420-Παράγρ. 1.1
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11.1 . 17 - 18 A 1. f(x) = 3 3 . f(1), f(2) , f(-1) f(1) = 13 3 1 = 13 = -2 f(2) = 23 3 2 = 8 6 = 2 f(-1) = (1)3 3 (1) = 1 + 3 = 2 2. (t) = t2 5t + 6, (0) (1). t (t) = 0; (0) = 02 5 0 + 6 = 0 0 + 6 = 6 (1) = 12 5 1 + 6 = 1 5 + 6 = 2 (t) = 0 =
t2 5t + 6 = 0 t=
52 4. 1 6 = 25 24 = 1,
5 1 5 1 = = =3 2 2 2.1 2h(0)
3.
h() = ,
h(
2
) .
[0 , 2] h( )= 0; h(0) = 0 0 = 1 0 = 1
h ( ) = 0 = 0 = , = 0 = 0 , 0, = 1 = 1 = 4 = + , 4 5 = + = [0 , 2], =4 4 44. f(x) =
h( ) = = 0 1 = 1 2 2 2
1 2 ln , f(1) f(e) 2 f (e) = 1 1 ln e2 = 2ln e = 11=1 2 22 ; ( 1 2) )(
f (1 = )5.
1 1 1 ln12 = ln1 = 0 = 0 2 2 2
( 1)( 2) 0
f () =
1 0 2 0 1 2 =
{1 , 2}
6. f() = ( 3)( 7) ;
2)( () = ( 3 7) ;
f() = ( 3)( 7)
f() = 2 10 + 21
f() x f(x )
+
3 0
7 0
+ +
f() < 0 3 < < 7 )( () = ( 3 7) f() 0 3 . 7. ( 3)( 7) 0 ) = ( , 3] [7 , + 7. f() = 32 2 1 g() = 2 1 ,
f (x) + g(x)
, f(x) + g(x)
,
f(x) g(x)
f g Af = R g = R R f() + g() = (32 2 1) + (2 1) = 32 2 1 + 2 1 = 32 2 2
R
f()g() = (3 2 1)(2 1) = 63 32 42 + 2 2 + 1 = 63 72 + 1 g() 0
f (x) , g(x)
2 1 0
2 1
1 2
f (x) 3x2 2x 1 = g(x) 2x 1
8.
i ) lim ( x 2 3x + 4),x 0
ii)
x -2
lim [( 2 x 1)( x + 4)],x 4
iii) lim ( x + 1 ) x 4 x
iv ) lim ( 2 x + 3 x )x 0
, v) lim (3 x + x )
) )
lim x2 3x + 4) = 02 3 0 + 4 = 4 (x 0x2
lim[(2x 1 x + 4)] = ([2(2) 1 2 + 4)) = (4 1 2 + 4) = (5) 2 = 10 )( ]( )(1 1 1 5 ) = 4+ = 2+ = 2 2 x 4
) lim( x + x4 )x 0
lim(2 + 3 = 2 + 3 0 = 2 0 + 3 1 = 3 ) 0
39.
i ) lim
x -2
x 2 -4 3(x-2)2
iv ) lim x 16 x 4 x 4 i)
2 ii) lim 5x x -1 x 2 + 1 2 v) limx x 25 x -5 x + 5
iii) lim[( x + 1) x ]x 0 2 vi) lim 2 x 3x 2 x 2 x2
x2 4 (2) 2 4 4 4 0 = = = =0 x2 3 x 2 ( ) 3(2 2) 12 12 lim
ii) iii)
5x2 5(1 2 ) 5 = = 2 x1 x2 + 1 (1 +1 2 ) lim
lim x + 1 ] (0 + 1 0 = 1 1 = 1 [( ) = ) x0
iv)
lim lim x 16 = x 4 x 4 x 4
2
0 0
( x 4)( x + 4) lim = x 4 ( x + 4) = 4 + 4 = 8 x4
v)
0 2 ( x + 5)( x 5) lim x 25 0 lim = lim ( x 5) = 5 5 = 10 x 5 x + 5 x 5 x+5 = x 5
vi)
lim
2x2 3x 2 0 = x2 x 2 02x2 3x 2 = (x - 2)(2x + 1)2 (x 2)(2x + 1 ) = lim(2x + 1 = 2 2 + 1 = 5 ) lim 2 x 3x 2 = lim x2 x2 x 2 x 2 x2
:
1.
f (x) =
1 1+ e1 x
,
f(x) + f(x) = 1
f (x) + f (x) =
11 1+ e x
+
11 1+ e x
=
11 1+ ex
+
1 1+ 1 1 ex1
1
1=1 1+ ex
+
11 ex
=
11 1 + ex
+1
x + e 1
1 + ex =
1+ ex1 1+ ex
= 1
1 ex
2. 100 m . , .
