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7/23/2019 diaforik1 logismos
1/215
E : 1
I
7/23/2019 diaforik1 logismos
2/215
E : 2
.
7/23/2019 diaforik1 logismos
3/215
E : 3
1. ; ; ;
)(tSS
t. H S t .
x=S(t)
M
O
A
S(t0)
S(t)
M0
1
, , 0t 0M h,
htt 0 , . (. 1). 0t t )()( 0tStS . ,
)()(
0
0
tt
tStS
.
t 0t ,
0t . , t
0t , 0t
)( 0t . :0
0
00
)()(lim)(
tt
tStSt
tt
.
7/23/2019 diaforik1 logismos
4/215
E : 4
, tttS 4)( 2 (.2),
x=S(t)
t=4t=2
4321O
t=0
() x=S(t)
2 4
4
O
x
t
2
()
11t , 22t
33t :
21
)3)(1(lim
1
34lim
1
)1()(lim)1(
1
2
11
t
tt
t
tt
t
StS
ttt
02
)2)(2(lim
2
44lim
2
)2()(lim)2(
2
2
22
t
tt
t
tt
t
StS
ttt
23
)3)(1(lim
3
34lim
3
)3()(lim)3(
3
2
33
t
tt
t
tt
t
StS
ttt.
, 0t
0)()(
0
0
tt
tStS, 0)( 0 t , ,
0t 0)()(
0
0
tt
tStS , 0)( 0 t .
2. ;
, . , , 2xy )1,1(A (. 4), 3xy )1,1(A (. 4).
O
3
7/23/2019 diaforik1 logismos
5/215
E : 5
x
A(1,1)
()
y=x2
y
()
x
y=x3
y4
A(1,1)
3.
, .
, , (. 5). MA, , AM. , ,
. , , . f ))(,( 00 xfxA
.
))(,( xfxM
, 0xx
, f M, :
O
5
xO
Cf
x
()
x0
M(x,f(x))
y
M
A(x0,f(x0))
6
xO
Cf
xx0
M(x,f(x))
A(x0,f(x0))
y
M
()
7/23/2019 diaforik1 logismos
6/215
E : 6
x 0x 0xx , (. 6). x 0x 0xx (. 6). f .
0
0 )()(xx
xfxf ,
fC ))(,( 00 xfxA
0
0 )()(lim0 xx
xfxf
xx
.
4. f
))(,( 00 xfxA .
: f ))(,( 00 xfxA fC .
0
0
0
)()(lim
xx
xfxf
xx
,
fC ,
.
, ))(,( 00 xfxA
)()( 00 xxxfy ,
0
0
0
)()(lim
xx
xfxf
xx
.
, 2)( xxf )1,1(A .
1
1
lim1
)1()(
lim
2
11
x
x
x
fxf
xx 2)1(lim
1
x
x,
fC
)1,1(A . 2 )1(21 xy .
A(1,1)y=x2
xO
y7
7/23/2019 diaforik1 logismos
7/215
E : 7
5. .
: f 0x
, 0
0 )()(lim0 xx
xfxf
xx
. f
0x
)( 0xf . :
0
0
0
)()(lim)(
0 xx
xfxfxf
xx
.
, 1)( 2 xxf , 10 x
2)1(lim1
)1)(1(lim
1
1lim
1
)1()(lim
11
2
11
xx
xx
x
x
x
fxf
xxxx.
, 2)1( f .
6. ;
, , 0
0
00
)()(lim)(
xx
xfxfxf
xx
hxx 0 ,
h
xfhxfxf
h
)()(lim)(
00
00
.
0xxh x , )()( 00 xfhxf )()( 00 xfxxf )( 0xf , :
x
xfxf
x
)(lim)(
0
00
.
Leibniz 0x
dx
xdf )( 0 0
)(xx
dx
xdf . )( 0xf
Lagrange.
7. f f ; .
7/23/2019 diaforik1 logismos
8/215
E : 8
, 0x f, :
f 0x ,
0
0
0
)()(lim
xx
xfxf
xx
,0
0
0
)()(lim
xx
xfxf
xx
.
,
0,
0,)(
2
2
xx
xxxf
0 0)0( f ,
00
lim0
)0()(lim
2
00
x
x
x
fxf
xx
00
lim0
)0()(lim
2
00
x
x
x
fxf
xx
,
0,5
0,)(
3
xx
xxxf
0,
00
lim0
)0()(lim
3
00
x
x
x
fxf
xx
505
lim0
)0()(lim
00
x
x
x
fxf
xx
.
8.
, 0t ; fC
f, ))(,( 00 xfxA ; fC f 0x .
, 0t ,
)(tSx 0t . ,
O x
y
y=x2
y=x2
8
O x
y
y=x3
y=5x
9
7/23/2019 diaforik1 logismos
9/215
E : 9
)()( 00 tSt .
fC
f, ))(,( 00 xfxA f 0x .
, )( 0xf ,
:
))(()( 000 xxxfxfy
)( 0xf ))(,( 00 xfxA
fC f 0x .
9. ||)( xxf 00 x ;
;
||)( xxf . f 00x , ,
1lim0
)0()(lim
00
x
x
x
fxf
xx
,
1lim0
)0()(lim
00
x
x
x
fxf
xx
.
, , f 0x .
10. ; ;
f 0x ,
.
O x
y 14
7/23/2019 diaforik1 logismos
10/215
E : 10
0xx
)()()(
)()( 00
0
0 xxxx
xfxfxfxf
,
)(
)()(lim)]()([lim 0
0
0
00
0
xxxx
xfxfxfxf
xxxx
)(lim)()(
lim 00
0
0
0
xxxx
xfxf
xxxx
00)( 0 xf ,
f 0x . , )()(lim 00
xfxfxx
, f
0x .
f 0x , , , 0x .
11.
,
0,1
0,)(
3
22
xxx
xxxxf
: i) 00 x ; ii) 00 x ;
i) f 00x ,
)0()(lim)(lim00
fxfxfxx
, ,
112 1 .ii) :
11, , f .
1 ,
0,1
0,1)(
3
2
xxx
xxxxf .
0x ,
7/23/2019 diaforik1 logismos
11/215
E : 11
1)1(11
0
)0()( 2
x
x
xx
x
xx
x
fxf ,
1)1(lim0
)0()(lim
00
xx
fxf
xx
.
0x
1)1(11
0
)0()( 223
x
x
xx
x
xx
x
fxf ,
1)1(lim0
)0()(lim 2
00
x
x
fxf
xx
.
0
)0()(lim
0
)0()(lim
00
x
fxf
x
fxf
xx
, 1 f 00x .
1 ,
0,1
0,1)(
3
2
xxx
xxxxf
0x ,
1)1(11
0
)0()( 2
x
x
xx
x
xx
x
fxf ,
1)1(lim0
)0()(lim
00
xx
fxf
xx
.
0x
1)1(11
0
)0()( 223
x
x
xx
x
xx
x
fxf,
1)1(lim0
)0()(lim 2
00
xx
fxf
xx
.
0
)0()(lim
0
)0()(lim
00
x
fxf
x
fxf
xx
, 1 f 00 x .
7/23/2019 diaforik1 logismos
12/215
E : 12
1
1. f x = 2x + 1
0x = 0
fD =
f (0) =0
limx
00
f x f
x
=0
limx
2 1 1xx =
0lim
xx = 0
2. f x = 21
x
0x = 1
fD = *
f (1) =1
limx
11
f x f
x
=1
limx
21 1
1xx
=1
limx
2
2
1( 1)
xx x
=1
limx
2
(1 )(1 )
( 1)
x x
x x
=1
limx
2
(1 )(1 )
( 1)
x x
x x
=1
limx
2
(1 )(1 )
( 1)
x x
x x
=1
lim
x
2
(1 )x
x
= 2(1 1)
1
= 2
7/23/2019 diaforik1 logismos
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E : 13
3. f x = 2x 0x = 0
f
D =
0
x = 0 00
f x f
x
=2 20
0xx
=2x
x = x
x x
f (0) =0
limx
00
f x f
x
=0
limx
xx
.0
limx
x = 1 . 0 = 0
4. ( ) f x = x x
0
x =0
fD =
x 0 00
f x f
x
=0
0
x x
x
= x
f (0) =0
limx
00
f x f
x
=0
limx
x = 0
5. = 2 x + x
= 0
=
= 0 =
== 1+
(0) = = (1+ ) = 1 + 0 = 1
f x x
0x
fD
0x
00
f x f
x
2 2x x x
x
( 1 )x x
x
x
f0
limx
00
f x f
x
0
limx
x
7/23/2019 diaforik1 logismos
14/215
E : 14
-+ +-
+
0 3
2
6. ( ) f x = 1x
0x = 1
fD =
x 1 11
f x f
x
=1 0
1
x
x
=1
1
x
x
1lim
x
11
f x f
x
=1
limx
11
xx
=1
limx
1 = 1 (1)
1lim
x
11
f x f
x
=1
limx
( 1)1
xx
=1
limx
(1) = 1 (2)
(1), (2) f 0
x = 1
7. ( ) f x = 2 3x x
0x = 1
fD =
2x 3x
0
x = 1 11
f x f
x
=2 2( 3 ) 1 3.1
1
x x
x
=2 3 2
1x x
x
= ( 1)( 2)
1x x
x
= 2 x
f (1) =1
limx
11f x fx = 1lim
x(2 x) = 2 1 = 1
7/23/2019 diaforik1 logismos
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E : 15
8. ( )
f x =2 1 0
1 0
x x , x
x , x
0x =0
0lim
x
00
f x f
x
=0
limx
2 1 (0 1)x xx
=0
limx
2x xx
=0
limx
( 1)x xx
=0
limx
(x + 1) = 0 + 1 = 1 (1)
0lim
x
00
f x f
x
=0
limx
1 (0 1)xx
=0
limx
xx
=0
limx
1 = 1 (2)
(1) (2) f (0) =0
limx
00
f x f
x
= 1
9. =
= 0
x 0 =
=
=
=
= =
(0) = = . = = =
f x
1 , 0
0 , 0
x xx
x
0x
0
0
f x f
x
1 0xx
x
21 x
x
2
(1 )(1 )
(1 )
x x
x x
2
21 )(1 )xx x2
2(1 )
x
x x
2x
x 1
1 x
f0
limx
00
f x f
x
0
limx
2
xx
0
limx
11 x
21 11 0
11 1
12
7/23/2019 diaforik1 logismos
16/215
E : 16
3
10. f 0 ,
g(x) = x f(x) 0.
f 0 0
limx
f(x) = f(0)
x 0 0
0
g x g
x
= 0. 0x f x f
x
= f(x)
0
limx
00
g x g
x
=0
limx
f(x) = f(0) g (0) = f(0)
11. 0
x = 0
f x =2
3
1 0
0
x , x
x , x
.
0lim
x f x =
0lim
x ( 2x + 1) = 1
0lim
x f x =
0lim
x
3x = 0
0
limx
f x , f 0,
.
