99課綱專刊 - 數學
192
普通高級中學課程 數學科課程綱要專刊 (99 年版) 教育部高中數學學科中心
Transcript of 99課綱專刊 - 數學
(99 )
93 8 31 94 1 20 95 6 23 97 1 24
97 1
1. - .......................................................................... 2. - ...................................... 3. - .................................... 4. (97 1 27 ).............. 5. Q&A (97 11 17 ) ....................... 6. (97 12 9 ).......................... 7. 8. 9.9
99 102 ...... 99 .................................... 99 ........................ 5
10. 11. 12. 13. 14. 15. 16. 17.
95 - .............................................................. 95 .................................... 95 (I) ................................ 95 (II)...............................
95 .................................................................... 138 95 () ................................................. 142 95 .................................................................... 153 95 () ................................................. 157
18. 19.
99 ........................................................ 175 95 ........................................................ 181
9 9
1 6 34 52 59 71 98 100 106
118 124 131 134
97 1 24 0970011604B
2* 2* 2* 2* 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4
2* 4 4 2* 4 4
1. AB A B 2. 4 3. 2
/
2
2
2
2
2
(2)
10 2
2
2
2
2
(2)
2
2 1 29 /31 0-2
2 1 29 /31 0-2
2 28/30
2 28/30
2 12 /14 0-19
2 12 /14 0-19 138 /150
0-3
0-3
1
-1-
2-4 4 33 / 35
2-4
2-5
2-5
2-21
1 2-21
433 35
533 35
533 35
2133 35
2133 35
60 198 210
* AB A B 160 () 120 () () 40 8
() 1. 2. 3. 2
-2-
4. () 1. 2. () 1. 2. () 1. 2. 3.
() 1. 2. 3. () 1. 2. 3
-3-
() 1. 2. 3. 4. 5. () 1. 2.
() 1. 2. 3. 4. () 1. 2.
() 1. 2. 3. 4. 5.4
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6. 7. 8. 9. () 1. 2. 3. 4.
5
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IIIIIIIV III IIIIV AB B A
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I 4 1. 1.1 1.2 1.2 1.3 2. 2.1 2.2 1. 1.1 1.2 1.3 1.3 4 2. 2.1 2.1 3. 3.1 3.1n
3.2 a 3.3 4. 4.1 4.1 1. 1.1 2. 2.1 3. 3.1 3.2 3.2 4. 4.1
44
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5. 5.1 5.1 5.2 5.3 5.4 2
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II 4 1. 1.1 1.2
1.1 1.2 / I
2. 2.1 1. 1.1 1.2 1.3 1.4 2. 2.1 2.1 2.2 3. 3.1 3.1 1. 1.1 2. 2.1 2.1 3. 3.1 1. 1.1 1.1 2. 2.1 2.1 1. 2.
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III 4 1. 1.1 2. 2.1 2.1 2.2 3. 3.1 4. 4.1 4.1 5. 5.1 5.1 5.2 1. 1.1 1.2 2. 2.1 2.2 3. 3.1 3.2 3.2 1. 1.1 1.2 2. 2.1 2.2 3. 3.1 3.2
cot, sec, csc I I
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IV 4 1. 1.1 1.1 2. 2.1 2.2 3. 3.1 4. 4.1 4.2 4.3 4.3 1. 1.1 2. 2.1 2.2 3. 3.1 3.2 1. 1.1 1.1 2. 2.1 3. 3.1 4. 4.1 4.2 4.2 1. 1.1 2. 2.1 3. 3.1
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1. 2. 3. 4. 5. 1 2 3 6. 7. 8.
49
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IIIIII IV
I
2 1. 1.1 50
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1.2 2 1 = 2 + 1 a 2 = a 2 1
a+b 2 1.3 ab ( a + b ) ( a + b ) ( a 2 ab + b2 ) ( a b ) ( a 2 + ab + b2 ) ( a + b + c ) 3 2
(1 x ) (1 + x + x 2 ) ( x + y 1)( x y + 2 )
x3 + y 3 + z 3 3 xyz = ( x + y + z )( x 2 + y 2 + z 2 xy yz zx) c 1 2ab 1 5+ 2 6 = 3 + 2 = = 2 2 11 1 a +b a 2 + b2 a b + + 2a b c cx 2 + x 2 + 2 = x + x 1 2. 2.1 a, b a, b 2:3 x 2.2 a + b a + b
x 3 < 2 x 1 < 1 1 < x < 2 x 1 < 2 x 3
n y = cx n=123451
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f ( x ) a f (x ) (x a ) ( x b ) (a, ) (b, ) f (x ) (x a ) ( x b ) I n 1. 1.1 y = f ( x ) y = mx + b = m ( x x0 ) m, x0 , b m y x 1.2 2 f ( x) = x + 2 x + 3,2 x 2 x 2 x + 4
