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Transcript of A § w f õ A § Ù Î ¶ ¼ º ¡ á!web.ntnu.edu.tw/~696080204/pdf/handout04.pdf · ¡ á! ô /...

1 09/08 09/14 m

2 09/15 09/21

3 09/22 09/28 ( )

4 09/29 10/05 ( )

5 10/06 10/12 ( )

6 10/13 10/19

7 10/20 10/26 ( )

8 10/27 11/02 (Cramer)

9 11/03 11/091 n

10 11/10 11/16

11 11/17 11/23

12 11/24 11/30

13 12/01 12/07

14 12/08 12/14

15 12/15 12/21 (PR ) ( )

16 12/22 12/28

17 12/29 01/04

18 01/05 01/11

.379

- - -(Sandwich Theorem)

19 01/12 01/18

20 01/19 01/25

21

1

m -

2014/09/08 2014/09/14

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a � b 2 35

a bP ��

3 25

a bQ ��

45

a bR �� __________

A(3, � 5) B(11,7) A P B : 1:3AP PB � P ____________

(2,3), (5,5)A B P y P ________

2 2PA PB� ____________

P1 (2 , 5) P2 (6 , k) 1 2PP 43

m � l (k , l) � ____________

a L ax � 5y � 16 � 0 (3 , 2) ____________

L ax � by � c � 0 ac � 0 bc � 0 L ____________

L A (3 , 4) B ( � 3 , 1) L x ____________

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2 1 0x ay� � � 3 0bx y c� � � 3 2 5a b� � � ____________

3 34

�13

(1) A (1 , 2) L1 ________________(2) L2 __________________

( � 4 , 1) ___________________

1x ya b

� � (2 , 3)

______

L 4x � 3y � 2 � 0 L (3,1) 2 L ____________

L1 4x � 3y � 3 � 0 L2 8x � 6y � 5 � 0 ____________

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2 35

a bP ��

3 25

a bQ ��

45

a bR �� P Q R� �

A P B : 1:3AP PB � A(3, � 5) B(11,7) 3 3 11 ( 5) 3 7( , ) (5, 2)4 4

P � � � � �� �

P y (0, )P y2 2 2 2 2 2 2(0 2) ( 3) (0 5) ( 5) 2 16 63PA PB y y y y� � � � � � � � � � � �

2 22( 8 16) 31 2( 4) 31y y y� � � � � � � 4y � (0,4)P2 2

PA PB� 31

5 46 2 3km �

� ��

� 313

k � 2 24 ( 5)l k� � � � 203

l �

(k , l) � 31 20( , )3 3

(3 , 2) L 3a � 10 � 16 � 0 � a � 2 25 5a

� � �

x 0ca

� � y 0cb

� � L ( )

L x (a , 0) 4 0 4 13 3 ( 3)a� �

�� � �

� a � � 5

1 : 2 1 0L x ay� � � 12m

a�

� 2 : 3 0L bx y c� � � 2 3bm �

1 2L L 1 2 1m m � � � 2 13b

a�

� � �3 2a b� 3 2 5 5a b� � �

1L , 2L , 3L 113

m � � , 234

m � � , m3 � 3

(1) L1 m � m1 � � 1 � m � 3 y � 2 � 3(x � 1) � 3x � y � 1

(2) 3( 0) ( 0)4

y x� � � � � 3x � 4y � 0

� 0 1x ya a

� � ( � 4 , 1) � a � � 3 x � y � 3 � 0

� 0 ( � 4 , 1) (0 , 0) � x � 4y � 0

a � 0 b � 0 2 3 1a b

� �2 3 62a b ab

� � 21 6( )2 ab

� � ab � 24 � 1 122

ab �

� 2 3 12a b

� � a � 4 b � 6 12

L 3x � 4y � k � 0 L (3,1)2 2

| 3 3 4 1 | 23 4

kd � � � �� �

� |k � 13| � 10 � k � 13 � ± 10 � k � � 3 � 23 L:3x � 4y � 3 � 0 3x � 4y � 23 � 0x,y L1 4x � 3y � 3 � 0 � L1 8x � 6y � 6 � 0

L2 8x � 6y � 5 � 0 L1 L2 2 2

| 6 ( 5) | 11108 ( 6)

