 () 0.3 () () 4 (3) (4) 3 (5) (6) 0 3 4 3 5 () 0.30 (). (3) 0.693 (4).30 5 (!) 6! 7 A, B a, b () a + b 3 = ( 5 3, 0 ) 3 () a + b 3 = ( 7 3, 4 ) 3 8 () 5, 3 (),, 3, 0, 5 5 (3) 5, 3, 0 (4), 3, 0 3 9 ().9 () 3.5 (3) 87 0 () 4 ().8 (3) 7.7 ().8 cm () 3 cm (3) ( ) () 3. 3.3, 0., 3. () 3.0 3.4, 0., 3. (3).6 3.8,, 3 3 (!)?...,,,,,...,! 3?......,,,,

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) *,. 999 9 3,,,,,,, *!!,,, *, ( credit ). 000 3 30,,,, 80 mg,, 80 ( 0 mg ),, 800 mg (),,,,, *3,, 0, (unit) (credit) 4?...,?,,, 0,?..., AB AB,?..., AB cm, m, km,, F, m...,?... F m m, m = 3 kg m = 3000 g, m, 3 3000 kg g t, m, 6 m t? t =3? t =3?..., t *3,,

0 t, t t = 3,,,,, 0 cm +.3 m=?, 0. m +.3 m =.43 m (.) 0 cm + 3 cm = 43 cm (.3),, 0 cm.3 m?, 0.3 0!!, (.), m, =(0. +.3) m,,.43 m, ( (.4))!, 0. m.3 m,,,!., 0 cm 5,,,, 0 cm, 5 4 ( )?,,, 0 cm 5, 0 5 5, 0 cm 5.3,, :.3 m 3 m, m 3 m = 6 m (.4), m,, m (), :.4, 00 m 0 00 m 0 ( ) = 5 m/ (.5),, m/ /,, m/ m OK () 5 /,, m/ OK?..., / ( ) 5 m/5 5...,... 5 5 m,......, 6 3,?, 6 (3 ),.......5 GDP, 4, 4, 4 /( ).6, 40 m, 40 m, 40 m/

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

,, (),,, 3,, 6,,...,,,, π )!,,, L, L = 5 cm cm ( cm), L/cm = 5 (.4), L cm,,, 5 (.4),,,, 5, () 5 500 () 5 500 4 m 3,? (3) 4900 ha,,000 t ( ),,? kg/m (4),, m /kg (5), 60 m,? (6) 60 kg 30,?.5,,, d S, d S, (cm m ) d S S/d,, 3.459... (,.6 SI,,,, SI ( ), SI,, 7 : : m () : kg ( ) : s ( ) : A () : K () ( ) : mol ( ) : cd () 7 SI 6 SI?, ( SI ), m, m s m s m/s, /, / ( )!, kg/m s, s,,, kg/(m s), kg m s, kg m s kg/m /s

.6 SI 3,,, kg m s m kg s OK, s kg m OK, ( ), (,, ),, SI, : 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) 7, 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, (SI ) 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 P p, M m,.0 µm (, ) ps (), 50 kg 50 km 50, cm, 5 mm 5, : h (hour), m,, hpa mg,, 8 SI 0? () G () M (3) k (4) h (5) d (6) c (7) m (8) µ 7 SI g ( )? kg g k, kg g... SI kg kg SI (m s ),, kg,, k kg, mg ( ), (kg) (g), 3, 3,, km, (km), k(m )!. km = (km) = (0 3 m) = 0 6 m (.5) dm 3 = (dm) 3 = (0 m) 3 = 0 3 m 3 (.6) cm = (cm) = (0 m) = 0 4 m (.7) 0.03 km = 0.03 (km) = 0.03 (0 3 m) = 0.03 0 6 m = 3 0 4 m (.8) () 5 km =000 m, cm 3 =0.0 m 3..., k c, m m 3, 8?, ab a (b ) km k (m )

4?..., km (km),...? km,,,, km square kilometer km (square) kilo meter kilometer?... km km 000 m, 000 m = 3.6 m? ha, 00 a a, 00 00 a=00 m, ha = 00 a=00 00 m =0000 m 9 ha, ka ( ) da ( )?..., ka, 000 00 m = 0 5 m = 0 0 m = 36.7 m,. (m km ) 9. () 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 = 0.0464 m 30 ( ) () 0.009 km m () 0.00003 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 ( ), SI SI, : : x = 5 OK, x= 5 : sin x OK, sinx : 5 m OK, 5 m : 5 m OK, 5m, : m s OK, ms : 3 kg OK, 3 k g, ms ( ) m s ( ) JIS *5, (, ), *5,,...

.7 5,,, [ ], 5.3 m/s, 5.3 [m/s].7. 7.5 km h, m s 7.5 km h = 7.5 km 000 m = 7.5 h 3600 s = 7.5 000 3600 m s =.08 m s. m s (.9).08 8 7.5,.3.0 g/s, kg/h.0 g/s =.0 g s =.0 0 3 kg 3600 =.0 (/3600) h 000 kg/h =.0 3.6 kg/h = 7. kg/h (.0) (), (h, s, km, m, kg, g ),,,,., 7.5 km h = 7.5 km h (.) = 7.5 km 000 m h h km 3600 s (.) = 7.5 000 km m h 3600 h km s (.3) = 75 m 36 s =.08 m s (.4). m s (.5), (.) (.), 000 m km h 3600 s (.6),, 000 m= km, km, (000 m)/km= h/(3600 s)=,, (.6),, 6 7. km/h = 590000 m/s,..., 3600, 3600,,,, 7. km/h, 500, 3,,.3 33, OK () 340 m/s km/h ( ) () 3.00 0 8 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, (),,, (),, 34, : ha..., L... PET () ml () m 3 (3) mg (4) g (5) kg (6) t (7) µm (8) 0000 km (9) 00000 km

