数学の基礎訓練I

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1 I ,, () () () A 5 B 6

2 ( 1 (1) γ () ϵ (3) σ (4) δ (5) η (6) λ (7) ρ (8) τ (9) ψ (10) ω (11) ϕ (1) χ (13) µ (14) ξ (15) ζ (16) φ (17) ε (18) κ (19) ν (0) Γ (1) Π () Σ (3) Θ (4) Λ (5) (6) Φ (7) Ψ (8) Ω 1. Σ S (Sum) Π P (Product) 3 n = = 14 n=1 3 n = 1 3 = 36 n=1, n 1, 3 n {1,..., 3},, 0, 1 ( ) (j + k) = (j + k) j=0 k=0 = j=0 (0 + k) + k=0 (1 + k) + k=0 k=0 k=0 ( + k) = ( ) + ( ) + ( ) = 18 ( ) (j + k) = (j + k) j=0 k=0 = = j=0 k=0 ((j + 0) + (j + 1) + (j + )) j=0 3(j + 1) j=0 = 3((0 + 1) + (1 + 1) + ( + 1)) = 18 1 ( (1) () (3) (4) 3 n + 1 n=1 3 (n + 1) n=1 0 n n=1 0 k k=3 (5) 3 1 n=1 ( 1 (1) γ () ϵ (3) σ (4) δ (5) η (6) λ (7) ρ (8) τ (9) ψ: (10) ω: (11) ϕ (1) χ (13) µ (14) ξ (15) ζ (16) φ (ϕ ) (17) ε (ϵ ) (18) κ (19) ν (0) Γ (1) Π () Σ: (3) Θ: (4) Λ: (5) (6) Φ (7) Ψ (8) Ω: 3 (6) n=1 5 (7) ( ) n n n=0 ( (1) 7 () 9 (3) 0 (4) 1 (5) 3 (6) 8 (7) 3 (8) 7 (9) 1 (10) 11 1

3 1.3 1 (8) (9) (10) 5 n= n + n j=0 k=0 k=0 l=1 j (j + k) k l n ( 3 (1) () (3) ( 4 r t+1 + γr t+ + γ r t γ T 1 r t+t = T +t 1 k=t T 1 γ (1) r k+1 = γ i r () i=0 T 1 = γ (3) r T +t i i=0 4, (1) () (3) N N n = n n=1 n=1 n=1 N N n = n i=1 j=1 n=1 N M M N f(i, j) = f(i, j) j=1 i=1 1.3 n (factorial) n! = n k = n(n 1) 1 k=1 0! = 1 n! = n (n 1)! n 1 1 ( 5 (1) () (3) n! n=0 n! n=0 3 n=0 (n + )! n! ( 1 (1) π () (3) (4) N M i=1 j=1 f(i, j) = M N j=1 i=1 (f(i, j) n n ( 3 (1) (k 1) () ( ) k 1 k (3) k=1 k=1 ( 4 (1) k t () i + t + 1 (3) T i 1 n (k + 1)k k=1 ( 5 (1) 4 () (3) 40 ( 1, (Eukleides, Euclid, BC365? - BC75?)

4 1.5 1 A E F D 1.5 B G a, b ab (1) ( 1) () ( xyz P (x 1, y 1, z 1 ) Q(x, y, z ) (distance) P Q P Q = (x 1 x ) + (y 1 y ) + (z 1 z ) ( (Pythagoras, BC58-BC496) C (function),, x y f y = f(x) or x f y y = f(x), f(x), y x f (image) x, y 1.5. (polynomial), f(x) x a 0,..., a n f(x) = a n x n + a n 1 x n a 1 x + a 0 (1) (1) f(x) f(x) = (x c)(b n 1 x n b 1 x + b 0 ) f(x) (factor) x c (factorization) f(x) x c f(c) = 0 1 (1) x 3 1 () x 3 + 3x + 4x + (3) x n 1, (1) x xy x + y () xy + 3ay a x 3a 3 3

