..

Vol.4 4 5 8

(diff L' Hospital)x 1lim

7 3x x+ 3

10 x 8x+1=

00

=x 1lim

1

2 73x(3) 1

2 x+3(1)

1

2 10x( 1) 1

2 8x+1(8)

=

32

12

13

83

=23

12k = 12 23 = 8 7 7 53

Con x = 2 f (2) function $% : f(2) = a b

x 2lim f (x) =

x 2lim x

3 3x 2x 2

=

x 2lim 3x

2 31

= 3 (4) 3 = 9

00

x 2+lim f (x) =

x 2+lim x2 + ax + 1 = 4 + 2a + 1 = 2a + 5

'() 2a + 5 = 9 a = 2

*( a b = 9 2 b = 9 b = 7

a2 + b2 = 22 + (7 )2 = 53

17 13 2

f (x) = 2x , x 02 , x < 0

f ( 5) = 2f (3) = 2(3) = 6

=

1

18 13 2' f (x) = d

dxf(x) =

h 0lim

f (x+ h) f (x)h

1 = d (sin x)dx h 0

limsin (x+ h) sin x

h

2 = d (x+ 1)dx h 0

lim((x+ h) + 1) (x+ 1)

h=

x0lim

((x+ x) + 1) (x+ 1)x

3 = d (e2x)

dx h 0lim e

2(x+h) e2x

h=

x 0lim e

2(x+ x) e2x

x

4 = d (2x2

1)dx h 0

lim(2 (x+ h)2 1) (2x2 1)

h

21 15 54.'/0123456278$290$2$: 4 1;( t = 0 => 2$?2%f(4) (0)

' f (t) = 4f (t) 3f (t) + 1 (1)

f (t) = 4f (t) 3f (t) (2)

f(4) (t) = 4f (t) 3f (t) (3)

f(4) (0) = 4f (0) 3f (0) (4)

*/2 t = 0 @2 (1) , f (0) = 4f (0) 3f (0) + 1 = 4(2) 3 (1) + 1 = 6

*/2 t = 0 @2 (2) , f (0) = 4f (0) 3f (0) = 4 (6) 3 (2) = 18

*/2@2 (4) , f(4) (0) = 4 (18) 3 (6) = 54 25 17 4

' xy = c y = cx y = cx1 y = cx2 = cx2

B667C'7B$6D$B> a , ca m =

yx =

2 25 1

= 1

m.>% 1;( x @G > y = c

x2

m.>% /3? > a , ca ca2

* m.>% /3? a , ca = m

c

a2= 1 c = a2 (1)

(1,2)(5,-2)

(a, )ca

y = cx

m

2

m ''7 $: (1 , 2)a , ca = m =

ca 2

a 1 1

= , */2 '()ca 2 a + 1 c = a2

= a2a 2 a + 1

2a = 3 a = 32

*/2@2 (1) , c = 32 2

=94

26 17 3*'7$ y = 2x2 + 8x + 6 (1)

y = 2x + 6 (2)

' (1) *( (2) , 2x2 + 8x + 6 = 2x + 6

2x2 + 10x = 0 2x(x 5) = 0 x = 0 , 5

*/2@2 (2) , 5 x = 0 ) y = 6 , 5 x = 5 ) y = 4

'7$ > (0 , 6) $: (5,4)@ y = f(x) = 2x2 + 8x + 6

f (x) = 4x + 8m1 = f (0) = 8

m2 = f (5) = 4(5) + 8 = 12

B6K : L1 y 6 = 8(x 0) y = 8x + 6 (3)

B6K : L2 y (4) = 12(x 5) y = 12x + 56 (4)

*/2 @2 (3) ) (3) (4), 0 = 20x 50 x = 52

x = 52

y = 8 52 + 6 = 26 (a , b) > 52, 26 a + b = 28.5

(a,b)

(0,6) (5,-4)

L : m = 81 1 L : m = -122 2

3

30 19 3 @ 8./. B63?6N2.>% = A

A = 12(20 2r)r = (10 r)r = 10r r2

dAdr

= 10 2r = 0

r = 5

= K%P2C6KAmax (10 5)(5) = 25

32 21 6' f (x) = 3 x + 5 = 3x

12 + 5

f(x) = f (x)dx = (3x12 + 5)dx = 3x

32

32

+ 5x + c = 2x x + 5x + c

.'/0: f(1) = 5 f(1) = 2 + 5 + c = 5 c = 2

f(x) = 2x x + 5x 2

x 4lim

f(x2) 2f(x) =

f(16) 2f(4) =

(32 16 + 80 2) 216+ 20 2

=20434

= 6

34 22 7'.'/0 ">6S$21%B2B$6D$B /3?'7 (2 , 19) /$: 19"*B% *( f(2) = 19f (2) = 19

f (x) = f (x)dx = (6x + 4)dx = 3x2 + 4x + c f (2) = 12 + 8 + c = 19 c = 1 f (x) = 3x2 + 4x 1