12. 0= 1 = x 1+ 1
.
=
= (x 1+ 1 ) = 1 = 1 1+ 1 = 1
= , 1
x < 1 = = 1
= 1 (1)
f x
fD
1lim
x f x
1lim
x 1f
1lim
x f x 1f f
11
f x f
x
( 1) 1 11
xx
1lim
x 1
1f x f
x
7/23/2019 diaforik1 logismos
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E : 17
x > 1 = = 1
= 1 (2)
(1) , (2) 1
13. 0 = 4 ,
: i) = 0 ii) (0) = 4
i) 0 = = [ x ]
= . x = 4 . 0 = 0
ii) (0) = = = 4
11
f x f
x
1 1 11
xx
1lim
x
11
f x f
x
f
f0
limx
f xx
0f f
f 0f0
limx
f x0
limx
f xx
0limx
f xx 0limx
f0
limx
00
f x f
x
0
limx
f xx
7/23/2019 diaforik1 logismos
18/215
E : 18
4
14. f x =1 0
1
1 0
, xx
x , x,
fC (0, 1) x
4
x < 0 00
f x f
x
=1 1
1 xx
= 1 (1 )(1 )
xx x
= 1 1
(1 )
x
x x
= 1
1 x
0
limx
00
f x f
x
=0
limx
11 x
= 1 (1)
x > 0 00
f x f
x
= 1 1xx
= xx
0
limx
00
f x f
x
=0
limx
xx
= 1 (2)
(1) , (2) f (0) = 0limx 0
0
f x f
x
= 1
, (0, 1) f
C
f (0) = 1 = 4
7/23/2019 diaforik1 logismos
19/215
E : 19
5
15. x + 1 f x 2x + x + 1 x ,
:i) 0f = 1
ii) 1 0f x f
x
x + 1 x < 0
1 0f x f
x
x + 1 x > 0
iii) f (0) = 1
i) x = 0 1 0f 1 0f = 1
ii) x + 1 f x 2x + x + 1
x + 1 1 f x 0f 2x + x + 1 1
x f x 0f = 2
x + x (1) x < 0 , (1) x
x
0f x f
x
2x xx
1 0f x f
x
x + 1 (2)
x > 0 , (1) xx
0f x f
x
2x xx
1 0f x f
x
x + 1 (3)
iii) 0
limx
( x + 1) = 1 , (2)
0lim
x
0f x fx
= 1 (4)
, (3)
0lim
x
0f x fx
= 1 (5)
(4) , (5) f (0) =0
limx
0f x fx
= 1
7/23/2019 diaforik1 logismos
20/215
E : 20
16. f 0
x = 0
x 2 x 4x x f x 2 x + 4x
: i) 0f = 0 ii) f (0) = 1
i) f
0x = 0
0lim
x f x = 0f (1)
0
limx
f x
x > 0 , 2 4x xx
f x 2 4x xx
2xx
- 3x f x 2x
x + 3x
xx
x - 3x f x xx
x + 3x (2)
0
limx
( xx
x 3x ) = 1 . 0 30 = 0.
(2) 0
limx
f x = 0(1)
0f
ii) x 0 2 4
2
x x
x
2xf x
x
2 4
2
x x
x
2
2
x
x
2x f xx
2
2
x
x
+ 2x
2
xx
2x 0f x fx 2
xx +
2x (3)
0
limx
[ 2
xx
2x ] = 21 0 = 1
(3) 0
limx
0f x fx
= 1
f (0) = 1
7/23/2019 diaforik1 logismos
21/215
E : 21
6
h
17. f(1+h) = 2 + 3h + 3h2+ h3 ,
h , : i) f(1) = 2 ii) f(1) = 3
i) h = 0 , f (1) = 2 + 30 + 302+ 03= 2
ii) h0 f(1+h)f(1)
h=
2+3+3+2
=3+3h+3h2
f(1) = 20
lim(3 3 3 )h
h h
= 3 + 0 + 0 = 3
18. , f
0x ,
i)0
limh
0 0f x h f xh
= f (
0x )
ii)0
limh
0 0f x h f x h
h
= 2 f (
0x )
f 0
x 0
limu
0 0f x u f xu
= f (
0x ) (1)
i)0
limh
0 0f x h f xh
=
0limh
0 0( )f x h f xh
( - h u 0 )
= 0limu 0 0f x u f x
u
(1)
f (0
x )
ii)0
limh
0 0f x h f x hh
=
0limh
0 0 0 0f x h f x f x f x hh
=0
limh
[ 0 0f x h f x
h
-
0 0f x h f xh
]
=0
limh
0 0f x h f xh
-
0limh
0 0f x h f xh
(1, )i
f ( 0x ) ( f ( 0x )) = 2 f ( 0x )
f
u
7/23/2019 diaforik1 logismos
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E : 22
7
19.
xx 0sec 8sec. :
8
765
42
O
t(sec)
x=S(t)
i) ;
ii) ;
iii) 2t sec, 4t sec 5t sec;
iv)
0sec 4sec;
v) ;
vi) ;
i) .ii) .
iii) 2t , A, 4t 5t
iv)
v)
vi) .
7/23/2019 diaforik1 logismos
23/215
E : 23
1. f ; f f ][ , ;
f . :
H f , , ,
Ax 0 . f
, ),(0 x .
f ],[ , ),(
x
fxf
x
)()(lim
x
fxf
x
)()(lim .
2. f ; f f ;
f 1A o . 1Ax )(xf ,
),(: 1
xfx
RAf
),(
),(
7/23/2019 diaforik1 logismos
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E : 24
f f. H
f dx
df
. )(xfy
))(( xfy . 1 , f , , f f .
f, 3 , )(f .
][ 1)()( ff , 3 .
: , , . , ( ).
3. cxf )( , c
R 0)( xf , :
0)( c
, 0x R, 0xx :
0)()(
00
0
xx
cc
xx
xfxf.
,
0
)()(
lim 0
0
0
xx
xfxf
xx , 0)( c .
4. xxf )( . f
R 1)( xf ,
1)( x
7/23/2019 diaforik1 logismos
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E : 25
, 0x R, 0xx :
1)()(
0
0
0
0
xx
xx
xx
xfxf.
, 11lim)()(
lim 00
0
0
xxxx xx
xfxf
,
1)( x .
5. xxf )( , }1,0{ .
f R 1)( xxf ,
1)( xx
, 0x R, 0xx :
100
21
0
1
00
21
0
0
0
0
0 ))(()()(
xxxxxx
xxxxxx
xx
xx
xx
xfxf
,
10101010100210
0
0
0
)(lim)()(
lim
xxxx
xxxxxxxxxx
xfxf ,
1)( xx .
6.
xxf )( ),0(
xxf
2
1)( ,
x
x2
1
, 0x ),0( , 0xx :
0000
00
00
0
0
0
0 1
)()(
)()(
xxxxxx
xx
xxxx
xxxx
xx
xx
xx
xfxf
,
000
0
0
0 2
11lim
)()(lim
xxxxx
xfxf
xxxx
,
x
x2
1
.
xxf )( 0.
7/23/2019 diaforik1 logismos
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E : 26
7. xxf )( . f
R xxf )( ,
xx )(
, x 0h
h
xhxhx
h
xhx
h
xfhxf )()()(
h
hx
h
hx
)1(
.
1
lim0
h
h
h
01lim0
h
h
h
,
xxxh
xfhxf
h10
)()(lim
0
.
, xx )( .
8. xxf )( . f
R xxf )( ,
xx )(
, x 0h :
h
xhxhx
h
xhx
h
xfhxf )()()(
h
hx
h
hx
1
,
h
hx
h
hx
h
xfhxf
hhh
lim
1lim
)()(lim
000
xxx 10 .
, xx )( .
7/23/2019 diaforik1 logismos
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E : 27
9.
1
lim0
x
x
x
,
01
lim0
x
x
x
, xxf )( , xxg )(
1lim0
x
x
x, 01lim
0
x
x
x,
xxf )( , xxg )( 00 x gf, ,
)0(0
0lim
lim
00f
x
x
x
x
xx
)0(0
0lim
1lim
00g
x
x
x
x
xx
.
10. xexf )( xxf ln)( .
xexf )( . f
xexf )( ,
xxee )(
xxf ln)( . f
),0( x
xf1
)( ,
xx
1)(ln
11. xxf ln)( ,
.
7/23/2019 diaforik1 logismos
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E : 28
x
xxf1
)(ln)( ,
fC
))(,( 00 xfxM
)(1
ln 00
0 xxx
xy .
)0,0(O ,
exxxx
x 0000
0 1ln)0(1
ln0 .
, )1,(eM .
12. 1
2
xxf )( 0)(0,O
,0)(A . :i) 1
2
ii)
1
,2
x.
i) xxxf )()( , 1)0( f
1)( f 1 , 2
xy )( xy .
ii)
1 , 2
2,2
.
, 422
1 2 .
1
y
O e
A(,0)
y
x
21
O(0,0)
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E : 29
1
1. f 0
x :
i) f x = 4x , 0x = -1 ii) f x = x , 0x = 9
iii) f x = x ,0
x =6 iv) f x = nx,
0x = e
v) f x =
x
e , 0x = 2n
i) xR f (x) = 4 3x f (1) = 4 3( 1) = 4
ii) x(0, + ) f (x) = 12 x
f (9) = 12 9
= 16
iii) xR f (x) = x f 6
= 6
= 12
iv) x(0, + ) f (x) = 1x
f (e) = 1e
v) xR f (x) = xe f ( 2n ) = 2ne = 2
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E : 30
2
2. , ,
f x =2 , 1
, 1
x x
x x
x < 1 f (x) = ( 2x ) = 2x
x > 1 f (x) = ( x ) = 12 x
1
lim
x
11
f x f
x
=
1
lim
x
2 11
xx
=
1
lim
x
(x + 1) = 1 + 1 = 2 (1)
1lim
x
11
f x f
x
=1
limx
11
xx
=1
limx
2
1
( ) 1
x
x
=1
limx
1
( 1)( 1)
x
x x
=1
limx
1
1x= 1
1 1= 1
2 (2)
(1) , (2) f 0
x = 1
3. , ,
f x =, 0
, 0
x x
x x
x < 0 f (x) = (x) = x x > 0 f (x) = (x) = 1
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E : 31
0lim
x
00
f x f
x
=0
limx
00
xx
=0
limx
xx
= 1 (1)
0lim
x
00
f x f
x
=0
limx
00
xx
=0
limx
1 = 1 (2)
(1) , (2) f (0) = 1
4. , ,
f x =3
4
, 2
, 2
x x
x x
x < 2 f (x) = ( 3x ) = 3 2x
x > 2 f (x) = ( 4x ) = 4 3x
2lim
x f x =
2lim
x
3x = 32 = 8 (1)
2lim
x f x =
2lim
x
4x = 42 = 16 (2)
(1) , (2) f 2 , .