(x a )(x a ) (x b )x 2 + 1x 2 + x + 1 f (x ) (x a ) f ( a ) (x a ) f ( x ) (x a )
52
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y = c ( x a )( x b )
y = ax 2 + bx + c y = a ( x h ) + k 2
1.3 y = x n n = 1, 2, 3, 4 [ 1.5,1.5]
n y = cx n c y = cx n y = c ( x h ) + k n
2. 2.1 ( x a ) ( x n 1 + x n 2 a + L + a n 1 ) = x n a n n = 2,3,4 ( x + a )( x 2 ax + a 2 ) = x 3 + a 3
f ( x ) = 2 x 3 5 x 2 + 6 x + 3 = a + b ( x 1) + c ( x 1) + d ( x 1) a, b, c, d 2 3
f (1.01)
f ( x ) = 2 x 3 5 x 2 + 6 x + 3 = a + b ( x 1) + c ( x 1)( x 2 ) + d ( x 1)( x 2 )( x 3) f ( x ) (x a ) ( x b ) ( a, f ( a ) ) , ( b, f ( b ) ) f a,b f f ( x ) = q ( x )( x a )( x b ) (1,1) , ( 2, 3) , ( 3, 7 ) f ( x ) = a + b ( x 1) + c ( x 1) ( x 2 ) a, b, c, d
1 a, b, c f 2
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(11,3), (12,5), (13,8) ( x 11)( x 12) ( x 11)( x 13) ( x 12)( x 13) + 5 + 8 f ( x) = 3 (13 11)(13 12) (12 11)(12 13) (11 12)(11 13)f (11.5)
(1,1) , ( 2, 3) , ( 3, 7 ) f ( x ) = a + bx + cx 2 a , b, c IV 3. 3.1 ax 2 + bx + c = 0 x 2 + 5 x + 3 = 0 2 + 2 3 + 3 1 1 3 + = x 1 x 2 2 n 3.2 a x n = a a>0 f ( x ) = x 3 + 2 x 2 + 3x + 4 n 3.3
f ( x ) = k ( x a1 ) 1 L ( x ak ) k ( x 2 + b1 x + c1 ) L ( x 2 + bm x + cm )r r s1
sm
f ( x ) = 5 x 4 21x 3 + 30 x 2 9 x + 7 x = 2 + i
4. 4.1 ( x 1)( x + 2 )2
( x 4 ) > 0 ( x 1)( x 2 )
3
(x
2
+ x + 1) > 0
x3 1 > 0 x4 2 x2 3 > 0 1 1 1 < 0 < 1 < 1 x x 1 x54
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10 x , x = 0.1,0.2,L ,0.9 a > 0, a 1 2 x 10 x x = log a b a x = b 10 log( xy ) = log x + log y log ( x / y ) = log x log y log x = log x
( )
1 10 log a x = log x log a y = log x y = log x 1. 1.1 n 10 2 10 3 = 10 6 2 3 33 = 6 3 10 < 10 ( x a b ) c a ( x b c ) a b ( x c a ) b c a 3 x + a 3 x a x + a x + 6 a 1 b 1 c 1
1
1
5
1
1
1
1 3
1 2
a 2 x = 2 + 3
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2. 2.1 2 x 10 x 3. 3.1 log( xy ) = log x + log y log ( x / y ) = log x log y log x = log x n log a b, ( log a b )( logb c ) = log a c, a log x b = blog x a m 1 3.2 log a x = log x log a 10 4. 4.1 y = a x x = log a y 1 log a x = log x = log a log a m b n =
( )
log a + log b a+b a +b log a = b 2 2 2 5. 5.1 5.2 ab 2100 (1.18 ) 10
5.3 5.4
56
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II
/ 1. 1.1 an +1 = an + d an +1 = ran an +1 = an + n an +1 = an + n 2 an +1 = ( n + 1) an
1.2 / II 2. 2.1 ( ak + bk ) = ak + bk cak = c ak k =1 k =1 k =1 k =1 k =1 n n n n n
ak = ak 1 k =1 k =2
n
n +1
k k 2 k =1 k =1
n
n
1 k =1 k ( k + 1)n
1. 1.1 57
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1.2 1.3 |S| S 3 6
10 8 3 5 7 21 4 1 10+8+3+5+7+21+4+1+3+6 = 68
51 51 50 1.4 A B |A B| = |A| + |B| A, B A B |A B| = |A||B| A, B, C 1 |A B| = |A| + |B| |A B| 2 |A B C| = |A| + |B| + |C| |A B| |B C| |C A| + |A B C| Principle of Inclusion and Exclusion PIE Inclusion and Exclusion PIE 2. 2.1 n n! 1 n 1 n n! n k
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n! 1 k 1 n (n k )! Pkn 50 1 n k nk 1 k 1 n nk 10 103 1 0 A={0,1} n An = A A . . . A n 2n 2.2 n! Ckn = k !( n k )! n k n! Ckn = k !( n k )! k 1 n Ckn n k C n + k 1 k Pkn = k 1 n Ckn + k 1 n k x1 + x2 + + xn = k Ckn + k 1 3. 3.1 n (x + y
1 x 2 + + 1 x x
5
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1. 1.1 3 2. 2.1 50 3. 3.1 ABC A 40%B 30%C 30% A 5%B 10% C 8% A 90% 90% 90% 2%
1. 1.1 1 1 n = xk ( x1 x2 L xn ) n n k =1 =
1 n 2 ( xk ) n k =1
xi
1 n 2 ( xk x ) 2 x= n k =1
xi
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2. 2.1 0 ( xk , yk ) , k = 1, 2,L , n , e(r ) = yk rxkn k =1 n n
(
)
2
r e(r ) = ( yi2 2 xi yi r + xi2 r 2 ) = r 2 2( xi yi )r + 1 i =1 i =1
r = xi yi i =1
n
y = rx e( r)
II 1. 2.