f(x) = f (x)dx = (3x2 + 4x 1)dx = x3 + 2x2 x + c f(2) = 8 + 8 2 + c = 19 c = 5 f(x) = x3 + 2x2 x + 5

f(1) = 1 + 2 1 + 5 = 7

2 39 22?%' a, b, c CK/3?B6 KUK3:/3: '(% a3, b3, c3

a3 = 1 + 3 4 3

= 1 + 3 3 4 + 3 3 16 + 4 = 5 + 3 3 4 + 3 16 b3 = 3 2 + 3 3

3= 2 + 3 3 4 3 3 + 3 3 2 3 9 + 3 = 5 + 3 3 12 + 3 18

r r

20 - 2r

4

8: *2G b > a $1 4 /C4%) b3 > a3

/3:$: c3 = 11 a3 = 5 + 3 3 4 + 3 16 1 G 2 G

8: a > c $1 1 *( 3 /C4%) : 1 2a3 > c3

1234@SBXK : (a + b)3 = a3 + 3a2b + 3ab2 + b3

5 41 1

= 27 y = 35x 9x2log3

log2log 5

log 4log 7

log 6

log3

log4log 5

log 6log 7

log 8

= = 35x (32)x2 33 log3log2

log5

log4log7

log6log4

log3log6

log5log8

log7

= = 35x+ 2x2 33 log8log2

=log23

log2=

3 log2

log2= 3

= 3 5x + 2x2 xy = 12 3

=18

= 02x2 + 5x 3

= 0(2x 1)(x + 3)

x = 12, 3

.'/0: x R+

7 42 1 1 ! 2Y)U132KZlogxx =

log2

2

2 log x = x log 2 log x2 = log 2x

x2 = 2x

>Y: > x = 2 , 4 2 "# 8K(6?2Y (1 , 0) *( */2@2 U[2'KC% (9,2) y = log 1

3

x

3 %&' 4 5X *2G

y

x

(4,16)

(2,4)

y = x2

y = 2 x

'723463> x U[2:)68C'K;

5

15 46 3log82 + log84 + log88 + log816 + ... + log82

n= 210

log232 + log2322 + log232

3 + log2324 + ... + log232

n= 210

13

+ 23

+ 33

+ 43

+ ... + n3

= 210

= 6301 + 2 + 3 + 4 + ... + n

= 630n(n+1)2

= n(n + 1) 1260 = 35 36

n = 35 16 46 1

a =n = 2

9 log 1 1n2 = n = 2

9 log n

2 1n

= n = 29 log (n 1)(n+ 1)n2

=n = 2

9 log n 1n n+ 1n

a = log 12 32 + log 23 43 + log 34 54 + ... + log 89 109 a = log 12 32 23 43 34 54 ... 89 109 = log 1018 = log 59

10a = 59

18 47 10@ '() logxy + logyx = 103 A = logxy ,

1A

= logyx

A + 1A

=103

3A2 + 3 = 10A 3A2 10A + 3 = 0 (3A 1)(A 3) = 0

A = 13, 3 logxy =

13, 3 y = x

13 , x3

() 1 , ' y = x13 xy = 16 x x

13 = 16 x

43 = 16

(x43 )

34 = (16)

34 x = (24)

34 = 23 = 8 y = x

13 = 8

13 = 2

$%2$42 x + y = 8 + 2 = 10

() 2 ' x = 2y = x3, xy = 16 x x3 = 16 x4 = 16

y = x3 = 23 = 8 x + y = 2 + 8 = 10

6

20 48 f1(x) = 12log x1x1

y = f(x) = 10x 10x10x + 10x

=

10x 110x

10x + 110x

=

102x 1

10x

102x +110x

=102x 1

102x + 1

inverse U3?2 x U[2 y , U3?2 y U[2 x '()x = 10

2y 1

102y +1 x(102y + 1) = 102y 1 x(102y) + x 102y = 1

102y(x 1) = x 1 102y = x 1x 1

2y = log x 1x 1 y = 1

2log x 1x 1 f1(x) = 12 log x 1x 1

22 49 2logc+ ba + logc ba = x(logc+ ba)(logc ba)

log a

log (c+ b) +log a

log (c b) =x log a

log (c+ b) log a

log (c b)

2Y >X;/78'20@2B6K log (c + b) log (c b)

log (c b) + log (c + b) = x log a

*(.'/0: log(c2 b2) = log ax a2 + b2 = c2

log a2 = log ax

x = 2 25 51 > a )66

1234U[21B:.> 6.S. P\?%.'/02'(DC8K('.'/0 log a = 2 log( x 2 + x + 2 ) log x

log a = log( x 2 + x + 2 )2 log x

log a = log

( x 2 + x+ 2 )2x

a = ( x 2 + x+ 2 )2

x (1)

'%?2)1$% % > 0 *( /Y@ log 1 x 2

%*/2 x 'Y22=6/3? @2 (1) 2*/2 @S)6) a %U[2'Y22KK(x = 2 : a = (0+ 2)

2

2= 2

.... ........

