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1 2552

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99/20 10300

0-2668-7974 0-2668-7123 2530

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371.95

691

: 2552

174

ISBN 978-974-559-718-1

1. -

2. - 3.

580 8 . 34 1

. 10230

0-2943-8373-4 0-2510-7753

. .2542 ( 2) . .2545

10

28

( 2547)

(Acceleration Program)

5 6

-

( . )

. . 2542 ( 2) . .2545

10 ( 4)

28

2543

School in school Program 2547

Gifted Education

(Acceleration Program)

(Enrichment

Program)

( 3

6 5 6 )

(Extension Program)

(mentor)

AP Program (Advanced

Placement Program)

18

(Acceleration

Program) 5 ( 6 )

( )

Pre-test

Post-test

1 1.

2.

3.

4.

10

6

24

38

.

4

2 5.

6.

7.

8.

38

30

48

6

200

1 9.

10.

11. 2 3

12.

27

20

36

24

5

2 13.

14.

15.

15

38

40

.

200

6

1

16.

17.

18.

(12 )

(25 )

(13 )

30

20

50

100

ส า ร บ ัญส า ร บ ัญ

เรือ่ง หนาแผนการจดัการเรยีนรทูี ่1 เรือ่ง ลมิติของฟงกชนั 1

ใบความรทูี ่ 1.1 8ใบความรทูี ่ 1.2 9ใบกจิกรรมที ่ 1.1 10ใบกจิกรรมที ่ 1.2 11ใบกจิกรรมที ่ 1.3 13ใบกจิกรรมที ่ 1.4 18แบบฝกหดั 21แบบทดสอบรายจดุประสงค 1 22แบบทดสอบเพือ่ประเมนิตนเองกอนเรยีน (1) 23

แผนการจดัการเรยีนรทูี ่2 เรือ่ง ความตอเนือ่งของฟงกชนั 27ใบกจิกรรมที ่ 2.1 31ใบกจิกรรมที ่ 2.2 33ใบกจิกรรมที ่ 2.3 35แบบฝกหดั 39แบบฝกหดัเสรมิ 40แบบทดสอบรายจดุประสงค 2 45แบบทดสอบเพือ่ประเมนิตนเองหลงัเรยีน (1) 48

แผนการจดัการเรยีนรทูี ่3 เรือ่ง อตัราการเปลีย่นแปลง 52ใบความรทูี ่ 3 58ใบกจิกรรมที ่ 3.1 61ใบกจิกรรมที ่ 3.2 63แบบทดสอบรายจดุประสงค 3 69แบบทดสอบเพือ่ประเมนิตนเองกอนเรยีน (2) 70

แผนการจดัการเรยีนรทูี ่4 เรือ่ง อนพุนัธของฟงกชนั 73ใบความรทูี ่ 4 76ใบกจิกรรมที ่ 4.1 77ใบกจิกรรมที ่ 4.2 78

ใบกจิกรรมที ่ 4.3 79ใบกจิกรรมที ่ 4.4 80ใบกจิกรรมที ่ 4.5 81ใบกจิกรรมที ่ 4.6 82แบบทดสอบรายจดุประสงค 4 84

แผนการจดัการเรยีนรทูี ่ 5 เรือ่ง ความเรว็และความเรง 85ใบความรทูี ่ 5 89ใบกจิกรรมที ่ 5.1 90ใบกจิกรรมที ่ 5.2 91แบบทดสอบรายจดุประสงค 5 92

แผนการจดัการเรยีนรทูี ่ 6 93ใบความรทูี ่ 6 98ใบกจิกรรมที ่ 6 99แบบทดสอบรายจดุประสงค 6 103

แผนการจดัการเรยีนรทูี ่ 7 เรือ่ง อนพุนัธอนัดบัสงูของฟงกชนัและอนพุนัธของฟงกชนัคอมโพสทิ 104ใบความรทูี ่ 7 107ใบกจิกรรมที ่ 7 108แบบฝกหดัเสรมิ 109แบบทดสอบรายจดุประสงค 7 112

แผนการจดัการเรยีนรทูี ่8 เรือ่ง การประยกุตของอนพุนัธของฟงกชนั 114ใบความรทูี ่ 8 118ใบกจิกรรมที ่ 8.1 119ใบกจิกรรมที ่ 8.2 122แบบทดสอบรายจดุประสงค 8 126แบบทดสอบเพือ่ประเมนิตนเองหลงัเรยีน (2) 128

แผนการจดัการเรยีนรทูี ่9 เรือ่ง กระบวนการตรงกนัขามกบัการหาอนพุนัธ 131ใบกจิกรรมที ่ 9 135แบบทดสอบรายจดุประสงค 9 136

เรือ่ง หนา

แบบทดสอบเพือ่ประเมนิตนเองกอนเรยีน (3) 137แผนการจดัการเรยีนรทูี ่10 เรือ่ง อนิทกิรลัไมจำกดัเขต 139

ใบความรทูี ่ 10 142ใบกจิกรรมที ่ 10 143แบบทดสอบรายจดุประสงค 10 146

แผนการจดัการเรยีนรทูี ่11 เรือ่ง อนิทกิรลัจำกดัเขต 147ใบความรทูี ่ 11 150ใบกจิกรรมที ่ 11 151แบบทดสอบรายจดุประสงค 11 154

แผนการจดัการเรยีนรทูี ่12 เรือ่ง พืน้ทีท่ีป่ดลอมดวยเสนโคง 155ใบความรทูี ่ 12 160ใบกจิกรรมที ่ 12 161แบบทดสอบรายจดุประสงค 12 162แบบทดสอบเพือ่ประเมนิตนเองหลงัเรยีน (3) 163

เรือ่ง หนา

⌦⌦ 1

1

5

4

y = f (x) a f a

( )

1.

1.1 y = f (x) a

1.2 y = f (x) a

( ) a f

.

.

1.3 y = f (x) a

( ) a f

.

.

1.4 y = f (x) a ( )

f

.

.

2.

f (x) x f (x)

3.

⌫ ⌫ ⌦⌫ 2

4.

4.1 (1)

4.2 1.1 (

)

4.3 x

2 2

2

x 2 2 x 2

2 2 x 2 2

x 2

4.4 a x x < a x

a x a -

x x > a x a x a

4.5 x a

f x x a

f (x) = 2x – 6 x 5

x a+

x a–

–2 –1 0 1 2 3 4

x 2+

x 2–

⌦⌦ 3

1 x > 5 x 5 1 x < 5 x 5

x 2x 2x – 6 x 2x 2x – 6

6.0 4

5.5 4.5

5.1 4.9

5.01 4.99

5.001 4.999

5.0001 4.9999

5.00001 4.00002

4.99999 3.99998

f (x) x

x 5 f (x)

4

“ f (x) x 5 4”

5x

lim f (x) = 4

f (x) x 5 4

“ f (x) x 5 4”

5x

lim f (x) = 4

f (x) x 5 4

⌫ ⌫ ⌦⌫ 4

4.6 f f (x)

L x a ( x > a x < a) f L

a f x = a L

xlim f (x) = L

ax

lim f (x) = L , ax

lim f (x) = L

ax

lim f (x) = L

4.7 1.1

4.8

f (x) = x + 3

f (x) = x + 3

2x

lim (x + 3) x 2

(2x

lim f (x) , 2x

lim f (x) ) f (x)

5

2x

lim (x + 3) = 5

x

y

2–3

⌦⌦ 5

f 1x

lim f (x) , 1x

lim f (x) , 1x

lim f (x)

1x

lim f (x) = – 1

1x

lim f (x) = 0

x 1 1 f (x)

1x

lim f (x)

4.9 1.2

4.10

a L f (x) L x

a f (x) L x a

ax

lim f (x) = L

4.11 f (x) = x + 2

1) 2x

lim x + 2

2) 2x

lim x + 2

x

y

1

–1

1

0

⌫ ⌫ ⌦⌫ 6

4.12 2

4.13 2

f (x)

4.14 1.2

4.15 11

4.16 1.3

4.17 1x

lim f (x)

f (x) = 6xx

4x2

2

1x

lim f (x)

2x

lim f (x)

2x

lim (x2 + x - 6) = 0

8 2x

lim6xx

4x2

2

f (x) g (x)

f (x) 8

f (x) = g (x) x 0 < x – a < h

h

ax

lim g (x) = L ax

lim f (x) = L

4.18 1.4

4.19 1.4

4.20 1

⌦⌦ 7

5.

1.1 1.2

1.1 1.4

1

(1)

6.

1.

2.

3.

4.

5.

6. /

90%

80%

90%

10

7.

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8.

