2005 7
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1 3 1...................... 3 2...................... 4 3.................... 6 4............................. 8 2 11 1........................... 11 2.................... 13 3...................... 13 3? 17 1.......................... 17 2?.......................... 18 3?...................... 19 4 23 1?............................. 23 2........................... 25 3?....................... 26 5 29 1?....................... 29 2?............... 30 3?................... 31 4.............. 32 iii
iv 5...................... 34 6? 37 1........................... 37 2 (logic)............................ 42 7 Taylor 47 1............................. 47 2............................. 48 3.............................. 50 4 Taylor..................... 51 5........................ 52 6............................... 52 8 53 1?...................... 53 2....................... 55 3............................ 57 9 59 1?................ 59 2?..................... 61 3?.............. 61 4............................. 63 5.......................... 64 6...................... 65 10 69 1.......................... 69 2............................ 70 3................... 71 4?.......... 72 5?....................... 74 11 77 1?............................. 77
v 2........................ 78
vi
1. 4. 1?.. 1. 1.,.,.. 1 1.,. 2.,.?, 1.. 2.... 1 calculus. 2 Smirnov. 5,,..
2..
1 1... 1.,,...... 1, 1,..... ( ). 3
4 1,.... lim{ f (x) + g(x)} = lim f (x) + lim g(x) x c x c x c.. lim(x + 2) = lim x + lim 2 = 3 + 2 = 5 x 3 x 3 x 3..?.,,.,.?.?. 2,?... lim x c { f (x) + g(x)} = lim x c f (x) + lim x c g(x)
2 5. d dx x2 = 2x.?.... f (x) g(x) f (x)+ g(x) x c. f (x) g(x) x c........ ( )..,...,.?.,,..?.?.., f (x) g(x).., f (x) + g(x). x c., lim{ f (x) + g(x)} x c
6 1...?..(.), f (x) g(x).., lim f (x) lim g(x) x c x c.., lim f (x) + lim g(x) x c x c..? f (x) g(x).,... ( ), ( )... ( ).. ( ) ( ) = ( ) ( ) 3.., 90, (1, 0) (1, 0), 90.
3 7..??. lim{a f (x)} = a lim f (x) x c x c lim f (x) g(x) = (lim f (x)) (lim g(x)) x c x c x c. 1.1.., ɛ δ,... a(b + c) = ab + ac. b c a. ab = a b (a, b > 0), A (B C) = (A B) (A C). lim (a n + b n ) = lim a n + lim b n n n n, ( f + g) = f + g, { f (x) + g(x)} dx f (x) dx + g(x) dx
8 1. (?).. 1.2. 1.. 2.. 4?. f (x) = 2x, g(x) = 2x + 1 1.. 2x 2x + 1 1.(.), (0 ) (0 ).. f (x) = 2x?, f?, f (a + b) f (a) + f (b)?. 2(a + b) = 2a + 2b.. g(x) = 2x + 1? g(a + b) = 2(a + b) + 1 = 2a + 2b + 1 g(a) + g(b) = (2a + 1) + (2b + 1) = 2a + 2b + 2. g. f ( )..(.), 1.
4 9 f lim x c f (x).... 1.3.,.
10 1
2. 90%... 1., f x = a 1. f x = a f (a) (, a f ), 2. f x = a lim x a f (x) 3.., lim x a f (x) = f (a)... 11
12 2 lim f (x) = f (lim x) x a x a.., x f x a (function).,., f, f... lim x 3 ex = e 3 lim(x + b) = a + b x a lim x π cos 2 x + 1 e x + e x = 2 e π + e π x 3, a, π.??....(?) 2.1.... 1 1,,,...
3 13 2. (point mass). (x(t), y(t)).. 2?, ( ). 2.2. ( ).. SF 3.,.. 4,.. 3. 2. 3. 4..
