l x a b l = ax + b l x x l a b l = ax + b 5 cm cm 1 x l l = 0.5x 5cm cm 1 x l l = 0.25x 1.25 値 x 値 値 x
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1 l x a b l = ax + b l x x l a b l = ax + b 5 cm cm 1 x l l = 0.5x 5cm cm 1 x l l = 0.5x 1.5 値 x 値 値 x 値 長さ l 長さ cm 長さ l 長さ cm 1 1 1
2 5cm /99 cm 1 x l l = 5/99x 5/99 値 x 値 長さ l 長さ cm 3 3 x l a b c x y y = a log c x + b log c 10 x l a b l = a x + b (1) x l x l cm x l l =.5 x 1 1 = = = cm =.605 値 x 値 x 値 長さ l 長さ cm 4 1
3 10.5 cm 4 0.1cm cm x l l = (5/ 5) x 5 値 x 値 x 値 長さ l 長さ cm 5.1 (1) x 6 x y y = ax + b y = ax b y = ab x y = a log e x + b 1 y = 90x () y = 0.8x (3) y = x (4) 1 log a a y = a log e x + b = a log e 10 x + b = a (.30 ) x + b = a x + b A. 3
4 y (linear) 5000 Y (= y) 3 1 y (logarithmic) x (linear) x (linear) (a) (b) y (linear) 5000 Y (= y) 3 1 y (logarithmic) x (logarithmic) x (logarithmic) 0 1 X (= x) 0 1 X (= x) (c) (d) 6 y = 000 log e x (5) x y 10 X Y X = x y x = 10 X 4
5 y = 10 Y X Y x y y = 0 log x y = 90 x y = 0.8 x y = x y (linear) y = 000 log e x x (linear) y 8 () y = 10 Y y = 90x Y = 90x (90x + 500) 90 + (x + 500/90) Y (x ) Y x 8-7 5
6 Y (= y) y (logarithmic) y = 90 x y = 0.8 x y = x y = 000 log e x x (linear) 8 (3) y = 10 Y y = 0.8x 10 Y = 0.8x Y = x Y x Y x 8-7 (4) y = 10 Y y = x 10 Y = x Y = x Y x Y x 8 10 Y 1 x 100 Y (= y) x 100 Y (= y) a b y = ab x 10 Y = ab x a + x b 6
7 b a a b (5) y = 10 Y y = 000 log e x Y = 000 log e x Y = 3 + (log e x + 1/) Y (log e x ) x y = 90 x y = 0.8 x y = x y = 000 log e x y (linear) x (logarithmic) 0 1 X (= x) 9 () x = 10 X y = 90x y = X y 10 X y X 9-7 7
8 (3) x = 10 X y = 0.8x y = 0.8(10 X ) y = 0.8(10 ) X y = X y X 9-7 (4) x = 10 X y = x y = X y = 0.9 ( ) X y X y X 9-7 (5) x = 10 X y = 000 log e x y = 000 log e 10 X y = 000X log e y X y 4606X y X 9 a b y = a log e x + b y = a log e 10 X + b y = (log e 10)aX + b (log e 10)a b a b 8
9 x y y = 90 x y = 0.8 x Y (= y) 3 1 y (logarithmic) y = x y = 000 log e x x (logarithmic) 0 1 X (= x) 10 () x = 10 X y = 10 Y y = 100x y = 90x Y = X (90 10 X + 500) (90 10 X + 500) 10 (3) x = 10 X y = 10 Y y = 0.8x 10 Y = 0.8 (10 X ) 10 {0.8 (10 X ) } (10 X ) Y X Y X a b 9
10 y = ax b 10 Y = a(10 X ) b 10 Y = a(10 X ) b a + 10 b X a + bx b a a b (4) x = 10 X y = 10 Y y = x 10 Y = X ( X ) X 1.1 Y X Y X - 7 (5) x = 10 X y = 10 Y y = 000 log e x y = 000 log e x Y = 000 log e 10 X Y = 000X log e (000X log e ) Y (4605X ) Y (X ) Y X y = ax + b 7 10
11 8 y = ab x 9 y = a log e x + b 10 y = ax b 1 X Y X 1 Y X = 0 Y b = a = 0.1 a = y = 0.8x 1 - y = ax + b a b - y = ab x b a - y = a log e x + b (log e 10)a b - y = ax b b a y = x (4) x 10 (6) y 000 (7) (8) (9) y = (x 10) (6) y = x (7) y = x (8) y = 0.9 x (9) 11
12 (6) y = 10 Y 10 y = (x 10) 10 Y = (x 10) (x 10) (x 10) 1.1 Y = x Y x (b) 1 (7) y = 10 Y 10 y = x Y = x Y = x ( x + 000) 11(b) x x 0 (8) y = 10 Y 10 y = x 10 Y = x (3 1.1 x ) 3 + x 1.1 Y x (b) (9) y = 10 Y 10 y = 0.9 x 10 Y = 0.9 x (0.9 x ) y = x 10 = x 1.1 ( 10) x x 1