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⌫ ⌫ ⌦⌫ 8

1.1

y = f (x) a f a

( )

1

160

7 ( 7 ) 1000

14

2

( f (x) x)

x a– , x a

+ , x a , x a

x a– x a x a (x < a)

a

x a+ x a x a (x > a)

a

⌦⌦ 9

1.2

y = f (x) a f a

( )

a , L M f g

ax

lim f (x) = L ax

lim g (x) = M

1. ax

lim c = c c

2. ax

lim x = a

3. ax

lim xn = a

n , n I

+

4. ax

lim c f (x) = c ax

lim f (x) = cL , c

5. ax

lim [ f (x) + g (x) ] = ax

lim f (x) + ax

lim g (x) = L + M

6. ax

lim [ f (x) – g (x) ] = ax

lim f (x) – ax

lim g (x) = L – M

7. ax

lim [ f (x) g (x) ] = ax

lim f (x) ax

lim g (x) = L M

8. ax

lim)x(g

)x(f =

)x(glim

)x(flim

ax

ax = M

L , M 0

9. limax

[ f (x) ]n = [ lim

ax

f (x)]n = L

n , n = I

+

10. limax

)x(fn

= )x(flimax

n

= Ln

, n = I+ L

n

R

11. f (x) limax

f (x) = f (a)

f (x) = x3 + 4x

2 – 3x – 5

⌫ ⌫ ⌦⌫ 10

1.1

y = f (x) a f a

( )

f (x) = 2x + 1 2x

lim f(x) , 2x

lim f(x)

2 (

6 )

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

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

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

⌦⌦ 11

1.2

y = f (x) a f a

( )

(1) (2)

1x

lim f (x) = …………………………….. 1x

lim f (x) = ……………………………..

4x

lim f (x) = …………………………….. 2x

lim f (x) = ………………………..……..

(3) (4)

1x

lim f (x) = …………………………….. 1x

lim f (x) = ……………………………..

1x

lim f (x) = …………………………….. 2x

lim f (x) = ……………………………..

3x

lim f (x) = ……………………………..

x

y

x

y

0 1 2 3 4 5 -4 -3 -2 -1 0 1

3

2

1 1

x

y

-3 -2 -1 0 1 2 3

4

3

2

1 x

y

-1 0 1 2 3 4 5

4

3

2

1

⌫ ⌫ ⌦⌫ 12

1 x > 5 x 5 1 x < 5 x 5

x 2x 2x – 6 x 2x 2x – 6

6.0 4

5.5 4.5

5.1 4.9

5.01 4.99

5.001 4.999

5.0001 4.9999

5.00001 4.00002

4.99999 3.99998

f (x) = 2x – 6 5x

lim f (x)

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

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⌦⌦ 13

1.3

y = f (x) a f a

( )

1. f (x) = x

1) 2x

lim f (x) …………………………………………………………………………..………..

2) 2x

lim f (x) ……………………………………………………………………………..……..

3) 2x

lim f (x)

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

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

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

2. f (x) = x + 2

1) 1x

lim f (x) ……………………………………………………………………………..……..

2) 1x

lim f (x) ……………………………………………………………………………..……..

3) 1x

lim f (x)

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

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

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

3. f (x) = 3 – x

1) 2x

lim f (x) …………………………………………………………………………..………..

2) 2x

lim f (x) ……………………………………………………………………………..……..

3) 2x

lim f (x)

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

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

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

⌫ ⌫ ⌦⌫ 14

4. f (x) = x2

1) 2x

lim f (x) …………………………………………………………………………..………..

2) 2x

lim f (x) ……………………………………………………………………………..……..

3) 2x

lim f (x)

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

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

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

5. f (x) = x

1) 4

limx

f (x) …………………………………………………………………………..………..

2) 4

limx

f (x) ……………………………………………………………………………..……..

3) 4

limx

f (x)

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

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

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

6. f (x) = x

1) 3

limx

f (x) …………………………………………………………………………..………..

2) 3

limx

f (x) ……………………………………………………………………………..……..

3) 3

limx

f (x)

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

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

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

7. f (x) = 2x

4x 2

1) 2

limx

f (x) …………………………………………………………………………..………..

2) 2

limx

f (x) ……………………………………………………………………………..……..

3) 2

limx

f (x)

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

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

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

⌦⌦ 15

8. f (x) = x

x

1) 0

limx

f (x) …………………………………………………………………………..………..

2) 0

limx

f (x) ……………………………………………………………………………..……..

3) 0

limx

f (x)

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

9. f (x) =

1) 2

limx

f (x) …………………………………………………………………………..………..

2) 2

limx

f (x) ……………………………………………………………………………..……..

3) 2

limx

f (x)

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

10. f (x) =

1) 3

limx

f (x) …………………………………………………………………………..………..

2) 3

limx

f (x) ……………………………………………………………………………..……..

3) 3

limx

f (x)

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

x + 3 x 2

5 x > 2

x + 1 x 3

2 x > 3

⌫ ⌫ ⌦⌫ 16

11. f (x) = 3x

)3x( 2

1) 3

limx

f (x) …………………………………………………………………………..………..

2) 3

limx

f (x) ……………………………………………………………………………..……..

3) 3

limx

f (x)

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

12. f (x) =

1) 2

limx

f (x) …………………………………………………………………………..………..

2) 2

limx

f (x) ……………………………………………………………………………..……..

3) 2

limx

f (x)

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

……………………………………………………………………………………………………………….

13.

1) 2x

lim (x) = ……………………………………………………………………….

2) 5x

lim x2 = ……………………………………………………………………….

3) 4x

lim x = ……………………………………………………………………….

4) 0x

lim (x + 3) = ……………………………………………………………………….

5) 5x

lim (x2 + 2x) = ……………………………………………………………………….

X2 x 2

2x x > 2

⌦⌦ 17

6) 3x

lim (x2 – 5x + 10) = …………………………………………………………………..

7) 1x

lim (x + 2) (x + 3) = ……………………………………………………………………

8) 1x

lim2x

3x = ……………………………………………………………………

9) 2x

lim 5x2x 2 = ……………………………………………………………………

10) 2x

lim2x

1x = ……………………………………………………………………

11) 5x

lim (x + 1)2 = ……………………………………………………………………

12) 0x

limx

x2x 2

= ……………………………………………………………………

13) 2x

lim2x

8x2x 2

= ……………………………………………………………………

14) 2x

lim2x

4x 2

= ……………………………………………………………………

15) 2x

lim2x

8x3

= ……………………………………………………………………

16) 1x

lim1x

1x = ……………………………………………………………………

17) 0x

limx

11x = ……………………………………………………………………

18) 0x

limx

24x = ……………………………………………………………………

19) 8x

lim2x

8x3

= ……………………………………………………………………

20) 4x

limx4

x2 = ……………………………………………………………………

21) 0x

limx

11x3

= ……………………………………………………………………

⌫ ⌫ ⌦⌫ 18

1.4

y = f (x) a f a

( )

⌦⌦ 19

( )

f (x)

x f (x)

)x(flimx

x f (x)

)x(flimx

)x(flimx

)x(flimx

a f (x)

ax x x

12 p

p

x x

1lim = 0

p

x x

1lim = 0

px

12

p

x x

clim =

px x

1c lim

= c (0) = 0

p

x x

clim =

px x

1c lim

= c (0) = 0

⌫ ⌫ ⌦⌫ 20

)x(flimx

)x(flimx

f (x) = xq

xp p (x) q (x)

1. x p(x) x

q (x)

)x(flimx

= 0

)x(flimx

= 0

2. a x p (x)

b x q (x)

x p (x) x

q (x)

)x(flimx

= b

a

)x(flimx

= b

a

3. x p (x) x

q (x) f (x)

1 4x5

3x72

2

xlim =

5

7 ( 2)

4x5

3x72

2

xlim =

5

7

2 5x7

9x26

4

xlim = 0 ( 1)

5x7

9x26

4

xlim = 0

3 4x7

1x32

5

xlim ( 3)

4x7

1x32

5

xlim

⌦⌦ 21

1. limx xx3

1x3x52

2

2. limx 5x2

4x7

3. limx 11x5x6

8xx223

2

4. limx 2x2

2x3x43

2

5. limx

3x4

1x3

6. limx 3x

2

7. limx 7xx

5xx32

2

8. limx 1x

x23

2

9. limx 5x8

9x26

4

10. limx 6x2

4x32

3

⌫ ⌫ ⌦⌫ 22

1

5

1 10

3x

3x 3x

1. f (x) =

0 3x

1.1 f (x)

1.2 lim3x

f (x)

1.3 lim3x

f (x)

1.4 lim3x

f (x)

2.

2.1 lim1x

(2x)

2.2 lim2x

(x2 – 4x + 1)

2.3 lim1x 1x

1x

3. ( )

3.1 limx 1x

x32

3

3.2 limx 14

122

2

x

x

3.3 limx 2

)1( 2

x

x

3.4 limx 52

42

x

x

3.5 limx 14

)12(3

3

x

x

⌦⌦ 23

(1)

“ ”

1.

. lim1x

f (x) = –1 . lim1x

f (x) = 0

. lim1x

f (x) . lim1x

= –1 , 0

1 x > 0

2. f (x) =

–1 x < 0

f (x) 0

. lim0x

f (x) = –1

. lim0x

= 1

. lim0x

f (x) lim0x

f (x)

. lim0x

f (x) = 1

y

x10

(1, –1)

⌫ ⌫ ⌦⌫ 24

3.