5. , ,
f x =
2
3
2,3
2,3
x x
x x
x < 23
f (x) = ( 2x ) = 2x
x > 23
f (x) = ( 3x ) = 3 2x
23
lim
x
f x =23
lim
x
2x = 2
23
= 49
(1)
23
lim
x
f x =23
lim
x
3x = 3
23
= 827
(2)
(1) , (2) f 23
,
.
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E : 32
3
6. y = 2x .
f x =3
x ;
g(x) = 2x , xR g(x) = 2x
1M (
1x , g(
1x )) ,
2M (
2x , g(
2x ))
1x
2x
1 ,
2
g
C .
1 //
2 g(
1x ) = g(
2x )
21
x = 22
x
1x =
2x
gC
xR f x = ( 3x ) = 3 2x
1
N (1
x , f(1
x )) ,2
N (2
x , f(2
x )) 1
x 2
x 1
,2
f
C .
1 //
2 1f x = 2f x
3 21
x = 3 22
x 2
1x = 2
2x
1x =
2x (
1x
2x )
, ,
7. = ( , ()) , 0
. ( , ()) ( , 0)
.
f x x f
f
f
fC
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E : 33
= [0 , + )
() = , 0 ,
> 0 . = =
:
y () = (x )
y = (x )
( , 0)
0 = ( ) = (2) = .
.
8.
= ( , ) , 0 .
.
= R = 3 = 3
y () = (x )
y = 3 (x )y = 3 x 3 +
y = 3 x 2 , ,
(2 ,8 )
= 3 = 12 = 4 . 3 = 4
fD
f fD
f x1
2 x f 1
2
fC
f f
12
12
12
f x3x 3
fC
fD f x2x f 2
fC
f f 3 2
2 3 32 3
fC
3
2 33 2
y x
y a x a
3
3 2 33 2
y x
x a x a
3
3 2 2 32 2 0
y x
x a x a x a
3
2 2 2( ) 2 ( ) 0
y x
x x a a x a
3
2( )( ) 2 ( ) 0
y x
x x a x a a x a
3
2( )[ ( ) 2 ] 0
y x
x a x x a a
3
2 20 2 0
y x
x a x ax a
3
2
y x
x a x=a x a
3y a
x a
38
2
y a
x a
3
2f a 2( 2 )a 2 2 f
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E : 34
9.
= . ,
, i) To M ii) ,
R*
= = = = =
: y () = (x ) y = (x )
y = 0 = (x ) = x x = 2
(2 , 0)
x = 0 y = (0 ) y = y =
i) : =
ii) (OAB) = (OA)(OB) = = 2
f x 1x 1,
x x y y
f limx f x fx
limx
1 1xx
limx
xx
x
limx
1x 2
1
f f 1 21
1 2
1
1 2
1
1
1
2
20,
202 0
2 2,
1, 12
12
2 2
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E : 35
4
10. f
.
(2 , 0)
f x = 0 22 0
= 1
(0 , 2) f x = 2 22 0 = 2
(2 , 4) f x = 0
(4 , 6) f x = 4 ( 2)6 4
= 6
2= 3
(6 , 9) f x = 0 49 6
= 43
f
11. f: [0, 8]R, , f(0) = 0
.842
1
1
2
O
y
x
y=f (x)
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E : 36
(0 , 2) f x = 2 .
f
C .
(2 , 4) f x = 1 .
f
C .
(4 , 84) f x = 1 . fC .
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E : 37
5
12. ,
f x = ,,
x xx+ x
0
x =
0
x = , ,
. lim
x f x = lim
x f x = f()
limx x = limx (x + ) = + = + 0 = + = (1)
f 0
x = limx
f x fx
= limx
f x fx
limx
( )xx
= limx
xx
limx
xx
= lim
x
( )xx
0lim
x ( )xx = limx 1 =
(1) =
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E : 38
1. .
1
gf, 0x , gf
0x :
)()()()( 000 xgxfxgf
0xx , :
0
0
0
0
0
00
0
0 )()()()()()()()())(())((
xx
xgxg
xx
xfxf
xx
xgxfxgxf
xx
xgfxgf
.
gf, 0x , :
),()()()(
lim)()(
lim))(())((
lim 000
0
00
0
00
0
0
xgxfxx
xgxg
xx
xfxf
xx
xgfxgf
xxxxxx
)()()()( 000 xgxfxgf .
2.
, .
gf, , x : )()()()( xgxfxgf .
., kfff ...,,, 21 , ,
)()()()()( 2121 xfxfxfxfff kk .
, xxx exxexxexx 2)3()()()()3( 22 .
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E : 39
3. .
2
gf, 0x ,
gf 0x :
)()()()()()( 00000 xgxfxgxfxgf
0xx :
0
00
0
0 )()()()())(())((xx
xgxfxgxfxx
xgfxgf
0
0000 )()()()()()()()(
xx
xgxfxgxfxgxfxgxf
0
0
0
0
0 )()()()()()(
xx
xgxgxfxg
xx
xfxf
.
gf, , 0x , :
0
0
00
00
0
00
0
0
)()(lim)()(lim
)()(lim
))(())((lim
xx
xgxgxfxg
xx
xfxf
xx
xgfxgf
xxxxxxxx
)()()()( 0000 xgxfxgxf ,
).()()()()()( 00000 xgxfxgxfxgf
4. , .
gf,
, x : )()()()()()( xgxfxgxfxgf .
,x
exexexexe xxxxx1
ln)(lnln)()ln( , 0x .
. , :
)()()()()()()()()())()()(( )()(])[( xhxgxfxhxgxfxhxgxfxhxgxf
)()()()()]()()()([ xhxgxfxhxgxfxgxf
)()()()()()()()()( xhxgxfxhxgxfxhxgxf .
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E : 40
, )(lnln)(ln)()ln( xxxxxxxxxxxx
xxxxxxxx
x
1lnln
2
1 , 0x .
5. f c R, )())(( xfcxcf
f c0)( c , :
)())(( xfcxcf
, 2233 1836)(6)6( xxxx .
6. .
gf, 0x 0)( 0 xg ,
g
f 0x :
2
0
00000
)]([
)()()()()(
xg
xgxfxgxfx
g
f
7. .
gf,
x 0)( xg , x :
2)]([
)(()()()(
xg
xgx)fxgxfx
g
f
.
,
2
2
2
222
)15(
5)15(2
)15(
)15()15()(
15
x
xxx
x
xxxx
x
x
2
2
2
22
)15(
25
)15(
5210
x
xx
x
xxx ,5
1x .
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E : 41
8. xxf )( ,N f
R* 1)( xxf ,
1
)(
xx
.
, R* :
1
2
1
2)(
)(1)1(1)(
xx
x
x
xx
xx .
, 55144 444)(
xxxx , 0x .
, , 1)( xx , 1 . ,
}1,0{ , 1)( xx .
9. xxf )(
}0|{Af xx x
xf2
1)( ,
xx
2
1)(
, }0|{Af xx :
x
xxxx
x
xxxx
x
xx
22
)()(
)(
xx
xx
22
22
1
.
10. xxf )(
}0|{Af xx x
xf2
1)( ,
xx
2
1)(
7/23/2019 diaforik1 logismos
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E : 42
, }0|{Af xx :
x
xxxx
x
xxxx
x
xx
22
-
)()(
)(
xx
xx22
22
1
- .
11.
1
ln)(
x
xxxf .
:22
)1(
ln)1)(1(ln
)1(
)1(ln)1()ln(
1
ln)(
x
xxxx
x
xxxxxx
x
xxxf
22 )1(
ln1
)1(
ln1lnln
x
xx
x
xxxxxx
12.
1
1)(
x
xf 1)( 2 xxxg
1)(0,A .
)0()0( gf . :
22 )1(
1
)1(
)1(1)1()1(
1
1)(
xx
xx
xxf
12)1()( 2 xxxxg ,
1)0( f 1)0( g .
)0(1)0( gf .
)1,0(A :1)0(11 xyxy .
13. . ;
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E : 43
g 0x f
)( 0xg , gf 0x )())(()()( 000 xgxgfxgf
, g f )(g , gf
)())(()))((( xgxgfxgf .
, )(xgu ,
uufuf )())(( .
Leibniz, )(ufy )(xgu ,
dx
du
du
dy
dx
dy
.
dx
dy .
, .
14. xxf )( , R-
),0( 1)( xxf ,
1)( xx (1)
, x exy ln xu ln , uey .,
1ln 1)( xuu x
x
x
xeueey .
15. xxf )( , 0
R xf x ln)( ,
xx ln)(
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E : 44
, xx ey ln xu ln , uey .,
eueey xxuu lnln)( ln .
16. ||ln)( xxf , R* R*
xx
1)||(ln
.
0x ,
xxx
1)(ln)||(ln ,
0x , )ln(||ln xx , , )ln( xy xu ,
uy ln . ,xx
uu
uy1
)1(1
1
)(ln
x
x1
)||(ln .
17.
, )(xfu .
, )(xfu , :
uuu 1)( uu
u 2
1)(
u
u
u 2
1)( u
u
u 2
1)(
uuu )( uee uu )(
uuu )( u uu ln)(
uu
u 1
)||(ln
7/23/2019 diaforik1 logismos
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E : 45
18.
i) 92 5)()( 3xxf ii) 12
)( xexg iii) 1ln)( 2 xxh .
i) 53 2 xu , )(xfy
9uy ,
uuuy 89 9)(
)53()53(9 282 xx
xx 6)53(982
82 )53(54 xx .
,
ii) )1()()( 21212 xeexg xx ( )12 xu
)2(12
xe x
12
2 xxe
iii) )( 11
1))1(ln()(
2
2
2
xx
xxh ( 12 xu )
)1(12
1
1
1 222
xxx
12
)1(2
122
x
xx
x.
19. 222: yxC
),( 111 yxM .
y,
22xy , 0y 22 xy ,
0y .
, C
1(x1,y1)
A(,0)A(-,0)
C
y
x
O
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E : 46
22
1 )( xxf 222 )( xxf
],[ ),( ., , )(xfy
),( 111 yxM ,
)( 1xf (1) 222 )( xfx (2)
, (2),
0)()(22 xfxfx
, 1xx ,
0)()( 111 xfxfx .
, (1) 011 yx
, 01y ,
1
1
y
x
.
, :
)( 1
1
11 xx
y
xyy ,
:2
11
2
11 xxxyyy
2
1
2
111 yxyyxx
2
11 yyxx , (3)
22121 yx . 01y , ),( 111 yxM )0,(A )0,( ,
fC
x x
. (3) )0,(),( 11 yx )0,(),( 11 yx .
.
