数学の基礎訓練I

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基礎数学I

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )


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Note.tex 2008/09/19( )


sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

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I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

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I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1.............. 3 1.5............. 3 1.5.3.............. 3 1.5.4........... 4 1.5.5............ 4 1.5.6.......... 5 1.5.7.......... 6 1.5.8.............. 6 1.5.9.............. 7 1.6.......... 7 1.6.1............ 7 1.6............ 8 1.7,,..... 8 1.7.1........... 8 1.7............... 9 1.8.............. 9 1.8.1............ 9 1.8....... 10 11.1 ()............. 11.1.1.......... 11.1....... 11.............. 1..1................ 1................. 1..3.............. 1..4........... 13..5................ 13..6............ 13.3.................. 14.3.1...... 14.3....... 14.3.3....... 15.3.4............ 15.3.5............ 16.4..... 16.4.1............ 16.4........... 17.4.3............ 17 3 18 3.1 ().......... 18 3. ().......... 18 4 19 4.1............. 19 4................... 0 4.3.......... 0 4.4.......... 0 5 5.1.............. 5................. 5..1....... 5........... 5.3.......... 5.4................... 3 5.5........... 3 5.5.1..... 3 5.5... 3 5.5.3.. 4 5.5.4..... 4 A 5 B 6

1 1 1.1 ( 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 (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 = 1 + + 3 = 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) = (0 + 1 + ) + (1 + + 3) + ( + 3 + 4) = 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

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) 1 + 3 + 5 + 7 + 9 + () 1 + 3 4 + 5 6 + 7 8 + 9 (3) 1 + 3 + 4 3 + 3 ( 4 r t+1 + γr t+ + γ r t+3 + + 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! 1.4 1.4.1 ( 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?)

1.5 1 A E F D 1.5 B G 1 1.4. a, b ab (1) ( 1) () 1.4.3 ( 1.4.4 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 1.5.1 (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 1 + + a 1 x + a 0 (1) 1.5.3 (1) f(x) f(x) = (x c)(b n 1 x n 1 + + 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

1.5 1 1.5.4 3 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]? 1.5.5 (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

1.5 1 - 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 = 1 1.5.6 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

1.5 1 11 y = x + c x + y = 1 (1) c = 1, x () c = 1, x (3) c x 1.5.7 (inequality), (3) x > x (4) x > x (5) x 1 0 (6) x > 1 x y 1 y = x x > 1 y = 1 - -1 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 1 3 1.5.8 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

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 ( 9 1.5.9 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) = 1 1.6 1.6.1 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, 4000 1 1, 1, 1000 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

1.7,, 1 x, y ( 1 () 1, ( 13 1.6. 0, 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 + 1000 / 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,, 1.7.1 1 ( ) ( 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

1.8 1 1.7., [ ] ( ) 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 ( 18 1.8 1.8.1, 1 y 1 O 1 1 4 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

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 1.8. 1 (1) R, r, ( 6 θ r R θ () R, R = 1.5 10 8 km, r 6.4 10 3 km θ 1, sin θ θ θ (3) 1 (4) θ (5) 109 100 (Eratosthenes, BC75 -BC195 ) 7. 18.5 km 50 ( 1 ( 6 ( 1 18.5 km/day 50 days 360 /7. = 10

.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 = 0.017454 ).1. 6 (Hipparchos, BC190-BC15 ) ( 8 ( 8 11

. (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 ( 10 0 0.. 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) ( ) 3 1 4 ( ) 1 3 4 ( ) 7 1 3 (6) 4 e e+ ( (1) 8 () 1/8 (3) 1/8 (4) 8 (5) 3 (6) e 1

...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) 16 1 4..6 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 1 + 1 n =.7188 n n) ( 4 (1) ± () (3) (4), 1 ± 3i (5) (6) (7) ±, ±i (8) (9) ( 11 e e (Jakob Bernoulli, 1654-1705, ) 13

