5A 128a

5A 128

© 1995 K

umon Institute of E

ducation

M_WS_5A_121_math 5A 121-130 8/5/10 10:25 AM Page 15

E116åTime : to : Date Name

E116

Subtraction of Fractions 2

90 80 70 69100(mistakes) 0 1 2 3 4~0

®

© 1996.10 K

umon Institute of E

ducation

` Subtract.

( 1 ) -1 5 5 58 = =

14

( 2 )

( 3 )

( 4 )

( 5 )

-1 58

58

58=

12

-188 =-

88

=-1 =-

-1 14 =

18

-1 13 =

16

-1 25 =

13

Ex. -1 38 = =

14

-1 38

28 =- 3

81 08

78

M_WS_E_111_Math English E 111-120 8/6/10 10:20 AM Page 11

Time : to : Date Name

Simultaneous Linear Equations in Three and Four Variables 1

90 80 70 69100(mistakes) 0 –– –– –– 1~0

®

H96åH96

© 1998.6 K

umon Institute of E

ducation

` Solve the following equations.

( 1 ) {2x-3y-z=12 '''13x+2y+z=-1 '''26x-y+2z=3 '''3

[Sol] Form two equations with z eliminated.

1*2 : 4x- y-2z= '''1`

3+1 :̀

1+2 :

5*2 : 10x- y= '''5`

4-5 :̀ y=

y=

Substituting this into 5,

5x+ =

+5x=

+5x= .

Substituting x= and y= into 1,

+ -z=12

+ -z=

+ -z= .

{10x- y= '''4

5x-y= '''5

Ans.(x,y,z)=( , , )

M_WS_H_091_Math English H 091-100 8/6/10 12:10 PM Page 11

J 43 b

2.

Factor the expression. Begin by arranging the expression as follows:

In exercise (1), arrange it in standard polynomial form, with as the variable.In exercise (2), arrange it in standard polynomial form, with as the variable.

(1) With as the variable,

(2) With as the variable,

5 s2b 2 3a 2 5d sb 2 a 1 2d5 32b 2 s3a 1 5d 4 3b 2 sa 2 2d 45 2b2 2 s5a 1 1db 1 s3a 1 5d sa 2 2d5 2b2 2 s5a 1 1db 1 s3a2 2 a 2 10d3a2 2 5ab 1 2b2 2 a 2 b 2 10

b

5 s3a 2 2b 1 5d sa 2 b 2 2d5 33a 2 s2b 2 5d 4 3a 2 sb 1 2d 45 3a2 2 s5b 1 1da 1 s2b 2 5d sb 1 2d5 3a2 2 s5b 1 1da 1 s2b2 2 b 2 10d3a2 2 5ab 1 2b2 2 a 2 b 2 10

a

ba

3a2 2 5ab 1 2b2 2 a 2 b 2 10

J 44 a Factorization 4J 44

®

NameDateTime : to :

© 2002 K

umon Institute of E

ducation

1. Factor the following expressions by using the difference of two squares.(i.e. by forming the given expression into two terms as )

(1)

(2)

(3)

5 s2a 1 b 2 3cd s2a 2 b 1 3cd5 32a 1 sb 2 3cd 4 32a 2 sb 2 3cd 45 4a2 2 sb 2 3cd25 4a2 2 sb2 2 6bc 1 9c2d4a2 2 b2 1 6bc 2 9c2

5 s2a 2 3b 1 2cd s2a 2 3b 2 2cd5 3s2a 2 3bd1 2c4 3s2a 2 3bd2 2c45 s2a 2 3bd2 2 4c25 s4a2 2 12ab 1 9b2d2 4c24a2 2 12ab 1 9b2 2 4c2

5 sa 1 3b 1 cd sa 1 3b 2 cd5 3sa 1 3bd1 c4 3sa 1 3bd2 c45 sa 1 3bd2 2 c25 sa2 1 6ab 1 9b2d2 c2a2 1 6ab 1 9b2 2 c2

A2 2 B2

100% 69%~90% 80% 70%(mistakes) – 1 – 2~0

Ex.

5 s2a 1 b 1 cd s2a 1 b 2 cd

5 3 s2a 1 bd1 c4 3 s2a 1 bd2 c45 s2a 1 bd2 2 c2

5 s4a2 1 4ab 1 b2d2 c2

4a2 1 4ab 1 b2 2 c2

M_WS_J_041.indd 7 8/11/10 10:17 AM

O 31 a Concavity and Tangent LinesO 31

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NameDateTime : to :

© 2006 K

umon Institute of E

ducation

1. In each exercise, determine whether the curve of the given function isconcave up or concave down.

(1)

[Sol] yr5 ,

Since ,The curve is concave over .

(2)

(3)

ys. 0

ys5 exyr5 ex

y 5 ex

0 , x , `

ys, 0

ys5 21x2yr5

1x

sx . 0dy 5 ln x

0 , x , `

ys . 0

12x2 1 6x 1 10ys54x3 1 3x2 1 10x 1 1

sx . 0dy 5 x4 1 x3 1 5x2 1 x 1 6

100% 69%~90% 80% 70%(mistakes) – 1 – 2~0

When f0(x) . 0, the curve of f(x) opens upward, and is called concave up.When f0(x) , 0, the curve of f(x) opens downward, and is called concave down.

Concave up Concave down

x

y5 f(x)

x

y5 f(x)

Note that the slopes of the tangentlines f9(x) uniformly increase.

f0(x) . 0

Note that the slopes of the tangentlines f9(x) uniformly decrease.

f0(x) , 0

M_WS_O_031.indd 1 8/11/10 1:50 PM