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E : 47
1
1. i) f x = 7x 4x + 6x 1 ii) f x = 2 3x + lnx 3
iii) f x =4
4x
3
3x +
2
2x x iv) f x = x 3 x + ln3
i) R f (x) = 7 6x 4 3x + 6
ii) 0, x f (x) = 6 2x + 1x
iii) R f (x) = 3x 2x + x 1iv) R f (x) = x 3 x
2. i) f x = ( 2x 1)(x 3) ii) f x = xe x
iii) f x =2
2
11
xx
iv) f x =1x x
x
v) f x = 2x x x
i) R f (x) = ( 2x 1) (x 3) + ( 2x 1)( x 3)
= 2x(x 3) + ( 2x 1). 1= 2 2x 6x + 2x 1 = 3 2x 6x 1
ii) R f (x) = ( xe )x + xe (x) = xe x + xe x
iii) R f (x) =
2 2 2 2
22
(1 ) (1 ) (1 )(1 )
1
x x x x
x
=
2 2
22
2 (1 ) (1 )2
1
x x x x
x
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E : 48
=
2 2
22
2 (1 1 )
1
x x x
x
=
22
4
1
x
x
iv) R 1 + x 0
f (x) = 2( ) (1 ) ( )(1 )
(1 )
x x x x x x
x
=2
( )(1 ) ( )( )
(1 )
x x x x x x
x
=2 2
2(1 )
x x x x x x x x
x
=2
1
(1 )
x x
x
v) R
f (x) = ( 2x )x x + 2x (x )x + 2x x (x)
= 2x x x + 2x x x + 2x x (x)
= 2x x x + 2x 2x 2x 2x
3.
i) f x =ln
xex
ii) f x = x + x
iii) f x = xx
e
iv) f x = 1
1xx
11
xx
i) x(0 , 1)(1, + ) f (x) =
2
ln ln
ln
x xe x e x
x
=
2
1ln
ln
x xe x ex
x
=
2
1ln
ln
xe xx
x
ii)
R x 0 x 0 f (x) = (x) + (x)=
2
1x
21
x
iii) R f (x) =
2
( ) ( )x x
x
x e x e
e
= 2x x
x
x e x e
e
= xx x
e
iv) x( , 1) (1, 1) (1, ) f x =
2 2( 1) ( 1)
( 1)( 1)
x x
x x
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E : 49
=2 2
2
2 1 2 11
x x x xx
=2
41
xx
f (x) = 4 2 1x
x
= 4
2
22
1.( 1) .2
1
x x x
x
= 4
2
22
1
1
x
x
= 4
2
22
1
1
x
x
4. = g(x) = + ,
, . = ;
=R , = [0 , 1) (1 , + )
x (x) = 2 = 2 =
x g(x) =
= = = 2
, x (x) = .
= , .
f x2( 1)
1xx
1
1
x
x
1
1
x
x
f g f g
fD 1 gD
f
D f2
1.( 1) ( 1).1
( 1)
x x
x
2
2( 1)x
2
4( 1)x
g
D2 2( 1) ( 1)
( 1)( 1)
x x
x x
2
2 1 2 1
( ) 1
x x x x
x
2 21
xx
11
xx
g
D g 24
( 1)x
f g f g
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E : 50
2
5. , ,
f x =22 3 ,
12 6 , 0
x x x 0
x x x
x < 0 f (x) = 4x + 3
x > 0 f (x) = 12 12 x
+ 6 = 6x
+ 6
0f = 12 0 + 6. 0 = 0
0lim
x
00
f x f
x
=0
limx
22 3 0x xx
=0
limx
(2x + 3) = 3
0lim
x
00
f x f
x
=0
limx
12 6x xx
=0
limx
12 6x = +
f 0
x = 0
6. , ,
f x =2 , 0
, 0
x x x
x x
x < 0 f (x) = 2x + x
x > 0 f (x) = 1
0f = 20 + 0 = 0
0lim
x
00
f x f
x
=0
limx
2x xx =
0lim
x xx x
= 0 + 1 = 1
0limx
0
0
f x f
x
= 0limx x
x = 0limx 1 = 1
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E : 51
f (0) = 1
7.
i) (x) = ii) (x) = (0 , 0),
.
i) = R
(x) = = =
x < 0 (x) = (x (x)
= (x (1)
= (x =
x > 0 , (x) = = =
ii) = R
(x) = = =
x < 0 (x) = (x (x)
= (x (1)
= (x =
x > 0 , (x) = = =
=
=
= = = 0 (1)
= = = = 0 (2)
(1) , (2) (0) = 0
f3 2x f
3 4x
fC
fD
f23 x
23x
23
23
( ) , 0
, 0
x x
x x
f 23
2 1
3)
23
13)
23
13)
3
2
3 x
f 23
2 13x 2
3
13x
3
2
3 x
fD
f43 x
43x
4
3
43
( ) , 0
, 0
x x
x x
f 43
4 13)
43
13)
43
13) 4
33 x
f 43
4 13x 4
3
13x 4
33 x
0lim
x
( ) (0)0
f x fx
0
limx
43( ) 0x
x
0lim
x
113( )x
x
0lim
x
13( ) ( )x x
x
0lim
x
3 x
0lim
x
( ) (0)0
f x fx
0
limx
43 0xx
0lim
x
13x
0lim
x
3 x
f
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E : 52
y (0) = (0)(x 0)y 0 = 0 x y = 0
3
8.
f x = x +4x
,
x.
fD = R*
f (x) = 1 24
x
=2
2
4x
x
f (x) = 0 2
2
4xx
= 0 2x 4 = 0 2x = 4 x = 2 x = 2
2f = 2 + 42
= 4 2f = 2 42
= 4
, , (2, 4) (2, 4)
9. f x
= xxe
, x.
fD = R
f (x) = 21. x x
x
e xee = 1 x
xe
f (x) = 0 1 x = 0 x = 1
1f = 1e
f f
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E : 53
o o o 11, e
10.
f x = 2 1xx ,
x.
fD = R*
f (x) =2
2
2 . ( 1).1x x x
x
=2
2
1xx
f (x) = 0 2x 1 = 0 2x = 1 x = 1 x = 1
1f =2
1 11 = 2 1f =2
( 1) 11 = 2
, , (1, 2) (1, 2)
11.
f x = 2x g(x) = 12x
+ 12
(1, 1), .
fD = R gD = R*
f (x) = 2x g (x) = 12 2
1x
= 21
2x
f (1) g (1) = 2. 12 = 1. fC , gC (1, 1), .
12. f x = xx
, R*.
, fC (0, 1)
12.
fD =R
f x = 1xx
f (x) =2
1.( ) ( 1).1
( )
x x
x
=2
1)
( )
x x
x
=2
1)
( )x
f (0) = 21)
(0 )
= 1
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E : 54
f (0) = 12
1
= 12
2 2 = = 2
13.
f x =3
x 3x + 5 :i) y = 9x + 1ii) y = x
fD = R
f (x) = 3 2x 3i) y = 9x + 1 f (x) = 9
3 2x 3 = 93 2x = 12
2x = 4 x = 2 x = 2 2f = 3( 2) 3. (2) + 5 = 8 + 6 + 5 = 3
f(2)= 32 3. 2 + 5 = 8 6 + 5 = 7
(2, 3) , (2, 7)
ii) y = x f (x) (1) = 13 2x 3 = 13 2x = 4
2x = 43 x = 23 x = 23
23f = 3
2
3
3. 23
+ 5
= 83 3
+ 63
+ 5
= 8 39
+ 6 33
+ 5
= 8 3 18 3 459
= 10 3 459
23f = 3
23 3. 23 + 5
= 83 3
63
+ 5
= 8 39
6 33
+ 5
= 8 3 18 3 459
= 10 3 459
( 23
, 10 3 459
) , ( 23
, 10 3 459
)
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E : 55
14. f x = 2x , (0, 1).
fD = R
f
C (0
x , 0f x )
y 0f x = 0f x (x 0x ) y 2
0x = 2
0x ( x
0x )
y = 20
x x 20
x (1)
1 = 2
0x . 0 2
0x
2
0x = 1
0x = 1
0x = 1
0
x = 1 , (1) y = 2x 1
0
x = 1 , (1) y = 2x 1
15. f x = 2x + x + , , , R.
, , f
C
(1, 2) y = x .
fC (1, 2) 1f = 2 21 + . 1 + = 2 + + = 2 (1)
f
C 0f = 0
= 0 (2)
f
C y - 0f = 0f (x 0)
f x = 2x + 0f = 2. 0 + = ,
y 0 = x y = x , y = x , = 1 (3)
(2) , (3) (1) + 1 + 0 = 2 = 1
16.
= g(x) = x + 1 ,
.
f x 1x
2x
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E : 56
f=R* Ag=R , R*
, : g(x) = x + 1 =
+ x 1 = 0
( x 1) + (x 1) = 0(x 1)( + 1) = 0
x 1 = 0 x = 1
g(1) = = = 1
(1, 1) ,
(x) = (1) = = 1
g(x) = 2x 1 g(1) = 2.1 1 = 1
(1) . g(1) = -1 , , (1, 1)
17. y = 3x 2 , =
.
y = 3x 2 g(x) = 3x 2 , R f=R* , :
g(x) = 3x 2 =
3x + 2 = 0x 2x + 2 = 0
x( 1) 2(x 1) = 0x (x 1) (x + 1) 2(x 1) = 0
(x 1) [x (x + 1) 2] = 0(x 1)( + x 2) = 0 x = 1 x = 2 x = 1
g(1) = = = 1
g(2) = = = 8
, (1,1) (2, 8)
(x) = 3 (1) = 3. = 3
(1,1) : y 1 = 3 (x 1)
y 1 = 3x 3y = 3x 2
fC
gC f x 2x 1
x3x 2x
2
x2x
1f 11
fC
gC
f2
1x
f2
11
ff
Cg
C
f x 3x
fC
gC
f x 3x3x
3x2x
2x 1f 31
2f 3( 2)
fC
gC
f 2x f 21
fC
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18. = + x + 2 g(x) = .
, R,
= 1.
f=R* Ag=R , R*
(x) = 2x + , g(x) =
= 1 = g(1) (1) = g(1)
+ + 2 = 1 2 + = 1 + = 1 = 1 2 1 2 = 1 = 1 2
= 0 = 1
19. = g(x) = x .
(0, 1) ,
.
f=R Ag=R , R(x) = , g(x) = 2x 1 (0) = = 1
(0, 1) : y = (0) (x 0)
y = 1(x 0)
y 1 = x y = x + 1 , (0, 1) ,
. ( , g( ))
: y g( ) = g( )(x ) y ( ) = (2 1) (x ) (1)
() (0, 1) 1 ( ) = (2 1) (0 )
1 + + = 2 +
= 1 = 1 = 1
= 1 () y ( 1) = (2.11) (x 1)
y + 2 = 3(x 1)y + 2 = 3x + 3y = 3x + 1
= 1 () y (1 + 1) = (2 1) (x + 1)
f x 2x 1x
0x
f2
1x
0x 1f f
f x xe 2x
fC
gC
f xe f 0e
fC 0f f
0e
gC
gC
0x
0x
0x 0x 0x 2
0x 0x 0x 0x
20
x0
x0
x0
x
2
0x
0x 2
0x
0x
2
0x
0x
0x
0x 21
0x
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y = x + 1 , .