.3 1 1 r = 1.0 1 r a 0 (1 + r) 1 (= a 0 ) r ( a 0 1 + r (=.5a 0 ) ) r ( 4 a 0 1 + r 3 (=.37a 0 ) 3 3)... 1/n r ( a 0 1 + r ) n n n log 16 log 8 ) 7 A 14 17 14 17 3.80735 4.08746 = 3.80735+4.08746 = 7.89481 38... 1 ( 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) 18 64 4096 () 13 3 8 (3) 18 138 184 a x log a x, John Napier, 1550-1617, ( 14 log logistic algorithm (logarithm), log 0 0.3.3.1 16 8 16 8 = 4 3 = 4+3 = 7 16 8 ( ( 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

.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.3.3 3 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 3.3.4 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

.4.3.5 4 (1) log log 4 () log 1 log 1 4 10 3 10 10 1 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 1 9 4 (4) log 1 x > log 1 9 4 (5) log 1 (x 1) > log 1 (3 x) 3 9.4.4.1 (x, y), x log x(),, y = x y = x 10 1 10 10 10 1 10 0 10 1 10 10 3 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

.4, 1 ( ) (1) M n ( 30 () E n ( 31.4. 3 E M log 10 E = 4.8 + 1.5M.4.3 5 3 0 10 (1) a 10 b x ) ( 3 () 3 0 10 ( 33 log 10 3 = 0.4771 1995 () 7.0, 011 ( ) 9.0 4 ( 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 ( 33 3 0 10 10 17

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, 1596-1650, ) ( 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

4 z 4 θ O φ r P y 4.1 1 (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

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 = 1 4.3 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

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

5 5 5.1 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 5. 5..1 1 (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, 6 5.. (1) y = ex + e x (y = ex y = e x ) () y = e x sin x (y = sin x y = e x ) 5.3 1 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)

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 ) 5.5 5.5.1 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 + 6 6 3 nπ (n ) ( 19 x [a, b] a x b ( 0 x (a, b) a < x < b ( 50 (1) y = x + 1 + 1 () y = ln x (3) y = e x 1 (4) y = 1 3 sin 1 x (5) y = 1 3 f 1 ( x ) 3

5.5 5 5.5.3 (r, θ, ϕ) (1) r = 1 () θ = π 3 r = ϕ (r > 0) (3) θ = π 3 5.5.4 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