6,, (!), 3 m 3 3 t, 3 m 3 = 3 t (.7), m 3 = t (!) (.8) m 3, t, (.8), m 3 t (.7),,,,, (.9), F force ( ), m mass ( ), a acceleration ( ),,,,,,, ( ),, 7 F = ma... F, m, a 35, ( ),,,, 0 kg 3 F a...... P.0,, ( ) 30 (.7) 3 m 3 =3 t, (.8)?... 3 m 3, 3 m 3 m 3 3,, 3 m 3,,, SI N ( *6 ) (.9), F N, m kg, a m s, N = kg m s (.30) N, F = ma,.8 3 m s,, ( ), ( ) m F,, a, F = ma (.9)!...,, ( ) m s, (, ) m s ( s), m s, *6,

.9 7? SI N,, /6, ( ) 0 N,,, kg, kg!?!?,,? (.9) (.9), F, m a, m a m,!, kg 0 kg,, kg?, (.9),,,,,,,,,, kg 9.8 N, kg, kgf (kg ), kgf := 9.80665 N (.3), kgf, kgf SI, kgf, kgf f kg,,, A 60 kg, A 60 kgf A 60 kg, kgf kg,, 33, 00 g = N...... 00 g, N,,, 00 g N ( 0.98 N), 00 g = N,,...... L ( ) kg L= kg, 36 60 kgf N?.9,, :,,, ( ), = SI, J ( *7 ) =, J, N, m, J = N m (.3) ( ) (.3), =, *7

8, SI J,,,, = / SI, Pa ( *8 ) = /, Pa, N, m, : Pa = N/m = N m (.33) Pa (.33), = /,,,,,,,, = / SI, W ( *9 ) = /, W, J, s, 37 () () (3) (4) (5) 38 () kg 3 m s? () N 4 m? (3) m 0 N? (4) 00 W,? 39 : () 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 40 J, Pa, W, SI (kg, m, s) W = J/s = J s (.34) 4 W (.34), = /, 8 W J... J W W=J/s, J=W s,, 9 W... W,, W=A V (A, V ), W *8 *9, () kw h J () hpa Pa, atm ( ) atm := 03.5 hpa (hpa h ) atm atmosphere atm 4 () atm Pa () atm kpa (3) Pa atm (4) (!) 895 hpa atm, P, V, n, T, R, P V = nrt ( ), R = 8.345 J mol K 43.0000 mol, 73.5 K,

.! 9 03.5 hpa, L 44 R = 8.345 J mol K, atm L mol K : J=Pa m 3 (cal), cal := 4.84 J (), cal g C,, J 45, J kg K 46 : () 500 kcal J (), 0.5 m 3! K? (3), 0 kg?.0 dimension check,,.4, m v, K : K := m v (.35) SI / m kg, v (m s ) =m s, kg m s J ( 40 ), K, SI J,, (!) (),, ( ), dimension check,, ( ), ( (.35) v ),,,,,.8.9,, dimension check 47,, m h, U * 0 U = mgh (.36) g, g 9.8 m s dimension check 48 r S V S = 4πr (.37) V = 4 3 πr3 (.38), A, (V = 4πr, S = 4πr 3 /3 ) dimension check, A,? *.!,,, 0 0, 0 m 0 ( ), 0 0,,,, *0,,, * S surface, V volume, r radius, ( ),

30!.5,,, 0 C = 546 K (.4), K ( ), 0 0 K,, 0 C 73 K, 0 K.6 ph, [H + ], : ph := log 0 [H + ] mol L (.39) (.4), 73 K = 546 K (.43)? 5,,,, SI : :,,,,, 49 : :,,,, :,,, [H + ] = 0 ph mol L (.40) 50 ph 0? 5 () 0.005 mol L ph? () ph=5.6, (ph=7)? (3) ph=5.0, ph=3.0,,, ph?,, C ph,,,,,,, (.4), dimension check,, 34,?..., C K, ph mol L : 6,,,,, 35...,,,,? 36,?...,,,,,, 37?...,, 4 0 C 73 K, 0 C = 73 K (.4)

.! 3 4 5 ().08 0 9 km/h (3).0 kg/l (4).0 t/m 3 (5).3 kg/m 3 (6).4 0 g/s 34 0.5 OK! () () (3) 500 5 5 500 4 m3 = = 00 5 500 4 03 = 40 000 t 4900 ha = 000 03 kg 4900 0 4 m =. 07 kg 4.9 0 7 m 0.43 kg/m () ml... (0.5 ml) () m 3... (3) mg... (4) g... (5) kg... (0.5 kg) (6) t... (7) µm... (8) 0000 km... (3000 km) (4) (3) /(0.43 kg/m ).3 m kg (5) 0.43 kg m 60 04 m 0 0 4 kg = 00 t (6) 60 kg m 30.3 kg 8000 m =.8 ha 6 () 7 ( ), kg, m, s 3 () 60 s () 3600 s (3) 00 m, 8 () 9 ( ) () 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 30 ( ) () 9 0 3 m () 3 0 4 m 3 3 ( ) () ml=0 3 L = 0 3 0 3 m 3 =0 6 m 3 cm 3 =(0 m) 3 =0 6 m 3, ml = cm 3 (), (3) (4) Gt=0 9 0 3 kg=0 kg Pg=0 5 g=0 kg, Gt=Pg 3.0 g/s =.0 g s =.0 g s =.0 g s 33 ( ) kg 3600 s = 000 g h kg 3600 s 000 g h.0 3600 kg = 7. kg/h 000 h () 0 km/h ( 3 4 4 km/h OK) (9) 00000 km... (83500 km ) 35 36 60 9.8 N 590 N 37 () 38 () kg 3 m s =6 kg m s = 6 N () N 4 m = 8 N m = 8 J (3) ( ) 5 Pa (4) ( ) 00 J 39( ) () J = N m m () Pa=N/m, N=J/m (3) 40 (.3) N (.30), J=kg m s m=kg m s (.44) (.33) N (.30), Pa=kg m s /m =kg m s (.45) (.34) J (.44), W=kg m s /s=kg m s 3 (.46) 4 () kw h = (0 3 W) (3600 s) = 3.6 0 6 W s = 3.6 0 6 J () hpa = 0 Pa = 00 Pa 4 () atm := 03.5 hpa =.035 0 3 0 Pa =.035 0 5 Pa

