., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P
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- ゆりか ねぎたや
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1 . ϵριµϵτρoζ perimetros 76 Jones, Euler. =.,.,,,, C, C n+ P, P,, P n P, P n P n, P P P P n P n n P n,, C P, P j P j j =,,, n P n P., C.,, C. f [a, b], f. C = f a b, C l l = b a + f d P j P j a b j j j j 3
2 ., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P,, P,., P, a + b = r { = b + r a a r a + r b = b r a a r a + r b { = a + r b b r b + r a = a r b b r b + r a, d/db ± r a = ±a / r a, = a+r/ a r/ + a r a d + b+r/ b r/ + b r b d : gt [α, β], g t., F [c, d] F f c gt d α t β., F gt b + r/ a, b d dt F gt = F gtg t = fgtg t t α β, F d F c = d c f d, b r/ a r/ a + r/ 4 gβ f d = F gβ F gα = β gα α fgtg t dt
3 ., gt = a + rt, f = + a /r a, a+r/ a r/ f d = / / + t t r dt = r / / t dt.. / = 4r dt / t / 3 = / / dt = 4 t t dt, a, b r. 6,. 8, θ, θ.,.,.,,.,., F, 5 F F = f a b 5 6 F., f [a, b] f a b, 6 S. r = S = r r b a f d r d = r t dt, gt = rt, f = r. 7 n,. r, 3
4 = r. n, r, r, r. =. t dt = 4 t dt B A r r O C θ. Archimedes 7 8 n L n, l n, l n < < L n 8, < θ < / OAC = sin θ / < OAC = θ/ < OAB = tan θ / sin θ < θ < tan θ. θ = /n, l n = n sin θ, = nθ, L n = n tan θ. 4 L n = l nl n l n + L n., L n = L n + l n, l n = l n L n l n L n = n sin n tan n = 4n sin n = l n Archimedes B.C. 87? n = 6 l 96 L = 6 4, < < L 6 = 3, l 6 = 3, L = 3, l = 3 6, L 4 = , l 4 = n = 4, L 4 = 4, l 4 =, L n, l n n =, 3,. a = =, a n = l n+ n =,, l 4 l n+, a n sin n+ n =,,,, n =. a n = = cos / sin n =,,. n+ n+ a n+ = cos n+3 = + cos = n+ + a n 4 n =,,,
5 . /l n+ = a a a n, lim n l n+ = n = lim cos n = k+ + k= Viéte., /a a a a 3 a 4 = θ < θ = sin θ cos θ cos θ cos θ 3 = sin θ / lim n k= n cos θ k Euler. 3. Arctan Gregor θ < /,, 9 AB = θ = sin θ t dt. f = /, gt = t/ + t, gt = / + t, sin θ = tan θ/ + tan θ g t = + t 3/,, A B θ sin θ 5 θ = = tan θ tan θ + t / + t 3/ dt + t dt 9., θ = /4 4 = + t dt 5
6 . / < < / = tan,, / < < /. Arctan. < <, = tan = Arctan Arctan Tan, tan. 5 / = tan / Arctan = + t dt < <. + r + r + + r n = r n+ / r r + t = t + t 4 t n t n + n+ t n+ + t Arctan = R n = t + t 4 t n t n dt + R n = , R n n+ tn+ + t dt t n+ + t dt + + n n+ n + + R n t n+ dt = n+3 n + 3 n + 3, n R n., 6 Arctan = = lim n n k= k k+ k +., 67 Gregor, Gregor. =, 4 = = lim n n k= k k +, Gregor 674 Leibnitz Leibnitz. 6 Arctan = Talor Maclaurin 6
7 . f = Talor f f +! f +! f + 3 3! f +., f. e e = lim n + /n n, e, 7 e = +! +! + 3 3! + 4 4! +., 8 9 cos =! + 4 4! 6 6! + sin =! 3 3! + 5 5! +. 7, 8, 9, e, cos, sin. e =! + 4 4! + +! 3 3! + = cos + sin Euler = / 3, Arctan / 3 = /6 6 = Sharp., / < α, β, α + β < /. tanα + β =., p = tan α, q = tan β tan α + tan β tan α tan β α + β = Arctan p + Arctan q = Arctan p + q pq, Arctan /5 =Arctan 5/, 4Arctan /5 =Arctan /9. 4Arctan 5 Arctan 39 = Arctan = 4 7
8 . 6, Machin 76 4 = = 4Arctan Arctan + Arctan Rutherford = 6Arctan + Arctan = Arctan Gauss = 5Arctan + Arctan Machin = Arctan 3 + Arctan 7 Euler = Arctan + Arctan 3 Euler = 3Arctan + Arctan Hutton = 8Arctan Gauss-Legendre : a =, b = cos α >, a n+ = a n + b n /, b n+ = a n b n n =,,, / / lim a n = lim b n = n n sin α sin θ dθ. a, b Gauss. α = /4 b = /, c = /, c n+ = a n a n+ n =,,, t n = /4 n k= k c k n =,,., A = lim n a n, T = lim n t n, = A /T. : A = : B = / : T = /4: X =, { }. { Y = A : A = A + B/ : B = BY : T = T XA Y : X = X }, Y = A Y A., A + B /4T A /T 976 Salamin, Brent. 5. Machin Gauss-Legendre,. : I n = / sin n d, = Wallis I n = n!! n!!, I n+ = n!! n +!!, I n+ < I n < I n 8
9 ., n!! = n n n 4, n+!! = n+ n 3. /4 = 3/3 + /4 d, = 3 3 4, !! 5 7! 3!! 8 9 3! 4 = = = a +a Newton a 3 +a Brouncker, Euler., S = a + a a + a a a 3 + = a., Arctan /n =Arctan /n +Arctan /n + 4 = Arctan + Arctan + Arctan 3 + Euler. /6 = / / d!! = 3 +! 3 + 3!! 3! 5 + 5!! 6 3! Euler. = = = + 3, Euler Euler : h = /n, k = kh, k = / + k k =,,,, n, 4 = + d h { + n n } h { + n n n } 3 Simpson /3 9
10 h, Simpson 8h 4 /5. k+ / + d A ABED + BCF E, Simpson A, B, C. 6. Simpson. Simpson,. Buffon : d, L < d. L/d.., k L sin d/ ABCO. d k D E B C k+ k+ F k k+ k+ A B = L sin C 5. :,.. 3, 6 = /3 3.., : O,, E,, P,,, P r r, r,,,., O, E, P,, P r. P,,,,,,,, r,,, r m,,,,, r,,, r m.