13 0.9 + x Y x (b) x y Y axis (linear) y = x y = x -10 y = x +000 y = x y = 0.9 x Y axis (logarithmic) y = x y = x -10 y = x +000 y = x y = 0.9 x X axis (linear) X axis (linear) (a) (b) y = 0.8x (3) x 10 (10) y 0 (11) (1) (13) y = 0.8(x 10) (10) y = 0.8x (11) y = 1.3x (1) y = 0.8x 3 (13) (10) x = 10 X y = 10 Y y = 0.8(x 10) 13
14 10 Y = 0.8(10 X 10) 10 {0.8(10 X 10) } Y = (10 X 10) Y X 1(b) X 3 x X X (11) x = 10 X y = 10 Y y = 0.8x Y = 0.8(10 X ) {0.8(10 X ) + 000} {0.8(10 X ) + 000} Y X 1(b) (10) X 3 y X X (1) x = 10 X y = 10 Y y = 1.3x 10 Y = 1.3(10 X ) 10 {1.3(10 X ) } Y = X Y X Y X 1(b) 1 x 1(b) (13) x = 10 X y = 10 Y y = 0.8x 3 10 Y = 0.8(10 X ) 3 10 {0.8(10 X ) 3 } Y = 3X Y 3X
15 Y X 1(b) 1 x 1(b) x y Y axis (linear) y = 0.8x y = 0.8(x - 10) y = 0.8x y = 1.3x y = 0.8x 3 Y axis (logarithmic) y = 0.8x y = 0.8(x - 10) y = 0.8x y = 1.3x y = 0.8x X axis (linear) X axis (logarithmic) (a) (b) 1 - A A.1 log log log b c c b c = b a c b log b c = log b b a = a a = 10 = = = 10 1 = = 10 3 = 3 15
16 0.1 = 10 1 = 1 1 = 10 0 = 0 log log 8 = log 3 = 3 log 64 = log 6 = 6 log 1 = log 0 = 0 1 log 0.5 = log = log 1 = 1 1 log 0.5 = log 4 = log 1 = log = A. log A..1 log log a b c = c log a b b = a n n log a b = log a a n = n log a b c = log a (a n ) c = log a a nc = nc log a b = n = c log a b log 8 = log 3 = 3 log 16
17 A.. log log a bc = log a b + log a c a B = b a C = c A B bc = a B a C bc = a B+C a log a bc = log a a B+C log a bc = B + C B = log a b C = log a c log a bc = log a b + log a c A..3 a b log b a d = a h (14) a log log a d = log a a h log a d = h (14) d = a log a d b log log b d = log b a log a d log b d = log a d log b a log b a a b A.3 e e a e d dx ax = a x (15) 17
18 (15) d a x+h a x a h 1 = lim = a dx ax x lim h 0 h h 0 h (15) (16) (16) a lim k 0 a h 1 lim = 1 h 0 h (16) h lim h 0 a h 1 = 1 h = log a(1 + k) h 0 k 0 log a h = 1 + k log a (1 + k) = 1 k lim log a (1 + k) k 1 = 1 k 0 lim(1 + k) k 1 = a k 0 (17) (17) a e e e = lim(1 + k) k 1 k 0 k = 1/m m ( e = lim m (18) m m) m (1 + 1/m) m 13 m (1 + 1 / m ) m m 13 e (18) a n = (1 + 1/n) n (18) a n = ( a + 1 n) n 18
19 = n ( ) 1 nc p p=0 p=0 n p n ( ) n! 1 = p!(n p)! n p = 1 + n 1 1! n + n(n 1)! C n! (n p)! 1 n(n 1) {n (p 1)} 1 n(n 1) p n p! n n! = n 1! n + 1 n(n 1) n(n 1) {n (p 1)} n(n 1) 1! n p! n p n! n n = n ! n! = ! ( 1 0 n n 0 n n n p! + 1 n (n 1) n (n ) n (n 1) n! n n n n ) + 1 ( 1 0 ) ( 1 1 ) + + 1! n n p! + 1 n! n 0 n 1 n (p 1) + n n n ( 1 0 ) ( 1 1 ) ( 1 n ) ( 1 n 1 ) n n n n ( 1 0 ) ( 1 1 ) ( 1 p 1 n n n n n ) + (19) (19) a n n a n a n (19) 1/1! 1/! 1/p! 1/n! (1 0/n) (1 0/n)(1 1/n) (1 0/n)(1 1/n) {1 (p 1)/n} (1 0/n)(1 1/n) {1 (n )/n}{1 (n 1)/n} {1 (p 1)/n} /1! 1/! 1/p! 1/n! a n 1/1! 1/! 1/p! 1/n! a n ! + 1! n! (0) 3! = 3 1 = 3 > = 4! = = 4 3 > = 3 p! = p(p 1)(p ) 3 > p 1 (0) a n ( ) n 1 n 1 1 = { ( ) n } 1 = = 3 1 n 1 < 3 a n a n (18) e 19
20 A.4 + = = ( = 1 + ) 100 ( = 1 + ) 100 = = % n n = n = 1.01 log n = log 1.01 n log [1] 0
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180 30 30 180 180 181 (3)(4) (3)(4)(2) 60 180 (1) (2) 20 (3)