3 () atm=0.35 0 3 Pa=0.35 kpa (3) (), Pa= /(.035 0 5 ) atm = 9.8693 0 6 atm (4) 895 hpa = 895 0 9.8693 0 6 atm = 0.883 atm 43 V = nrt P =.0000 mol 8.345 J mol K 73.5 K.035 0 5 Pa = 8.345 73.5 J.035 0 5 Pa =.44 0 J/Pa.44 0 m 3 =.44 0 0 3 L =.44 L, J/Pa m 3, 39(4) 44 39(3), J=Pa m 3, m 3 =0 3 L, J=0 3 Pa L, 4, Pa = 9.8693 0 6 atm, J=0 3 9.8693 0 6 atm L=9.8693 0 3 atm L, R = 8.345 J mol K = 8.345 9.8693 0 3 atm L mol K = 0.08058 atm L mol K 45, Q, T, m, 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 46 (, ): () 0 7 J () 5 K (3) 00 km 47 SI J, N m, kg m s, m kg, g m s, h m, mgh kg m s (kg m s ), 48 r, SI m A, V r m V, A, S 49 (.39) ( ), ph = log 0 [H + ] mol L (.47), 0 ph = [H+ ] mol L 50 (.40) ph=0, 5 [H + ] = 0 0 mol L = mol L () [H + ]=0.005 mol L, [H + ] ph = log 0 mol L = log 0.005 mol L 0 mol L = log 0 0.005.3 () (.40), ph=5.6 [H + ]=0 5.6 mol L ph=7.0 [H + ]=0 7.0 mol L, 0 5.6 /0 7 = 0.4 5., 5 (3) x L H +, ph= 5.0, 0 5 x mol ph= 3.0, 0 3 x mol, H +, 0 5 x mol + 0 3 x mol x L + x L = 0 5 + 0 3 mol L = 5.05 0 4 mol L, ph= log 0 (5.05 0 4 ) 3.3

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),, ( 3 ) 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, 5 0 a, b, a = b a + b ab (3.4) 0,, 0 :, <, >, 0 > a 0 < a

34 3 3. 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)! ) 38 0!=? 0!=0,... (3.9), n! = (n )! n, n =,,! = ( )!,! = 0!! = = 0!, 0! =, (3.9) 3. 53 3! 3, 6 () () 4! () 5! (3) 0! (4)! (5) ( 5)! 3.4 3. a, b, c 3, ( )?, 6 : abc, acb, bac, bca, cab, cba, 6? a, b, c 3,,,, 3 = 3! 6 () 3., n, n! 54 3.3 5? a, b, c, d, e 5, 3,? 5 4 3, 5 4 3 60 () 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 = 55 56 n! (n m)! (3.) n n P n = n! 8,? 3.4, a, b, c 3,,, aa, cb?,,

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 57 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 9 8 3 = 0 *, ( )! 58 (n 3 ) () 4C () 5C (3) 5C 3 (4) nc 3 (5) nc 0 (6) nc n 59 60 n, m, n > m nc m = n C n m (3.6) 40, 3 ( 3 ) 3.5 ( ) ( ) 3.6 () 3x xy abc (polynomial) ( ) 3.7 + 3x + x, x + y + xy,, +x, + x + x, () (term), ( ), n, n ( n ) 3.8 x 3 + x, x x 3, 3, 3, 3 3.9 : x 3 y + x y + (3.7), x y x 3 y, 4, 4, 4 (),,,

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 6 ( x χ ): (x + χ)(x + χ)(x χ) (3.8) 39 x χ... x 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), : (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) 6 () (x + ) 7, x 3 () (x 3) 6, x 3 3.7 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 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) (), x, x ( x + b ) = x + b ( b ) a a x + (3.38) a

3.8 37, 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) 63 : () x + 4x + 5 () x + x + (3) x x (4) x + 4x + 3 (5) 4x + x + 40?...,,,, () x =,... P.5 (.33)? (),, : 3.5 x x + = 0 (3.44), (x ) = 0, x = ( ) (), ( x ),, ( ), 64 () x x 6 = 0 () x 3x + = 0 (3) x 4 5x + 4 = 0, 65 x ax + bx + c = 0 (3.45) 3.8 3.3 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, 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),, (3.48), D,, D := b 4ac (3.49)

38 3 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 = 3 4 5 = < 0, (3.48), ( = i ) () 4,,?...,, D, ( ), ± D, a, b i, a + bi 4?... a + bi, b 0 b = 0 a,,,, n (n ),, n n, n,, 66 () x + x + = 0 () x + 3x + = 0 3.9,,, 3.7 x = (3.55), x = x =, x = 3 ± i (3.54) x = (x )(x + ) (3.56)

3. 39, x, (),, = 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 (3.57) a = 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,,, {6, 3, 3, 4,, 5, 6,, 3,, } (3.6), (),,, 3., ( ), 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), ( ), a n, n a n+ = r a n (r ) (3.65), r ( ), (3.59), r a n, a n = a r n ( a ) (3.66) a n = a 0 r n ( a 0 ) (3.67)? 67, 3.9, (), () 0, (3) 3, 3

