E-learning 25 -02-55

1

( )y f x

0

( ) ( )limh

f x h f x

h f x

( )f x dydx

y ( )df xdx

0

( ) ( ) ( ) ( ) lim

h

dy d f x h f xy f x f x

dx dx h

0

( ) ( )lim h

f x h f x

h

0 0

( ) ( ) ( ) ( )lim lim h h

f x h f x f x h f x

h h

f x a ( )f a x adydx ( )y a

0

( ) ( )( ) lim

h

f x h f xf x

h

f 0x 0 00 0

( ) ( )( ) lim

h

f x h f xf x

h

0x x h 0h 0x x f 0x

0

00

0

( ) ( )( ) lim

x x

f x f xf x

x x

E-learning 25 -02-55

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

1. y x x x h ( ) ( )f x h f x

h

2. y x x

0

( ) ( )lim

0hf x h f x

( )f x

( )y f x f x 1. ( ) ( , ( ))x f x ( , ( ))x h f x h

( ) ( )f x h f xh

2. ( , )x y 0

( ) ( )lim h

f x h f x

h

( )f x

( )s f t t 1. [ , ]t t h ( ) ( )f t h f t

h

2. t ( )v 0

( ) ( )lim h

f t h f t

h

( ) dsf tdt

( )u f x ( )v g x c 1. 0dc

dx

2. 1dxdx

3. 1( )n nd x nxdx

n

4. ( )d du dvu vdx dx du

E-learning 25 -02-55

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5. ( )d du dvu vdx dx dx

6. d ducu cdx dx

7. ( . )d dv duu v u v

dx dx dx

8.

2du dv

v ud u dx dxdx v v

0v 9.

( )y g u ( )u f x ( ( ( )) ( )( )y g f x gof x dy dy du

dx du dx

( )( ) ( ( ))y gof x g f x ( ) ( ) ( ( )) ( )gof x g f x f x

10. ny u ( )u f x 1.

nndu dunu

dx dx n

( )y f x x , dyydx

( )f x

f x 2

2,d x

ydx

( )f x

f x 3

3,d y

ydx

( )f x

( )nf n f ( )dyy f x

dx

2

2( )

d yy f x

dx

3

3( )

d yy f x

dx

( ) ( ) ( )

nn n

n

d yy f x

dx n

E-learning 25 -02-55

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( LHospital Rule)

f g ( ) 0g x a R lim ( ) 0

x af x

lim ( ) 0

x ag x

( ) ( )lim lim

( ) ( )x a x af x f x

g x g x

( )lim

( )x af x

g x

( )lim ( )x a

f x

g x

00

1 1.

4

3

81lim

3xx

x

2. 3

1

1lim

1xx

x

1. 4 3

3 3

81 4lim lim

3 1x xx x

x

108

2. 2

33

11 12

11 3lim lim

112

x x

xx

xx

23

# 2

3

22

8lim

4xx

x

1. 0 2. 1 3. 3 4. 4

3 2

22 2

8 3lim lim

4 2x xx x

x x

2

3 lim

2xx

3 #

E-learning 25 -02-55

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3 2

2

2lim

2hh h

h

1. 3 2. 2 3. 0 4. 1

2

2 2

2 1 2lim lim

2 1h hh h h

h

2 lim(2 1) 3h

h

2 2 2

2 2 2

2 ( 2) ( 2)( 1)lim lim lim

2 ( 2) ( 2)h h hh h h h h h

h h h

2 lim (h+1) = 3h

#

1.

f I 1 2,x x I 1 2x x 1 2( ) ( )f x f x f I 1 2,x x I 1 2x x 1 2( ) ( )f x f x 1 f ( , )a b ( ) 0f x x ( , )a b f ( , )a b ( ) 0f x x ( , )a b f ( , )a b 2.

f 0x x ( , )a b 0 ( , )x a b 0( ) ( )f x f x ( , )x a b 0( )f x f 0x x ( , )a b 0 ( , )x a b 0( ) ( )f x f x ( , )x a b 0( )f x

E-learning 25 -02-55

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2 f 0x x 0 fx D 0( ) 0f x 0( )f x f 1x 3x f 2x 0x f 0( ) 0f x 0( )f x 0x (Critical value) f 0x f 0 0( , ( )x f x (Critical point)

3 f 0x f ( )f x 0x x

1. ( )f x 0x f 0x x 2. ( )f x 0x f 0x x

3. ( )f x f 0x x

0x

0x

( ) 0f x

3( )f x

E-learning 25 -02-55

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1

1. ( )f x 2. 0x ( ) 0f x ( )f x 0(x ) 3. 0x 2 ( ) 0f x ( ) 0f x

0x x

2

0x 0( ) 0f x 0( )f x 1. 0( ) 0f x 0x x 0( )f x 2. 0( ) 0f x 0x x 0( )f x 3. 0( ) 0f x 1

0x

0x

0x

0x