7

*/2 x = 3 : a = (1+ 5 )2

2=

6+ 2 5

3

*/2 x = 4 : a = ( 2 + 6 )2

4=

8+ 2 12

4= 2 + 3

*/2 x = 5 : a = ( 3 + 7 )2

5=

10+ 2 21

5

8:5K*/2 x 'Y22=6)UK?G '()> a 66> 1 63 544

B66C>K:>K$/3? 1 : >K:>K$/3? 2 : A,B, a, b M,N, m, n 8, *6 X 8, *6 X

'$8, *6 /$4% 2 >K:>K$2$?%2 *2YXG 1B3:() 1 : A, B XK%16

- M, N B$:$2) 2! C93- X>K:>K$/3? 1 1B3:@2/3?% 4 S%) C934 3- X>K:>K$/3? 2 1B3:@2/3?% 4 S%) (1S%3$:>K:>K$*K))

/Y) C934 3

K;3/3? 1 /Y) C932! (4 3) (4 3) = 288

() 2 : A, B C$2

A

B

M N

LOCK

A

M

N B

LOCK1 2

34

8

- B XPK11% A /Y) 2 C93- B$: M, N ) 2! C93- X>K:>K$/3? 1 1B3:) 2 *:: >

S% 1, 2, 3 - >2(S% /Y) C93 S%/3?3 2

K6 8 C93- S%3$2 /Y) 2 C93

S% 2 /2$42*(B$: a, b ) 2 C93- X>K:>K$/3? 2 /Y62>K:>K$*K /Y) 8 C93

K;3/3? 2 /Y) C932 2! 8 8 = 256

K6 2 K;3) C93288 + 256 = 544 4 64 3

__ __ __ __ __ __ __

x ) 7 $' 3 9 B$: 4 $

() 1 x )6@S 0, 1, 2 /Y) C937 4! = 168

() 2 x U[2 0, 1 K 2 /Y) C933 4!2!

= 36

x ' 0, 1, 2 B$:*::63PY4 1 >X

C93/$4%6 C93= 168 + 36 = 204

7 66 3521 1 - 10 1( 4 >(*22, 1 11 - 20 1( 1 >(*22 K6 50 >(*22%K/Y) 45 >(*22 *U /YDC 5 >(*22() 1 DC1 4 >(*22 1 1, 1 1 >(*22 1 1

/Y) C93 101 101 = 100() 2 DC1 1 >(*22 5 1 /Y) C93 105 = 252 C93/$4%6 C93= 100 + 252 = 352

3 9 2

1 C93 0, 1, 2, x

9

14 69 7' /7 *B% *( f(x) x x A f(1) 1 f(2) 2

A B () 1 1 /&0 2 *( 2 /&0 1, 3, 41 1 1 C93 3 C93 = 3 C932 2 $@$2\?%/3? 1 )6

3 () 2 1 /&0 3, 4 *( 2 /&0 1, 3, 44 2 C93 2 C93 = 4 C93

C93/$4%6 C93= 3 + 4 = 7 16 70 0.7

n(s) = 53 = 10E : D:U[2'Y22>XKD>X;U[2'Y22>3? 63 2 K;3

K;3/3? 1 : >X, >3?, >3? 63 C93 21 32 = 6 K6 C937

K;3/3? 2 : >3?, >3?, >3? 63 C93 33 = 1 P(E) = n(E)

n(S) =710

= 0.7

17 71 1n(s) = 104 = 210E : () 1 : 63 C93+,+,+, 53 51 = 50

K6 C93100

63 C93+,,, 51 53 = 50 P(E) = n(E)

n(S) =100210

=1021

19 72 2n(s) = 72 = 21E : C:)6)B3Y *U %C:'B3*% + 1/2$42 P\?%63 5 X

n(E) = 52 = 10 P(E) = 1021

10

21 73 196 49pi196

n(s) = 8./. = 14 14n(s) = 8./. *K% = 14 14 pi(72)

P(E) = 196 49pi196

24 74 3n(s) = 6 6 = 36E : (1, 6) (6, 1) (2, 5) (5, 2) (4, 3) (3, 4) (6, 2) (2, 6)

P(E) = n(E)n(S) =

836

=29

29 77 0.55 P((A B) ) = 0.55

7

7

7 7

7

7

77

A B

0.15 0.15 0.15

0.55

11

30 77 1S : 2Y a, b, c, d, e, f 6B$:/3?)U6 C93n(s) = 6! = 720E : *U $*K)6@S a *( b *($B7/)6@SBK(A B

__ __ __ __ __ __ c, d, e, f CBK( b, c, d, f

8:$*K*($B7/ 63 c, d, f /3?@SK6$2'\%%*K;34!