40 3 (4), / (3.6) (3.65),,, 3.,,, 3.0 {,, 3, 4, 5, } (3.68) () (4) (5) (6) 0 (7) {a n = /n} n,,,, 3.3 ( ) n (3.73) {a n },, n, a n < a n+ (3.69),,,,, 3. {,, 0,,, 3, 4, } (3.70) () {a n },, n, a n > a n+ (3.7), * 3. {,,,,,, } (3.7) () 68,,,, () () (3) * (3.69) (3.7),, < >,,, {, 3 4, 7 8, 5 6, } (3.74),,, () 3.3,, a, a 3.4 {,, 3, 4, },,, ( ), () 69,,?? () () (3) (4) (5) 0 (6) 0 (7) {a n = /n}

3.3 4 3.3,,, {a, a, a 3, }, a + a + + a n (3.75), n 3.5 {a n } = {, 3, 5, 7, }, 3, + 3 + 5 = 9 (3.76) () (3.75) n a k (3.77) k=, n a k := a + a + a 3 + + a n (3.78) k= ( ) Σ Σ, S S sum Σ k = n,, k n a k 43..., k, 3 (n ) (3.8) n=, 3 (p ) (3.8) p= (3.80), (3.8), (3.8), 3 (n ) (3.83) k=, (k), ( a n ) n, k =, k =, k = 3 n, a, n a = a + a + + a (a n ) k=, : n a = na (3.84) k= 70 () () 4 (3) k= (4) 3 k= k a k, k, 3 p=0 p 4 (n + ) n= 4 k = + + 3 + 4 = 0 (3.79) k=,, (3.76), k ( ) k, (3.76) 3 (k ) (3.80) k= 3.6, P.36 (3.3) : n (a + b) n = nc n k a n k b k (3.85) k=0 () 7 (3.33), Σ

4 3,, {a k }, {b k }, ( ) α, ( ) n n n (a k + b k ) = a k + b k (3.86) k= k= k=, n =, (3.88), =, = (3.89), =, N, n = N (3.88), n n αa k = α a k (3.87) k= k= (3.86), N k = k= N(N + ), (3.90) (a + b ) + (a + b ) + + (a n + b n ) = (a + a + + a n ) + (b + b + + b n ) (3.87), αa + αa + + αa n = α(a + a + + a n ),,,,,, 3.4, {,, 3, }, ( ) n k = k= n(n + ) (n ) (3.88)?, n = =, = / = n = =+=3, = 3/ = 3 n = 3 =++3=6, =3 4/ = 6..., n n n = 00, n = 0 n = 0000 n,, (3.88) : N+ k= = k = N k + (N + ) (3.9) k= N(N + ) = N + N + N + = N + 3N + + N + (3.9) = (N + )(N + ) (3.93) (3.94), n = N + (3.88),, (3.89) n = n = + = (3.88), n = + = 3 (3.88), n = 3 + = 4 (3.88),,, n (3.88),..., n, () n = () N, n = N (3) n = N +, ( 3 ) 7 A,, (3.90) N N +

3.5 43,...( )... N k = k= N(N + ) N N +, N+ k= k =...( )... (N + )(N + + ) = (N + )(N + )? A 0 (3.9), (3.9)...,, (3.90), (3.90), (3.9), N k= k, (3.9), (3.9) N(N+),!, ( ) ( ),,,,,,? (), n = N..., n = N,! 44...,,,,,,,,,,,,, 45...,,,,,,,? ( ) 73 n, r, () () n k=0 r k = rn+ r n k = k= n(n + )(n + ) 6 (3.95) (3.96) 73, (3.95) (3.96)...,,..., 46?... 74 : () 0 k k=0 47 () 0 k k=,?...,, 3.5, ( ),,

44 3!,,,,,,,,, *3,, Excel, *4 LibreOffice-calc, LibreOffice-calc *5, LibreOffice Excel 48?...,,,, ( ), B 3 3,, 3, B3, A, : =*3 *3,,, *4 *5 LibreOffice, ( Linux, Windows, Mac, A B C =*3 3, A, 6, *3, 3,, * ( ),,,,, A 4, B, = (!) A, B =A, *5, B =A*5 6 A B C 4 =A*5 3, B, 0, A (4) 5 49..., B, =A*5,,, A 7, B 35!,, 3.6, (,,, 3.7 : 0! +! +! + 3! + + n! + (3.97)

3.6 45,,,,, (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 0 3 4 5 3.. 0.. B, (/0!), B3 /!, B4 /!,...,, B3, =B/A3, B3, B ( ) A3 (, B3, B4 B *6 : A B C k /k! sum 0 3 4 0.5 5 3 0.666... 0 0.000. k (/k!) B 50 B!.0877E-009...,.0877 0 9,, 0.000000000877 0,, 0, E, B, C ( C =B, C3 =C+B3, C4 C, A B C k /k! sum 0 3 4 0.5.5 5 3 0.666.6666.... 0 0.000.788 C, (3.97) n = 0.788,, (.44)! (),,, *6, (LibreOffice

46 3,, A, A,,, C3 =C+B3, C4, =C3+B4?, (,!), (,,, (,, 3, OK 75 n k= : k = + + 3 + 4 + + n (3.98),, n π /6 =.64493... *7 3 n ( 060987 n = 987), (3.98), 7 : 3.7, A n k ( k ), B, /k B, =/(A*A), B3 ( A n ) (,!) ( 3.7 ) 3, ( ) =/A*A..., *7 II, 5 =/(A*A) =/Aˆ?...,, ˆ, ˆ,, ˆ,, = 3ˆ 9, 9?,, ( ), ( 3),,,,, *,, (, ) 4 5 *, * 5... 7,, ( ; ),,,,,,,,, #,, #,,, #,, #,, #,, 7 5, ( 3.),,