20. , y = x + 1 y = 3x 1 (0, 1) (1, 2) .
= + x + 0 .
1 = (1)
2 = + + + = 2 (2)
(x) = 2x + , (0) = (3) (1) = 2 + (4)
y = x + 1 (0) = 1 = 1 (5)
y = 3x 1 (1) = 3
2 + = 3
2 = 2 = 1 (6) (6), (5) , (1) (2) 1 + 1 = 2 1 0 = 1
21. (x) = 2x 2 , x [0, 2] ,
x. x (x) = 0 2x (2x)4x
(x) = 02x . 2 4x x = 0
22x 22x = 02x = 2x2x = 1 (1)
0 x 2 0 2x 4(1) 2x = , + , 2 + , 3 +
2x = , , ,
x = , , ,
22. R (1) = 1 g g(x) =
f
f x 2x
f
C
f
C
f f f
fC f
(3)
fC f
(4)
(5)
f 2x
f
4
4
4
4
4 5
4 9
4 13
4
8 5
8 9
8 13
8
f
f
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E : 59
( + x + 1) 1 , xR.
(1, (1)) (0, g(0)) .
(1, (1)) : y (1) = (1)(x 1)
y (1) = 1.(x 1)y = x 1 + (1) (1)
g(0) = ( + 0 + 1) 1 = (1) 1g(x) = ( + x + 1) ( + x + 1) = ( + x + 1) (2x + 1)
g(0) = ( + 0 + 1) (2 .0 + 1) = (1) . 1 = 1 (0, g(0)) :
y g(0) = g(0)(x 0)y [ (1) 1] = 1. x
y (1) + 1 = xy = x + (1) 1 (2)
(1) , (2) , , , (0, g(0)) .
23. (1, 1),
(x) = x , x (/2 , /2)i) (0)
ii) (0, (0))
.
i) x = 0 (0) = 0 (0) = 1 . 1 = 1 (x)(x) = x - x
(x)x = (x x) (0) = (0 0) =
(0) = 1 . (1 0)
(0) = 1ii) H (0, (0))
: y (0) = (0)(x 0)y 1 = 1. x
y = x + 1 (1)
y = 0 , (1) x = 1.
() (1, 0)
f 2xf
C
fg
C
fC f
f f
f
f
f 20 f
f 2x 2x f 2x
f 20 f
gC
f
f
f
gC
f
f xe f
fC f
f 0e f
f xe xe
f xe
f 0e
f
f
fC f
f f
x x
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E : 60
x = 0 , (1) y = 1. () (0, 1)
() = () = 1 .
4
24. :
i) f x = (3 4 34x x 2) ii) f x = (x 123)
iii) f x = 211 x iv) f x = ln 1 xx
v) f x =2xe
i) 3 4 34x x 0 3x (3x +4) 0 x 0 x 43
f (x) = 2(3 4 34x x 2 1) (3 4 34x x ) = 2(3 4 34x x 3) (12 3x +12 2x )
= 2
34 3
1
3 4x x12 2x (x + 1)
= 24
39
1
3 4x x 2x (x + 1)
= 24
37
1
3 4
x
x x
ii) x 1 > 0 x > 1
f (x) =2
3 (x 1
2 13
)
(x 1) =2
3 (x 1
13
)
iii)
fD = R
f (x) = 211 x 21
1 x
= 211 x2
2 2
(1 )
(1 )
x
x
= 211 x 2 22
(1 )xx
iv) x 0 1x
x > 021 x
x > 0
x (1 x)(1 + x) > 0 x < 1 0 < x < 1
y y
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E : 61
x 1 0 1 + +
f (x) = 11 xx
1 xx
= 21
1 xx 2
1 1x
=21
xx
2
2
1 xx
=2
2
1( 1)x
x x
v)f
D = R
f (x) =2xe ( 2x ) =
2xe (2x) = 2x 2xe
25. f 0
x :
i) f x = 2x 31 x , 0x = 2 ii) f x = (2x13) + (2x
23) ,
0x = 4
iii) f x = 3x 3 (x) ,0
x = 16 iv) f x =
2 22
xx
,0
x = 3
i) xR
f (x) = ( 2x ) 31 x + 2x ( 31 x ) = 2x 31 x + 2x3
1
2 1 x3 2x
f (2) = 4. 3 + 4 1
2.3
12 = 12 + 8 = 20
ii) x > 0
f (x) = 13
(2x1 13)
(2x) + 23
(2x2 13)
(2x)
= 13
(2x23)
.2 + 2
3(2x
13)
.2
f (4) = 13
(2. 423)
.2 + 2
3(2. 4
13)
.2 = 1
3
238
.2 + 2
3
138
.2
= 13
22 . 2 + 23
12 . 2
= 13 . 12 + 23 = 16 + 46 = 56 iii) xR :f (x) = ( 3x ) 3 (x) + 3x ( 3 (x)) =
= 3 2x 3 (x) + 3x 3 2 (x) ((x)) =
= 3 2x 3 (x) + 3x 3 2 (x) (x) (x) =
= 3 2x 3 (x) + 3x 3 2 (x) (x).
f (16
) = 3 2
16
3 (16
) + 3
16
3 2 ( 16
) ( 16
).
= 3. 136 . 3
12 + 1216 . 3 .
2
12 . 32 .
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E : 62
= 112
. 18
+ 172
. 14
. 32
. = 196
+ 3576
iv) x 2 f (x) =2
22 (2 ) ( 2)( 1)(2 )x x xx
=2 2
2
4 2 2(2 )
x x xx
f (3) =2 2
2
4.3 2.3 3 2(2 3)
=2
12 18 9 2( 1)
= 5
26. :
i) f x = lnxx ii) f x = 5 32 x
iii) f x = (lnx )x , x > 1 iv) f x = x. xe
i) x > 0
f x = ln .lnx xe =2ln xe
f (x) =2ln xe 2ln x = lnxx 2lnx (lnx) = 2 1
x lnxx lnx
ii) xR
f x = (5 3)ln 2xe
f (x) = (5 3)ln 2xe [(5x-3)ln2] = 5 32 x 5ln2
iii) f x = ln(ln )x xe
f (x) = ln(ln )x xe [xln(lnx)] = (lnx )x [1.ln(lnx) + x [ln(lnx)]]
= (lnx )x [ln(lnx) + x 1lnx
(lnx)]
= (lnx )x [ln(lnx) + x 1lnx
. 1x
]
= (lnx )x [ln(lnx) + 1lnx
]
iv) xR
f (x) = (x) xe + x.( xe )= x . xe + x.. xe ( x)
= x . xe + x.. xe ( x)
= xe (x 2x )
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5
27. f x = 2x , f (x) + 4 f x = 2
xR f (x) = 2 x (x) = 2x x = 2xf (x) = 2x (2x) = 22x
f (x) + 4 f x = 22x + 4 2x
= 2(1 2 2x ) + 4 2x = 2 4 2x + 4 2x = 2
28. , f(0) = 4,f(1) = 2 ,f(2) = 4 f(3)(1) = 6.
f(x)= x3+ x2+ x + , 0 . f(x) = 3x2+ 2x + , f(x) = 6x + 2 , f(3)(x) = 6
f(0) = 4 = 4 = 4 = 4f(-1)= 2 3 2 + = 2 3 2 + = 2 3.1 2 + = 2f(2) = 4 12 + 2 = 4 6 + = 2 6.1 + = 2
f(3)(1) = 6. 6 = 6 = 1 = 1
= 4 = 4 = 42 + = 1 2(4) + = 1 = 9 = 4 = 4 = 4 = 1 = 1 = 1
f(x) = x3 4x29x + 4
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E : 64
7
29. f:RR
0x = ,
i) limx
( ) ( )xf x fx
= ( ) ( )f f
ii) limx
( ) ( )x ae f x e f x
= e ( ( ) ( )f f )
i) x ( ) ( )xf x f
x
= ( ) ( ) ( ) ( )xf x f xf xfx
= [ ( ) ( )] ( )( )x f x f f xx
= x ( ) ( )f x fx
+ ( )f
limx
( ) ( )xf x fx
= limx
[ x ( ) ( )f x fx
+ ( )f ]
= limx
x . limx
( ) ( )f x fx
+ limx
( )f
= ( )f + ( )f
ii) x ( ) ( )x ae f x e f
x
= ( ) ( ) ( ) ( )x a x xe f x e f e f e f
x
= [ ( ) ( )] ( )( )x xe f x f f e e
x
= xe ( ) ( )f x fx
+ ( )f xe ex
(1)
h(x) = xe ,
h() =limx
xe ex
(2)
h(x) = xe , h() = e (2) e = limx
xe ex
(1) limx
( ) ( )x ae f x e f x
= limx
[ xe ( ) ( )f x fx
+ ( )f xe ex
]
= limx
[ xe ( ) ( )f x fx
] + limx
[ ( )f xe ex
]
= limx
xe .limx
( ) ( )f x fx
+ ( )f limx
xe ex
= e
( )f
+ ( )f e
= e ( ( ) ( )f f )
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1. S t t0
t0
S t t0.
2. y x x0.
x , y y = f(x) , f
x0, y x x0 f (x0) .
3. ( ) , ; , ;
, t t0 (t0) ,
t t0. (t0) t0 (t0). (t0) = (t0) = S(t0) , , x ., (x0) x, x = x0 x0. , x0 x0.
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4. . t0 r 50 cm, r 20cm/sec. , t0.
= r2 r t,
(t) = r2(t).
(t) = 2r(t) r(t).
, (cm2/sec).
5.A x (x) E(x), P(x) = E(x) K(x)
.i)
.
ii) .
i) P(x) = E(x) K(x) .
,
P(x) = 0 E(x) K(x) = 0 E(x) = K(x).
ii)
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E : 67
1
1. . , cm r = 4 2t , t
sec . V , t = 1 sec .( = 4 2r
V = 43 3r )
t.r = r(t) . = (t) .V = V(t) r(t) = 4 2t (1) , V(1)(t) = 4 .[r(t) 2] = 4 .(4 2t 2) (t) = 4 .2(4 2t ).(4 2t )
= 8 (4 2t ).(2t)= 16 (4 2t ) t
(1) = 16 (4 21 ) .1 = 48 c 2m /sec
V(t) = 43
[r(t) 3] = 43
.(4 2t 3) V(t) = 43
3.(4 2t 2) . (4 2t )
= 4 .(4 2t 2) .(2t)
= 8. (4 2t 2) t
V(1) = 8 (4 21 2) .1 = 8. 23 = 72 c 3m /sec
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2. V 100 c 3m /sec . r
0t , 9 cm;
t.r = r(t) .V = V(t) V(t) = 100 c 3m /sec
r(0
t )
V(t) = 43
[r(t) 3] V(t) = 43
3 [r(t) 2] . r(t)
100 = 4 [r(t) 2] . r(t)
t = 0t 100 = 4 [r( 0t ) 2] . r( 0t )100 = 4 29 r(
0t )
25 = 81 r(0
t )
r(0
t ) = 2581
3. 10 c /sec , , r = 85
cm .
t.r = r(t) = (t) .V = V(t) (t) = 10 r( ) = 85
V( )
(t) = 4 .[r(t) (t) = 4 .2 r(t). r(t)10 = 8 r(t). r(t)10 = 8 r( ). r( )
10 = 8 .85. r( ) r( ) =
V(t) = [r(t) V(t) = 3[r(t) r(t)
V( ) = 4 [r( ) r( )
V( ) = 4 .