. lim2x

f (x) = 3 . lim2x

f (x) = 2

. lim1x

f (x) = 3 , 2 . lim2x

f (x)

4. lim1x

5x2 – 2x

. 3 . –1

. –3 . 1

5. lim5x

(x4) (x– 1)

. 4(5)3 . 5

. 54(4) . 5

7

6. lim3x 5x2

1x

. 4 . 4

1

. 5

4 .

4

5

7. lim3x x1

9x2

. . 0

. 6 . –6

y

x

0 1 2 3

3

–3 –2 –1

5

4

⌦⌦ 25

8. lim1x x1

x1

. 2

1 . 1

. 2 . 0

9. lim0x

3 2 )1x(

. 1 . –1

. 0 . 2

10. limx 3x2

4x5

. 5

2 . 0

. 4

3 .

2

5

11. limx 6x2

4x32

3

. 2

3 .

3

2

. 2 .

12. limx 4x

1x22

. 0 .

. 2

1 . 2

13. f (x) =

. lim1x

f (x) . f (1)

. lim1x

.

1x

1x2

x 1

4 x = 1

⌫ ⌫ ⌦⌫ 26

14. f (x) =

. lim3x

f (x) = lim3x

(x2 + 3x + 9)

. f (3) = 9

. lim3x

f (x) f (3)

. f x = 3

15.

. f (x) = x2 + 1 . g (x) = sin x

. f (x) = xn . h (x) =

4x

16x2

3x

27x3

x 3

9 x = 3

f (3) = 9

⌦⌦ 27

2

5

3

y = f (x) x = a

1.

y = f (x) x = a

2.

(Continuity) f (x) x = a

1) f (a)

2) limax

f (x)

3) f (a) = limax

f (x)

3.

f (x) x = a

4.

4.1 f (x) = 2x

4.2 2.1 2.1

2.1 x = a

⌫ ⌫ ⌦⌫ 28

4.3 f (x) =

x = 4 1

3 x = 4

x = 4

4.4 . .

x = a

. .

. .

x0 a

y = f3 (x)

y

x0 a

y = f4 (x)

y

y

x0 a

y = f1 (x)

y

x0 a

y = f2 (x)

2x x 4

7 x = 4

⌦⌦ 29

4.5 2.2 1

4.6 2.2 f

limax

f (x) f (a) x = a

4.7

x = a

4.8 a

x = a

x = a

x = a

4.9 x = a

4.10 2.3

4.11

4.12 2

4.13 (1)

5.

2.1 2.3

2

(1)

⌫ ⌫ ⌦⌫ 30

6.

1.

2.

3.

4.

5.

6. /

80%

90%

8

13

7.

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

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

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

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

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

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

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

8.

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

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

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

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

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

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

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

⌦⌦ 31

2.1

y = f (x) x = a

1. 6 x = a

x = a

1 2

3 4

5 6

f ...................................................................................... x = a

f ............................................................................... x = a

y

x0 a

f (x)

y

x 0 a

f (x)

x 0 a f (x)

x0 a

f (x)y y

y

x0 a

f (x)

y

x 0 a

f (x)

⌫ ⌫ ⌦⌫ 32

2. x = a

(a)

(b)

(c)

(d)

(e)

(f)

........................................................................................................................................ x = a

................................................................................................................................. x = a

c .......................................................................................................................

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

a a

b a

a a

a

⌦⌦ 33

2.2

y = f (x) x = a

x = a f (a)

limax

f (x) f (a)

limax

f (x)

f (a)

.

f1

.

f2

y

x 0 a

y = f1 (x)

y

x 0 a

y = f2 (x)

⌫ ⌫ ⌦⌫ 34

limax

f (x) f (a)

limax

f (x)

f (a)

.

f3

x 0 a

y = f4 (x)

y

.

f4

x 0 a

y = f3 (x)

y

⌦⌦ 35

2.3

y = f (x) x = a

⌫ ⌫ ⌦⌫ 36

f (x) = x

f (x) = cx

f (x) = x2

f (x) = xn

f (x) =

f (x) = ( )

f (x) = x , xn

f (x) = sin x

f (x) = cos x

f (x) = ax

f (x) = xlog a

f (x) = 5x

f a

1. limax

f (x)

2. f (a)

3. limax

f (x) = f (a)

3 limax

f (x) x = a f (a) f (a)

⌦⌦ 37

1. lim2x

4x = 4(2) = 8

2. lim1x 1x

1x10x5 2

…………………………………………………………………

3. lim3x

(4x – 1)(2x + 3) …………………………………………………………………

4. lim1x

(x + 8) …………………………………………………………………

5. lim2x 2x

6xx 2

= lim2x )2x(

)3x)(2x( = lim

2x

(x + 3) = 5

6. lim1x 1x

x1x 2

…………………………………………………………………

7. lim2x

6 …………………………………………………………………

8. lim1x 1x

1x 2

…………………………………………………………………

9. lim1x

x3x 2 …………………………………………………………………

10. lim0x x

39x …………………………………………………………………

11. lim3x

)1x)(3

x1( 2

3

…………………………………………………………………

12. lim1x x1

1x2

…………………………………………………………………

13. lim10x

)x1(

x

x …………………………………………………………………

14. lim2x

)x1(

x

x …………………………………………………………………

15. lim3x 3x

3x2x2

…………………………………………………………………

16. lim4x

4

1

x

1

4x …………………………………………………………………

17. lim2x

f (x) f (x) =

18. lim

3

1x

xx2 …………………………………………………………………

19. lim0x

)x1(x

x …………………………………………………………………

1 – x2

, x > 2

5 – 2x , x < 2

⌫ ⌫ ⌦⌫ 38

f f x = a

f 3

1. a f f (a)

2. limax

f (x)

3. limax

f (x) = f (a)

f f (x) = x2 + 4

f x = 2

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

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

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

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

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

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

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

f f (x) = 3x

9x2

f x = 3

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

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

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

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

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

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

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

⌦⌦ 39

1. f

3x

9x2

x 3

f (x) =

6 x = 3

f x = 3

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

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

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

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

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

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

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

2. f

x2 x < 0

f (x) =

1 x 0

f x = 0

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 40

1. f (x) = x + 5 f x = –5

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

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

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

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

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

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

2. f (x) = 2

1

x f x = 2

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

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

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

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

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

3. f (x) =

f x = 2

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

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

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

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

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

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

2x

4x2

x 2

4 x = 2

⌦⌦ 41

4. f (x) = 1x

1x f x = 1

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

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

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

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

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

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

5. f (x) =

f x = 5

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

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

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

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

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

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

6. f (x) = 9x

3x2

f x = 3

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

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

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

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

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

5x

25x 2

x 5

5 x = 5

⌫ ⌫ ⌦⌫ 42

7. f (x) =

f x = 1 k

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

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

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

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

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

8. f (x) = 2x

8x3

x 8 f x = 8

f (8)

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

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

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

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

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

9. g (x) =

g x = 0 a

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

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

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

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

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

1x

1x x 1

k x = 1 (k R)

x

11x x 0

a x = 0 (a R)

⌦⌦ 43

10. g (x) =

g x = 5 m

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

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

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

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

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

11. f (x) =

f x = 2

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

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

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

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

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

12. f (x) =

f x = 1 a b

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

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

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

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

5x

15x2x 2

x 5

m2 – 1 x = 5 (m R)

x2 x 2

2x x > 2

ax x < 1

4 x = 1

x + b x > 1

⌫ ⌫ ⌦⌫ 44

13. g (x) =

a f x = 2

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

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

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

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

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

14. f (x) =

1) f x = 1

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

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

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

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

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

2) f x = 2

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

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

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

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

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

3x + a x = 2 a R

2x

4x2

x 2

0 x 1

1x

1 1 < x 2

3 - x x > 2

⌦⌦ 45

2

5

30 10

10

1 ( )

1. f (x) = 3x

9x2

, lim3x

f (x)

. lim3x

f (x) . lim3x

f (x)

. 3 . 6

. , ,

2. lim3x

f (x)

. –1

. 1

. 0

. 1

.

x3

6

–3

3

0

x

y

(1, –1)

y

⌫ ⌫ ⌦⌫ 46

3. lim3x

f (x)

.

. 4

. 0

. –2

. 2

4. lim2x 6xx

9x2

2

. . 4

1

. 3

2 . 0

. f (x) 2

5. lim3x 5x2

1x

. –1 . 0

. 1 . 3

. 4

6. f (x) = x

x

. lim0x

f (x) = –1 . lim0x

f (x) = 1

. lim0x

f (x) = 1 . 0

. , ,

7. lim3x

(x2 + 2x – 4)

. –1 . 5

. 9 . 11

. 19

y

x4 0 –2 –1

1 2 1

2

2 1

⌦⌦ 47

8. lim4x 4x

16x2

. – 8 . – 4

. 0 . 8

.