III
61
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360 r 0 360 X 1. 1.1
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2. 2.1 X = 150, = 30; = 225, = 45; = 300, = 60 cos , sin 2.2 r, 0 r < , 0 < 360 3. CBC = CD + BD = b 2 sin 2 A + ( c b cos A)2 2 2 2
= b 2 + c 2 2bc cos A
A D B ABC B ( c, 0 ) C
( b cos A, b sin A) 1 ABC c b sin A 2 1 ABC a c sin B 2 1 ABC a b sin C 2 ABC A B ( c, 0 ) C
( b cos A, b sin A) 2 2 a 2 = ( b cos A c ) + ( b sin A) = b2 + c 2 2bc cos A 4. cos ( 2 1 ) = cos 1 cos 2 + sin 1 sin 2
4.1 B ( cos 2 ,sin 2 ) A,B A ( cos 1 ,sin 1 ) ( cos 1,sin 1 ) , ( cos 2 ,sin 2 ) OAB cos ( 2 1 ) = cos 1 cos 2 + sin 1 sin 2 O1,0
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cos (1 + 2 ) sin (1 + 2 ) tan (1 + 2 ) cos 2 sin 2 cos
2
sin
2
2
cos15o 5. 5.1 5.2
1. 1.1 1.2 2. 2.1 2.2 ax + by = k 3. 3.1 3.2 64
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v v a b v v a b cos = a1b1 + a2 b2 v v a b sin = a1b2 a2 b1 a x + b1 y = c 1 1 a2 x + b2 y = c2 v v v v v c = xa + yb a b 0 v v a b 1. 1.1 1.2 1 2 x + y 0 x 1 , 0 y 1 1 2 4 1 2 5 2 1 2. 2.1 65
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r r a = ( a1 , a2 ) , b = ( b1 , b2 ) 2
r2 r2 r r ( a1 b1 ) + ( a2 b2 ) = a + b 2 a b cos r r a1b1 + a2b2 = a b cos 2
( b1 , b2 )r b
r r r r a, b a b r r r r r r r r r r r r r r2 a b = ba a + b c = a c + bc a a = a
r a
( a1 , a2 )
(
)
2
ax + by ( a 2 + b 2 )( x 2 + y 2 ) r r r r a + b a + b 4, 51, 2 2.2 3. 3.1 1.OABO0,0 A ( a1 , a2 ) B ( b1 , b2 ) yB ( b1 , b2 ) A ( a1 , a2 )
OAB = =
1 1 1 b1b2 + ( a2 + b2 )( a1 b1 ) a1a2 2 2 2
O
x D C r r 2. a, b 1 v v 1 v v 1 v2 v2 v v 2 OAB = a b sin = a b 1 cos2 = a b a b 2 2 2 1 1 2 2 2 2 2 = ( a1 + a2 )( b1 + b2 ) ( a1b1 + a2b2 ) = 2 a1b2 a2b1 2 a b a a b a1 a + c b1 a1 b1 c1 1 1 = 1 2 = 1 1 1 = + a2 b2 b1 b2 b2 a2 a2 + c2 b2 a2 b2 c2
1 ( a1b2 a2 b1 ) 2
(
)
b1 b2
ca1 ca2
b1 a =c 1 b2 a2
b1 b2
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3.2 a x + b1 y = c 1 1 a2 x + b2 y = c2 v v v v v c a b a b 0
IV 0 0 1. 1.1 2. 2.1 2.2 {(1, 2,3) + t ( 0,1, 1) | 0 t 1} {s (1, 2,3) + t ( 0,1, 1) | 0 s 1,0 t 2}
3. 3.1 ( a1 , a2 , a3 ) ( b1 , b2 , b3 ) v v v v a1b1 + a2b2 + a3b3 = a b cos a b a1b1 + a2b2 + a3b3 4. 4.1 67
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v v v v A =| a |2 | b |2 a b2
(
)
2
=
a2 b2
a3 b3
2
+
a3 b3
a1 b1
2
+
a1 b1
a2 b2
2
4.2 v v v v v v a , b , c a (b c ) 4.3 a2 + 1 ba b2 + 1 bc ca cb c2 + 1
abac b+c a a b a+c b c c a+b
1. 1.1 2. 2.1 x 1 z L : = y + 1 = E : 2 x + y z = 3 2 3 2.2 3. 3.1 3.2 a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d 2 a x + b y + c z = d 3 3 3 3 u v v v v v v v d a b c a b c 0
68
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1. 2. 2.1 Caley-Hamilton 3. 3.1 4. 4.1 4.2 1 2 1 2 1 0 A = A 0 1 2 1 2 1
1 3 1 2 2 4 2 1
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1. 1.1 x 2 + ( y - c ) = y + c 2. 2.1 2
( x c)
2
+ y2 +
( x + c)2
2
+ y 2 = 2a
x = x h ( x h ) + ( y k ) = 1 x 2 + y 2 = 1 a2 b2 a 2 b2 y = y k x2 y2 + = k a 2 b2 x2 y2 x y x2 y2 x = , y = 2 + 2 = t 2 2 + 2 = 1 a b a b t t ax 2 + cy 2 + dx + ey + f = 0 2
a ( x h ) + c ( y k ) = ah 2 + ck 2 f 2 2
3. 3.1
( x c)
2
+ y2
( x + c)
2
+ y 2 = 2a
70
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I II I II
I II
I II 1 I II II 2 III
4 I 4 II 3 I 3 II I III71
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I II 1 1
2~3 1
I II 3~4 3~4 2 2 2 2
III I4 1. 1.1 1.2 2. 2.1 3. 3.1 3.2 3.3 3.1 2.1 n 72
1. 1.1 1.2
1.3 2. 2.1 2.2 3. 3.1 3.2
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II4 1. 1.1 1.2
1.2 1.4
1.3 1.4 2.
2.1 3. 3.1 3.2 1. 1.1 1.2 2. 2.1 2.2 3. 3.1 3.2 3.3 3.3 4. 4.1
I3 1. 1.1 2. 2.1 3. 3.1 4. 4.1
73
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5. 5.1 5.1 5.2 5.3 1. 1.1 2. 2.1 2.2
II3 1. 1.1 1.2
1.2
2. 2.1 2.2 2.3
2.3
3. 4.
3.1 4.1 4.2
1. 2. 3. 4. 5. 74
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1 2 3 6. 7. 8.