� ��

� �

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2014/09/15 2014/09/21

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y � f (x) f (0) � 6 x 3 y 6 f (x) �________

y � 2x2 � 4x � a 1 4 y � bx2 � cx � 5a � b � c � ____________

3 1( )2 1 4 1x xfx x� �

�� �

f(1) ___________

2 0( )

2 0x x

f xx x

� ��

� ��

2 0( )

2 0x x

g xx x

� ��� � ��

( ( 1))g f � � ____________

f (x) � x2 � ax � b (3,6) ( , )a b �____________

f(x) � ax2 � bx � 7 x � 2 f(x) 11 a � b �____________

y � � x2 � 2x � 2 � 3 � x � 2 y ____________

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( )f x (0) (1) 2f f� � ( 1) 0f � � ____________

20 3005 p p ____________

x (a � 1)x2 � 2x � 3 0 a ____________

y � mx2 � 4x � 2 x m ___________

a y ax2 a y x� � a _______________

2 2 2( ) ( ) 8( ) 12y f x x x x x� � � � � � x _____________

( ) 2f x x� � ( ) 3 2g x x� � ____________

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f (x) � ax � b (1) f (0) � 6�b � 6 f (x) � ax � 6 (2) f (x � 3) � f (x) � � 6 � [a(x � 3) � 6] � (ax � 6) � � 6�a � � 2 f (x) � � 2x � 6

y � 4 � 2(x � 1)2 � 4(x � 1) � a y � 2x2 � (a � 6)y � bx2 � cx � 5 2b � 0c � 6 5a � � a � b � c � 11 � 2 � 0 � 13 3 1

2 1xx�

��

� x � 2 2 1 1(1)4 2 1 9

f �� �

� �2( ( 1)) (( 1) ) (1) 2 1 3g f g g� � � � � � �

(3,6) f (x) � (x � 3)2 � 6 � x2 � 6x � 15 a � � 6 b � 15 f(x) � ax2 � bx � 7 � a(x � 2)2 � 11 a � 0 � � 4a � � b 4a � 11 � 7 � a � � 1 b � � 4 � a � b � � 5

y � � (x � 1)2 � 3 x � � 3 y � 13 2( )f x ax bx c� � �

(0) 2 2(1) 2 2( 1) 0 0

f cf a b cf a b c

� �� �� �� � � � � � �� � � � �� �

���

� � � a � b � 0 �

� � � a � b � � 2

� � � 2a � � 2 a � � 1 � b � 1 f(x) � � x2 � x � 2

( )f x f (x) � (20 � p)(300 � 5p) � � 5p2 � 200p � 6000 � � 5(p � 20)2 � 8000 � 8000 20

x a � 1 � 0

1 � 3(a � 1) � 0 a � 1 a � 43

a � 43

y � mx2 � 4x � 2 x�b2 � 4ac � 0 � ( � 4)2 � 4m � 2 � 0 � 16 � 8m � 0 � m � 2

ax2+a �x�ax2+x+a 0� 2

001 11 4 02 2

aaa aa

���� �� � � �� �� ��

�a � 12

2 2( ) 0 ( ) 6 ( ) 2 0y f x x x x x� � � �� � � � � � � �� � � � � �� �( 3)( 2) ( 2)( 1) 0x x x x� � � � � �

3 2 1 2x� � � � x 4

( ) 2f x x� � ( ) 3 2g x x� � �2 3 2x x� � � 24 4 3 2x x x� � � �

2 7 6 0 ( 1)( 6) 0x x x x� � � � � � � � 1x � 6x �6x � (6) 4f � � (6) 4g � ( ) (1, (1)) (1,1)f �

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2014/09/22 2014/09/28

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(1) 103

�� ____________ (2) ___________

4 ____________

5sin cos tan6 3 4� � � � __________

cot135 tan 45 sec150 2 tan 30� � � � � � � ______________

sin( 210 ) tan135 cos( 300 )� � � � � � � _______________

tan22� � k sin2002� � ____________

� tan 3� � 1 sin 1 sec1 cos 1 csc

� �� �

� �

� �_______________

12tan5

� � cos 0� � sin cos� �� __________

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tan 2� � cos 3sin2sin 3cos

� �� ��

��

__________

(A)sin 2 2x � (B) sin 22

x � (C) 2sin x � (D) 2sin2

x �

__________

( ) 2sin 3 1f x x� � M m T ( , , )M m T �_______________

(1) sin 47 , tan 47 , sec 47a b c� � � � � � _______________(2) a � sin130� b � cos( � 20�) c � tan250� a b c _______________