9. f (x) =

f (x) x = 2

. 2 Df f (2)

. lim2x

f (x)

. lim2x

f (x) = f (2)

. –

. lim2x

f (x) f (2)

10.

f (x) (–3, 3)

. lim0x

f (x) = 0

. lim0x

f (x) = 2

. lim0x

f (x) = lim0x

f (x)

. f (2) = 0 , f (–2) = 2

. f x = 0

2x

4x 2

, x 2

4 , x = 2

y

x

2

3-3

⌫ ⌫ ⌦⌫ 48

“

”

1.

. lim1x

f (x) = –1 . lim1x

f (x) = 0

. lim1x

f (x) . lim1x

= –1 , 0

2. f (x) =

f (x) 0

. lim0x

f (x) = –1

. lim0x

= 1

. lim0x

f (x) lim0x

f (x)

. lim0x

f (x) = 1

y

x10

(1, –1)

1 x > 0

–1 x < 0

(1)

⌦⌦ 49

3.

. lim2x

f (x) = 3

. lim2x

f (x) = 2

. lim1x

f (x) = 3 , 2

. lim2x

f (x)

4. lim1x

5x2 – 2x

. 3 . –1

. –3 . 1

5. lim5x

(x4) (x– 1)

. 4(5)3 . 5

. 54(4) . 5

7

6. lim3x 5x2

1x

. 4 . 4

1

. 5

4 .

4

5

7. lim3x x1

9x2

. . 0

. 6 . –6

y

x

0 1 2 3

3

–3 –2 –1

⌫ ⌫ ⌦⌫ 50

8. lim1x x1

x1

. 2

1 . 1

. 2 . 0

9. lim0x

3 2 )1x(

. 1 . –1

. 0 . 2

10. limx 3x2

4x5

. 5

2 . 0

. 4

3 .

2

5

11. limx 6x2

4x32

3

. 2

3 .

3

2

. 2 .

12. limx 4x

1x22

. 0 .

. 2

1 . 2

13. f (x) =

. lim1x

f (x)

. f (1)

. lim1x

.

1x

1x2

x 1

4 x = 1

⌦⌦ 51

14. f (x) =

. lim3x

f (x) = lim3x

(x2 + 3x + 9)

. f (3) = 9

. lim3x

f (x) f (3)

. f x = 3

15.

. f (x) = x2 + 1 . g (x) = sin x

. f (x) = xn . h (x) =

4x

16x2

3x

27x3

x 3

9 x = 3

⌫ ⌫ ⌦⌫ 52

3

5

5

y x t1 t2

y x

1.

1.1

1.2 y = f (x) y x

x x + h x

2.

3.

t

12

12

tt

SS , t2 > t1

y x x x + h

y x

⌦⌦ 53

4.

4.1 (2)

4.2 .

4.3 3

4.2 ( )

S = f (t)

t1 f (t1) t2

f (t2)

t1 t2 = 12

12

tt

)t(f)t(f

21 tt

4.4

S = 5t2

( )

/

0 – 1

1 – 2

2 – 3

3 – 4

4 – 5

5

15

25

35

45

3

3 4 3

⌫ ⌫ ⌦⌫ 54

t = 3 t = 3 + h ( 3 )

t = 3 – h t = 3 ( 3 ) h

1) t = 3 – h t = 3 = 30 – 5h /

2) t = 3 t = 3 + h = 30 + 5h /

h 0 1) 2)

30 / 3 = 30 /

4.5 t h

t = 3 t = 3 + h

= h

)3(f)h3(f h < 0

h

t = 3 t = 3 + h

= h

)3(f)h3(f h > 0

3 = h

)3(f)h3(f h 0

3 = lim0h h

)3(f)h3(f

S = f (t)

t = lim0h h

)t(f)ht(f

4.6 t

1)

2)

12

12

tt

SS , 12 tt

3)

4) t

5) t

t

⌦⌦ 55

4.7 3.1

4.8 – 1.2

y = f (x) = x2 + 2

x = 1 y = f (1) = 3

x = 3 y = f (3) = 11

x 1 3 x 2

y 3 11 y 8

x 3 5 y 11 27

x 2 y

y x

4.9 y = x2 x 1 2

y 2 4 x y

y x

4.10 4.8 x 2 y

8

x 1 y 2

8 = 4 4

y x x = 1 x = 3

4.11

y = f (x)

x = x1 y = f (x1) ….. (1)

x = x1 + h y = f (x1 + h) ….. (2)

x h y f (x1 + h) – f (x1)

y x

x = x1 x = x1 + h h

)x(f)hx(f 11

⌫ ⌫ ⌦⌫ 56

4.12 y x x x + h

h

)x(f)hx(f y x x

t

y = f (x)

4.13 2 – 3

x

4.14 3.2

4.15 3.2

y = f (x) x y f

y = f (x) x

y y

9.00 . 10.00 . – 3 . .

4.16 3

5.

3

3.1 3.2

3

(2)

⌦⌦ 57

6.

1.

2.

3.

4.

5.

6. /

80%

90%

7.

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

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

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

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

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

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

8.

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 58

3

1.1

1.2 y = f (x) y x

x x + h y x

S = f (t)

t1 t1 + h t1 t1 + h

h

)t(f)ht(f 11

t t + h h

)t(f)ht(f

t = lim0h h

)t(f)ht(f

y = f (x) x x + h

h 0 y f (x) f (x + h)

y x x x + h

h

)x(f)hx(f

y x x

lim0h h

)x(f)hx(f

S t

S = f (t)

S = tt 22

1 3

⌦⌦ 59

t = 1 t = 2 t = 3

S = 0

S = 8

S = 2

5

S = 2

39

1

2

t = 1

t = 2

t = 3

t = 0

S = 8

S = 2

5

S = 2

39

S = 0

S = t2t2

1 3

t S

t = 0 S = 0

t = 1 S = 2

5

t = 2 S = 8

t = 3 S = 2

39

t = 0

⌫ ⌫ ⌦⌫ 60

t = 1 t = 3 2

17 = 8.5

t = 2 t = 6

26

8120 = 28

S = f (t)

t1 t1 + h t1

t1 + h h

)t(f)ht(f 11 t1

t1 + h h

)t(f)ht(f

⌦⌦ 61

⌫ ⌫ ⌦⌫ 62

2. S = t3 – 6t

2 + 9t + 4 S

t

1) t = 1 t = 3

2) t = 1

3) t = 1.5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

⌦⌦ 63

3.2

y = f (x) y

x x x + h y x

1. y = f (x) y x x x

x + h

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

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

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

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

2. y = f (x) = 3x + 1 y x x

1 3

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

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

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

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

3. y = f (x) = x2 y x x

1 5

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

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

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

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

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

⌫ ⌫ ⌦⌫ 64

4. y = f (x) = x2 + 2x + 5 y x x

1 4

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

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

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

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

5. y = f (x) = x

1 y x x x

x + h

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

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

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

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

6.

8 10

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

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

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

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

7.

10 12

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

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

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

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

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

⌦⌦ 65

8.

10 12

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

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

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

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

9.

8 10

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

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

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

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

10.

5 8

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

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

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

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

11.

1)

8 10

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

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

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

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

⌫ ⌫ ⌦⌫ 66

2)

6 8

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

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

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

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

12. S = f (t) S t

t t + h S t t

t t + h

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

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

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

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

13. S = f (t) = t2 + 2t + 5 S

t t = 2

t = 5

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

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

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

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

14. y = f (x) = x y x x

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

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

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

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

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

⌦⌦ 67

15. y = f (x) = x2 y x x

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

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

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

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

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

16. y = f (x) = x2 + 2x + 5 y x x

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

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

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

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

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

17. y = f (x) = x

1 y x x

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

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

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

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

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

18. y = f (x) = x2 + 1 y x x = 2

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

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

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

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

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

⌫ ⌫ ⌦⌫ 68

19. y = f (x) = x2 – 5x + 1 y x x = 3

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

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

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

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

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

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

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

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

20. S = f (t) = t2 + 2t + 5 S t t = 2

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

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

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

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

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

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

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

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

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

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

⌦⌦ 69

3

5

30 10

1. S = t2t3

1 3 S

t

1) t

2) t = 2

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

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

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

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

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

2. f (x) = 4x + 1 x1 = 2 x1 + h = 5

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

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

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

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

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

3. f (x) = x2 – 2x x = 1

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

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

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

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

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

⌫ ⌫ ⌦⌫ 70

⌦⌦ 71

6. S = t3 – 3t

2 + 4t + 8 t = 2

. –4 . 2

. 4 . 12

7. S = t5 – t

4 + t

3 + 2t – 1 t = 1

. –4 . 6

. 11 . 14

8. y = 3x3 – 4x

2 – x + 9 (–2, 5)

. 51 . 20

. 12 . –21

9. y = 3x2 – 5x + 1 (1, –1)

. y = x . y = x – 1

. y = x – 2 . y = x + 1

10. xy = 4 (–1, –4)

. 4x – y + 8 = 0 . 4x + y – 8 = 0

. 4x + y + 8 = 0 . x – 4y – 15 = 0

11. f (x) = 5x3 – 3x

5 f’(x) f”(x)

. 5x2 – 3x

4 , 5x – 3x

3

. 15x2 – 15x

4 , 30x – 60x

3

. 5x4 – 3x

6 , 5x

5 – 3x

7

. 15x2 – 15x

4 , 15x – 15x

3

12. y = 7 – 6x – x2

. 16 . 7

. –3 . –20

(x) f"f ' (x)

⌫ ⌫ ⌦⌫ 72

13. f (x) = x3 – 6x

2 + 9x – 8

. –4 8 . 4 –8

. –8 –4 . –4 –8

14. 12

. 7 , 5 . 9 , 3

. 8 , 4 . 6 , 6

15. 10 c.c

x f (x) = x8x33

x 23

0 x 6 6 6

. 2 2 2

. 2 2 2

. 2 4

. 2 4

⌦⌦ 73

4

5

4

1.