75
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III III
I I II II III
IIvariables n n n X Y 0 1. 1.1 X Y X X=0,1,2,3 Y 1.2 76
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2. 2.1
1 1/2 n k C 2 np np (1 p ) n k
n
3. 3.1 3.2 3.3 p n p Ckn ( p ) (1 p )k n k
, p = 0.4, n = 100
1 e 2
( x )2 2
2
, = np, = np (1 p ) , e = 2.718L
0.1 0.08 0.06 0.04 0.02 00 10 20 30 40 50 60 70 80 90 100
77
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1 n 1. 1.1 1.2 1.3 cos2 + sin 2 = 1 1 + tan 2 = sec 2 2. 2.1 A sin ( t + 0 ) A t + 0 a cos (t ) + b sin (t ) = A sin (t + 0 ) 2.2 3. 3.1 3.2 n ( cos + i sin )( cos + i sin ) = cos ( + ) + i sin ( + ) n 1 + + 2 + L + n 1 1 1 1 + +L+ 2 1 1 1 n
II II 1. 1.1 78
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1.2 1.3 ABCD 1 ABCD
A
B
C D 1.4 n 2. 2.1 y = ( x h ) y = x 3 y = x 3 3
y = ( x h ) 3
c y = x y = n n=1,2 x y = x , y = ax 2 + bx + c x 2 + y 2 = 1 3. 3.1 3.2
1. 1.1 1.2 2. 2.1 79
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III III
I I II II III
IIvariables n n n X Y 0 1. 1.1 X Y X X=0,1,2,3 Y 1.2 76
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2. 2.1
1 1/2 n k C 2 np np (1 p ) n k
n
3. 3.1 3.2 3.3 p n p Ckn ( p ) (1 p )k n k
, p = 0.4, n = 100
1 e 2
( x )2 2
2
, = np, = np (1 p ) , e = 2.718L
0.1 0.08 0.06 0.04 0.02 00 10 20 30 40 50 60 70 80 90 100
77
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1 n 1. 1.1 1.2 1.3 cos2 + sin 2 = 1 1 + tan 2 = sec 2 2. 2.1 A sin ( t + 0 ) A t + 0 a cos (t ) + b sin (t ) = A sin (t + 0 ) 2.2 3. 3.1 3.2 n ( cos + i sin )( cos + i sin ) = cos ( + ) + i sin ( + ) n 1 + + 2 + L + n 1 1 1 1 + +L+ 2 1 1 1 n
II II 1. 1.1 78
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1.2 1.3 ABCD 1 ABCD
A
B
C D 1.4 n 2. 2.1 y = ( x h ) y = x 3 y = x 3 3
y = ( x h ) 3
c y = x y = n n=1,2 x y = x , y = ax 2 + bx + c x 2 + y 2 = 1 3. 3.1 3.2
1. 1.1 1.2 2. 2.1 79
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2.2 3. 3.1 k k 2 xdx, x 2 dx a a k =1 k =10 0
n
n
3.2 3.3 4. 4.1 2 rdr = R 2 0R
X 2 R 4 R 2 x 2 dx = R 3 R 3 A Z 2 H Az 1 dz = AH H 2 : z 2 = A : A ( z ) 2 0 H 3 z0 v0 t t 1 v ( t ) = v0 gt t z0 + v ( t ) dt = z0 + v0t gt 2 0 2 II
(
)
a a
1 n
I I II II III
IIvariables n n n 80
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X Y 0 1. 1.1 X Y X X=0,1,2,3 Y 2. 3. 4. 4.1 1/2 n k
1 Ckn 2 np np (1 p ) 5. 5.1 5.2 5.3 81
n
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p n p Ckn ( p ) (1 p )k n k
, p = 0.4, n = 100
1 e 2 0.1 0.08 0.06 0.04 0.02 00 10
( x )22 2
, = np, = np (1 p ) , e = 2.718L
20
30
40
50
60
70
80
90
100
1. 1.1 2. 2.1 2.2 cos2 + sin 2 = 1 1 + tan 2 = sec 2
II II
1. 1.1 82
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1.2 2. 2.1 ABCD 1 ABCD B A
D
C
2.2 2.3 n 3. 3.1 y = ( x h ) y = x 3 y = x 3 3
y = ( x h ) 3
c y = x y = n ,n=1,2 x y = x , y = ax 2 + bx + c x 2 + y 2 = 1 4. 4.1 4.2
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95 23 95 http://140.116.223.225/98course 98 12 98 121518 95 http://mathcourse.ck.tp.edu.tw 98 95 4 1
1
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2
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95 1. 2. 3. 4. 5. 6. 1. 2. 3. 1. IV a. II b. c. 2. I II 3. II a. III / I b. c.
3
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95 4. I a. II II b. 5. I III I III 6. I III 7. I IV 8. a. II b. c. 1. 2.
3. 4.
5.
6.
4
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95 1. 2 10 2. 4. 3.
AP III 1. 2. 3.
94 1 6 http://140.116.223.225/98course/ 121518 94 12 95 4 95 2 151617 121518 http://140.116.223.225/98course/ 5
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95 8 9 () http://140.116.223.225/98course/ 95 http://140.116.223.225/98course/ 95 4/84/225/6 5/205/276/178/1 8/10 8/15 11/11 308 11/1912/212/17 96 4/74/154/21 5/126/246/30 96.5.5 ()
96.5.12 () 96.5.19 ()
96.5.19 () 96.6.2 () 96.6.2 () 96.6.9 ()
308 95 95
6
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96.6.9 ()
7
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QA Q1 A1 Q2 A2
1
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Q3 A3 2
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Q4 A4 I, II III, IV I, II I, II Q5 A5 III III
3
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Q6 95 A6 I, II II Q7 A7 Q7Q7-1 sin, cos, tan cot, sec, csc A7-1 sin, cos tan cot, sec csc
4
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Q8 A8 tan 1 Q9 A9 Q10 2 1 3 5.3 A10 Q11 A11 Q12 A12 B
5
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Q13 A13 95 1. 2. 3. 4. 5. 6. 1. 2. 3.