5 xy (1) y = x + 1 () y = x x (3) x = 1 (4) x = 1 y = 1 (5) x + y = 1 (1) (a) (x, y)? (b)? (c) (Origin)? () (a) (b) 4 3 xyz (1) x = 1 () x = 1 y = 1 (3) x + y + z = 1 (4) x + y + z y = 1 (5) x + y = 1 3, x, y z 5 x + y = z (1) z = 0 xyz () z = 1 (3) z = (4) x = 0 xyz (5) 6 y = x + x (1) x [0, 1]? () x [, 0]? (equation ( 3 ), 7 ( 6 (1) x + 1 () x + 1 = 0 (3) x + 1 = 0 (4) f(x) (5) f(x) (6) y = x +1 (x y ) (7) y = x + 1 (x y ) (8) y = x + 1 (x y ) (9) y = x + 1 (x y ) 8 (1) x = 1 x = x () x = 1 x 1 (3) x = a, x = b, x = c x 1 3 ( 3 equation equ ( 6 (1) () (3) (4) (5) (6) (7) (8) (9) 4

6 y 1 y = x y = 1 x = 1-1 O 1 1 x -1-9, (1) x x = 0 () x = x (3) x = x (4) x 1 = 0 x = f(x) = g(x) y = f(x) y = g(x) x, x = 1 () x ( ) y = x y = 1 () x + 1 = 0 (3) x y = x + 1 y = 0 () (3) (5) x = 1 x (6) x 3 3x + x = 0 y = x (7) x + y = 1 10 y = x + x + 1 y = x + c (1) x + x + 1 = x + c () x + x + 1 c = 0 (3) (i) () (ii), (iii) (a) (b) (c) 5

7 y = x + c x + y = 1 (1) c = 1, x () c = 1, x (3) c x (inequality), (3) x > x (4) x > x (5) x 1 0 (6) x > 1 x y 1 y = x x > 1 y = O 1 1 x -1-3 x > 1 f(x) > g(x) ( ) (7) x 3 3x + x < 0 l, m l : y = f(x) m : y = g(x) (*), l y m x, x > 1 y = x y = 1 y x ( 3) 1, xy x (1) x 1 () x x > 0 13 ( 7 (1) x < 1 < x x () x = 1 x 1 (3) a < b < c x < a, b < x < c x x f, g y, y = g(f(x)) or y = (g f)(x) 14 f(x) = x, g(x) = x + 1, ( 8 ( 7 (1) (x 1)(x ) > 0 () (x 1) 0 (3) (x a)(x b)(x c) < 0 ( 8 (f g)(x) = f(g(x)) = f(x + 1) = (x + 1), (g f)(x) = g(f(x)) = g(x) = x + 1 (1) (f g)() = 6 () (g f)() = 5 6

8 1.6 1 (1) (f g)()? () (g f)()? 15 f(x) = g(x ) 1, g(x) = x + 1, f(g(x)) = g(f(x)) x ( x, y xy ax + by + pxy + qx + ry + c = 0 (1) : y = ax, x = ay (), : ax + by = c (a, b, c > 0) (3) : ax by = c (a, b, p, q, c: ) (1), () (3) (4) 16 ( ) ( 10 (1) (, ) () (1, 0) ( 1, 0) 4 (3) (1, 1) x (4) 3 (a, b, c) 1 ( 9 f(g(x)) = f(x+1) = (x+1), g(f(x)) = g(g(x ) 1) = g((x + 1) 1) = x + 1 x x = 0 ( 10 (1) (x ) + (y ) = () x + y = 1 (3) 4 3 y = 1 (x 1) + 1 (4) (x a) + (y b) + (z c) = f(x) x (optimization problem) f (objective function or cost function) x g(x) = 0 (constraint function) 1 ( 11 (1) x + y = 4 x + y (a) (b) xy x, y () x + y = 1 x + y x, y ( ) 1, 5, , 1, , 34, 10, 6 (1) 1 x y 1 r ( 11 (1)(a) z = x + y, x + y = 4 (1)(b) (x, y) = (, ), () min(x + y ) = 1/, (x, y) = (1/, 1/) 7