3.6 47, a b = 0 a = b 53 () 4 3 = 4 () 5 4 3 = 0 (3) (4) (5) 3. 54 5! = 0 55 (3.), m = n, np n = n!/(n n)! = n!/0! = n!,, (), 440 Hz? (),? (3), 88, 88,? (4),, 0Hz 0000Hz? (5),? (6) 440 Hz,.5,,? (7),,,, 4:5:6, 5 0, (3.3), ( ) ( ) = a + ab + b ab 4 = a ab + b ( a b ) = 0 4, {(a b)/} = 0, (a b) = 0 (.33) 56 8 3,, 8 3, 8 P 3 = 336 57 8 = 56 58 () 4 () 0 (3) 0 (4) n(n )(n )/6 (5) n C 0 = n!/(0!n!) = n!/n! = 0! = (6) 59 (3.4), nc m = n! m! (n m)!, (3.4) m n m, nc n m = =, n C m = n C n m 60 40 C 3 = 9880 n! (n m)! {n (n m)}! n! (n m)!m! = n! m!(n m)! 6 x 3 + 3x χ 3xχ χ 3 6 () (3.3) a = x, b =, n = 7, (x + ) 7 = 7 C 7 x 7 + 7 C 6 x 6 + 7 C 5 x 5 x 3, 7 C 3 x 3 = 35x 3, 35 () (3.3) a = x, b = 3, n = 6, 63 x 3 6 C 3 (x) 3 ( 3) 3 = 430x 3, 430

48 3 () (x + ) + () ( x + ) 3 + 4 7 n (a + b) n = nc k a n k b k (3.99) k=0 (3) (5) (x ) ( 4 x + ) 5 + 8 6 (4) (x + ) + 64 ( ): () x =, 3 () x =, (3), (x 4)(x ) = (x + )(x )(x + )(x ), x = ±, ± 65 ( ) 66 : () x = ( ± 7 i)/ () x = ( 3 ± 5)/ 67 () n () n (3) 3 n (4) (/) n 68 () : {,, 3, 4, } () : {4, 3,,, } (3) : {,, 4, 8, } (4) : {,, 4, 8, } (5) : {,, 4, 8, } (6) : {, /, /4, /8, } (7) 69 () ( ) () ( ) (3) ( ) (4) ( ) (5) 0 (6) 0 (7) 0 70 () + + + = 8 () + 4 + 9 = 4 (3) + + 4 + 8 = 5 (4) 3 + 5 + 7 + 9 = 4 7 ( ) (3.90) N, N +,, 73 () 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 + 74 () (3.95), ( )/( ) = 047 () (3.96), (0 )/6 = 385 75

49 4 4. y (function),,, ( ) y = x, x, y, x = 3, y = 3 = 9 *, 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 () 4. 4 - - 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 - - - - 4. - x y= -x y O x - - - y = x y = x, a, y = ax (4.3) *,,,,,, a, a, a, x y

50 4,, a, x y a,, y = ax x y, y x ( ) * x y, x y y x, y = x (4.4), x = 0,,, 3,, y = 0,, 4, 9, 4.3,, lim x ( 4.3 ), x y,, ( x y ) ( ), a, y = a (4.8) x x y, y x ( ),, y=x 4 3 O x -4-3 - - - 3 4 - -3-4 y y=/x y 4 3 O x -4-3 - - - 3 4 4.3 : y = x, : y = /x, y = x - -3-4 (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 +, ( (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 ) 76 (, 3),, xy () x 5 (7, 3) () y 4 (, 7) (3) x 3 (6, 3)

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 ) 76, 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) y 0 = f(x 0 ) (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), (x, y ), P (4.) 53 (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) 3) y a, y = af(x) 4) x, y = f(x) 5) y, y = f( x) 6), y = f( x) 77 4,,,,, 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)

5 4 4.5 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?, (4.9) y=x- y 3-3 - O - - x 3 - -3 y= -x y 3-3 - O - - x 3 - -3 y= -/x y 3 y= x /4 y 3-3 - O - - x 3 - -3 y= x +4x+ y 3-3 - O - - x 3 - -3 y=/(x-)+ y 3 y = x + x = + x (4.0) -3 - O - - x 3 - -3-3 - O - - x 3 - -3 y = /x y, x, y ( 3, )! 4.4 () 4.4, 78 y = x, () x, y () x, y 3 (3) x, x (4) x, x 79 (!), () 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),,,, (4.), y = ax ( ) y b x = 0 y = b (0, b) y y (b ), x y a,,, (x 0, y 0 ), a?, O a

4.5 53 y = ax x x 0, y y 0, (4.) : y = a(x x 0 ) + y 0 (4.) 4.8 y = x + x (4.4)? x x 0 y = y 0 (x 0, y 0 ) x 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, y = y y 0 x x 0 (x x 0 ) + y 0 (4.3), 0 < x y = x y = /x,, x, x = y = x y = /x y =,, (, ), (, ) x y = /x 0, y = x, x 0 y = /x, y = x 0,, 4.5 () y 80 3 () 3, (), (, ) (3) (, ) (4, 3) 8, ( ) 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, O 3 y=x y=/x y=x+/x 4.5 y = x + /x ( ) y = x y = /x! 8, y = x + /x 4.5, 4.6 x y, A, x y?,,, A x 0 30, y 00 300 300 ), x

54 4 y 300 00 A B,, y x, a y, y ( ), y = ax + b (x, y), 83 A, b, a = 0.6, b = 550! 4.6, 00 0 30 40 x x, y,... f(x, y,...), f(x, y,...) = 0 ( ) P.37, f(x, y,...) 4.6, A, y = 00 y = 300, P, Q PQ PQ PA PA (mm cm ) A y, PA (300 00) + 00 (4.5) PQ!, A,, x = 0 x = 30,, x 83 A x y!, :, x = 4, y = 85 (x 0.5, y )! 84 4.6,, y = ax + b (a, b ) : (),, 54 (3.4) x y +xy +xy = x+, f(x) = 0... x y+xy +xy x = 0? f(x, y), f(x, y) = 0, 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, 4.9 y = x 3 x x 3 x = 0, x =, 0, 3 x x ( ), x y, x y,,,, 4.7 y = x 3 y = x, () f(x) = 0, y = f(x) x 4.0 y = x + x + x +x+ = 0 (P.37), D = 4 = 3 < 0 (