1.
2.
3. 4.
5.
6
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95
6. 1. 2.
2 10
3. 4. a. 1. IV b. II c. 2. I II a. b. 3. II III V c. a. 4. I II II b. I III 5. I III 6. I III 7
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95 7. I IV 8. a. II b. c. Q14 A14 Q15 A15
8
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Q16 A16 2 1 a b =a b (a, b 0 )
3 2 ( 3) ( 2 )
Q17 A17 II
Q18 88 95 A18
Q19 A19
9
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Q20 95 A20 Q21 A21 Q22 A22 Q23 I 1 2 A23 2
Q24 A24 93 6 95 95 95 4 1 96 4 30 10
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Q2595 A25 3 4
Q26 A26 121518 Q27 A27
Q28 Q28 A28 7-a-09 7-a-10 7-a-11 7-a-12 9-a-02 11
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99 p.52 1.1 y = f ( x ) 95
12
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97 7 28
93 4
94 1 6 12 95 A 94 12 95 4 121518 95 8 9 95 95 4 1 http://www.mathcenter.ck.tp.edu.tw 17 96 6
4-1
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I II III IV 1 2 3 10 +
4-2
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x n a n n = 2,3,4
(a + b)
n
n = 2,3 4
I x + 1 2
x 2 3 x I ( x1)( x 2)( x 3) >0 x II y = f ( x )
4-3
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1 I
I II
4-4
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2
II
I
3 III IV
4-5
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97 1 24
97 1 24 pp. 4370pp. 7184
5.1 p.72 p.12 2
5.2
5.3
4-6
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5.4
5.5
5.6 1. p.46 2. p.48 95 3. p.52 x y 4. p.52 5. p.52 6. p.52
f ( x ) ( x a ) ( x b ) ( a , f ( a ) ) ( b, f ( b ) ) [a, b] 7. p.53 f ( x ) g ( x ) 3 f ( a ) = g ( a ) f ( b ) = g ( b )
f ( c ) = g ( c ) f ( d ) = g ( d ) a, b, c, d f ( x) = g ( x)
4-7
-78-
h (x ) = f
( x ) g ( x ) h ( a ) = h ( b ) = h ( c ) = 0 h (x ) = (x a )(x b )(x c ) x = d h ( d ) = ( d a )( d b )( d c ) = 0 = 0 h ( x ) 0
n xi 8. p.53 ( x, y ) ( x, y )
( x, y )
y
y = c( x h) n + k n = 1,2,3,4 123 4 9. p.5253 1 2 3 A. B. C. D. 10. p.54
4-8
-79-
11. p.54 12. p.54 a | bc , (a, b) = 1 a | c 13. p.56 14. p.56 1 + a + a 2 + + a n a 4 b 4 = (a b)(a 3 + a 2 b + ab 2 + b 3 ) II 1 rn 1 + r + r 2 + + r n 1 = 1 r 15. p.5758 a n {a n } n( A) A A
16. p.57 an +1 = an + d an +1 = ran an +1 = an + n an +1 = an + n 2 an +1 = ( n + 1) an
1357 9 d = 2 0151430 an +1 = an + n 2 17. p.60
4-9
-80-
18. p.62 19. p.62 ( x, y ) ( x, y ) ( x, y ) ( x, y ) x y 20. p.63 60 120 cos 120 sin 120 -cos 60 sin 60 cos 120 sin 120 21. p.64 22. p.64 23. p.69 24. p.7273 25. p.7374 95 26. p.7374 27. p.7374 95% 28. p.7782 e e e 29. p.7983
4-10
-81-
x y x y x
III IIIIV I/II A / I II 4-11
-82-
a 4 b 4 = (a b)(a 3 + a 2 b + ab 2 + b 3 ) x=y
4-12
-83-
4-13
-84-
linguistic processes Cazden1988
4-14
-85-
A
12
1234 92
4-15
-86-
:
8 8-9 8-9 8-9 10
1. 2. 3. 6-7 7-8 7-8 8-9 10 10 10 11 11 10 1012 11 11 11
4-16
-87-
92 95 1-8 9 10-11 1. 1 2 3 4 5 6 2. 1 2 3 4 3. 2 1 2 3 4 5 6 7
4-17
-88-
1. 2. 3. 4. 5-6 7 7 8 8 11 11 11 11 11 11 11 11
9-10 10 9 9 10 I 10 I 10 II 12 II 12
4-18
-89-
1. 2. 3. 5-6 8-9 10 10 10 12
1. 2. -14 3. 4. 5. 1 24-19
-90-
3 1. 2. 3. 4. 5. 6. 7-8 10 8-9 10 10 10 11-12 12 12
4-20
-91-
B
p.51 line 15