2.

y = f (x) lim0h h

)x(f)hx(f

“ f x” f (x)

3.

3.1 y x x h

)x(f)hx(f h 0

3.2 f (x) = lim0h h

)x(f)hx(f

3.3

4.

4.1

1) y = f (x)

lim0h h

)x(f)hx(f

2) t

S = f (t) t lim0h h

)x(f)hx(f

⌫ ⌫ ⌦⌫ 74

4.2 lim0h h

)x(f)hx(f

h

)x(f)hx(f h

t S = f (t)

lim0h h

)t(f)ht(f

4.3 y = f (x) lim0h h

)x(f)hx(f

x f

}h

)x(f)hx(flimz)z,x(

0h

lim0h h

)x(f)hx(f x 1

lim0h h

)x(f)hx(f

h

)x(f)hx(flimz)z,x(

0h

f f

f ( )

f = h

)x(f)hx(flimz)z,x(

0h

fz,x z f x

Z = )x(f = lim0h h

)x(f)hx(f

)x(f f x

4.4 x

4.5 dx

dy

4.6 dx

dy,

dx

df(x) , )x(f y y = f (x)

4.1

4.7 t

S = f (t)

t

t

4.8

⌦⌦ 75

4.9

4.10 4.2 4.6 4

4.11 4.2 4.6

4.12 4

5.

4

4.1 4.6

4

6.

1.

2.

3.

4.

5.

6. /

80%

90%

5

–

7.

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

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

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

8.

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

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

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

⌫ ⌫ ⌦⌫ 76

4

1 y = c c dx

dy = 0

dx

)c(d = 0

2 y = x dx

dy = 1

dx

dx = 1

3 y = xn n

dx

dy = nx

n-1

dx

)x(d n

= nxn-1

4 y = f (x) + g (x)

dx

dy =

dx

df (x) +

dx

dg (x)

5 y = f (x) – g (x)

dx

dy =

dx

df (x) –

dx

dg (x)

6 y = cf (x) c

dx

dy = c

dx

df (x)

7 y = f(x) g(x)

dx

dy = f (x)

dx

dg (x) + g (x)

dx

df (x)

8 y = )x(g

)x(f g (x) 0

dx

dy =

2)x(g

)x(gdx

d)x(f)x(f

dx

d)x(g

9 y = )x)(fg( = ))x(f(g

dx

dy = ))x(f(g

)x(df

d)x(f

dx

d

⌦⌦ 77

4.1

f (x) , dx

dy , y , g (x)

1. f (x) = x2 f (x)

2. y = 4x + 5 y

3. y = 2x2 + x – 4

dx

dy

4. g (x) = x

1 g (x)

⌫ ⌫ ⌦⌫ 78

4.2

S = t2 – 2t + 5 s t

V t = 3

= h

)t(f)ht(f 0h

= 0h

= h

5t2t5h2t2hth2t 22

0h

= h

h2hth2 2

0h

=

V = 2t – 2

t = 3

V = /

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

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

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

1

2

3

2

1

2

3

⌦⌦ 79

4.3

dx

dy

1. y = 5 , dx

dy = ………………………………………………………………

2. y = 7 …………………...……………………………………………………..

3. y = –2 …………………...……………………………………………………..

4. y = 2

1 …………………...……………………………………………………..

5. y = 0.7 …………………...……………………………………………………..

6. y = 4

3 …………………...……………………………………………………..

7. y = x …………………...……………………………………………………..

8. y = x4 …………………...……………………………………………………..

9. y = x10

…………………...……………………………………………………..

10. y = x12

…………………...……………………………………………………..

11. y = x9 …………………...……………………………………………………..

12. y = x–5

…………………...……………………………………………………..

13. y = x–11

…………………...……………………………………………………..

14. y = 2

1

x …………………...……………………………………………………..

⌫ ⌫ ⌦⌫ 80

4.4

dx

dy

1. y = x–7

, dx

dy = …………………...…………………………………………

2. y = 2

1

x …………………...……………………………………………………..

3. y = 4

3

x …………………...……………………………………………………..

4. y = x–8

…………………...……………………………………………………..

5. y = 4

1

x …………………...……………………………………………………..

6. y = x …………………...……………………………………………………..

7. y = x

1 …………………...……………………………………………………..

8. y = x–10

…………………...……………………………………………………..

9. y = 3 x …………………...……………………………………………………..

10. y = x2 …………………...……………………………………………………..

⌦⌦ 81

4.5

dx

dy

1. y = 2x3 ,

dx

dy = …………………...…………………………………………..

2. y = 3x4 …………………...……………………………………………………..

3. y = 4x5 …………………...……………………………………………………..

4. y = 2x6 …………………...……………………………………………………..

5. y = 5x2 …………………...……………………………………………………..

6. y = 4x3 …………………...……………………………………………………..

7. y = –2x6 …………………...……………………………………………………..

8. y = 6x–5

…………………...……………………………………………………..

9. y = 2

1

x6 …………………...……………………………………………………..

10. y = 4x

5 …………………...……………………………………………………..

⌫ ⌫ ⌦⌫ 82

4.6

dx

dy

1. y = 3

2

x3 , dx

dy = …………………...…………………………………………

2. y = 7x7

3 …………………...…………………………………………

3. y = 2x2 + x – 7 …………………...…………………………………………

4. y = 3x5 + 2 …………………...…………………………………………

5. y = 3x2 + 2x – 1 …………………...…………………………………………

6. y = 3x4 + 2x

3 – 2x

2 + x – 7 …………………...…………………………………..

7. y = 5x4 + 7x

3 – 8x

2 – 3x + 25 …………………...……………………………...

8. y = 6x7 – 8x

5 + 3x

2 – 4x – 18 …………………...……………………………...

9. y = 4x2 + 3x –

x

2x4 …………………...…………………………………..

10. y = –2 + 3x – 3

2

x3 …………………...…………………………………..

11. y = 5x4

dx

dy

12. f (x) = 3x-5 + 2 )x(f )0(f

13. y = 3x2 – 2x + 15

dx

dy

14. y = 5x4 + 7x

3 – 8x

2 – 3x + 25

dx

dy

⌦⌦ 83

⌫ ⌫ ⌦⌫ 84

4

5

30 10

1. y = 3x2 + 7

2.

2.1 y = 2x2 + x

2.2 y = x–3

+ 4x–5

2.3 y = 68

7

2.4 y = (x + 3)5

2.5 y = 1x4

x

⌦⌦ 85

5

5

3

1.

2.

S = f (t)

)t(f t

)t(f t

1.

S = f (t) t

S t

1) t1 t2

t2 > t1

12

12

tt

)t(f)t(f

2) t1 t2

t2 > t1

12

12

tt

)t(f)t(f

3) t = lim0h h

)t(f)ht(f = )t(f

V(t) t

V(t) = dt

ds = )t(f

4) t )t(V

5) t dt

dv a (t) t

a (t) = dt

dv =

2

2

dt

dS = )t(f

6) t )t(a

3.

⌫ ⌫ ⌦⌫ 86

(1)

(2)

(3)

(4)

(5)

(6)

4.

4.1 t

V S = f (t)

V = dt

ds = )t(f t

4.2

)t(f > 0 f (t) f

)t(f < 0 f (t) f

V > 0 S

V < 0 S

4.3 V = g (t)

t1

t1 + h h

)t(g)ht(g 11

t t + h t t + h

⌦⌦ 87

V = g(t) t t + h

h

)t(g)ht(g

h 0 t t + h t

h

)t(g)ht(g h 0 t

h

)t(g)ht(glim

0h

= t

)t(g = t

S = f (t)

V = )t(f t

V = g (t) = )t(f

)t(g = )t(f t

a t

S = f (t)

a = )t(f

= t

4.4 5

4.5 dt

ds

dt

dv

4.6 5.1 5.2

4.7 5

5.