() 1 /Y) C931 __ __ __ __ 4 1 4! 4 = 96 K6

4! 312 C93() 2 /Y) C933 __ __ __ __ 3 3 4! 3 = 216

P(E) = 312720

=1330

32 78 2' =Tr+ 1 nr an r br

= 10r 2x3 10 r

x3

y2

r

= 10r 210 r x30+ r (x3)r

y2r

= x

6

y12r = 6

B.U.B. 1% > x6y12

106 210 6 = 3, 360 2 93 3

u = 4a + 2b c = 4 21 + 2 32 10 = 18 >6S$21% u = mu = 8

1= 8

Vector /3?$4%j$: %63>6S$2 u = 18

'$/$4% 4 8: 6313/3? >1 3m = 18

3 94 10' a + b 2 = a 2 + 2a b + b 2

132 = 102 + 2a b + 72 a b = 10

e

c, d, f

b, c, d, f

b, c, d, f /3?)6PY4$:$*K

12

12 98 1' a b a b = 0 3x + 2y = 0 y = 3

2x (1)

' a = 4 x2 + y2 = 4 x2 + y2 = 16 (2)

*/2 @2 (2), y = 32x x2 + 32x

2= 16 x2 + 9

4x2 = 16 13x

2

4= 16

*/2@2 (1) 8?> y13x2 = 64 x2 = 6413

x = 813

5 x = 813

y = 32

813

= 1213

5 x = 813

y = 32

813

= 1213

*(.'/0: a c > 0 2x + 3y > 0 2x > 3y

K;3 )6'KC%x = 813

, y = 1213

2 813

> 3

1213

K;3 'KC%x = 813

, y = 1213

2 813

> 3

1213

x y = 813

12

13=

20

13

14 99 1U[2B6:$C12\?%1% vector 3 6CC

17 101 1.5@ AB = u , AD = v

AM = AB + BM = u + 13v

AN = AD + DN = v + 13u

AM + AN = u + 13v + v + 1

3u

AM + AN = 43(u + v) = 4

3(AC)

AC = 34(AM) + 3

4(AN)

'() = 34, = 3

4 + = 3

4+ 34

=32

= 1.5

A

D C

B

1

2

21

u

v

N

M

13

4 111 31 3 DC S2 A = {1, 2} B = {1} C = {2}

P(A) = {, {1}, {2}, {1, 2}}

P(B) = {, {1}}

P(C) = {, {2}}P(B) P(C) = {, {1}, {2}}8: P(A) P(B) P(C)

6 112 4' set B : x2 4x + 3 = 0 (x 3)(x 1) = 0 x = 3, 1

B = {3, 1} , C = {1, 2, 3, .....}

A C = {4,3,2,1, 0}

63'Y22B6SC 7 $(A C) B = {4,3,2,1, 0, 1, 3}

7 113 2'.'/0 *B%C A B

1 1 (C A) (C B) C1 2 (A C) (B C)

= (A B) C = C1 3 (A C) (B C)

= (A B) C C1 4 (A B C) (A B)

10 114 9'B6SC 1, 2, 3, ....., 10 8: P\?%K 5 %$1 + 2 + 3 + ..... + 10 = 55

$%2$42 KB6SC6 8 $ * DK6$%>%K 5 %$'\%62$:KB6SC 2 $ ' 10 $ .

D:1% 2 $2$42%K 5 %$S2$2 '() B6SC 2 $/3?) 63$%234

K6 9 C93(1, 4) (1, 9) (2, 3) (2, 8) (3, 7) (4, 6) (5, 10) (6, 9) (7, 8)

A B

C

14

11 115 48C'K;B6SC *($@2 8: *($'(63/%A B = {1, 2, 3, 4} A B

3 %> )UX@2P A %3, )UX@2P B %3K)UX@2P A *( B /$4%B%P (*'()6X@2 A K B )6)) '(BK%) C933 3 3 3 = 81*@2 81 C93/3?)%$ 2 C93 >

1. 1, 2, 3, 4 )UX@2 A 6 B '(U[2 2. 1, 2, 3, 4 )UX@2 B 6 A '(U[2

C93KBK% A *( B > C9381 2 = 79 14 116 26

n(UUUU) = 66 2$K32/3?B:)CS3 = 6 x + y + z = 6' n(UUUU) = 66(x + y + z) + 10 + 11 + 17 + 13 + m = 66 m = 9

'Y22>2/3?B:)CS$%mn*(>;C >2= 17 + 9 = 26

******************************

)/ $%mn

>;C

x

z

y1017

11 m

13

15