x 2 + x 2 + 2 = x + x 1 x 2 + x 2 + 2 = x + x 1p.54 line 17 f ( x) = x 3 + 2 x 2 + 3x + 4 f ( x) = x 3 + 2 x 2 + 3x + 4 p.56 line 10 p.60 line 3 = = p.61 lines 46 1 n 2 ( xk ) n k =1 1 n 2 ( xk ) n k =1
( x k , y k ) , k=1, 2, , n e(r ) = ( y k rx k ) 2 k =1
n
2 2 r e(r ) = y i 2 xi y i r + xi r 2 = r 2 2 xi y i r + 1 i =1 i =1 n
(
)
n
r = xi y i i =1
n
4-21
-92-
( x k , y k ) , k=1, 2, , n e(r ) = ( y k rx k )k =1
n
2
n n 2 2 r e(r ) = y i 2 xi y i r + xi r 2 = nr 2 2 xi y i r + n i =1 i =1 n 1 r = xi y i n i =1
(
)
p.77 line 3 Cn ( p )k (1 p )n k p=0.4 n=100 k Cn ( p )k (1 p )n k p=0.4 n=100 k p.77 line 2 1 2 e 2 , = np, = np (1 p ) , e = 2.718 2 p.80line 14 X x
( x )2
p.80line 16 Z z p.82 line 3 Cn ( p )k (1 p )n k p=0.4 n=100 k Cn ( p )k (1 p )n k p=0.4 n=100 k
4-22
-93-
p.82 line 4 1 e 2
( x )22 2
, = np, = np (1 p ) , e = 2.718
4-23
-94-
C
1. 5.6 14 2. 5.6 2 3.? ? 8 4. I II III IV 5.5 2 5." """ 5.6 20 6." """ 5.6 20 7. 5.4 8.? 5.6 11 9. 5.1 10. 5.1 11. 5.6 5 12. 5.6 1 4-24
-95-
13. 5.6 14 14. 5.6 4 15. y = c( x h) n + k 5.6 8 16. 5.6 9 () 17. a | bc , (a, b) = 1 a | c 5.6 12
18. 5.6 11 19. pH 5.2 20. 5.4 21. n(A)A an{an} 5.6 15 22. 2 1.96 5.6 27 23. 95 5.6 24 24. e 5.6 28 25. 5.3 5.6 1, 7, 11, 28 26. 5.6 4 27. 5.6 6, 8, 9
4-25
-96-
28. (x 1)1001 (x 2)2003 (x 4 )2005 > 0 5.6 10 29. 5.6 7 30. 5.6 11 31. x n a n = (x a ) x n 1 + ax n 1 +
(
+ a n 1
)
5.6 14
32. 5.6 1 33.
4-26
-97-
2011-01-11
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99102 AB97124 B 9899 94120 B
9999
1 1 A (1) A
2
1.95 2.
102 10
1 ceec.edu.tw//99.htm
BI
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2011-01-11
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99AB (99.10.29) 99.09.30
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99
99 999712499 IV IVABBA
99 A 99
1
-100-
4 I 1. 1.2 1.2 1.3 2. 2.1 2.2 1. 1.1 1.2 1.3 1.3 4 3. 3.1 3.2 n a 3.3 4. 4.1 4.1 3.1 2. 2.1 2.1 1.1
2
-101-
1. 2. 2.1 4. 4.1 5. 5.1 5.1 5.2 5.3 5.4 2 3. 3.1 3.2 3.2 1.1
3
-102-
4 II 1. 2. 1.1 1.2 1.1 1.2 / I 2.1
1. 1.1 1.2 1.3 ( ) 1.4 2. 2.1 2.2 2.1 3. 3.1 3.1
1. 1.1 2. 2.1 3. 3.1 1. 1.1 1.1 2.1
2. 2.1 2.1 1.
2.
4
-103-
4 III 4. 4.1 5.1 5.2 1. 1.1 2. 1.2 () 2.1 2.2 () 3. 3.1 3.2 3.2 1. 1.1 1.2 2. 2.1 2.2 3. 3.1 3.2 4.1 5. 5.1 2.2 3. 3.1 I I
1. 1.1 2. 2.1 2.1 cot, sec, csc
5
-104-
4 IV IV AB B A 1. 2. 2.1 2.2 3. 3.1 4. 4.1 4.2 1. 1.1 2. 2.1 3. 3.1 1.1 1.1
1. 1.1 1.1 2. 3. 1. 2. 3. 2.1 3.1 1.1 2.1 3.1 6
-105-
99
99 99 97 1 24 99 IV IV AB B A 102 B A 98
99
1
-106-
99
2
-107-
99 4 I 1. 2.
1.1 1.2 1.2 1.3 2.1 2.2
1. 1.1 1.2 1.3 1.3 4 3. 3.1 3.2 n a 3.3 2. 2.1 2.1 3.1
3
-108-
4. 4.1 4.1 1. 1.1 2. 2.1 3. 5.2 5.3 5.4 2
3.1 3.2 3.2 4.1
4.
5. 5.1 5.1
4
-109-
4 II
1.
1.1
1.1
1.2
1.2 / I
2.
2.1
1. 1.1
1.2
1.3
1.4
2.
2.1 2.2
2.1
3.
3.1 3.1
1. 1.1
2. 2.1
2.1
3. 3.1 5
-110-
1. 1.1 1.1
2. 2.1 2.1
1. 2.
6
-111-
4 III
1. 1.1
2. 2.1 2.1 cot, sec, csc I 2.2 I 3. 3.1 4. 5.
4.1 4.1 5.1 5.2 5.1
1. 1.1 1.2 ( ) 2. 2.1 2.2
3. 3.1 3.2 3.2 1. 1.1 1.2
2. 2.1 2.2 3. 3.1 3.2 7
-112-
4 IV IV AB B A
1.
1.1 1.1
2. 2.1
2.2
3. 3.1
4. 4.1
4.2 4.3 4.3
1.
1.1
2. 2.1 2.2
3. 3.1 3.2
8
-113-
1. 1.1 1.1
2. 3.
2.1
3.1 4.2
4. 4.1 4.2
1. 2. 3.
1.1
2.1 3.1
9
-114-
I4
1.
1.1 1.2 2.1
2.