4.7 55 y - - O - x - 4.7 y = x 3 x ( ), y = x 3 y = x ) y = x + x + x, 4.8 () 4.9 y = x 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 () 4.7,, y = f(x), x, y, f(x), (x, y) P.44, 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!), ( ) (,, x ) 4.0, *4 () *4,

56 4 4.0 55 A? A?... B, B,, 5,...,, :, :,,,,, 85,, 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 4. y = x, y = x, y = x 3, 3 4. 6,...... 4.8 4.3 x + < x x? 3 < x, x 3 4.4 x x < 0 (4.6) ( ) (x )(x+) < 0 (3.), (x ) (x+), < x < x+,, x < 0 0 < x +, < x <, 4.3, 4.4,, (4.6) y = x x

4.9 57, 4.8 (4.6), y < 0, x x < x <,, : ax + bx + c = 0 x = α x = β (0 < a, α < β ), ax + bx + c < 0 (4.7) α < x < β, ax + bx + c > 0 (4.8) x < α β < x? y = ax + bx + c, x = α x = β x, a > 0 x ( (4.7)), ( (4.8)), (4.7), (4.8), <, <..., (, + i 5i,?), (4.30),,, ( x + ) 3 + 4 (4.3) (x + ) x 0, 3/4,, =0, (4.3) 87 x () x + x + > 0 () x + x + < 0 (3) x + x > 0 (4) x 4x + 4 < 0 86 x (5) x 4x + 4 0 () x + x 6 < 0 () x + 5x + 6 0 (3) x + 3x + 0 4.5 x + x + > 0 (4.9)? y = x + x +, 4.8 (4.9), y > 0, x x, x x,, x + x + < 0 (4.30)?,! 7 : 3 i (4.30) < x < + 3 i! (6) x 4x + 4 0 (7) x 4x + 4 > 0 (3) 0 x, (x ), (x ), x (P.33 (3.7) ) (x ) 0, (4) (7), (x ) 4.9 f(x), f( x) = f(x) (4.3), f(x) ( ), f(x) f( x) = f(x) (4.33), f(x) ( ),,,

58 4! : (4.33), f(x) = f( x),,, (4.33) 88??, y = f(x) y y = f( x) (4. 5), f( x) f(x),, y,, y, y = x, ( 4.3 ) y, 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) 89 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 ±? 56?...,,, 8 y,...,? () () (3) (4) (5) (6), 9 ( : ),,,, f(x) = { ( + x + x 4 ) 3 } 8 + x + x 4 (4.34),,,! 57,?...,,, (4.9) ax n, n, n, 90,,,,, 58, 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 : f(x) = x +, g(x) = x 90 x =, f() = + =, g() = = 4

4. 59, 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) (4.35) (4.37)! () g(f(x)),,,, 9 f(x) = x, g(x) = + x, g(f(x)) f(g(x)) y = g(x), g(x) f(x),, f(x) g(x), 4.8 y = x x = 3, y = 6 x y = x/ ( y = x 4.7 ), y = 6/ = 3 x (), : f(x) g(x), g(f(x)) = f(g(x)) = x (4.38)? f(x), x, g(x) x, g(f(x)), x,, g(f(x)) x x, g(f(x)) = x, f(g(x)) = x 4.9 4.7, y = x y = x/ f(x), g(x) f(x) = x, g(x) = x/ 4., y = f(x) x y y = g(x), f(x) ( ) 4.7 y = x, x y x = y y y = x/, y = x y = x/ () 93, a, b () y = x + () y = /(x ) (3) y = a/x (4) y = x + b?, y = f(x), x y, x, y, x y,,, y = f(x), f(g(x)) = g(x) = x = x (4.39) g(f(x)) = f(x) = x = x (4.40) (4.38)! () 94 0 x f(x) = x, g(x) = x, (4.38) (x x ) f(x) g(x), y = g(x) x = f(y), y = f(x) x y x = f(y), y = g(x) ( 4.) x y, y = x y = x x y, y = x y = f(x) x = f(y), y = g(x),

60 4 y y=f(x) y=x, *5 (4.) a b x=f(y) y=g(x),? O x 59?... 4. y = f(x) y = g(x) y = x,,, y = x (, ),,, 95 y = x y = x, 9 y = x, y... y = x x y = x, y!,, 4.7, y = x y = x/, x /(x), 93,, 96,, 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 : ˆ A3 3 =A3ˆ3 A3 /3, =A3ˆ(/3)!, n, y = x n y = x /n, y = x,, y = x α,, α α, x y 4.,,, y = f(x), x y, x, y, y = x + xy + y = 0, x y 0,, x y, x y F (x, y), F (x, y) = 0 x y, (implicit function) ( ), 97 r x + y r = 0 (4.4) ( r > 0 ), r y = f(x), x, y, (4.4), y = ± r x (4.4) *5,

4.3 6, ± x = 0 y = r y = r, x y ( ),,,, 4.0 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 () 98 (x, y), (3, ), 4.3, : x /x, 0, x ( x + y = x ) x = 0 y (y ) () y y = 0 x (x ) () x (x x y y ),,, x y, x ( ) y, 60 III,,?...,,,,, 0 x,,,,,,,,, 8,, : 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] 9 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)