9 1.7,, 1 x, y ( 1 () 1, ( , 1 McCulloch Pitts 1943 ( ) x 1 x w 1 w θ z z = 1, z = 0 (x 1, x ) z 1 (w 1 x 1 + w x θ ) z = 0 (w 1 x 1 + w x < θ ) w 1, w (), θ ( 1 ( ) 1 / x + 1 / y 10[ ] 1 / y 6[ ] 5hr/ x + yhr/ 34[hr] r = 4000[ / ]x / y { 0 x 0 y ( 13 ( ) 6, 4 () x, y r r 3 (1) AND, (w 1, w, θ) ( 14 () (w 1, w, θ) (x 1, x ) = (1, 0) z = 0 ( 0.1 ) ( 15 Rosenblatt 1957 (Perceptron) ( ) 1.7,, ( ) ( 16 (1) x y () x y (3) 60 (4) 60 (5) 80 (6) ( 14 ( ) (w 1, w, θ) = (1, 1, 1.5) ( 15 ( ) w 1: w : θ: ( 16 (1) y = ax + b (a, b ), ax + by + c = 0 (a, b, c ) () y = kx (k ) (3) E taro < 60 (E taro ) (4) E jiro 60 (E jiro ) (5) E sabu 80 (E sabu ) (6) F = k/n (F n, k ) 8

10 , [ ] ( ) 5 H, S H = (S 5) ( ) (1) (), (3) (4) [ ] 5, 3 10, ( 17 3, v 100 m 100 [m] v ( 17,, T,H,S { T 5 = 3(H 5) T 10 = (S 10)/ ), 5, x, y, z (1) 100 v () (3) 100 [m] v [m/s] 100 [m] v [s] (1) v v, () v (3) [ ] 15 km, 6 km, 9 km ( , 1 y 1 O θ 1 x 1 (radian)[rad] ( 4 ( 18 v b, v w { 15[km] v b = 6[km] v w = v w + 9[km/hr] v b ( 4 (radian) radius 9

11 1.8 1 ( 4) x r θ x = rθ θ θ = x r [rad] 360 (1 1[deg] ) (SI ( 5 ) (1) θ θ [rad] ( 19 () (1) 45 () 360 (3) 70 (4) 90 (5) 5 (6) 180 (7) 10 (8) 60 (9) 30 (3) 1 (1) 30 () 60 (3) 90 (4) 180 (4) ( 0 (a) m 1 rad (b) 10 cm π [rad] (c) r θ [rad] ( 5 Le Système International d Unités The International System of Units. [s], [m], [kg], [A], [K], [mol], [cd] π θ 360 ( 19 θ [rad]= ( 0 (a) m (b) 10π cm (c) rθ (1) R, r, ( 6 θ r R θ () R, R = km, r km θ 1, sin θ θ θ (3) 1 (4) θ (5) (Eratosthenes, BC75 -BC195 ) km 50 ( 1 ( 6 ( km/day 50 days 360 /7. = 10

12 .1 ().1.1 y 1 sin θ 1 O 1 P : (x, y) = (cos θ, sin θ) θ cos θ 1 x y O 1 θ[rad] 1 x 6 θ 1 sin θ θ (1) sin θ + cos θ = 1 () 1 + tan θ = 1 cos θ y θ 5 () x + y = 1 (1, 0) θ P = (x, y) ( 5) sin θ = y cos θ = x tan θ = y x (sine) (cosine) (tangent) (trigonometric function) csc θ = 1 y = 1 sin θ sec θ = 1 x = 1 cos θ cot θ = x y = 1 tan θ ( 7 (cosecant) (secant) (cotangent) (inverse trigonometric function) 1 θ ( 7 csc cosec 3 (1) sin π 6 = 1 () sin π 4 = (3) sin π 3 = 3 4 a sin x a cos x (a ) ( ) (1) sin( x) () cos( x) (3) sin(x + π) (4) sin(x + π ) (5) cos( x + π) (6) cos(x + 3 π) (7) sin( x + π ) (8) cos( x π ) 5 θ 1 6 y = sin θ θ [rad] θ sin θ θ (θ 1) sin 1 ( sin 1 = ).1. 6 (Hipparchos, BC190-BC15 ) ( 8 ( 8 11