3. 3.1 3.2 3.3
3.1
1. 1.1 1.2
1.3
2. 2.1
2.1
2.2
3. 3.1
3.2 n
10
-115-
II4
1. 1.1 1.2
1.2
1.3 1.4
1.4
2. 3. 1.
2.1
3.1 3.2 1.1 1.2
2. 2.1
2.2 3. 3.1 3.2 3.3 3.3
4.
4.1
11
-116-
I3
1.
1.1
2. 2.1 3. 4. 3.1 4.1
5. 5.1 5.1 5.2 5.3
1.
1.1
2. 2.1 2.2
II3
1. 1.1 1.2
2. 2.1 2.2 2.3
1.2
2.3
3. 4.
3.1 4.1 4.2
12
-117-
93.8.31 0930112130
1
-118-
2* 4 4 4 2 2 2 2* 4 4 4 2 2 2 2* 4 4 2* 4 4
2* 2* 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 2-3 2-3 2 2 2 2 2 2 2 2
2
2
2
2 2
2 1 1 30 / /32 0~3 0~3 30-33 / /32~35
2 2 2 2 1 1 1 1 1 1 30 26~27 26~27 14 /32 /28~29 /28~29 /16
14 /16
140~142 /152~154
0~3
6~7
6~7 14~19 14~19
0~3
6~7
6~7 14~19 14~19
40~58 180-198 /192-210
30-33 32-33 32-33 28-33 28~33 /32~35 /34-35 /34-35 /30-35 /30~35
2
-119-
*
3
-120-
6-10 8 8 6-8
4 4 4-6
4 4
4 48
48 94222 0940013084B
4
-121-
5
-122-
6
-123-
31
-124-
1. 2. 1-1 2-1 n n 2 2-2 18 = 3 2 3. 3 n m n, m 3-1 3-2 4-1 i 0 4-2 1-1 6 = 2 3 2 = 3 2
4.
1.
2. 2-1 3. 1. 2.
3-1 1-1 2-1
3. 3-1 4. 5. 6. 4-1 5-1 6-1
32
-125-
1.
2.
3. 5. 1.
4.
4-1 5-1 1-1 30 45 2-1
2.
3. 3-1 4. 5. 6. 1. 2. 2. 3. 4. 5. 4-1 5-1 1 n 1. 1-1 2-1
33
-126-
1. 2. 3. 4.
1-1 2-1 3-1 4-1 1-1
1. 2.
3. 3-1 4. 5. 4-1 5-1 6-1 6-2 6-3 6-4
6.
1. 2. 3. 4.
34
-127-
1.
2. 3. 4. 5. 1. 2.
4-1
1-1
3. 4. 5. 6. 5-1 a a f ( n) 6-1 n = n 1 + a n = a n 1 + a n 2
, , f(n) 3 1. 2.
1-1 1-2
3. 5. 4. 4-1 5-1
6. 6-1 68-95-99.7
35
-128-
I
p(1 p ) n
1. 2. 36
-129-
3. 4. 5.
37
-130-
-I
I I
1. 4. 4-1 1-1 3. 2. 2-1 transition matrix 3. 3-1 2. 2-1
1.
333
-131-
4.
4-1 5-1 6-1 6-2 1-1 2-1 2-2 2-3
5. 6.
1. 2.
3.
3-1
20 6 14 0.9586 5 15 0.9882 10 2.236 1. 2. 334
-132-
-I
I
335
-133-
-
1. 1-1 1-2 2-1 x 3. 3-1 y y x 3-2 3-3
2.
337
-134-
4.
4-1 4-2 4-3 1-1 1-2 2-1 3-1
1. 4. 1. 3. 2.
1-1 2 1-2 y = x
x = a x = b y = 0
2. 2-1
3. 3-1
338
-135-
-
1. 2. 3. 4. 1. 2. 3. 4.
339
-136-
340
-137-
-138-
() ( ) ()
-139-
1
120 100
()
* *()
-140-
2
() ()
-141-
3
-142-
........................................................................................................... 1 ........................................................................................................... 1 ........................................................................................................... 2 ....................................................................................... 4
-143-
( ) () ()
-144-
1
1 A{ z : z = r (cos + i sin ), 0 r 1, y
3 5 } 4 4x
D{w w = z 3 , z A } D
(1)
y
(2)
y
(3)
y
x
x
x
(4)y
(5)y
x
x
93 3 D
-145-
2
2 M AB BM = 2 AM N BC
cos M O N =
17 15 16
10
95 B
3 V(0, 3)F(0, 6) P(a, b) Q(a, 0) P x FPQ = 60 b = 96 H 38 39
-146-
3
1. 2.
1-1 2-1 n n 2
2-2 18 = 3 2
6 = 2 3
2 = 3
2 3
n m
n, m
3.
3-1 3-2
4.
4-1 i 0
4-2
1. 2.
1-1 2-1
3.
3-1
-147-
4
1. 2. 3.
1-1 2-1 3-1
4. 5.
4-1 5-1
6.
6-1
1. 2. 3.
4.
4-1
5.
5-1
-148-
5
1.
1-1 30 45
2. 3.
2-1 3-1
4. 5. 6. 1.
1-1 2-1
2. 3. 4. 5.
4-1 5-1 1 n
-149-
6
1. 2.
1-1 2-1
3.
3-1
4.
4-1
1.
1-1
2. 3.
3-1
4. 5.
4-1 5-1
6.
6-1 6-2 6-3
6-4 7
-150-
1. 2. 3. 4. 1.
2.() 3.() 4.() 5. 1. 2.
4-1
1-1
3. 4. 5. 6. 5-1 6-1 an = an 1 + f (n) an = an 1 + an 2 , , f (n) 3
-151-
8
1.