14 2...,..,.. m 0. x(t) = (x(t), y(t), z(t)). F = ma. m = m 0.. ɛ. m = m 0 + ɛ. m.... 0.1g..??..? ɛ.. m 0 m 0.,
3 15,,., CD volume,, 1. 2.3.., ( ). 2.4..(.)
16 2
3?,... 1 1, 4.?,. :,??. 1.....,..?? 17
18 3?.,..,,,,.,,. 1,,. 2?,. [0, 1] f (x) = x 2 x,.. f (x) f (x), f (x) 0 f (x)....?.. 3 2. 4, 3 1...,.
3? 19? 2 6. ( ) 3. ( ).,?.... 4.,... 3?...... 2 3. 3 Abel 5. 4. Weierstrass f (x) = k=0 sin(πk 2 x) πk 2. Hardy(1916).
20 3?.( 1. 1.),.??. [0, 1] f (x) = x 2 x,..... f (x) f (x), 0 f (0), f (1),..,., [0, 1] f (x) = x 2 x,, [0, 1],.,. 0.. 0.,, 0. ( ) ( )
3? 21 0 ( ). 0 ( ) ( )..(!).??,,... 5...,.. 6 5. 6. 2..
22 3?
4 4. ( )...? 1?....,.? a > 0 a n a n = a a a (n ).. n a n. a 0 = 1... 23
24 4., m, n a m/n. n > 0 a 1/n x n = a,..,?...?... 2 = 1.4142, π = 3.141592 (= ). {b n } x x = lim n b n a x = lim n a b n. b n a b n...
2 25... lim a x = a c x c?? a x x. a x.,,... 2. d dx ln x = 1 x ln x := x 1 1 t dt. e. 1/x.. 1. e x. 1.
26 4.,....,... 3?,,?... e x... e n = e e e e x. n,.. lim e x = e c x c.,..
3? 27.(.)...
28 4
5...,.. 1? ( ) (radian)... 1 = π ( ) 108 180 rad, 1 rad =, 180 = π rad. π : (rad). 29
30 5.. ( )??..?..,.,.??. 360......... 2??.,...
3? 31.... (sine), (cosine).,.(.)., ( y = 2x ). sin 90 = 1 sin π 2 = 1?,.....,..,.. sin 90 = 1... 3??
32 5...... lim x 0 sin x x = 1. d dx sin x = 1 x=0? 5.1.?.?.. 4?... sin x cos x. d 2 dx 2 sin x = sin x, d 2 cos x = cos x dx2
4 33 d 2 f (x) = f (x) dx2 f (x)? f (x) = a sin x + b cos x.? f (x) = a sin x+ b cos x. S(x), C(x) S (x) = S(x), S(0) = 0, S (0) = 1 C (x) = C(x), C(0) = 1, C (0) = 0 S(x) = sin x, C(x) = cos x. f = f. 5.2. S C,, f + f = 0 f (x) = a S(x) + b C(x). d S(x) = C(x) dx ( : T = S T + T = 0, T(0), T (0) a, b.), (x n ) sin x = x 1 3! x3 + 1 5! x5, cos x = 1 1 2! x2 + 1 4! x4
34 5... 5.... e x = 1 + x + 1 2! x2 + 1 3! x3 + 1 4! x4 + 1 5! x5 +.(.). e x ix e ix = 1 + ix + 1 2! (ix)2 + 1 3! (ix)3 + 1 4! (ix)4 + 1 5! (ix)5 +, 1 e ix = (1 1 2! x2 + 1 4! x4 ) + i(x 1 3! x3 + 1 5! x5 e ix = cos x + i sin x. 2 e ix = cos x + i sin x, e ix = cos x i sin x 1.,,. 2 Euler... Euler, Euler.
5 35 cos x = eix + e ix, sin x = eix e ix 2 2i.., cosh x = ex + e x, sinh x = ex e x 2 2 cosh x = 1 + 1 2! x2 + 1 4! x4 +, sinh x = x + 1 3! x3 + 1 5! x5 +.