6 4 0 F (x, y) = 0, : () x a, y b, F (x/a, y/b) = 0 () x, F (x, y) = 0 (3) F (x, y) = F (x, y), x x y =, ( x ) y = (4.45) ( 4.3 ) (4) y = x ( 4.3 ), x, y = (x ) (4.46) ( 4.3 ) x, x ) y =( (4.47) (x /4) + (y /9) = ( 4.3 ) y= x y=(x/) y=((x-)/) 3 3 3 76 ( xy ) 77 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, () y = x (y ) (x 4, y 4 ), x 4 = x 0, y 4 = y 0 () y = (x + ) +, y = x (x, x 0 = x 4, y 0 = y 4 ), y 0 = f(x 0 ), y 4 = f(x 4 ), (3) y = /x (y ) y 4 = f(x 4 ), P 4 y = f(x) (4) y = /(x + ), y = /x,, (x ), 78 () y = (x ) + () y = 3(x/) = 3x /4 (3) y = x ( 4.3 ), x, ( x ) y = (4.44) - - O 3 x 3 y= x - - O 3 x 4.3 - - O 3 x 3 y=(x-) - - O 3 x - - O 3 x y=((x/)-) 3 - - O 3 x y = x y = x 79 4.4 80 x = 0 y = 0, () y = 3x () (4.), y = (x + ) +, y = x + 3 (3) (4.3), ( 4.3 ) x y = 3 (x ) + (4.48) 4

4.3 63 y=x+ y 4 3 O - - - 3 - y=+/x y 4 3 x y=x +x+3 y 4 3 O - - - 3 - y=x/(+x) y 4 3 x x (), ( ) (3) ( ) x <, < x (4) ( ) (5) ( ) x = (6) ( ) (7) ( ) O - - - 3 - x O - - - 3 - x 88,! 8 4.4 y = x + 5 79 () (3, 0) (, 00) () y 0 = 00 0 (x 3), 3 y = 5 9 (x 3) = 5 9 x 60 9 89 4.5 () 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 =! x = 9 5 y + 3 (3) 8 83 84 y=4x 4-5x + y - O - - x y=x+x y - O - - x 85 86 - - () =0 x = 3, 3 < x < () =0 x = 3, x 3, x (3) =0 x = 3± 5, y=x-x 3 y y= /(+x ) y x 3 5, 3 + 5 x O - - - x O - - - x 87 - - () =(x + /) + 3/4 x,, 4.5 89

64 4 90 f (x), f (x), g (x), g (x), f ( x) = f (x), f ( x) = f (x) g ( x) = g (x), g ( x) = g (x), a/x (4) x y x = y + b, y = x + b : (3), (4) () 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), 95 4.6 94 g(f(x)) = f(x) = x = x x 0, x, f(g(x)) = (g(x)) = ( x) = x 96 4.7 y y=x y=sqrt(x) y=x F ( x) = f ( x)g ( x) = f (x){ g (x)} = f (x)g (x) = F (x) O x, F (x) (4) F (x) = f (x) + f (x), F ( x) = f ( x) + f ( x) = f (x) + f (x) = F (x) 4.6 95, sqrt(x) x y = 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) y 0.5 y=x y=x, y=x / y=x 3, y=x /3 y=x 4, y=x /4, F (x) (6) y = y = x : y = +x, 9 : f( x) = f(x) : f( x) = f(x) ( ), 9 g(f(x)) = + x ( x 0), f(g(x)) = + x O 0.5 4.7 96 y = x, x /, x /3, x /4 (, ) 97 x + y, (x, y), r, r r, r x 93 () x y x = y +, y = (x )/ () x y x = /(y ), y = + /x (3) x y x = a/y, y = 98,, (4.4) r =, x + y 4 = 0 x 3, y ( 4.0 ), (x 3) + (y + ) 4 = 0

65 5 :, 5. y = f(x), 5., x 0, (x 0, f(x 0 )) P, x 0 x 0 x, (x, f(x)) Q y f(x 0 + x) f P f(x 0 ) x y=f(x) Q Q R x x 0 x =x 0 + x 5. y = f(x) (), P (), y = f(x) P, Q (), P ( P ; 5. ) a, P(x 0, f(x 0 )), a, y = f(x 0 ) + a(x x 0 ), x 0 x,, ( 5. ), P ( 5. ), : f(x) f(x 0 ) + a(x x 0 ) (5.), (5.) Q y, Q y Q Q, =,, a ( ),,, x x 0 x x 5. PR * ( P.6, x, x ), x = x 0 + x (5.) (5.), (5.), f(x 0 + x) f(x 0 ) + a x (5.3), x x 0 5. R P, Q Q P ), x 0 x 0,, 0, 0 ( ) x, dx *, (5.3), f(x 0 + dx) = f(x 0 ) + a dx (5.4), =, x 0, 5., Q Q, P, P, (5.3) * R P * d x, dx

66 5, x 0 0, 0, =, *3 0 0,, dx,, 0, 0 x,, dx x d, difference *4 x = x 0 x = x 0 + dx x = x 0 + x, (5.4), a f (x 0 ) f(x) x = x 0, () f(x) dx, f (x 0 ), x = x 0 f(x) f(x 0 + dx) = f(x 0 ) + f (x 0 )dx (5.5) (5.3) : f(x 0 + x) f(x 0 ) + f (x 0 ) x (5.6),,,, (5.5) (5.5), f (x 0 ) = f(x 0 + dx) f(x 0 ) dx,, f (x 0 ) f(x 0 + x) f(x 0 ) x (5.7) (5.8), x 0 () f f(x 0 + x) f(x 0 ) (x 0 ) := lim x 0 x (5.9) =,, lim (P.50!) dx, lim (5.9),, ( x h ), (5.5), (5.9),,,! 99 6 (5.5) (5.9) 0 0...,... 6... 0? 0 63,,... ( ), 00 g(x), x = a, f(x 0 + dx) f(x 0 + x), f(x 0 ), f(x 0 ), df f, df := f(x 0 + dx) f(x 0 ) (5.0) f := f(x 0 + x) f(x 0 ) (5.) f 5. RQ *5 df f,,, df dx, f x,, (5.5), f(x 0 + dx) f(x 0 ) = f (x 0 )dx,, df = f (x 0 )dx (5.) *3,,, x dx, f df,, d, = *4, D ( (5.5) f(x 0 ) (5.0) ) (5.), *5 Q R