4
+ 2 . 100 m, + 2 = 100 (1) . = . (1) =
100 2 100 100 2 ( ) = = 2 2
3. , , 20 cm . h, h . r r 2r . h + 2r = 20 (1) V=
r=(2)
20 h 2
(2)
r 2 h
V ( h ) = 20 h 2 = + . (1) =2rh =r2 . = 2rh + r2 . (3)
(
)
2
h
0 < h < 20
h = 20 2r (3) (r) = 2r(20 2r) + r20 < 2r < 20
(1) 4.
0 < r < 20 0 < r < 10 . 2 1 0
(r) = 2r(20 2r) + r2 0 < r 90,
(180 ) = ,
(1800 ) = .
() = () = 10
=
5 = 90, = () = () = 10 = 10. 1 = 10. 90 = 10 () = 10 00 < < 1800
=
1 ( ) ( ) = 1 10 10 = 50 , 2 200 < < 1800 .
() = 50,
5. i)
limx 5
x 5 = 1 x 5 2 5
ii) limh 0
1+ h 1 1 = h 2
x 5 0 lim ( x 5 )( x + 5 ) 0 = x 5 x 5 = x 5 ( x 5)( x + 5 ) x 5 1 1 1 lim = = lim = . x 5 x 5 ( x 5)( x + 5 ) 5+ 5 2 5 x+ 5i) lim
ii)
limh 0
1 + h 1 0 lim ( 1 + h 1)( 1 + h + 1) 0 h = h 0 h ( 1 + h + 1)=
h ( 1 + h + 1) 1+ h 1 = lim h o h ( 1 + h + 1) 1 1 1 = lim = = h 0 2 1+ h +1 1+ 0 +1h 0
lim
( 1 + h 1)( 1 + h + 1)
1.2 . 26 27 A 1. i) ii) f() = 3 + 1 g() = 2 + 5 = 3 = 2 = 4 9 + 3h + 1 9 1 = lim h 0 h
iii) () = 2 + 2 i) lim
f (3 + h ) f (3) 3(3 + h ) + 1 3 3 1 = lim h 0 h 0 h h
3h = lim = lim 3 = 3 h 0 h 0 h ii) lim g ( 2 + h ) g ( 2) ( 2 + h ) 2 + 5 ( 2) 2 5 = lim h 0 h 0 h h f (3)=3
6 4 4h + h + 5 4 5 = lim h 0 h 2 = lim 4 h + h h 0 h h ( 4 + h ) = lim = lim ( 4 + h ) = 4 h 0 h 0 h2
iii)
g(2)=-4
( 4 + h ) ( 4) ( 4 + h ) 2 + 2( 4 + h ) 16 8 lim = lim h 0 h 0 h h 2 16 + 8h + h + 8 + 2 h 16 8 = lim h 0 h 2 h + 10h = lim h 0 h h ( h + 10) = lim = lim ( h + 10) = 0 + 10 = 10 h 0 h 0 h (4) = 101 t =1 t +1
2.
f (t) =
1 1 f (1 + h ) f (1) lim = lim 1 + h + 1 1 + 1 h 0 h 0 h h 1 1 = 2+h 2 lim h 0 h 22h 2( 2 + h ) = lim h 0 h h 1 1 = lim 2 h ( 2 + h ) = lim 2( 2 + h ) = 4 h 0 h 0 f (1) = 1 4
3.ii)
i) L r L = 2r . L r r = 3 . r = r2 . r r = 2 .
i) L (3 + h ) L (3) 2 (3 + h ) 2 3 = lim h 0 h 0 h h 6 + 2 h 6 = lim 2 h = = lim h 0 h 0 h h 2 h = lim = lim (2) = 2. h 0 h 0 h L(3) = 2, L(3) = 2 lim
7 ii) E ( 2 + h ) E ( 2) ( 2 + h ) 2 22 = lim h 0 h 0 h h ( 4 + 4h + h 2 ) 4 = lim h 0 h 4 + 4 h + h 2 4 = lim h 0 h 4 h + h 2 = lim h 0 h h ( 4 + h ) = lim = lim ( 4 + h ) = 4 + . 0 = 4 h o h 0 h (2) = 4 lim
4.i) , , = 5 ii) V , , = 10 . i) lim () = 2
E (5 + h ) E (5) (5 + h ) 2 52 25 + 10h + h 2 25 = lim = lim h 0 h 0 h 0 h h h 2 h (10 + h ) 10h + h = lim = lim = lim (10 + h ) = 10 . h 0 h 0 h 0 h h , = 5, (5) = 10 ii) V() = 3 V (10 + h ) V (10) (10 + h ) 3 10 3 = lim = lim h 0 h 0 h h 2 3 3 h (300 + 30 h + h 2 ) lim 1000 + 300h + 30 h + h 10 = lim . h 0 h 0 h h 2 = lim (300 + 30 h + h ) = 300 + 30. 0 + 02 = 300 h 0 V(10) = 300 .