36 5
6?..?. 1.. ɛ δ ɛ N... 1.,. 1..,.. 37
38 6? {a n = 2 1/n} 2,? a n 2. a n 2 a n 3.?...,. (.)? Cauchy(, 1789 1857)..... {a n } 0? {1/n}, { 1/n} {( 1) n /n} 0. {1 + (1/n)} 0, 1 0. a n 0. (.). a n = 1/n, b n. b n a 1 a n 0. 10, a 1. 11 b 11 a 1, 12 b 12 a 2. 100., b 110 a 100, 111 b 111 a 1 a n 1000. {b n } = { 1 2, 1 3,..., 1 10, 1 2, 1 3,..., 1 100, 1 2, 1 3,..., 1 1000, 1 2, 1 3, 1... 1000, 1 2, 1 3,... 1 10000, 1 2, 1 3,... 1 100000, 1 2, 1 3,...... }
1 39 {b n } 0? {b n } 0 (?) 0. b 11 1 b 111 1, b 1111 1. 0 1. 0, 0 1 11, 111, 1111 0...? 0 1 1.., 0 1 0... {b n } {b n /2}.. 0 1. b n /2 0.? 0 1,. 0 1/2. 0..?., 0 1/2 0.. 1/2 1/3
40 6?, 1/100, ɛ.. ɛ ɛ 0 ɛ 0., ( ) ɛ > 0 ( ) a n 0 ɛ n., ( ) ɛ > 0, a n 0 ɛ n. n( a n 0 ɛ n) n N. ( n N.) n N, n a n 0 ɛ., n N n a n 0 < ɛ., ( ) ɛ > 0, ( ) N N n (, n N ) a n 0 < ɛ. 136..,..,...
1 41.... ɛ > 0 ( ). ( ) ɛ > 0 ( )... ( ) ( ). N ( N) ( 2). N ( 2).. ( ) ɛ > 0, ɛ N [n N a n < ɛ]., ( ) ɛ > 0, ɛ N [ n N a n < ɛ]., ( ) ɛ > 0, ɛ N n( N) ( n N) a n < ɛ.(, a n ɛ.). {b n } 0 1. ɛ 0.5. (, ɛ = 0.5.)
42 6? N N ( ) n b n 1,, b n, b n 0 = 1 0.5. (.) 136 138. 6 9... 6.1. 3. 2 (logic) (mathematical logic). (truth table).(.). : (negation), (and), (or), (for all), (there exists). 2. 1.. 2. H C, C C (statement) H,. ( ) 3. [ S] S.(= ) 4. [H C] H [ C]. 5. [S T] [ S] [ T]. 2. Russell Whitehead Principia Mathematica., (Russell Whitehead)..
2 (LOGIC) 43 6. [ x S(x)] x [ S(x)]. 7. [ x S(x)] x [ S(x)]. 8. P Q P, Q. 9.. P [P Q], Q [P Q], [P Q] P, [P Q] P, [[ Q] [ P]] [P Q], [[P Q] [Q R]] [P R] 10. P, P P.. 2 (, reductio ad absurdum, RAA) ( ). 1, 3, 8, 10. 5, 6, 7, 9. 10 (, dichotomy). ( )., ( ). 9,. ( ;contraposition).... (implication(, ) )... (1) (A ) ( ).,, A, A.
44 6? (2) (A )... (1) (2). (2). A, A. (2) (1)... 4. 4. implication([h C]),, [H C] H C...,.,.., [H C] H. [H C] H H..,,,.. (H C). ( [H [ C]])., H C [H [ C]]
2 (LOGIC) 45.. 6.2.... (1). (2). (3) 2.. Marvin J. Greenberg, Euclidean and non-euclidean geometries, chapter 2.
46 6?