5

5.1 5.2

5

⌫ ⌫ ⌦⌫ 88

6.

1.

2.

1.

2.

3.

4. /

80%

90%

–

7.

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

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

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

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

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

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

8.

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

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

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

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

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

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

1.2.3.4.5.6.

⌦⌦ 89

5

S = f (t)

)t(f t

)t(f t

S = t3 – 3t

2 – 9t + 5 S

t

1. t = 1

2. t = 2

S = t3 – 3t

2 – 9t + 5

V = dt

ds = 3t

2 – 6t – 9

t = 1

3(1)2 – 6(1) – 9 = –12

= a = dt

dv = 6t – 6

t = 2 (6 × 2) – 6 = 6 2

⌫ ⌫ ⌦⌫ 90

5.1

1 – 6 (

)

1. S = t2 – 2t + 7 , t = 3

S = 32 – 2(3) + 7 = 9 – 6 + 7 = 10

S

2. S = t2 + t – 5 , t = 1

3. S = t3

- 2t2 + 7t + 6 , t = 1

4. S = t3 – 5t

2 + 4 , t = 3

5. S = t3 – 4t

2 – 2t – 1 , t = 2

6. S = t4 + 2t - 1 , t = 1

⌦⌦ 91

5.2

1 – 3

0

1. S = t2 – 4t + 2

S = t2 – 4t + 2

V = dt

dS = 2t – 4

t = 2t – 4

V = 2t – 4

a = dt

dv = 2

t = 2

V = 0 2t – 4 = 0 , t = 2

t = 2 = 0

2. S = 6 – 2t – 16t2

3. S = 3

1t3 – 2t

2 + 3t – 5

⌫ ⌫ ⌦⌫ 92

5

5

30 10

1. S = t3 – 3t

2 + 4t + 8 t = 2

2. S = 2

1t3 – 2t S

t t = 2

3. S = t3 – 6t

2 + 9t + 4 S t

1) t = 2

2) t = 2

3) 0

⌦⌦ 93

1.

2.

P P

y = f (x) x = x1 y = f (x)

p (x1 , y1) )x(f 1 = y = f (x) x = x1

)x(f = y = f (x) (x , y)

3.

3.1 y = mx + C m C

y – y1 = m (x – x1)

3.2

3.3 P (x , y) P (x , y)

4

4.1 tan x

e = tan

e

5

3

6

.

⌫ ⌫ ⌦⌫ 94

4.2

tan

4.3

4.4

P P

P

P

4.5

PQ = 11

11

xhx

)x(f)hx(f

= h

)x(f)hx(f 11

c

P

P h h

Q1

Q

y = f (x)

(x1 + h , f(x1 + h))

P (x1 , f(x1))

⌦⌦ 95

Q Q1 P

PQ P Q P h O

Q P y = f (x)

1. PQ P

2. PQ x 0 )0h(

PQ

P 0h

h

)x(f)hx(flim 11

0h = P

h

)x(f)hx(flim 11

0h = )x(f

y = f(x) x = x1

y = f (x) P = (x1, y1)

)x(f = y = f (x) (x , y)

4.6

y = x2 – 4x (4 , 0)

(4 , 0)

4.7 6

4.8

1) (a, b)

0 a b

y

0 a b

y

x x

⌫ ⌫ ⌦⌫ 96

2) (a, b)

4.9 6

4.10 6

4.11 6

5.

6

6

6

0 a b

y

0 a b

y

x x

0 a

y

0 a

y

x x

+ + + - - -

+ + + -- -

⌦⌦ 97

6.

1.

2.

3.

4.

5.

6. /

80%

90%

–

7.

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

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

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

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

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

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

8.

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 98

6

y = f (x) f a

a. (a, b) )a(f

b. (a, b) )a(f

c. f a (a, b)

d. (a, b)

L1 y = f (x) (a , b)

f a L1 = )a(f

f a L1

(L1)

1. y – b = )a(f (x – a) )a(f

2. x = a )a(f ( y)

⌦⌦ 99

6

1. y = x2 + 2x – 1 (1 , 2)

y = x2 + 2x – 1

dx

dy = 2x + 2

(1, 2) dx

dy = 2(1) + 2 = 4

(1, 2) = 4

2. y = x2 – 4x + 3 (–1 , 8)

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

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

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

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

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

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

3. y = x2 - 2x + 1 (1, 0)

y = x2 - 2x + 1

y = 2x + 1

(1,0) y = 2(1) + 1 = 3 = 3

y – y1 = m(x – x1), (x1, y1) = (1, 0)

y – 0 = 3 (x – 1)

y = 3x - 3

y – 3x + 3 = 0

⌫ ⌫ ⌦⌫ 100

4. y = x3 - 2x

2 + 3 (2, 3)

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

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

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

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

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

5. y = f (x) = x2 + 2x + 5 y = f (x) (2, 13)

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

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

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

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

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

6. y = 2x2 – x + 5 y = f (x) (1, 6)

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

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

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

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

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

7. y = x2 – 3x + 5 (4, 9)

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

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

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

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

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

⌦⌦ 101

8. xy = 1 (2, ½)

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

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

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

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

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

9. y = 2x2 + x + 3 (1, 6)

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

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

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

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

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

10. 2x + y + 4 = 0 y = x2 – 4x + 5

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

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

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

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

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

11. x + 4y – 5 = 0 y = x2 + 6x + 10

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

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

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

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

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

⌫ ⌫ ⌦⌫ 102

12. y = x2 – 3x + 5 x + y – 5 = 0

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

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

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

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

13. y = x

4 x – y – 5 = 0

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

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

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

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

14. y = ax + b y = 2x2 – x + 6 (1, 7) a

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

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

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

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

15. 4x – y + 2 = 0 y = -x2 + 1 (a, b) a + b

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

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

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

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

16. (2, 3) y = 2x – x2 (-1, -3)

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

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

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

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

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

⌦⌦ 103

6

5

30 10

1. y = -x3 + 3x + 1 (2, –1)

2. y = x3 – 2x

2 + 4 (2, 4)

3. y = x2 – 2x + 1

(2, 1)

⌫ ⌫ ⌦⌫ 104

7

5

3

1.

2.

f f { fDx f x }

f x f x 2 f x

)x(f

f x f x 3 f x

)x(f )3(

)x(f )3( x )3(f x 4 f x

)x(f )4(

3n f (n)

x f (n)

x

(n + 1) f (x) f(n+1)

(x)

n f x

f(n)

(x) , n

n

dx

yd , ))x(f(

dx

dn

n

, y(n)

, y(n)

(x)

3.

1

⌦⌦ 105

4.

4.1 f (x) f x

f f f x

4.2 f (x) = x2 – 2x )x(f

)x(f = 2x – 2 )x(f = 2x – 2

g (x) = )x(f g (x) = 2x – 2 )x(g )x(g = 2

)x(g f (x)

4.3 7 2

f (x)

)x(f ))x(f(dx

d2

2

4.4 f 2 f

4.5 2 3 f

)x(f )3(

4.6 n – 1 n

f f(n)

(x)

4.7

dx

dy =

du

dy

dx

du

4.8 7

4.9

4.10

4.11 7

5.

7

7

7

⌫ ⌫ ⌦⌫ 106

6.

1.

2.

3.

4.

5.

6. /

80%

90%

4

–

7.

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

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

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

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

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

8.

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

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

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

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

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

⌦⌦ 107

7

f f { x Df f

x } f

f x f x 2 f x

f (x) 2

2

dx

yd

f x f x 3 f x

f(3)

(x) 3

3

dx

yd

f(3)

(x) x f(3)

(x) x 4 f x

f(4)

(x) 4

4

dx

yd

n 3 f(n)

x f(n)

x

(n + 1) f (x) f(n + 1)

(x) n f

x f(n)

(x) n

n

dx

yd , )x(f

dx

dn

n

, y(n)

, y(n)

(x)

(The Chain Rule) f a g

f (a) fg a

)fg( (a) = g (f (a)) f (a)

f = )x(fu)u,x(

g = )u(gy)y,u(

)fg( (x) = g (f (x)) f (x)

dx

dy =

du

dy .

dx

du

n)x(f

dx

d = ))x(f(

dx

d)x(fn

1n

⌫ ⌫ ⌦⌫ 108

7

1. y = x4 – 3x

3 + 5x

2 + 3 2. y = 2

1

x + 3

1

x – 5

1

x

3. y = 2 + 3x 4. y = x

5. y = (x + 2) (3x – 1) 6. y = x2 – 2x + 5

7. y = 2x2 – 3x + 4 8. y = x

2

9. y = 2x 10. y = 2x2 + x

11. y = x32 12. y = 2x5

13. y = (3x – 1) (2x + 5) 14. y = (x – 1)3 (x + 2)

4

15. y = (x3 – 3x)

4 16. y =

2x

1x2

17. y = 1x

1x 18. y = (3 + 4x – x

2)