V( ) = 4 . V( ) = 425 c /sec
2m
0t
0t
2
]
0t
0t
0t
0t 10
8 . 8543
3] 4
32
]
0t
0t
2]
0t
0t
285 10
8 . 85
0t
285 10
8 . 85
0t
3m
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E : 69
2
4. (x) (x) , x
(x) = 13
3x 20 2x + 600x + 1000 (x) = 420x .
(x) = (x) (x) .
(x) = (x) (x) (x) = 420x ( 1
3
3x 20 2x + 600x + 1000)
(x) = 420x 13
3x + 20 2x 600x 1000)
(x) = 13
3x + 20 2x 180x 1000)
(x) = 2x + 40x 180 = 240 4 . 180 = 1600 720 = 880
x = 40 8802
= 40 2 2202
= 20 220
(x) > 0 20 220 < x < 20 + 220
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3
5. 1
2
.
1 15 km /h
2
20 km /h.
i)
1
2
ii) d = (
1
2 ) ,
.
t.x = x(t)
1
y = y(t) 2
d = d(t) (1
2
)
x(t) = 15 y(t) = 20 i) x(t) y(t)
ii) d(t)i) x(t) = 15t y(t) = 20t
ii) [d(t) 2] = [x(t) 2] + [y(t) 2]
= (15t 2) + (20t 2)
= 225 2t + 400 2t
= 625 2t d(t) = 25t d(t) = 25 km /h.
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6.
y = 14
2x , x 0 .
x
y , x(t) >0 t 0 .
t.x = x(t) y = y(t) x(t) > 0
0 ( x(
0t ) , y(
0t )) , x(
0t ) = y(
0t )
y = 142
x y(t) = 14 [x(t) 2] y(t) = 14 2 x(t) x(t)
y(0
t ) = 12
x(0
t ) x(0
t ) (1)
x(0
t ) = y(0
t )(1)
x(0
t ) = 12
x(0
t ) x(0
t )
1 = 12
x(0
t )
x(0
t ) = 2
y(0
t ) = 14
[x(0
t ) 2] y(0
t ) = 14
22 = 1
0 (2 , 1)
7. (0 , 0), (x , 0), (0 , lnx) x > 1. x 4 cm/sec , , x = 5 cm.
t.
x = x(t) x = (t) x(t) = 4 x(
0t ) = 5
T (0
t )
(t) = 12
() () = 12
x. lnx = 12
x(t).ln(x(t))
(t) = 12
[ x(t).ln(x(t)) + x(t) 1x(t)
x(t) ]
(0
t ) = 12
[ x(0
t ).ln(x(0
t )) + x(0
t ) ]
(0
t ) = 12
[4. ln5 + 4] = 12
4 (ln5 + 1) = 2(ln5 + 1) c 2m /sec
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8.
y = , x 0
() .
(t) = (t), , 3 .
t. = (t) = (t) (t) = (t) ( ) = 3
( )
(x) = , x 0
, < 0
y () = () (x ) (x) =
() =
: y + = (x )
y + = x +
y = x +
y = 0 0 = x + 0 = 3 x + 2
3 x = 2
x = ,
= (t) = (t)
(t) = (t)
( ) = ( ) = [- ( )] = (3) = 2
31 x3
0t
0t
f31 x
3
31, 3
fC f f f 2x
f 213
3 2
13
3 2 3
2 23
3
2 23
3 2 3
2 3
23
2 , 03
23
23
23
0t 2
3 0t 2
3 0t 2
3
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9. + = 1.
, y
3 . x .
t.x = x(t) xy = y(t) y
x( ) = , y( ) = , y( ) = -3
x( )+ = 1 [x(t) + [y(t) =
( [x(t) + [y(t) )= 0
2 x(t) x(t) + 2 y(t) y(t) = 0x(t) x(t) + y(t) y(t) = 0x( ) x( ) + y( ) y( ) = 0
x( ) + .(-3) = 0
x( ) = 3 /sec
2x
2y
1 3,2 2
0t
0t 1
2 0t 3
2 0t
0t2
x 2y 2] 2] 21
2]
2]
0t
0t
0t
0t
12 0
t 32
0t 3
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4
10. 3 m/sec . , y.
t.x = x(t) () ( )y = y(t) () x(t) = 3 y(t) ,
y5
= x20
4y = x 4 y(t) = x(t)
4 y(t) = x(t)4 y(t) = 3
y(t) = 34
m/sec
11. 100 m,
50 m/min. 100m;
t.h = h(t) .
= (t) . h(t) = 50 h(0
t ) = 100
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(0
t )
= h( )
(t) = 1100
h(t)
((t))= 1100
h(t)
2
1
(t) (t) = 1
100. 50
(t) = 12
2 (t)
(0
t ) = 12
2 (0
t ) (1)
0
t , () = () = 100
(0
t ) = 45
(1) (0
t ) = 1
2
2
2
2
= 1
2
. 2
4
= 1
4
rad/min
12. 1,60 m 8 m 0,8 m/sec . ;
t.x = x(t) ()s = s(t) x(t) = 0,8 s(t) ,
ss x
= 1,68
ss x
= 0,2
s = 0,2 (s + x)
s = 0,2 s + 0,2x
s(t) = 0,2 s(t) + 0,2 x(t)
0,8 s(t) = 0,2 x(t)
0,8 s (t) = 0,2 x(t)
0,8 s (t) = 0,2 . 0,8
s(t) = 0,2 m/sec
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13. 3 m . 0,1 m/sec .
0t ,
2,5 m, :
i)
ii)
.
t.x = x(t)
y = y(t) = (t) x(t) = 0,1 y(
0t ) = 2,5
i) (0
t )
ii) y(0
t )
x = 3 x(t) = 3(t)x(t) = (3(t)) x(t) = 3((t)). (t)
0,1 = 3(t). (t) 0,1 = 3(0
t ). (0
t ) (1)
, 0t ( 0t ) =0
y(t )
3 =2,53
(1) 0,1 = 3 2,53
(0
t ) 0,1 = 2,5 (0
t ) (0
t ) = 125
[x(t) 2] + [y(t) 2] = 23 ( [x(t) 2] + [y(t) 2] )= 0
2 x(t) x(t) + 2 y(t) y(t) = 0 x(t) x(t) + y(t) y(t) = 0
x(0
t ) x(0
t ) + y(0
t ) y(0
t ) = 0 (2)
[x(0
t ) 2] + [2,5 2] = 9 [x(0
t ) 2] + 6,25 = 9
[x( 0t )
2
] = 2,75 x( 0t ) = 2,75 (2) 2,75 . 0,1. + 2,5 y(0
t ) = 0
2,5 y(0
t ) = 0,1 2,75 y(0
t ) = 2,7525
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1. Rolle , .
(Rolle)
f : [, ] (, ) f() = f()
, , (, ) , : f () = 0
, , , (,) , Cf M(, f()) x.
, f(x) = x2 4x + 5, x [1,3]. (. 19)
f [1,3], (1,3), f (x) = 2x 4 f(1) = 2 = f(3), Rolle, (1, 3) , f () = 0.
, :f () = 0 2 4 = 0 = 2 .
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2. Rolle , .
( ...) f :
[, ] (, )
, , (, ) , :
, , , (,) ,
f M(, f()) .
,
f [0,4] (0,4),
, , (0,4) ,
,:
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3. N :i) f(x) = x3+ x2 (+1)x, R*, Rolle [0,1].
ii) f(x) = 3x2+ 2x (+1) = 0, R* ,, (0,1).
i) f Rolle [0,1] [0,1] (0,1) f (x) = 3x2+ 2x (+1) f(0) = f(1) = 0.
ii) , , f(x) = x3
+ x
2
(+1)x,
R
*
, Rolle, (0,1) , f () =0 , , 32+ 2 (+1) = 0. , (0,1) 3x2+ 2x (+1) = 0.
4. f(x) = x2+ x + , 0 [x1,x2], x0 (x1, x2), ,
[x1, x2], .
f(x) = x2+ x + [x1,x2] (x1,x2), f (x) = 2x +. , x0(x1, x2), ,
:
, (1) :
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5. 200 2,5 . , , 80 .
x = S(t), t [0, 2,5] . t0 [0, 2,5] , (t0) = S(t0) = 80. S [0, 2,5] (0, 2,5)., t0 (0, 2,5),
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1
ROLLE
1. f(x) =2
x 2x + 1 Rolle [0 , 2] , ( , ) f () = 0.
f [0 , 2]
(0 , 2) Rolle ( , )
f(0) = 1 = f(2) , f () = 0.
f (x) = 2x 2f () = 0 2 2 = 0 = 1
2. f(x) = 3x
Rolle 20,3
, ,
( , ) f () = 0.
f 20,3
20, 3 Rolle 20, 3
f(0) = 0 = f 23 , f () = 0.
f (x) = 33xf () = 0 33 = 0 3 = 0
0 < < 23 0 < 3
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3. f(x) = 1 + 2x Rolle 0, , ,
( , ) f () = 0.
f [0 , ]
(0 , ) Rolle ( , )
f(0) = 2 = f() , f () = 0. f (x) = 22x
f () = 0 22 = 0 2 = 0
0 < < 0 < 2 < 2 , 2 = = 2
4. f(x) = x
Rolle [1 , 1] ( , ) f () = 0.
f(x) =x , x 0
x , x 0
x 0
lim
( ) (0)0
f x fx
=x 0
lim
0 xx
=x 0
lim(1) = 1
x 0
lim
( ) (0)0
f x fx
=x 0
lim
0xx
=x 0
lim1 = 1
f 0 ,
[1 , 1] ,
Rolle [1 , 1]
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2
5. f(x) = 2x + 2x
[0 , 4] (0 , 4)
f () = f f
f [0 , 4].. (0 , 4)
(0 , 4) , f () = 4 04 0
f f
=24 04 =6
f (x) = 2x + 2
f () = 6 2 + 2 = 6 2 = 4 = 2
6. f(x) = 32x
0,2
, (0 ,2 )
f () = 02
02
f f
f 0,2
0, 2
, ..