7 Taylor x. a n x n (power series). 1, Taylor Taylor. Taylor computer. 1 sin x Taylor x = 0. sin x sin x x = 0 Taylor 1, 3, 4, 7, 9 19 7. 1, power series. x n x n-th power.. 47
48 7 TAYLOR 4 2-10 -5 5 10-2 -4? sin x ( ). sin x.. cos x. x = 0, 0, 2, 4, 6, 8 20.. 4 2-10 -5 5 10-2 -4 2 ln x. x = 1 1. 0 < x < 2. ln x. x = 1 Taylor. 3,
2 49 5, 7. 2 1 0.5 1 1.5 2 2.5-1 -2 (0, 2).. x = 5 Taylor
50 7 TAYLOR 7.5 5 2.5 2 4 6 8 10 12 14-2.5-5 -7.5 5, (0, 10).. (0, 2) Taylor.. 3 Taylor. f (x) = 10 (x 1)(x 5)(x 10) x = 3 x = 7 x = 1, 5, 10. ( x = 1, 5, 10 4.)
5 TAYLOR 51 x = 3 (1, 5), x = 7 (5, 9).. f (x) x = 3, x = 7. 4 2 2 4 6 8 10 12-2 -4 4 Taylor Taylor.. Taylor (order). (order) Taylor...
52 7 TAYLOR 5 Taylor. (remainder term).(.) Taylor,.... Taylor ( 0 ). 2.(.). Taylor, 0 Taylor (analytic function).. 0. 0. 0. 6 Mathematica.. ( download.),. 2..
8... dy dx = dy du du dx, {g( f (x))} = g ( f (x)) f (x) (g f ) = g f.. 1?... 1.(.). 1. y x.( 1) 1. 53
54 8..( 2).. 1 y x (= ) 1.. x y. x x y.., x y x x y f (x).( 3 )
2 55 f,,, f.? y x x y,. 2.. 2 f g.(.) g f.?. 2? bifocal., smooth.
56 8 ( ). x u y. f g g f.?. f u x, f (x) g y u, x y x., y x = y u u x = g ( f (x)) f (x).(.),.
3 57. (g f ) = g f.?. f g f x, g f x f (x). (g f ) (x) = g ( f (x)) f (x). dy dx = dy du du dx....,. 3.. 8.1. cos 2 x + sin 2 x. x = 0 cos 2 x + sin 2 x = 1.
58 8. d dx { f (x)}a = a{ f (x)} a 1 f (x) (8.1) d 1 dx f (x) = f (x) { f (x)} 2 (8.2) d dx ln f (x) = f (x) f (x) (8.3) d dx e f (x) = e f (x) f (x) (8.4) d dx f (ax + b) = a f (ax + b) (8.5).. (, ) a, b ab.. (chain rule).
9. 1.,. 2.. 3. ( ). 4.. 5.... 1?. f [a, b], (a, b), c [a, b] : f (c) = f (b) f (a) b a. 59
60 9 (smooth)..??.....,...,,.?. 1? (technic).. 1..
3? 61. 2?...?....,,.. Venn diagram.. ( )?...,. topic. 3???
62 9.?. 1.... 2..(.) 3.,,. match. 4. step. step.. 2?........,. ( ) 2.,. ( ).
4 63. 3.(.). 4 4. technic version.. version f (a) = f (b). 5 f (a) = f (b) = 0. (Rolle)....(.).. ( ). 0. 3..... 4..(.).(.) 5..
64 9... 6...,,,.? 5.?,?.. 7.. 0..?.?.( technic.)? 0??.,....?(.)? 6.,.? 7 Taylor.
6 65...?........ 9.1.. 1. f (x) 0, f (x). 2. x > 0 f (x) lim f (x) = 0 x +. lim [ f (x + 1) f (x)] = 0 x + 6.. 8 8
66 9. f g [a, b] (a, b) x (a, b) g (x) 0. c (a, b). f (c) f (b) f (a) g = (c) g(b) g(a)?,...?.. ( f (x), g(x)).(.) x [a, b] ( f (x), g(x)). f (x) = 0 g (x) = 0, (smooth).,, http://gallica.bnf.fr/ l hospital.. Analyse des infiniment petits, pour l intelligence des lignes courbes.