0 2� �� � 22cos 11cos 5 0� �� � � � _______________

25sin 8cos 1 0� �� � � cos� � __________

23tan sec 7� �� � sec� � __________

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(1) 23� (2) 4

3��

2r s� � 12rs�

1 22rs r s� � � 2 8 8s s s� � � � � �8 4 2� �� � �

sin cos tan6 3 4� � �

� �1 1 312 2 4

� � � �

( cot 45 ) tan 45 ( sec30 ) 2 tan 30� � � � � � � � � �2 3 31 1 2 0

3 3� � � � � �

sin 210 tan135 cos300� � � � � � � ( sin 30 ) ( tan 45 ) cos 60� � � � � � � � �1 11 22 2

� � � �

tan22� � k2

sin 2002 sin(180 11 22 ) sin 221

kk

� � �� � � � � � � ��

� tan 3� � , 60� � �

31 1 221 21 12 3

� ��

� �

2 3 3 33 3 2�

� �

3�

12tan 05

� � � �� , cos 0� � �� , �

12 5 17sin cos ( ) ( )13 13 13

� � � �� � � � �

3sin1 1 3tan 5cossin 2 tan 3 72 3cos

���

� ��

� �� � � �

��

2sin 22

x � �� � , 2sin 2x �� 2sin x ��

1 sin 3 1x� � � � 2 2sin 3 2x� � � � 1 2sin 3 1 3x� � � � �23� 2( , , ) (3, 1, )

3M m T �

� �

(1) sin 47 1a � � � , sin 47tan 47 1 (tan 45 1)cos47

b �� � � � � �

� , 1sec47 1

cos47c � � � �

�1c b a� � �

(2)a � sin130� � sin(180� � 50�) � sin50� b � cos( � 20�) � cos20� � sin70�c � tan250� � tan(180� � 70�) � tan70� � tan45� � 1 tan70� � sin70� � sin50� c � b � a

22 cos 11cos 5 0� �� � � (2 cos 1)(cos 5) 0� �� � � �1cos2

�� � cos 5� � ( )3��� �

53�

25(1 cos ) 8cos 1 0� �� � � � � 25cos 8cos 4 0� �� � � � � (5cos 2)( cos 2) 0� �� � � �

2cos cos 2( )5

� �� � �

2 2sec 1 tan� �� � 2 2tan sec 1� �� � � 23(sec 1) sec 7� �� � � � 23sec sec 10 0� �� � �

� (sec 2)(3sec 5) 0� �� � � �5sec 2, 3

� � �

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13

2014/09/29 2014/10/05

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, � � 5 3cos , sin13 5

� �� � sin( ) cos( )� � � �� � � _______________

2� � �� �

2� � �� � 1sin

5� �

3cos10

� � � � �� � ____________

sin100 sin( 160 ) cos200 cos( 280 )� � � � � � � � __________

(1) 14

� � �� � (1 � tan�)(1 � tan� ) �__________

(2) (1 tan 35 )(1 tan100 )� � � � � __________

sin� � 2cos� � 0 tan2� � ____________

23cos 5cos 2 0� �� � � cos2� � ____________

2 (tan cot ) 4 0x x� �� � � � 5 1� sin 2� _______________

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15

( ) 5cos 12sin 7f x x x� � � M m 3M m� ______________

2( ) sin sin 1f x x x� � � ( )f x _______

( ) sin cos sin cos 2y f x x x x x� � � � � x y ____________

0 2x �� � 2 2( ) 3sin 4sin cos 5cosf x x x x x� � � _______ ________

( 2 21 cos 2 1 cos 2sin 2 2sin cos ,sin ,cos2 2

� �� � � � �� �� � � )

(3, 2) 60� _______________

1 2: 2 1 0, : 3 5 0L x y L x y� � � � � � _______________

1 2: 2 3 0, : 2 1 0L x y L x y� � � � � � ______________________________

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16

12 4 5 3 33sin( ) sin cos cos sin13 5 13 5 65

� � � � � �� � � � � �

5 4 12 3 16cos( ) cos cos sin sin13 5 13 5 65

� � � � � � �� � � � � �

33 16 17( )65 65 65

�� � �

2� � �� �

2� � �� � 1sin

5� �

3cos10

� � � � 1 2cos 15 5

� � � � � �9 1sin 1

10 10� � � �

6 1 1 2cos( ) cos cos sin sin25 2 5 2 2

� � � � � �� � � � � � � � � � � � � 2� 74

� �� � �

sin100 sin200 cos200 cos80 sin80 ( sin20 ) ( cos20 )cos80� � � � � � � � � � � � � �

1(sin80 sin20 cos20 cos80 ) cos(80 20 ) cos602

� � � � � � � � � � � � � � � � �

(1) 14

� � �� � � 1tan( ) tan4

� � �� �tan tan 1

1 tan tan� �

� ��

��

� tan tan 1 tan tan� � � �� � �

(1 tan )(1 tan ) 1 tan tan tan tan 1 (1 tan tan ) tan tan 2� � � � � � � � � �� � � � � � � � � � �

(2) tan 35 tan100tan(35 100 ) tan135 11 tan 35 tan100

� � �� � � � � � � �

� � �� tan 35 tan100 tan 35 tan100 1� � � � � � �

(tan 35 tan100 ) 1 tan 35 tan100 2� � � � � � � � � �

sin 2cos� �� � sin 2cos

��

� � tan 2� � 2 2

2 tan 2 2 4tan 21 tan 1 2 3

���

�� � � �

� �

� (3cos 1)(cos 2) 0� �� � � � 1cos3

� � cos 2� � � ( ) 2 1 7cos2 2cos 1 2 19 9

� �� � � � � � �

� ( 5 1) tan cot ( 5 1) 4� � � �� � � � � � � 4 5 15 1

� � � ��

tan cot ( 5 1) ( 5 1)� �� � � � � 2 5�1tan cot 2 5

sin cos� �

� �� � � �

5sin 2 2sin cos5

� � �� �

5 12( ) 5cos 12sin 7 13( cos sin ) 7 13cos( ) 713 13

f x x x x x x�� � � � � � � � � ( )

1 cos( ) 1x�� � � � 20M � 6m � � 3 20 3 ( 6) 2M m� � � � �

2 21 3 1 3( ) (sin sin ) (sin )4 4 2 4

f x x x x� � � � � � � 1 sin 1x� � � sin 1x � ( )f x 3

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17

sin cost x x� � 2 2t� � � � 2 2 2sin cos 2sin cost x x x x� � � �2 1sin cos2

tx x � �

� � � �2

22 2 21 1 1 1 1 3 12 2 2 1 1 12 2 2 2 2 2 2

tt t t t t t�� � � � � � � � � � � � � � � 1t � � 1

1 cos2 1 cos2( ) 3 2(2sin cos ) 5 2sin 2 cos2 42 2

x xf x x x x x� �� � � � � �

2 2 2 22 1 4 2sin 2 cos2 4 2 1 4x x� � � � � � � � � �4 5 2sin 2 cos2 4 4 5x x� � � � � �

( )f x 4 5� � 4 5� �

3tan 60 31

� � � 2 3 ( 3)y x� � � � � 3 (2 3 3) 0x y� � � �

1 2

1 1,2 3L Lm m� � � �

1 1( )2 3tan 11 11 ( )

2 3

�� �

� �� �

� 45� � � 135�

�2 2 2 2

2 3 2 12 1 1 2

x y x y� � � ��

� ��2 3 ( 2 1)x y x y� � � � � � � 2 0 3 3 4 0x y x y� � � � � �

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2014/10/06 2014/10/12

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19

ABC 45 , 75 , 10A C BC� � � � � � � AC _______________

ABC 6, 3 1, 105AB AC B� � � � � � C� =_______________

ABC � � � � � �: : 10 :8 : 7b c c a a b� � � � sin : sin : sinA B C =_______________

ABC 2 4 3 0a b c a b c� � � � � � sin : sin : sinA B C � ________________

ABC : : 2 : 3 : 7A B C� � � � 6b �

(1) sin : sin : sinA B C � __________ (2) c � ____________

ABC ( ) : ( ) : ( ) 4 :5 : 6b c c a a b� � � � cos : cos : cosA B C =_______________ ( )