13 . (1) ( 9 θ x R x R θ x () 89 (3) 40,000km km 38.4 km...1 a a ( ) (power) a n a n a n a (base), n (exponent) ( 10 n a n = (a 1 ) n = 1 a n 1 (1) (x n ) m = x n m () x n x m = x n+m (3) x 0 = 1 ( 9 ( y = ax k (a, k )..3 n y = a 1 n, (a > 0) a n m = y n = a, (y > 0) ( a 1 m ) n x f(x) = a x, (a > 0) (exponential function) a > 0 x f(x) > 0 ( (1) 4 3 () 4 3 (3) (4) (5) ( ) ( ) ( ) (6) 4 e e+ ( (1) 8 () 1/8 (3) 1/8 (4) 8 (5) 3 (6) e 1

14 ...4 a, a x, x = a x a (square root), a n, n a x, x n = a x a n (n-th root), n ()..5 a, a 1 a (= a 1 ) a a = a = a i () (3) x ( 4 (1) 4 () 4 (3) 4 1 (4) 8 3 (5) 3 8 (6) ( 8) 1 3 (7) 16 4 (8) 4 16 (9) x () n a (n > ) (1) a(> 0) n a = a 1 n () a(< 0) n a = n a n : n : ( n ) ( 3 (1) ( )( ) ( 3 (1) () (3) x (1 + x) 1 x/( ), ( 1 + x ) 1 (n ) lim n ( 1 + x n) n (4) x = 1 (100%) ( 11 e ( e = lim n =.7188 n n) ( 4 (1) ± () (3) (4), 1 ± 3i (5) (6) (7) ±, ±i (8) (9) ( 11 e e (Jakob Bernoulli, , ) 13

15 r = r a 0 (1 + r) 1 (= a 0 ) r ( a r (=.5a 0 ) ) r ( 4 a r 3 (=.37a 0 ) 3 3)... 1/n r ( a r ) n n n log 16 log 8 ) 7 A = = ( 5 e (4) ( e x = lim 1 + x n (5) n n) ( 1 a 3 a x = e x ln a (6) (1) (4) (5) () (6) e ( 13 (1) () (3) a x log a x, John Napier, , ( 14 log logistic algorithm (logarithm), log = 4 3 = 4+3 = ( ( 1 e ( 13 e y (x) = y(x), y(0) = 1 y = exp x e = exp 1.3. x a( 1) log a x f(x) = log a x ( 5 (1) () 338 (3) 96 ( 14 14

16 .3 y = log a x x = a y a > 0 a = 1 1 a a > 0, a 1 x = a y > 0 x x > 0 y = log a x y = a x a (1) y = a x (a > 0) () y = log a x (a > 0, a 1) e log e (natural logarithm) ln 10 log 10 log (log 10 ) (log e ) log log a, b, c, x, y > 0 1 (1) log a 1 = 0 () log a a = 1 (3) log a b c = c log a b (4) log a xy = log a x + log a y (5) log a c = log a b log b c (log b c = log a c log a b ) ( 6 (1) log 3 15 log 3 5 () log ( 3 + 1) + log ( 3 1) (3) log 3 log 3 4 (4) log 3 (5) e ln (6) log log a = (a = ) log a = log a ( = ) (1) (log a x x > 0) () (3) (4) ( 7 (1) log (x 4) = 3 () = ln(x + e ) (3) log x + 3 log 8 (x 1) = 1 (4) log 3 (x + 3) log 9(3x + 9) = 1 (5) log 4 x + 6 log x = 5 ( 6 (1) 1 () 1 (3) (4) 3 (5) (6) 9 ( 7 (1) x = 1 () x = 0 (3) x = () (4) x = 0 (5) x = 4, 8 15

17 (1) log log 4 () log 1 log y 10 0 y = x 5 p, q ( 8 log a p > log a q, (a > 0, a 1) ( 9 (1) log x > log 4 () log 1 x > log 1 4 (3) log x > log (4) log 1 x > log (5) log 1 (x 1) > log 1 (3 x) (x, y), x log x(),, y = x y = x x 7 y = x X, Y y = x 7 1 (1) y = x 3 () y = 1 x M E 1 n log y = log x X = log x, y = log y Y = X ( 8 0 < a < 1 p < q, 1 < a p > q ( ) ( 9 (1) x > 4 () 0 < x < 4 (3) x > 1 (4) 0 < x < 3 (5) 3 1 < x < () () () 16