0, 2 ,
f () = 02
02
f f
= 3 3 0
2
= 0
f (x) = 62x f () = 0 62 = 0 2 = 0
0 < 0 xR (1)
h(0) = 02 + 20 2. 0 1 = 1 1 = 0 h(1) = 12 + 21 2. 1 1 = 2 + 1 2 1 = 0
, , f
C ,g
C
(, ) , , > 0 1. f() = g() = , h() = 0 h , . Rolle [0, 1] ,
1 (0 , 1) h(
1 ) = 0
, [1, ] , 2
(1, ) h(2
) = 0
h , . Rolle [ 1 , 2 ] , (
1 ,
2 ) h() = 0 (1)
f
C ,g
C .
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1.
f , f f (x) = 0 x , f .
x1,x2 f(x1) = f(x2) . x1= x2 , f(x1) = f(x2). x1< x2, [x1,x2] f . , (x1,x2) ,
(1)
, f () = 0, , (1), f(x1) = f(x2). x2
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f g
x
(f g)(x) = f (x) g(x) = 0
, , f g . , C , x f(x) g(x) = c, f(x) = g(x) + c.
( S O S )
.
,
, f (x) = 0 x (,0) (0,+), f (,0) (0,+).
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3. f f(x)=f(x)
i)
ii) f, f(0) = 1.
i) x R :
, R .ii) , c R , (x)= c x
R , , x R.
f(x) = cex x R.
f(0) = 1, 1 = c ,
f(x) = ex x R.
4. f, : f (x) > 0 x , f . f (x) > 0 x , f .
f (x) > 0.
x1 x2 x1 < x2. f(x1) < f(x2). , [x1,x2] f ... , (x1,x2) ,
,
f(x2) f(x1) = f () (x2x1)
f () > 0 x2 x1> 0, f(x2) f(x1) > 0, f(x1) 0 x (0,+).
f(x) = x2 2x [1,+), [1,+) f (x) = 2(x 1) > 0 x (1,+), (,1], (,1] f (x) = 2(x 1) < 0 x (,1).
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(,0), (0,+), x (,0) x (0,+).
7. N f(x) = 2x3 3x2+ 1 , .
f f (x) = 6x2 6x = 6x (x 1). f
E, f : (,0], (,0]
f (x) > 0 (,0). [0,1], [0,1]
f (x) < 0 (0,1).
[1,+), [1,+)
f (x) > 0 (1,+).
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f f (,0] ,[0,1] [1,+) :
8. i) f(x) = x x 2, x [0,] .
ii) x = x 2 [0,].
i)
f (x) = (x x 2)= 1 + x > 0 x [0,].
, f [0,]. f , 1.8, [f(0), f()] = [ 3, 1].ii) :
f [3, 1], 0, x0(0,), f(x0) = 0. f [0,], x0 f(x) = 0 . , 28, y = x 2 y =x.
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1
1. f, g : f (x) = g(x) g(x) = f(x) xR,
(x) = [ f(x) 2
] + [g (x) 2
] . R (x) = ([ f(x) 2] + [g (x) 2] )
= ([ f(x) 2] ) + (g (x) 2] )= 2 f(x) f (x) + 2 g (x) g(x)= 2 f(x) g (x) + 2 g (x) [ f(x)]= 2 f(x) g (x) 2 g (x) f(x) = 0 .
(x) = c
2. R
(x) (y) (x y x, y R,
.
x ,
(x) ( ) (x (x) ( )
= 0,
= 0 ( ) = 0
, x (x) = 0,
.
f
f f 2)
f
0x 0x
f f 0x 0x 2) f f 0x
2
0x x
00
f x f x
x x 0x x 0
0
f x f x
x x 0x x
0x x 0
0
f x f x
x x
0x x
0
limx x
0x x
0
limx x
00
f x f x
x x f 0x
0x f f
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2
3. f(x)= 3x +3x4
R f (x) = 3 2x + 3 > 0. f R
4. f(x) = 2 3x 3 2x 12 x
R f (x) = 6 2x 6x 12 = 6( 2x x 2)
= 1 + 8 = 9 , f :1 3
2
= 1 2 f , f
5. f(x) =2
x
x 1
R f (x) = (2
x
x 1) =
2 2
2 2
x (x 1) x(x 1)
[x 1]
=2
2 2
x 1 x.2x
[x 1]
=2
2 2
1 x
[x 1]
f (x) = 0 1 2x = 0 x = 1 x = 1
f (x) > 0 1 2x > 0 1 < x < 1
x 1 2 + f (x) + 0 0 +
f(x)
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f , f
6. (x) = lnx x
(0 , + )
(0 , + ) (x) = - 1 =
(x) = 0 1 x = 0 x = 1(x) > 0 1 x > 0 x < 1
,
7. (x) =
R
(x) = ( ) = = =
(x) = 0 1 x = 0 x = 1
(x) > 0 1 x > 0 x < 1 ,
3
f
f
f 1x
1 xx
f
f
f f
f xx
e
f
fx
x
e
x x
2x
e xe
e
x
2x
e (1 x)
e
x
1 x
e
f
f f f
x 1 1 + f (x) 0 + 0
f(x)
x 1 +(x) + 0
(x)
x 1 +(x) + 0
(x)
f
f
f
f
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8.
f(x) =24 x , x 1
x 2 , x 1
f ( , 1) (1 , + )
x 1lim
f(x) =
x 1lim
(4 2x ) = 4 1 = 3
x 1lim
f(x) =
x 1lim
(x + 2) = 1 + 2 = 3
f(1) = 4 1 = 3 1 f R.
f (x) =2x , x 1
1 , x 1
( , 1) : f (x) = 0 2x = 0 x = 0
f (x) > 0 2x > 0 x < 0
f (x) < 0 2x < 0 0 < x < 1
(1 , + ) , f (x) = 1 > 0
f , f
x 0 1 + f (x) 0 +
f(x)
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9. f(x) = 2x 1
f
R .
2x
1 :
f f(x) =2
2
x 1 , x 1 x 1
1 x , 1 x 1
f (x) =2x , x 1 x 1
2x , 1 x 1
f , f
10. f(x) = x + x, x[0, 2]
f [0 , 2]
f(x) =x x , 0 x
x x , x 2
f(x) =
2 x , 0 x
0 , x 2
f [0,], [, 2]
. f (x) = 2x
f (x) = 0 2x = 0 x = 0 x =2
f (x) > 0 2x > 0 x > 0 0 x 0 ,
.
g [0, + ) (0, + )
g(x) = 2 12 x
+ 1 > 0 , .
ii)xlim
f(x) =xlim
5x = xlim
f(x) =xlim
5x = +
f , R.
g(0) = 3 xlim
g (x) = + g
, [3, + )iii) 5x + 5x 6 = 0 f(x) = 0
f(1) = 0 , 1 f.
f , .
2 x + x 3 = 0 g(x) = 0
g(1) = 0 , 1 g.
g , .
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12. :i) f(x) = xe 1 + ln(x + 1) .
ii) xe = 1 ln(x + 1) x = 0.
i) fD = (1, + ) , x + 1 > 0
x(1, + ) f (x) = xe + 1x 1
> 0, f
ii) xe = 1 ln(x + 1) xe 1 + ln(x + 1) = 0 f(x) = 0
f(0) = 0e 1 + ln(0 + 1) = 1 1 + ln1 = 0, 0 f.
f , .
13. i) f(x) = 3x 3x + [1, 1]
ii) f [1, 1]
iii) 2 < < 2, 3x 3x + = 0
(1, 1).
i) f [1, 1] .
x(1, 1) f (x) = 3 2x 3 = 3( 2x 1) < 0, f
.
ii) f f([1, 1])=[ f(1), f(1)]
f(1) = 31 3.1 + = 2 f(1) = 3( 1) 3.( 1) + = 1+3 + = + 2
f( [1, 1] ) = [ 2 , + 2]
iii) f(1) f(1) = ( + 2)( 2) = 2 4 < 0 2 < < 2 .
f , Bolzano , f(x) = 0 ,
3x 3x + = 0 , (1, 1).
f , .
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,
( , 1), (1, 1) , (1, + ) , ( , + ),
y = .
(x) = . ,
.
15. :i) (x) = x xx
ii) x xx > 0 , x
iii) (x) =
i) (x) = x (x xx) = x x + xx = xx > 0
,
ii) x 0 < x <
, (0) < (x) 0 00 < x xx
0 < x xx
iii) x (x) = < 0 ( ii).
.
16. :
i) (x) = 2x + x 3x , x
ii) 2x + x 3x , x
f
fC
ff
f
0,2
0, 2
f x
x
0, 2
f
0,
2
f 0,2
0,
2
0, 2
2 f
0,2
f f
0, 2
f2
x. x x
x
f 0, 2
f 0, 2
0, 2
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i) (x) = 2x + - 3 = (1)
2 x 3 x + 1 = 2 x 2 x x + 1= 2 x ( x 1) ( x 1)
= 2 x ( x 1) ( x 1)( x + 1)= ( x 1) (2 x x 1)
( x 1) 2 ( x + )( x 1)
= ( x 1 (2 x + 1) > 0
= 1 + 8 = 9 , x = = = 1
(1) (x) > 0 , ,
ii) x 0 < x <
(0) (x)
20 + 0 3. 0 2x + x 3x0 2x + x 3x
3x 2x + x
f2
1
x
3 2
2
2 x 3 x 1
x
3 2 3 2 22 2
2 2
12
2) 0, 2
1 94
1 34 1
2
f 0, 2 f 0, 2f 0, 2
0, 2
2 f
0, 2
f f
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5
17. t : x = S(t) = 4t 8 3t + 18 2t 16t + 160,0 t 5. :i) ;ii) ;iii) ;
(t) = S(t) = 4 3t 24 2t + 36t 16 = 4( 3t 6 2t + 9t 4)
(t) = (t) = 4(3 2t 12t + 9) = 12( 2t 4t + 3)
i) (t) = 0 3t 6 2t + 9t 4 = 0 ( Horner) (t 1 2) (t 4) = 0 t = 1 t = 4
ii) (t)
(4, 5),
(0, 4)
iii) (t)
(0, 1), (4, 5) (1, 3).
t 0 1 4 5
(t) 0 0 +
t 0 1 3 5
(t) + 0 0 +
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18. V ( ) , t
, V = 50 2
2
25t
(t 2).
, , .
V(t) = 50 2
2
25t
(t 2) = 25 [2
2
2
t
(t 2)] , t[0, + )
V(t) = 25 [2 2
2
t
(t 2)]
= 25 [0
2 2
4
2t(t 2) t 2(t 2)
(t 2)
]
= 254
2t(t 2)(t 2 t)
(t 2)
= 253
4t
(t 2)< 0
V , .
V(0) = 50 2
2
25.0
(0 2) = 50
tlim
V(t) =tlim
[50 2
2
25t
(t 2) ]
= 50 25tlim
2
2
t
(t 2)
= 50 25tlim
2
2
t
t 4t 4 = 50 25
tlim
2
2
t
t = 25 = 1
2V(0)
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6
19. R* f(x) = 3x + 3 2x + x + 1
R.
f (x) = 3 2x + 6x + 1 = 36 12 = 12(3 ) = 3 , = 0
f (x) x = 2
= 62(3 )
= 1
= 13
3 = 3. 3 = 9, x < 13
x > 13
.
f ( , 13 ]
, [ 13
, + ) , R.