12 12 72 (1) (2) (3) 12 (1) (2) (3) (1) (2) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) (2) 180 30 30 180 180 181 (3)(4) (3)(4)(2) 60 180 (1) (2) 20 (3) 30 16 (1) 31 (2) 31 (3) (1) (2) (3) (4) 30
... 6 1) 2) No. 01 02 03 04 05 06 07 08 09 10 11 12 No. 1 2 2 3 3cm 4 untitled
1....1 2....2 2.1...2 2.2...2 3....14 3.1...14 3.2...14 4....15 4.1...15 4.2...18 4.3...21 4.4...23 4.5...26 5....27 5.1...27 5.2...35 5.3...54 5.4...64 5.5...75 6....79 6.1...79 6.2...85 6.3...94 6.4...
113 120cm 1120cm 3 10cm 900 500+240 10 1 2 3 5 4 5 3 8 6 3 8 6 7 6 8 4 4 4 4 23 23 5 5 7
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( ) () () 3 cm cm cm cm cm cm 1000 1500 50 500 1000 1000 1500 1000 10 50 300 1000 2000 1000 1500 50 10 1000 2000 300 50 1000 2000 1000 1500 50 10 1000 2000 300 30 10 300 1000 2000 1000 1500 1000
2002 (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (6) (7) (8) (1) (2) (3) (4) (1) (2) (3) (4) (5) (6) (7) (8) No 2,500 3 200 200 200 200 200 50 200 No, 3 1 2 00 No 2,500 200 7 2,000 7
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-25- -26- -27- -28- -29- -30- -31- -32- -33- -34- -35- -36- -37- -38- cm -39- -40- 1 2 3 4 4 3 2 1 5 5-41- -42- -43- -44- -45- -46- -47- -48- -49- -50- cm -51- -52- -53- -54- -55- -56- -57- -58- -59- -60-
II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1
II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2
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i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
(1) (2) (1) (2) 2 3 {a n } a 2 + a 4 + a a n S n S n = n = S n
. 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d = 99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n
さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a
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φ + 5 2 φ : φ [ ] a [ ] a : b a b b(a + b) b a 2 a 2 b(a + b). b 2 ( a b ) 2 a b + a/b X 2 X 0 a/b > 0 2 a b + 5 2 φ φ : 2 5 5 [ ] [ ] x x x : x : x x : x x : x x 2 x 2 x 0 x ± 5 2 x x φ : φ 2 : φ ( )
5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P
![5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P](/thumbs/90/103736623.jpg "5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P")
p P 1 n n n 1 φ(n) φ φ(1) = 1 1 n φ(n), n φ(n) = φ()φ(n) [ ] n 1 n 1 1 n 1 φ(n) φ() φ(n) 1 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 4 5 7 8 1 4 5 7 8 10 11 1 13 14 15 16 17 18 19 0 1 3 4 5 6 7 19 0 1 3 4 5 6 7
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
橡魅力ある数学教材を考えよう.PDF
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日経テレコン料金表（2016年4月）
1 2 3 4 8,000 15,000 22,000 29,000 5 6 7 8 36,000 42,000 48,000 54,000 9 10 20 30 60,000 66,000 126,000 166,000 50 100 246,000 396,000 1 25 8,000 7,000 620 2150 6,000 4,000 51100 101200 3,000 1,000 201
122011pp.139174 18501933
122011pp.139174 18501933 122011 1850 3 187912 3 1850 8 1933 84 4 1871 12 1879 5 2 1 9 15 1 1 5 3 3 3 6 19 9 9 6 28 7 7 4 1140 9 4 3 5750 58 4 3 1 57 2 122011 3 4 134,500,000 4,020,000 11,600,000 5 2 678.00m