18 .4, 1 ( ) (1) M n ( 30 () E n ( E M log 10 E = M (1) a 10 b x ) ( 3 () ( 33 log 10 3 = () 7.0, 011 ( ) ( 15 N dn dt = λn λ N(t) = N 0 e λt (7) N 0 t = 0 (1) τ τ N(t) () τ = ln λ (3) (7) τ N(t) N 0 = ( ) t 1 τ ( 30 n = a10 bm, (a, b ) ( 31 n = ae b, (a, b ) ( 15 ( 3 10 b 1 a < 10 b (

19 3 3 n n (rectangular coordinate system) (Cartesian coordinate system) ( 16 r (, radius) (, argument) θ 1, θ,..., θ n 1 (polar coordinate system) 3.1 (), ( 35 (1) (1, 0) () (0, 1) (3) ( 1, 1) (4) (, ) (5) ( 1, 3) 3, ( 36 (1) (, π 3 ) y O r θ P x () (4, π 4 ) (3) (1, π) (4) (, 7 6 π) (5) (0, 3) 8 P O r ( ) θ( ) θ OP 0 θ < π ( π < θ π ) (r, θ), (circular polar coordinates), θ x 1 (r, θ) (x, y) ( 34 ( 16 ( René Descartes, , ) ( 34 (x, y) = (r cos θ, r sin θ) 4 ( 37 (1, (1) ) (1, () ) ( (3), 1) (, (4) ) ( (5), 0) (, (6) ) ( (7), 1) ( (8), π ) (1, (9) ) ( (10), π 4 ) 3. () P 9 O r( ), OP xy x ( 35 (1) (1, 0) () (1, 3 π) (3) (, 5 π) (4) (, 7 π) (5) 4 4 (, π) 3 ( 36 (1) (1, 3) () (, ) (3) ( 1, 0) (4) ( 3, 1) (5) (0, 0) ( 37 (1) 0 () 0 (3) ± 3 (4) π 6 (x = 3 ), 5 6 π (x = 3 ) (5) ± (6) 0 (x = ), π (x = ) (7) 0 (8) 1 (9) 1 (10) 18

20 4 z 4 θ O φ r P y (imaginary unit) i ( 17 i ( 18 x 9 i = 1 i = 1 ϕ, OP z θ (r, θ, ϕ) 3 (spherical polar coordinates) 0 θ π, 0 ϕ < π 3, 3 θ θ (r, θ, ϕ) (x, y, z) ( 38 ( 39 * (1, 0, 0) (1) (0, 1, 0) () (0, 1, 1) (3) (4) (, π 6, π 4 ) (5) (, π 4, π) (real number) a, b i z = a + bi (complex number) z (real part) Re(z), (imaginary part) Im(z) z = a + bi Re(z) = a Im(z) = b Re(z) = 0, Im(z) 0 z (purely imaginary number) 1 (1) x + x + = 0 () x 3 = 1 (3) x 4 = 1 z = a+bi (, complex conjugate) z(= a bi) ( 38 (x, y, z) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) ( 39 (1) (1, π, 0) () (1, π, π ) (3) (, 3 4 π, π ) (4) ( 1, 1, 3) (5) (, 0, ) ( 17 i j ( 18 1, i, i 19

21 4. 4 z 1, z (distance)p Q z 1 z = z 1 z P Q = z 1 z = (a 1 a ) + (b 1 b ) ( ) (a + bi ) ( 40 (1) ( + i) () + i i (3) 1 + i 1 i 4. 1 (1) z = z z () (*) z 1, z P Q z 1 z z 1 z Im b z z = a + bi (1) z 1 = 1 z = 1 + i O a 10 Re z = a + bi (a, b) ( 10) (complex plane) (a) z 1, z (b) ( 41 (i) z 1 z (ii) z z (iii) z 1 () (a) x 3 = z () (absolute value, modulus) z z z = a + bi z = a + b, P : z 1 = a 1 + b 1 i Q : z = a + b i ( 40 (1) 3 + 4i () 1 i (3) i (b) x 4 = 1 (c) x + x + 1 = 0 (3) x +x+1 ϵ = 0 ϵ 4.4 z = a + bi arg z ( 11) z ( z, arg z) ( 41 z 1 z = 1, z z = z 1 = 5 0