2

19. y = 2x2

1+

x

4 20. y = 2

1

x2 – 2

3

x + 2

1

x2

21. y = 2x23x 22. y = x

1– x

2

23. y = 2

3

xx2 24. y = (x + x

1)

2

25. y = 2

3

22

x2

)3x( 26. y =

x

x3x5 2

27. y = 2xx2 28. y = f (x) = x2(x

5 – 1)

29. y = 3 3 x3x 30. f (x) = 1x2

3x

⌦⌦ 109

1. 2

1) f (x) = 3x + 1

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

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

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

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

2) f (x) = x2 – 2x + 5

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

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

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

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

3) f (x) = x + x

1

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

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

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

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

4) f (x) = x3 – 2x

2 + 5x – 6

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

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

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

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

⌫ ⌫ ⌦⌫ 110

5) f (x) = x

1

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

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

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

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

2. 3

1) f (x) = x3 – 2x

2 + 3x + 4

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

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

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

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

2) f (x) = x4 – x

3 + 2x

2 + x – 6

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

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

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

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

3) f (x) = x2 +

2x

1

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

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

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

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

⌦⌦ 111

3. y = (x + 1)3

2

2

dx

yd

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

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

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

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

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

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

4. y = 1x 2

2

dx

yd

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

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

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

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

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

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

5. f (x) = (1 – 3x)3 )1(f

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 112

7

5

30

10

1. f (x) = x2 + 2x + 5

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

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

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

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

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

2. f (x) = x

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

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

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

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

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

3. f (x) = 10x3x2

1x

3

2x

4

1 234

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

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

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

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

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

⌦⌦ 113

4. y = (1 – 5x)6

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

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

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

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

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

5. y = 1x

2

2

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

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

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

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

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

⌫ ⌫ ⌦⌫ 114

8

5

4

1.

1.1

1.2

1.3

2.

3.

f S

1. f (x) < 0 x S f S

2. f (x) > 0 x S f S

f x = c (a , b) Df

c (a , b) f (c) > f (x) x (a , b)

x c

f x = c (a, b) Df

c (a , b) f (c) < f (x) x (a , b)

x c

f S

c f f (c) = 0

1. f (c) > 0 f (c)

2. f (c) < 0 f (c)

⌦⌦ 115

f x = c

f (c) > f (x) x f x c

f x = c

f (c) < f (x) x f x c

4.

4.1 “ ”

1) y = f (x) = x2 y f (x)

2) y = – x2 + 5 y

3) y = x2 + 4x + 6 y

y = (x + 2)2 + 2

x = –a2

b , y = c –

a4

b 2

4.2

y = 3x4 – 2x

3 + x –

5

1

“ ” “ ”

4.3

4.4

y

x0 a

y

x0 a

y

x0 a

y

x0b a b

⌫ ⌫ ⌦⌫ 116

4.5

“ ” “ ”

4.6

4.7

4.8

4.9

1 2

4.10

4.9 8

4.11 8.1

4.12 “ ” “ ”

4.13

4.14

y

x 0 a b c d e f g

⌦⌦ 117

4.15 8.2

4.16 8.2

4.17 8

4.18 (2)

5.

8

8.1 8.2

8

(2)

6.

1.

2.

3.

4.

5.

6. /

80%

90%

4

12

7.

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

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

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

8.

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

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

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

⌫ ⌫ ⌦⌫ 118

8

y = f (x)

1. f (x)

2. f (x) = 0 x

3. x f (x)

4. 2 3 x

3

5.

2

1

(1)

(2)

(3) (1) (2)

2

f (x) x x f (x)

(1) f (x) > 0

(2) f (x) < 0

(3) f (x) = 0

1

⌦⌦ 119

8.1

( )

1. f (x) = x2 + 3

f (x) = x2 + 3

f (x) = 2x ( )

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

x = 0 ........

f (x) = 2x f (x) x

........

x = 0

x = –1 f (x) = 2 (–1) = –2

x = –1

x = 1 f (x) = 2 (1) = 2

x = 1

f (x) = x2 + 3

x = 0 f (x) = 3

= 3

1

2

1 2

⌫ ⌫ ⌦⌫ 120

2. f (x) = – 3x2 – 2

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

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

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

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

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

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

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

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

3. f (x) = 2x3 + 3x

2 + 6x – 2

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

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

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

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

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

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

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

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

4. f (x) = 2x3 3

2

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

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

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

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

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

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

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

⌦⌦ 121

5. f (x) = x2 + 2x – 5

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

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

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

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

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

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

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

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

6. f (x) = –x2 + x + 6

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

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

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

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

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

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

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

7. y = 2x3 – 3x

2 – 12x + 5

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

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

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 122

8.2

(1) y = f (x)

y

(2) x

(3) y y = 0

(4) 8

1. 20

x 20 – x

y =

y = x (20 – x) = 20x – x2

y

y = 20x – x2 y = 20 – 2x

20 – 2x = 0 x = 10

10 10

2. 200

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

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

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

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

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

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

⌦⌦ 123

3. 10 16

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

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

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

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

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

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

4. 90, 120 150

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

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

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

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

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

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

5.

192

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 124

6. 5

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

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

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

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

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

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

7. 80

4

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

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

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

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

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

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

8. 25 500

24 600 23

700

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

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

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

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

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

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

⌦⌦ 125

9. 120 40

41 118 2 42

116 4 43 114 6

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

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

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

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

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

10. 5,000 10,000

5,000 75 5,000

0.01

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

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

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

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

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

11. 9.8 19.6

t s = –4.9t2 + 19.6t + 9.8

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 126

8

5

50 10

( )

1) f (x) = –3x2 – 2

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

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

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

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

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

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

2) f (x) = 2x3 + 3x

2 + 6x – 2

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

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

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

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

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

3) f (x) = 3

2

x3 + 2

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

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

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

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

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

1

⌦⌦ 127

1) 8

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

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

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

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

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

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

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

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

2) 96

80

(1)

(2)

(3)

s = tV0 – 16t2 V0

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

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

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

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

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

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

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

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

2

⌫ ⌫ ⌦⌫ 128

⌦⌦ 129

6. S = t3 – 3t

2 + 4t + 8 t = 2

. –4 . 2

. 4 . 12

7. S = t5 – t

4 + t

3 + 2t – 1 t = 1

. –4 . 6

. 11 . 14

8. y = 3x3 – 4x

2 – t + 9 (–2 , 5)

. 51 . 20

. 12 . –21

9. y = 3x2 – 5x + 1 (1 , –1)

. y = x . y = x – 1

. y = x – 2 . y = x + 1

10. xy = 4 (–1 , –4)

. 4x – y + 8 = 0

. 4x + y – 8 = 0

. 4x + y + 8 = 0

. x – 4y – 15 = 0

11. f (x) = 5x3 – 3x

5 f (x) f (x)

. 5x2 – 3x

4 , 5x – 3x

3

. 15x2 – 15x

4 , 30x – 60x

3

. 5x4 – 3x

6 , 5x

5 – 3x

7

. 15x3 – 15x

4 , 15x – 15x

3

x

⌫ ⌫ ⌦⌫ 130

12. y = 7 – 6x – x2

. 16 . 7

. –3 . –20

13. f (x) = x3 – 6x

2 + 9x – 8

. –4 8 . 4 –8

. –8 –4 . –4 –8

14. 12

. 7 , 5 . 9 , 3

. 8 , 4 . 6 , 6

15. 10 cc.

x f (x) = 3

3

x+ 3x

2 + 8x

0 x 6 6

6

. 2 2 2

. 2 2 2

. 2 4

. 2 4

⌦⌦ 131

9

5

4

1.

1.1

1.2

1.3

2.

dx

dy = )x(f

y = f (x)

3.

3.1

3.2 ( dx

dy = x

n , n –1

y = c1m

x 1n

)

3.3

4.

4.1 (3)

4.2

y = f (x)

)x(f

f (x) = x2 )x(f = 2x f (x) = x

5 )x(f = 5x

4

⌫ ⌫ ⌦⌫ 132

4.3

y = f (x) f (x)

y = f (x)

)(xf

4.4 xn x

n = nx

n–1 (

x 1 ) 1

2x x2

x 2

2x

3x2 x

3

x2 + 0 x

2 + 1 x

2 + 7 x

2 – 9

2x 2x 2x 2x

x2 + 0 , x

2 + 1 , x

2 + 7

x2 – 9

C C

2x x2 + C

xn

1

1

n

x n + C ; n -1

C

1

1

n

x n + C x

n

y = f (x)

f (x)

xn

1n

x 1n

; n -1

2x

x2

⌦⌦ 133

4.5 f

f (x) = xn f (x)

4.6

4.7

t S = t3 + 3t

2

S t

dt

dS = V = 3t

2 + 6t /

( t 3t2 + 6t / )

t V /

V = 3t2 + 6t 4.6 t

4.8 9

4.9 9

4.10 9

f(x) = 1

1

n

x n + C C n -1

f (x)

y = F (x)

F (x)

F (x) = f(x)

y = F(x) + C

f

⌫ ⌫ ⌦⌫ 134

5.