6 67. ( f (x), g (x)). ( f (c), g (c)), ( f (b) f (a), g(b) g(a)).. ( f (a), g(a)) ( f (b), g(b)) (x = c) ( f (a), g(a)) ( f (b), g(b))........ 9.2. ( f (x), g (x)). ( :.)
68 9
10..,.. 1. f (x) g (x) dx = f (x) g(x) f (x) g(x) dx b a f (x) g (x) dx = [ f (x) g(x) ] b a b.. { f (x) g(x)} = f (x) g(x) + f (x) g (x) a f (x) g(x) dx,,, Leibniz... x e x dx = x (e x ) dx = x e x (x) e x dx = x e x e x dx = x e x e x 69
70 10 (.). guess and try.. 2.. f (g(x)) g (x) dx = F(g(x)) F f., f (u) du = F(u).?... f (g(x)) dg(x) dx dx = F(g(x)) g(x) = u f (u) du dx = F(u) = f (u) du dx. f (u) ( ). 1, du dx = du dx.... ( ). 1.
3 71., du dx = du dx Leibniz. u = g(x). g (x) dx = dg(x) g (x) = dg(x) dx dx. dg = g (x) dx (differential) (differential form).. 3.. Leibniz d(uv) dx = udv dx + du dx v. dx. { d(uv) dx dx = u dv dx + du } dx v dx d(uv) = u (dv) + (du) v. d(uv) = u dv + v du. ( ). u dv = uv v du
72 10.... ln x dx = x ln x x.. ln x dx = ln x (x) dx = x ln x x (ln x) dx = x ln x x. g(x).. g (x) = 1... g (x). u = ln x, v = x., ln x dx = (ln x) x x d(ln x) = x ln x x 1 x dx.... tan 1 x dx = x tan 1 x = x tan 1 x. x d(tan 1 x) x 1 + x 2 dx = x tan 1 x 1 2 ln(1 + x2 ) 4?. b f (x) g(x) dx a. f (x). g(a) = g(b) = 0 g(x) f (x).
4? 73.. h(x).(.). g(x) b a h(x) g(x) dx h(x) ( ). b a h 1 (x) g(x) dx = b a h 2 (x) g(x) dx h 1 (x) h 2 (x) h 1 (x) h 2 (x). 2. g(x) b a (h 1 (x) h 2 (x)) g(x) dx = 0 h 1 (x) h 2 (x) 0., b a (h(x)) g(x) dx = 0 h(x) 0.. : h(x) 0,, h(x) 0 0 g(x). 2. g(x) h(x) h(x).. h(x).
74 10? h(x).. h(x) x = c 0. h(x) x = c h(x) 0.( h(x) 0 x = c.) 3 h(x) > 0 h(x) < 0. g(x) > 0 ( g(x) = 0 ) 0.? b a h(x) g(x) dx g(x) h(x).(.) 5?. b a f (x) g (x) dx = [ f (x) g(x) ] b a b a f (x) g(x) dx..,....? f (x)g(x) x = a, b 0 0. 3.. x = c h(x) 0, h(x) 0 h(x). x = c h(x)?
5? 75? 0. g(x)., g(a) = g(b) = 0. b f (x) g (x) dx = a b a f (x) g(x) dx., g(x)., g(x) f (x).. f (x). : b f (x) g (x) dx a g(x) f (x).. f (x) f (x).?. f (x)? b a f (x) g (x) dx? f (x)?.. 0, x < 0 f (x) = 1, x 0 x = 0. x = 0? a < 0 < b, b a f (x) g(x) dx = b a f (x) g (x) dx = b 0 g (x) dx = g(0) g(b) = g(0)
76 10., f (x) b a f (x) g(x) dx = g(0). f (x)? g(0) f (x) x 0 f (x) = 0. x = 0 f (x) 0, g(0) 0 f (x) x = 0. ( x = 0 black hole.),. f (x),. 4 idea..... 4 20 Laurent Schwartz.