5. 67 f(x),, (df dx),,, 64...,, (5.7), (5.8), (5.9), f (x 0 ) = df dx f (x 0 ) f x f f (x 0 ) = lim x 0 x (5.3) (5.3) (5.4) (5.5) d dx f (5.6) (5.3) (5.4), f(x) df f, x dx x ( ),, 5., RQ PR,,, PQ ( PQ ) (RQ /PR), x 0 (R P ), Q Q, PQ PQ, PQ,,,, (5.5), (5.9),,,, (5.5), (5.9)!, 65?...,,,,,,?...,,,,,,?...,,, dx df, f (x 0),,,,,,, f(x), (derivative), f (x) (differentiate),, (5.5) x 0 x 5. f(x) = x + : f(x + dx) = (x + dx) + = x + + dx = f(x) + dx (5.5), dx f (x), f (x) = (5.7) () (5.7), : (x + ) = (5.8) (5.8), ( ), ( ),

68 5 ( ), d (x + ) = (5.9) dx, d/dx 66... 5., f(x) = x f (x) = x x = 3 f (3) = 6, x = 3 0 p, q f(x) = px + q, f (x) = p, (px + q) = p (5.0), y = q, 0, (q) = 0 (5.), 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, dx, d x, (dx),, dx d x dx ( ), dx 0, dx,, 0, dx!, dx, x x x 0, x 0, x dx,,,, f x, ( ) ( ) 67, dx?... dx, dx, 68,,... dx,, 69 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 + (5.3) dx (5.5), dx nx n, f (x) = nx n (5.4) () f(x + dx) = f(x) + xdx (5.) 5.4 f(x) = /x, (5.5), dx x, f (x) = x () f(x + dx) = x + dx (5.5)

5.3 69, x dx, f(x + dx) = x dx (x + dx)(x dx) = x dx x dx (5.6), dx, f(x + dx) = x dx x = x x dx = f(x) dx (5.7) x (5.5), dx /x, f (x) = x (5.8), (5.4) n = (5.4), n, n = () 5.,,,,, (5.9), f f(x + x) f(x) (x) := lim x 0 x (5.9), x 0, 0,, f (x) f(x + x) f(x) x (5.30) f(x) B : A B C x f(x)=x f (x) - 3-0.95 0.905 4-0.9 0.8 5-0.85 0.75 (C ) f (x), C, =(B3 B)/(A3 A) B3 B f(x + x) f(x), A3 A x, ( ), x =, x = x = 0.95, x ( x = 0.05),,, C, C3 C, C,, f (x), 0, f(x) = x, x,, f(x), *6 f (x) = x, x 5. 5.3, ( ) (5.5),,, ( ), f(x), g(x), ( ),, f(x) (f(x + x) f(x), x ( x, f(x) = x, x 0.05 A, *6 (

70 5 : a, {af(x)} = af (x) (5.35) : F (x) = af(x), F (x + dx) = af(x + dx) = a{f(x) + f (x)dx} = af(x) + af (x)dx = F (x) + af (x)dx (5.36) 5. y = x (dy/dx), y = x x 0.05 y = x, 5.!, dx,, F (x) = af (x), Σ (P.4)? ( ) 5.6 f(x) = 3x, f (x) = (3x ) = 3(x ) (5.37) : {f(x) + g(x)} = f (x) + g (x) (5.3) : 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 5., (x ) = x, f (x) = 3(x) = 6x (5.38) () 5.7 f(x) = x 3 + x + 3x +,, f (x) = (x 3 ) + (x ) + 3(x) + () (5.39), () ( x = F (x) + {f (x) + g (x)}dx (5.3) ), (5.), dx,,, 0 (x 3 ) (x ), (x) (5.4) F (x) = f (x) + g (x), = 3x + 4x + 3 (5.40) 5.5 f(x) = x +x, () f (x) = (x + x) = (x ) + (x) (5.33) 5., (x ) = x, (5.0), (x) =, f (x) = x + (5.34) () 03 : () 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

5.3 7 3: {f(x)g(x)} = f (x)g(x) + f(x)g (x) (5.4) : 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 dx,, F (x) = f (x)g(x) + f(x)g (x) (5.4) f(x) (x),, 3, : (f + g) = f + g (5.46) (af) = af (5.47) (fg) = f g + fg (5.48),, 4: {g(f(x))} = g (f(x))f (x) (5.49) : 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, 3, F (x) = (x + ) (x + x + 3) + (x + )(x + x + 3) = x(x + x + 3) + (x + )(x + ) = 4x 3 + 3x + 0x + (5.43),, F (x) = (x 4 + x 3 + 5x + x + 6) = (x 4 ) + (x 3 ) + 5(x ) + (x) + (6) = 4x 3 + 3x + 0x + (5.44),, () 04 : f(x) = (x + x + )(x x ) (5.45), () : x + x + x x, () ( ), 5.9 (x + ) 3, g(x) = x 3 x, f(x) = x + g (x) = 3x, 4, {(x + ) 3 } = {(f(x)) 3 } = 3(f(x)) f (x) = 3(x + ) {(x + ) } = 3(x + ) (x) = 6x(x + ) (5.50), (5.50), 6x 5 + x 3 + 6x (5.5), (x +) 3, {(x + ) 3 } = (x 6 + 3x 4 + 3x + ) = 6x 5 + x 3 + 6x (5.5) (5.5), (5.5) ( ) () 4,,, 4 : 5.0 x + (5.53) f(x) = x +, g(x) = /x, 4