ABC 2 2 1a � � 3 2b � � c � 1 ABC ___________

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20

ABC 2 2( )b c a ca� � � B� =______________

ABC 6AB � 9AC � �A � 120� �A BC D AD � __________

5 6 7 __________

30� 10____________

15� 20____________

A 45� 60� B 30�

A,B 70 ____________

A,B,C 60� , 45� , 30�

40AB � 60BC � ____________

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21

60B� � �10 10 2 5 3

sin 45 sin 60 22 32 2

AC AC AC� � � � �� �

5 6AC �

6 3 1 6 3 1sin sin sin sin105 sin 6 2

4

AB ACC B C C

� �� � � � �

� �

�2 3( 3 1)( 3 1)sin

4C �

� �3sin

2C �

60C� � � 120� ( AC AB� B C� � � )

10 (1)8 (2)7 (3)

b c rc a ra b r

� ��� � � � � ��

������

� (1) (2) (3) 25: (4)2 2

a b c r� �� � � � 5(4) (1)

2a r� �

9(4) (2)2

b r� �11(4) (3)2

c r� � sin : sin : sin : : 5 : 9 :11A B C a b c� �

2 04 3 0a b c

a b c� � ��

� � ��

����

�� 2 9 5 0a c� � � �� � , 5 7

9 9a c b c� ��

5 7: : : : 5 : 7 : 99 9

a b c c c c� � sin : sin : sin 5 : 7 : 9A B C �

(1) 2 180 302 3 7

A� � � � � �� �

3 180 452 3 7

B� � � � � �� �

180 30 45 105C� � � � � � � � �

sin : sin : sin sin 30 : sin 45 : sin105A B C � � � �1 2 6 2: :2 2 4

�� 2 : 2 2 : ( 6 2)� �

(2) 6 6sin 45 sin105 2 6 2

2 4

c c� � �

� � �3( 3 1)c � �

4 , 5 , 6b c k c a k a b k� � � � � �7 5 3, ,2 2 2

a k b k c k� � �

2 2 225 9 4914 4 4cos 5 3 22 ( )( )

2 2

k k kA

k k

� �� � �

2 2 29 49 2533 114 4 4cos 3 7 42 142 ( )( )

2 2

k k kB

k k

� �� � �

2 2 249 25 965 134 4 4cos 7 5 70 142( )( )

2 2

k k kC k k

� �� � �

1 11 13cos : cos : cos : : 7 :11:132 14 14

A B C � � � �

b � a � c �B � �A � �C2 2 2(2 2 1) 1 (3 2) 8 4 2 1 1 9 6 2 2cos

2 (2 2 1) 1 2(2 2 1)B � � � � � � � � � �

� �� � � �

(2 2 1) 122(2 2 1)

� �� � �

��B � 120�

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22

2 2 2 2 2 2 2 2( ) 2b c a ca b c ca a ca c a b ca� � � � � � � � � � � �

�2 2 2 1cos

2 2 2c a b caB

ca ca� �

� � � 60B� � �

ABD � ACD � ABC

� 1 1 16 sin60 9 sin60 6 9 sin1202 2 2

AD AD� � � � � � � � � � � � � 15 54AD � � 185

AD �

5 6 7 92

s � �� � 9 (9 5)(9 6)(9 7) 6 6� � � � � � 9 6 6r s r� � � �

2 63

r �

BC x AB x� � � 10 3x x� � � 10 5( 3 1)3 1

x � � ��

5( 3 1)�

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2014/10/13 2014/10/19

蔡奇霖 編著

24

(tan120 , cos180 )a � � ��

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27

( )

2014/10/20 2014/10/26

蔡奇霖 編著

28

3 2 2 2( ) (2 4 ) (3 6 ) 3 4f x a x x b x x x x a� � � � � � � x ( )f x � ______________

5 4 3 2( ) ( 2) 8( 2) 7( 2) 10( 2) 11( 2) 7f x x x x x x� � � � � � � � � � � ( )f x(1) _________(2) __________ (3) _________

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29

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蔡奇霖 編著

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4 7( 23) ( 22)x x� � � 21x � ____________

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蔡奇霖 編著

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(Cramer)

2014/10/27 2014/11/02

蔡奇霖 編著

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( 1)( 3)( 5)( 7) 15 0x x x x� � � � � � ________________________

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2014/11/03 2014/11/09

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20 205 10 2 25 5

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50 40 50020 40 300 900

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(1)1

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2 2 1xa � �3 3x x

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