22 4.4 4 Im b z z = a + bi (), Re{z} = ( 43 O θ = arg z a Re 3 z π/, z 11 z = 1 + i, z =, arg z = π + nπ, (n: ) 4 (1) ( 44 () z z = ( 45 z = 1 + i (, π 4 + nπ ), (n: ) 1 z 1 = 1 z = 1 + i (1) z 1, z () z 1, z (a) (b) i (c) i (d) z (3) z z II z i z (1) ( 4 ( 4 z i = z ( 43 z = + i ( 44 iz = z ( 45 z = ± (1 i) 1

23 x f(x) = f( x) f(x) (even function) f(x) = f( x) f(x) (odd function) 1 (a) (b) (c), ( 46 exp x e x (1) x () sin x (3) cos x (4) exp x (5) sin x (6) ln x (7) exp( x ) (8) e x e x sin x (9) x (1) y = 1 + e x () y = ex e x e x + e x x 0 x < 0? (1) () x ±? ( x OK) x = 0? ( 46 (a) 1, 3, 5, 7, 9 (b), 8 (c) 4, (1) y = ex + e x (y = ex y = e x ) () y = e x sin x (y = sin x y = e x ) y = sin x ( 47 (1) x π () x (3) y 1 (4) y (5) x x π (6) x π x ( 48 (1) y = f(x) x a () y = f(x) x c (3) y = f(x) y b (4) y = f(x) y d 3 ( 47 (1) y = sin(x π ) () y = sin x (3) y 1 = sin x (4) y = sin x (5) y = sin x π (6) y = sin( x π x π ) = sin ( 48 (1) y = f(x a) () y = f(x/c) (3) y b = f(x) (4) y/d = f(x)

24 5.4 5 (1) y = x x + 1 x 1 () x + 4y = 1 y 1 4 (1) y = ex 1 e x+1 e x 1 + e x+1 + (5. () ) 1 () y =, (h, T : ) 1 + e x h T T (> 0) () y = cos x (0 x π) 3 x cos 1 x = sin 1 x+a, (a : ) a, 0 a < π 4 ( 50 (1) y = x x (x 1) () y = e x (3) y = ln(x + 1) (x > 1) (4) y = sin 3x (5) y = f(3x) 5.4 y = f(x) y x x = f 1 (y) x, y y = f 1 (x) f(x) y = sin x, y = cos x y = sin 1 x, y = cos 1 x y = arcsin x, y = arccos x 1 ( 49 (1) sin 1 1 ( [ π, π ] ( 19 ) xy x = t (1) (t ) y = t + x = cos θ () (θ ) y = sin θ 5.5. (r, θ) () sin 1 1 ( (, ) ( 0 ) (1) r = 1 (3) cos 1 1 ( (, ) ) () r = θ (θ 0) (3) r = cos θ ( π θ π ) (1) y = sin x ( π x π ) ( 49 (1) π/ () π + nπ, 5π + nπ (n ) (3) ± π nπ (n ) ( 19 x [a, b] a x b ( 0 x (a, b) a < x < b ( 50 (1) y = x () y = ln x (3) y = e x 1 (4) y = 1 3 sin 1 x (5) y = 1 3 f 1 ( x ) 3

25 (r, θ, ϕ) (1) r = 1 () θ = π 3 r = ϕ (r > 0) (3) θ = π z = x + iy (1) Re(z) = 1 () z = 1 (3) z = i (4) z = 1 (5) z 1 = 1 (6) z i = (7) z = (1 + i)t (t ) 4

26 A A n log n n log n n log n n log n n log n n log n n log n

27 B B (1) yamaguchi-u.ac.jp () (3), 6

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