1-1 1-2
2.
3. 4. 4-1
5.
5-1
6.
6-1 68-95-99.7
-152-
9
-153-
() () ( )
-154-
1
80
()
(I) (II) (I)
() ()
-155-
2
-156-
3
-157-
........................................................................................................... 1 ........................................................................................................... 2 ........................................................................................................... 3 ....................................................................................................................... 8
-158-
1
-159-
( ) ()() I () II (I) I () ()
-160-
2
1 x 5 = 1 1, 1 , 2 , 3 , 4 (3 1 )(3 2 )(3 3 )(3 4 ) =
(1) 81
(2) 162
(3) 121
(4) 242
93 1 2 sin + cos 12 =0 (1 + cos ) x y x + (1 + sin ) y = 0 93 x, y (1 + cos )(1 + sin ) 1 = 0
-161-
3
a0 , a1 , a2an +1 = an + bn ,
b0 , b1 , b21 bn +1 = bn 2
(n = 0,1, 2, ).
(1) A
an +1 an = A bn +1 bn
( n = 0,1, 2, )(2 )
(2) A2 A3 Ak k (6 ) Ak (1) (2) Ak Ak
3
0.5 0. 8 9 10 96 A 0.5
-162-
4
4 P y = 2 + x x 2 P x log3 10 P
(1) (2) (3) (4) (5)
95 2 P 5 20 21 21 (1 ) 1200 1000 600 500 400 200 400 600 800 750 800 30000 20000 10000 4600 10 20 30 40 50 60 10 20 30 40 50 60 10000 17400 29250 900 40000 32200 38100 60000 50000
()
()
(1) 30 (2) 30 94 5
-163-
6
SARS 7 ()5 7
(1) 3 (2) 1 (3) 3 2 (4) 3 2 (5) =1 492 8
7 x, y ABC f ( x, y ) = ax + by a, b A(12,3) 42 B (6,1) 20BCDE D E (10, 4) (9, 2)
(1) a, b (4 ) (2) A, D , C ? (3) f ( x, y ) = ax + by (4 ) (1) a, b (2) E AB BCDE D AC (3) f ( x, y ) = ax + by
-164-
6
x, y ABC f ( x, y ) = ax + by a, b A(12,3) 42 B (6,1) 20BCDE D E (10, 4) (9, 2)
(1) a, b (4 ) (2) f ( x, y ) = ax + by (4 ) (3) ABC C ( s, t ) s + 2t = 18 1 s < 10 (5 ) (1) a, b (2) BCDE f ( x, y ) = ax + by (3)(3) f ( x, y ) = ax + by C C AD s + 2t = 18 1 s < 10
-165-
7
1. 2. 1-1 2-1 n n 2 2-2 18 = 3 2
3.
6 = 2 3
2 = 3
2 3
n m n, m 3-1 3-2 4. 4-1 i 0 4-2 1. 1-1 2. 2-1 3. 3-1
-166-
8
1. 1-1
2. 2-1 3. 3-1 5. 5-1 6. 6-1 1. 2. 4. 4-1 5. 5-1
4. 4-1
3.
-167-
9
1. 1-1 30 45 5. 6. 1. 5. 5-1 1 n 4. 4-1 3. 2. 1-1 2-1 4. 3. 3-1 2. 2-1
-168-
10
1. 1-1 2. 2-1 3. 3-1 4. 4-1 1. 1-1 2. 3. 3-1 6. 6-1 6-2 6-3 6-4
4. 5.
4-1 5-1
-169-
11
2. 3. 4. 1.
1. 4.() 5. 1. 2. 3. 6. 6-1 an = an 1 + f (n) 5. 5-1 4.
2.() 3.() 4-1
1-1
an = an 1 + an 2
, , f (n) 3
-170-
12
1. 1-1 1-2 2. 5. 5-1 6. 6-1 68-95-99.7 ( ) 4. 4-1 3.
-171-
13
-() 1. 2. 2-1 3. 4. 3-1
4-1
1. 1-1 2. 2-1 ( transition matrix ) 3. 5. 6. 4. 4-1 5-1 6-1 6-2
-172-
14
1. ( 1-1 )
2. ( 2-1 )
2-2
2-3 ()
3.
3-1
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15
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1.
1-1 1-2
2. 3.
2-1 x 3-1 y y x 3-2 3-3
4. 4-1 4-2 4-3 4. 1. 1-1 1-2 y = x 2 2. 2-1 () x=a x=b y=0 3. 3-1 2. 3. 2-1 3-1
1.
1-1 1-2
-174-
16
97/08/20
97 B A B
1 1~2
1 2 1 3 14-1
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4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 DVD 24 25
1 1 1 1 1 1 1 1~2 1 1 1 1 1 1 5 1 1
1 2 3 4 5 6 7 8 9 10 11 4-2
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12
zoom tool
1 1 2 1 3 1 4 1 61 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1
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1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11
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12 13 14 15 16 17 18 1 Geometers Sketch PadGSP 2 Cabri 3D 3 Autograph 4 Mathematica 5 Maple 6 Derive 7 Matlab 8 SPSS 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
20 1~2
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300 0-9 EXCEL Visual Basic Delphi
http://tms.math.ntu.edu.tw/main.htm http://www.math.sinica.edu.tw/www/ http://www.nsc.gov.tw/SCI/ http://www.nioerar.edu.tw/ http://mathcenter.ck.tp.edu.tw
4-6
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() () () () () () () () () () ()
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41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 () 1
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DVD 1. 2. 3. 4. 5. 20~50
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(99 )
[10066] 56 02-2303-4381 212 02-2301-7961 [email protected] http://mathcenter.ck.tp.edu.tw 99 12 1