9

9

(3)

6.

1.

2.

3.

4.

5.

6. /

80%

7.

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

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

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

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

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

8.

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

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

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

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

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

⌦⌦ 135

9

1. f

1.1 f (x) = x6 f (x) = ………………………………………………………………….

1.2 f (x) = x10

f (x) = ………………………………………………………………….

1.3 f (x) = x-8 f (x) = ………………………………………………………………….

1.4 f (x) = x-2 f (x) = ………………………………………………………………….

1.5 f (x) = 2

3

x f (x) = ………………………………………………………………….

2.

2.1 f (x) = 5x

f (x) = 2

5 2x+ C ; C

2.2 f (x) = x2

2.3 f (x) = xx 3

2.4 f (x) = x-3

2.5 f (x) = 7

1

x

⌫ ⌫ ⌦⌫ 136

9

5

20 10

1.

(1) dx

dy = 200x

4

(2) f (x) = 6x – 24

2. F F (x) = 3x2

3. f f(x) = x

⌦⌦ 137

(3)

“ ”

1. x-8

. -8x-9 + C . -8x

-7 + C

. Cx

7

7

. C8

x 8

2. dx

dy = 4x

3 + 2 , y

. y = x4 + C . y = x

4 + 2x + C

. y = 4x2 – x + 4 . y = x

4 + 2x + C

3. f (x) = 3x , f(x)

. 2

3 2x + C . 3

. 3x2 + C .

3

2 3x + C

4. f(x) = 2x3 – 3x

2 + 1 dx)x(f

. 6x2 – 6x + C . Cxxx

2

1 34

. Cxx2

3x

3

2 24 . Cxx2

3x

4

1 34

5. dx)3x3( 2

. x3 + 3x

2 + C . 3x

3 + 3x + C

. x3 + 3x + C . 3x

2 + 3x + C

6. f (x) = x f(0) = 0 , f(x)

. xx . x2

x

. x2

1 . x

7. dx

dy = 2x – 3 y = 2 x = 3 y x = 5

. 12 . 13

. 10 . 11

_

⌫ ⌫ ⌦⌫ 138

⌦⌦ 139

10

5

2

1.

2.

f f (x) = xn f (x) f (x)

= 1n

x 1n

+ C C n –1 dx)x(f = f (x) + C dx)x(f

3.

4.

4.1

f (x) f (x) 1n

x 1n

+ C ; n –1

4.2

xn

1n

x 1n

+ C; n –1

⌫ ⌫ ⌦⌫ 140

dxxn = 1n

x 1n

+ C ; n –1

dx)x(f

4.3

4.4 10

4.5

4.6

4.7

(1) nxdx

dy

C1n

xy

1n

; 1n 1n

x

1

bax

1 a , b 0a

(2)

dx)x2x( 2 = 2

2

1

3

c2

x2c

3

x

= 21

23

C2C2

x2

3

x

= Cx3

x 23

C1 + 2C2 = C

dxx2 C1 xdx2

2C2 C

C1 + 2C2

4.8 10

4.9 10

4.10 10

dx)x(f = f(x) + C

⌦⌦ 141

5.

10

10

10

6.

1.

2.

3.

4.

5.

6. /

90%

90%

5

–

7.

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

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

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

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

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

8.

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

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

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

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

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

⌫ ⌫ ⌦⌫ 142

⌦⌦ 143

10

1.

1) dx7 = dx7 = 7x + C

)Cx7(dx

d = )x7(

dx

d + )C(

dx

d = 7

2) dxx6 = ………………………………………………………………………………..

3) dtt8 3 = ………………………………………………………………………………..

4) dx)1x2( = ………………………………………………………………………………..

5) 3x4

dx = ………………………………………………………………………………..

2.

1) dx2 ………………………………………………………………………………..

2) xdx ………………………………………………………………………………..

3) 2xdx ………………………………………………………………………………..

4) x2dx ………………………………………………………………………………..

5) 3x2dx ………………………………………………………………………………..

6) x3dx ………………………………………………………………………………..

7) (x+5)dx ………………………………………………………………………………..

8) (3x2+2x-5)dx ………………………………………………………………………………..

9) (4x3-3x

2+2x-4)dx ………………………………………………………………………

⌫ ⌫ ⌦⌫ 144

10) dxx

12

………………………………………………………………………………..

11) dxx

13

………………………………………………………………………………..

12) dxx ………………………………………………………………………………..

13) dxx3 ………………………………………………………………………………..

14) dxx

1 ………………………………………………………………………………..

15) dx)x

1x(

2 ………………………………………………………………………………..

16) dx)x

1x( ………………………………………………………………………………..

17) dx)x2

1

x

1(

2 ………………………………………………………………………………..

18) dx)x

2x(

3 ………………………………………………………………………………..

19) dx)4x(x ………………………………………………………………………………..

20) dx)4x(x 22 ………………………………………………………………………………..

21) dx)2x(x ………………………………………………………………………………..

22) dx)xx( 3

2

2

3

………………………………………………………………………………..

3. f (x) = 4x f(1) = 3 f(x)

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

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

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

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

4. f (x) = 6x – 8 , f (0) = 2 f (1) = 4 f (x)

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

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

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

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

⌦⌦ 145

5. y = f (x) (x, y) 2x + 4 y = f (x)

(1, 6) y = f (x)

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

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

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

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

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

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

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

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

6. 6t2 /( )

2 t

s t = 2

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

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

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 146

10

5

30 10

1. x10

dx

2. 7x dx

3. dx)5x4x3x( 2

1

2

4. dx)x

3x(

2

3

5. y = f (x) (x, y) 2

y = f (x) x = 0 -2 (1, 4) y = f (x)

6. 15

0.8 /( )2

⌦⌦ 147

⌫ ⌫ ⌦⌫ 148

⌦⌦ 149

6.

1.

2.

3.

4.

5.

6. /

90%

–

7.

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

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

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

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

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

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

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

8.

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

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

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

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

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

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

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

⌫ ⌫ ⌦⌫ 150

⌦⌦ 151

⌫ ⌫ ⌦⌫ 152

⌦⌦ 153

⌫ ⌫ ⌦⌫ 154

⌦⌦ 155

⌫ ⌫ ⌦⌫ 156

⌦⌦ 157

⌫ ⌫ ⌦⌫ 158

⌦⌦ 159

5.

12

12

12

(3)

6.

1.

2.

3.

4.

5.

6. /

90%

12

7.

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

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

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

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

8.

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

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

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

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

⌫ ⌫ ⌦⌫ 160

⌦⌦ 161

12

y = f (x)

1. x

(1) y = 2x + 4 x = 1 x = 3

(2) y = 3x2 x = 1 x = 2

(3) y = x2 + 2 x = –1 x = 0

(4) y = 4 – x2 x = –1 x = 2

2. y = f (x) = x x = 0 x = 4

3.

4.

1

6

2 3 4

5

4

3

2

1

x

y

y + x2

= 4

-4 -5 -6 -2 -1-30

y – x = 6

y

(4, 2)

0

2

1

21 3

x

4

y = x

y = x–2

⌫ ⌫ ⌦⌫ 162

12

5

30 10

1. y = x + 1 x = –2 x = 1

2. y = 9 – x2 x = –3 x = 3

3. f (x) = x2 – 2x x x = –1 x = 2

4.

y = x – x2 y = –x

y = –x

x

y

0

y = x–x2

⌦⌦ 163

(3)

“ ”

1. x-8

. –8x-9 + C . –8x

-7 + C

. –7

x 7

+ C . –8

x 8

+ C

2. dx

dy = 4x

3 + 2 y

. y = x4 + C

. y = x4 – 2x + C

. y = 4x2 – x + 4

. y = x4 + 2x + C

3. f (x) = 3x , f (x)

. C2

x3 2

. 3

. 3x2 + C . C

3

x2 3

4. f (x) = 2x3 - 3x

2 + 1 f(x) dx

. 6x2 – 6x + C

. Cxxx2

1 34

. Cxx2

3x

3

2 24

. Cxx2

3x

4

1 34

⌫ ⌫ ⌦⌫ 164

5. (3x2 + 3) dx

. x3 + 3x

2 + C . 3x

3 + 3x + C

. x3 + 3x + C . 3x

2 + 3x + C

6. f (x) = x f (0) = 0 , f (x)

. x x . 2

xx

. 2

1x . x

7. dx

dy = 2x – 3 y = 2 x = 3 y x = 5

. 12 . 13

. 10 . 11

8. (x, y) 4x – 1 (–1, 2)

. y = 4x2 – x – 4

. y = 2x2 – x + 1

. y = 4x2 – x + 4

. y = 2x2 – x – 1

⌦⌦ 165

⌫ ⌫ ⌦⌫ 166

:

.

. .

.

.

:

:

.

:

:

:

:

:

⌦⌦ 167

( .)

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