11 839 Wantzel, Galois. n. Gauss n n = a p p p s, p,, p s Fermat n = 3, 4, 5, 6, 8,,, 5, 6, 7,,. p Fermat, p p = m +. 3,. 767, Lambert.,.,. 88, Lindemann. a,, a n a + + a n =, r,, r n, a e r + + a n e rn., Euler e + e =,., 3.,,,,, 973.,,, 974.,,, 99. Hobson, E. W., Squaring the Circle, Oford Univ. Press, 93.,,,, 964. = ,,
12 B.C. = 5/8 = 6/9 B.C. Archimedes 3/7 < 3 < / < < Arabhata, Viete 596 Ludolph 3, /6 Newton 5 67 Gregor Arctan 674 Leibnitz Arctan 75 Sharp 7 76 Machin 76 Jones 79 de Lagn Lambert 794 Vega Dase 877 Clausen Rutherford Richter 5 873/4 Shanks Lindemann 945 Shanks ENIAC *M 954, 39 NORC M Pegasus 958 IBM74 M IBM74 M 96, 65 IBM79 S 966, 5 IBM73 G 967, 5 CDC66 G 973,,,5 CDC76 G 98,,,36 FACOM M- K 98, 8,388,576 HITAC M-8 GL 983,, CRAY- 986, 67 HITAC S-8/ GL 987,, 3 NEC SX- GL 988, HITAC S-8/8 GL IBM39 989, 5 HITAC S-8/8 GL 989 IBM39 989,,73,74,799 HITAC S-8/8 GL 99,,8 FM-TOWNS S, M: Machin, S:, G: Gauss, K:, GL: Gauss-Legendre
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1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
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9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
() 800 ( p.38) 50 8 8 8 64 64 r ( r r ) ( ) 6 = r = 56 8 r r = 56 8 = 3.60438 3.6 3.6 3.6 800 8 AD BC ( ) ( 4 3 3 ) = 8 8 = 63 63 64 = 8 ( ) ( 3 = 8 )
3 (r) (r ) 3.4 r l l r dx x 0 εριϕέρεια (William Oughtred : 573 660?) 8 (Leonhard Euler (707783) (Introductio in Analysin Innitorum : 748 ) ( 8 ) a 0x n + a x n + + a n x + a n = 0 (algebraic number) (transcendental
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =
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(000) < > = = = (BC 67» BC 1) 3.14 10 (= ) 18 ( 00 ) ( ¼"½ '"½ &) ¼ 18 ¼ 0 ¼ =3:141596535897933846 ¼ 1 5cm ` ¼ = ` 5 = ` 10 () ` =10¼ (cm) (1) 3cm () r () () (1) r () r 1 4 (3) r, 60 ± 1 < > µ AB ` µ ±
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名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
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x : = : x x x :1 = 1: x 1 x : = : x x : = : x x : = : x x ( x ) = x = x x = + x x = + + x x = + + + + x = + + + + +L x x :1 = 1: x 1 x ( x 1) = 1 x 2 x =1 x 2 x 1= 0 1± 1+ 4 x = 2 = 1 ± 5 2 x > 1
Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
di-problem.dvi
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#A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/
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f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f
208 3 28. f fd f Df 関数 接線 D f f 0 f f f 2 f f f f f 3 f lim f f df 0 d 4 f df d 3 f d f df d 5 d c 208 2 f f t t f df d 6 d t dt 7 f df df d d df dt lim f 0 t df d d dt d t 8 dt 9.2 f,, f 0 f 0 lim 0 lim
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5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4
... A F F l F l F(p, 0) = p p > 0 l p 0 P(, ) H P(, ) P l PH F PF = PH PF = PH p O p ( p) + = { ( p)} = 4p l = 4p (p 0) F(p, 0) = p O 3 5 5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 =
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熊本県数学問題正解
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chap1.dvi
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di-problem.dvi
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