< 3 , > 0 f (x)
, f .
> 3 , < 0 f (x) 3 ,
x , f R.
, 3.
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1. .
f
],( .
0xx 0x . f 0x . 1x 2x
. :
f, , Ax 0
, 0 ,
)()( 0xfxf ),( 00 xxAx .
0x , )( 0xf
f.
A )()( 0xfxf Ax , , , f
Ax 0 , )( 0xf .
2.
.
y
O
A(x0,f(x0))
Cf
a x0 x2x1
x
29
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0xx
0x . f 0x . 1x . , :
f, , Ax 0 , 0 ,
)()( 0xfxf , ),( 00 xxAx .
0x , )( 0xf
f.
)()( 0xfxf Ax , , f Ax 0 , )( 0xf .
3. ; ; .
f , f . f .
,
1,1
1,
)(
2
xx
xx
xf
:i) 0x , 0)0( f ,
ii) 1x , 1)1( f .
f , () .
y
O
Cf
a x0 x1 x
y
1
1
O
Cf
x
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4. ; .
i)
(.32).y
x4x3x2x1(a)
O x
()
y
O
min
max
a
x
32
x4x3x2x1
ii) f ,
, , , . (. 32). . (. 32).
5. Fermat.
(Fermat)
f 0x . f 0x
, :
0)( 0 xf
f 0x . 0x
f , 0 ,
xx ),( 00
)()( 0xfxf , ),( 00 xxx . (1), , f 0x ,
0
0
00
0
0
0
)()(lim
)()(lim)(
xx
xfxf
xx
xfxfxf
xxxx
.
,
y
O
f(x0)
x0 x0+x0 x
33
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),( 00 xxx , , (1), 0)()(
0
0
xx
xfxf,
0)()(lim)(0
0
0
0
xx
xfxfxf
xx
(2)
),( 00 xxx , , (1), 0)()(
0
0 xxxfxf ,
0)()(lim)(0
0
0
0
xx
xfxfxf
xx
. (3)
, (2) (3) 0)( 0 xf . .
6.
f ; .
Fermat, , f
, .
, 29 30, f :
1. f.
2. f .3. ( ).
f , f .
y
O
A(x0,f(x0))
Cf
a x0 x2x1
x
29 y
O
Cf
a x0 x1 x
30
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,
1,)2(
1,)(
2
3
xx
xxxf .
f R
R }1{
1,)2(2
1,3)(
2
xx
xxxf .
0)( xf 0 2.
E f 0 2, 1,
f 0, 1 2. , , 1 2 , 0 . f.
7. f
),( , 0x , f :
i) 0)( xf ),( 0x 0)( xf ),( 0 x , )( 0xf f. (. 35)
ii) 0)( xf ),( 0x 0)( xf ),( 0 x , )( 0xf f. (. 35)
iii) A )(xf ),(),( 00 xx , )( 0xf
f ),( . (.
35).
i) E 0)( xf ),( 0xx f 0x , f
],( 0x .
)()( 0xfxf , ],( 0xx . (1)
0)( xf ),( 0 xx f 0x , f ),[ 0 x . :
)()( 0xfxf , ),[ 0 xx . (2)
y
O
Cf
2
1
1 x
34
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y
O
f(x0)
f 0
a x0 x
y
O
f 0
a x0 x
35a
f(x0)
, (1) (2), :
)()( 0xfxf , ),( x ,
)( 0xf f ),( .ii) .
y
O
f 0
a x0 x
y
O
f 0
a x0 x
35
iii) 0)( xf , ),(),( 00 xxx .
y
O
f >0
f >0
a x0 x
y
O
f >0
f >0
a x0 x
35
f 0x ],( 0x ),[ 0 x . , 201 xxx
)()()( 201 xfxfxf . )( 0xf f.
, , f ),( . , ),(, 21 xx 21 xx .
],(, 021 xxx , f ],( 0x , )()( 21 xfxf .
),[, 021 xxx , f ),[ 0 x , )()( 21 xfxf .
, 201 xxx , )()()( 201 xfxfxf .
, )()( 21 xfxf , f
),( ., 0)( xf ),(),( 00 xxx .
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: )( 0xf f ),( ,
)( 0xf f ),( .
8. .
34 4)( xxxf
f R, 23 124)( xxxf .
0)( xf 0x () 3x , f :
x 0 3 )(xf 0 0 +
, f ]3,( , ),3[ , 3x ,
27)3( f .
9. ;
.
:
1. f.
2. f .
3. f.
, 1924152)( 23 xxxxf , ]5,0[x . 24306)( 2 xxxf , ]5,0[x . 0)( xf 1x , 4x .
, f 1x , 4x . f ]5,0[
30)1( f , 3)4( f , 19)0( f 14)5( f ., f ]5,0[ 30
1x , f 3 4x .
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E : 119
x 0 1 3
E(x) + 0
E(x) max
min
0
4
min
0
, 4 1x .
12. xxxf ln1)( .
i) .
ii) 1ln xx , 0x .
i) x
xx
xf 111)( , ),0( x . 0)( xf ,
1x . f :
x 0 1 +f (x) 0 +
f(x)
min0
ii) f 1x , ),0( x : 0ln1)0()( xxfxf
1ln xx .
1x .
13. )(x
, x , xx 640000)( .
4000 . 1200. , , .
x
xxxxxxxE 400006)640000()()( 2 .
x xxK 4000)( .
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E : 121
1
1. H f f (x)=3(x1 3) (x2 2) (x3)
x f
;
fD =R, .
x
f (x) = 0 3(x 1 3) (x 2 2) (x 3) = 0
x 1 = 0 x 2 = 0 x 3 = 0
x = 1 x = 2 x = 3
f (x) > 0 3(x 1 3) (x 2 2) (x 3) > 0
3(x 1 2) (x 1)(x 2 2) (x 3) > 0
(x 1) (x 3) > 0
x < 1 x > 3
f , f
2. (x) = , x > 0
x > 0 (x) = ( ) = ( ) = (xlnx) = (1 + lnx)
(x) = 0 1 + lnx = 0 lnx = 1 x =
fx
x
f xx x lnxe x lnxe xx
f 1e
x 1 2 3 +
f (x) + 0 0 0 +f(x) . .
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E : 122
(x) > 0 1 + lnx > 0 lnx > 1 x >
,
3.
(x) = x
= R
(x) = 1, x R
,
f 1e
f f
f
x
e
fD
f xe
f f
x +(x) 0 +
(x)
.
x 0 +
(x) 0 +(x)
1.
1e f
f
1
e1
e
f
f
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E : 123
2
4. ) f(x) = 3x 3 2x + 3x + 1
) 3x 3 2x + 3x + 1 = 0
) fD = R, .
f (x) = 3 2x 6x + 3 = 3( 2x 2x + 1) = 3(x 1 2) f f
f 1, R.) f(x) = 0
f R
f(R) = (xlim
f(x),xlim
f(x))
xlim
f(x) =xlim
3x =
xlim
f(x) =xlim
3x = +
f(R) = R, , 0 f(R), . ,
f() = 0 , , f .
5. ) g(x) = 3x 3x + 2
) 3
x 3x + 2 = 0
x 1 + f (x) + 0 +
f(x)
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E : 124
) gD = R, .
g(x) = 3 2x 3 = 3( 2x 1) g , g
g (1) = 3( 1) 3(1) + 2 = 1 + 3 + 2 = 4g (1) = 31 3. 1 + 2 = 1 3 + 2 = 0) g(x) = 0 g ( , 1]
g (( , 1]) = (xlim
g(x), g(1)] = ( , 4]
xlim
g(x) =xlim
3x =
0( , 4), . , g(x) = 0 ( , 1]
g [1, 1] g ([1, 1]) = [g(1), g(1)] = [4, 0]
g(x) = 0 [1, 1] , x = 1
g [1, + ) g ([1, + )) = ([g(1),
xlim
g(x) ) = [0, + )
xlim
g(x) =xlim
3x = +
g(x) = 0 [1, + ) x = 1
, g(x) = 0 .
6. ) h(x) = 2 3x 3 2x 1
) 2 3x 3 2x 1 = 0
) hD =R, : h(x) = 6
2x 6x = 6x(x 1)
h , h
x 1 1 + g(x) + 0 0 +g (x)
4 0
. .
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E : 125
x 0 1 + h(x) + 0 0 +h (x) 1 2
. .h(0) = 2 30 3 20 1 = 1h(1) = 2. 31 3. 21 1 = 2 3 1 = 2
) h(x) = 0 h ( , 0]
h (( , 0]) = (xlim
h(x), h(0)] = ( , 1]
xlim
h(x) =xlim
2 3x =
h(x) = 0 ( , 0]
h [0, 1] h ([0, 1]) = [h(1), h(0)] = [2, 1] h(x) = 0 [0, 1]
h [1, + ) h ([1, + )) = [h(1),
xlim
h(x)) = [2, + )
xlim
h(x) =xlim
2 3x = +
h(x) = 0 [1, + ), h(x) = 0
7. (x) = 2x x + 3 , x [0, ]i) .
ii) x = x
(0, )
i) (x) = 2x 1
(x) = 0 2x 1 = 0 x = x =
(x) > 0 2x 1 > 0 x > 0 < x 0
() = 2 + 3 = 3 < 0 ( ) ().
ii) H 2x = x 32x x + 3 = 0
(x) = 0
( [0, ] ) = [ (0), ( )] = [3, + 3] 0
, (x) = 0 [0, ]
( [ , ] ) = [ (), ( )] = [3 , + 3] 0
, (x) = 0 [ , ] ,
.
8. i) (x)=lnx+x1 .
ii)
(x) = 2xlnx + 4x + 3iii)
g(x) = xlnx h(x) = + 2x
.
i) = (0, + )
(x) = + 1 > 0, , .
x = 1 , (1) = ln1 + 1 1 = 0. .
x < 1 (x) < (1) (x) < 0
x > 1 (x) > (1) (x) > 0ii) = (0, + )
(x) = 2(lnx + 1) + 2x 4 = 2(lnx + 1 + x 2) = 2(lnx + x 1) = 2 (x)
,
f
f3
3
3 3
3
ff f
3
f
f
f3 f f
3 3
3
f3
f 3 f f 3 3 3
f3
f
f
2
x
12
2x 3
2
f fD
f 1x
f
f
f
f f f
f
f f f
D
f
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E : 127
iii) ,
g(x) = h(x)
xlnx = + 2x
2 xlnx = + 4x 32 xlnx + 4x + 3 = 0 (x) = 0 x = 1, (1) = 2. 1 ln1 + 4. 1 + 3 = 0
ii), .g(x) = (xlnx ) = lnx + 1 g(1) = ln1 + 1 = 1
h(x) = ( + 2x )