A A n log n n log n n log n n log n n log n n log n n log n 1 0 51 5.6743 100 6.64386 151 7.384 01 7.65105 3300 11.6885 4051 11.98406 1 5 5.70044 101 6.6581 15 7.4793 0 7.6581 3301 11.68869 405 11.9844 3 1.58496 53 5.779 10 6.6743 153 7.5739 03 7.66534 330 11.6891 4053 11.98477 4 54 5.75489 103 6.6865 154 7.6679 04 7.6743 3303 11.68956 4054 11.98513 5.3193 55 5.78136 104 6.70044 155 7.761 05 7.67948 3304 11.69 4055 11.98549 6.58496 56 5.80735 105 6.7145 156 7.854 06 7.6865 3305 11.69043 4056 11.98584 7.80735 57 5.8389 106 6.779 157 7.946 07 7.69349 3306 11.69087 4057 11.986 8 3 58 5.85798 107 6.74147 158 7.30378 08 7.70044 3307 11.69131 4058 11.98655 9 3.16993 59 5.8864 108 6.75489 159 7.3188 09 7.70736 3308 11.69174 4059 11.98691 10 3.3193 60 5.90689 109 6.76818 160 7.3193 10 7.7145 3309 11.6918 4060 11.9876 11 3.45943 61 5.93074 110 6.78136 161 7.3309 11 7.711 3310 11.696 4061 11.9876 1 3.58496 6 5.954 111 6.7944 16 7.33985 1 7.779 3311 11.69305 406 11.98797 13 3.70044 63 5.9778 11 6.80735 163 7.34873 13 7.73471 331 11.69349 4063 11.98833 14 3.80735 64 6 113 6.8018 164 7.35755 14 7.74147 3313 11.6939 4064 11.98868 15 3.90689 65 6.037 114 6.8389 165 7.3663 15 7.74819 3314 11.69436 4065 11.98904 16 4 66 6.04439 115 6.84549 166 7.37504 16 7.75489 3315 11.69479 4066 11.98939 17 4.08746 67 6.06609 116 6.85798 167 7.3837 17 7.76155 3316 11.6953 4067 11.98975 18 4.16993 68 6.08746 117 6.87036 168 7.393 18 7.76818 3317 11.69566 4068 11.9901 19 4.4793 69 6.1085 118 6.8864 169 7.40088 19 7.77479 3318 11.6961 4069 11.99046 0 4.3193 70 6.198 119 6.8948 170 7.40939 0 7.78136 3319 11.69653 4070 11.99081 1 4.393 71 6.14975 10 6.90689 171 7.41785 1 7.7879 330 11.69697 4071 11.99117 4.45943 7 6.16993 11 6.91886 17 7.466 7.7944 331 11.6974 407 11.9915 3 4.5356 73 6.1898 1 6.93074 173 7.43463 3 7.8009 33 11.69784 4073 11.99188 4 4.58496 74 6.0945 13 6.9451 174 7.4494 4 7.80735 333 11.6987 4074 11.993 5 4.64386 75 6.88 14 6.954 175 7.4511 5 7.81378 334 11.6987 4075 11.9958 6 4.70044 76 6.4793 15 6.96578 176 7.45943 6 7.8018 335 11.69914 4076 11.9994 7 4.75489 77 6.6679 16 6.9778 177 7.46761 7 7.8655 336 11.69957 4077 11.9939 8 4.80735 78 6.854 17 6.98868 178 7.47573 8 7.8389 337 11.70001 4078 11.99365 9 4.85798 79 6.30378 18 7 179 7.4838 9 7.839 338 11.70044 4079 11.994 30 4.90689 80 6.3193 19 7.0113 180 7.49185 30 7.84549 339 11.70087 4080 11.99435 31 4.954 81 6.33985 130 7.037 181 7.49985 31 7.85175 3330 11.70131 4081 11.99471 3 5 8 6.35755 131 7.0334 18 7.50779 3 7.85798 3331 11.70174 408 11.99506 33 5.04439 83 6.37504 13 7.04439 183 7.5157 33 7.86419 333 11.7017 4083 11.99541 34 5.08746 84 6.393 133 7.0558 184 7.5356 34 7.87036 3333 11.7061 4084 11.99577 35 5.198 85 6.40939 134 7.06609 185 7.53138 35 7.8765 3334 11.70304 4085 11.9961 36 5.16993 86 6.466 135 7.0768 186 7.53916 36 7.8864 3335 11.70347 4086 11.99647 37 5.0945 87 6.4494 136 7.08746 187 7.54689 37 7.88874 3336 11.7039 4087 11.99683 38 5.4793 88 6.45943 137 7.09803 188 7.55459 38 7.8948 3337 11.70434 4088 11.99718 39 5.854 89 6.47573 138 7.1085 189 7.564 39 7.90087 3338 11.70477 4089 11.99753 40 5.3193 90 6.49185 139 7.11894 190 7.56986 40 7.90689 3339 11.705 4090 11.99789 41 5.35755 91 6.50779 140 7.198 191 7.57743 41 7.9189 3340 11.70563 4091 11.9984 4 5.393 9 6.5356 141 7.13955 19 7.58496 4 7.91886 3341 11.70606 409 11.99859 43 5.466 93 6.53916 14 7.14975 193 7.5946 43 7.9481 334 11.7065 4093 11.99894 44 5.45943 94 6.55459 143 7.15987 194 7.59991 44 7.93074 3343 11.70693 4094 11.9993 45 5.49185 95 6.56986 144 7.16993 195 7.60733 45 7.93664 3344 11.70736 4095 11.99965 46 5.5356 96 6.58496 145 7.17991 196 7.61471 46 7.9451 3345 11.70779 4096 1 47 5.55459 97 6.59991 146 7.1898 197 7.605 47 7.94837 3346 11.708 4097 1.00035 48 5.58496 98 6.61471 147 7.19967 198 7.6936 48 7.954 3347 11.70865 4098 1.0007 49 5.61471 99 6.6936 148 7.0945 199 7.6366 49 7.96 3348 11.70908 4099 1.00106 50 5.64386 100 6.64386 149 7.1917 00 7.64386 50 7.96578 3349 11.70951 4100 1.00141 5

B B (1) nishii@sci. yamaguchi-u.ac.jp () (3), 6

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