5. i) f(x) = x2 (3, f(3) ) ii) f(x) = 2 x (4 , f(4)) ( x 0 , f( x 0 )) y f( x 0 ) = f( x 0 )(x x 0 ) i) f (3 + h ) f (3) (3 + h ) 2 9 lim = lim h 0 h 0 h h
8 9 + 6h + h 9 = lim h 0 h h (6 + h ) = lim = lim (6 + h ) = 6 + 0 = 6 h 0 h 0 h f( x 0 ) = f(3) = 6, f( x 0 ) = f(3) = 32 = 9 2
y 9 = 6(x 3)
y 9 = 6x 18
y = 6x 9
ii) f( x 0 ) = f(4) = 2 4 = 2 2 = 4 f ( 4 + h ) f ( 4) 2 4+h 4 0 0 lim = lim h 0 h 0 = h h 4( 4 + h ) 16 ( 2 4 + h 4)( 2 4 + h + 4) = lim = lim h 0 h 0 h ( 2 4 + h + 4) h ( 2 4 + h + 4) 16 + 4 h 16 4h 4 = lim = = lim = h 0 h 0 h ( 2 4 + h + 4) h ( 2 4 + h + 4) 2 4+h +44 4 1 = = 4+ 4 8 2
f(4) =
1 2
y4=
1 ( x 4) 2
y= 1x+2 2
1.3 . 35 38 A ( 1 18)
1.i) f(x) = 5 ii) f(x) = x4 iii) f(x) = x9 i) f () = (5) = 0 ii) f () = (4) = 43 iii) f () = (9) = 98
2..i)
f() = 2'
3
ii)
f() = -3
iii)
f() = -5
3 1 3 3 1 3 i) f (x) = 2 = 2 = 2 2 2
ii) f() = ( -3) = 3 -3 1 = 3 - 4 iii) f () = ( -5) = 5 -5 - 1= 5 - 6
3.
9 i) f (x) = 3 x ii) f(x) = x5 ' 2
,
x>0
i) f () =
( )3
x
'
1 2 1 1 = x3 = 1 x3 = 1x 3 3 3 '
ii) f ( ) =
(
5
x
2
)
'
3 2 2 5 2 x5 1 = 2 x 5 = x = 5 5
1 4. i) f ( x ) = x '
ii)
f(x) = 1 3 x
iii) f(x) = 1 , 5 x2
>0
' 3 1 1 21 =x = 1 x 2 i) = 1 2 x x2 ' ' 1 ' 4 1 1 3 1 3 x ii) 3 = 1 = x = 3 x 3 x '' ' 7 1 1 52 = 2 = x = 2x 5 iii) 5 2 5 x x5 '
5.i) f(x) = 4x3 i) (43)= 12x2 ii) (6 -5) = 30x - 6 iii) - 2 x 20 5
ii) f(x) = 6x -5
2 20 iii) f ( x ) = x 5
(
) = 40 x 5'
19
= 8 x19 ii) f(x) = 6x x
6 6.i) f ( x ) = 4 i)
x'
, >0
5 1 ' 1 -6 = 6 = 6 x 4 = 6 x 4 1 = 3 x 4 4 1 4 2 x x4 '
ii) 6 x x
(
)
'
1 1 1 3 = 6x x 2 = 6x 2 = 18 x 2 = 9 x 2 2 2 iii) f(x) = x + 2x-1 x
'
'
7.i) f(x) = x4 + 3x2 i)
3 2 ii) f ( x ) = x + 5 + x
f(x) = (x4 + 3x2) = (x4) + (3x2) = 4x3 + 6x
ii) f(x) = x 2 + 5 + 3 = ( x 2 + 5 + 3x 1 ) x = (x2)+ (5) + (3x -1) = 2x + 0 3x -2 = 2x 3x -2 2 2 iii) f(x) = x + 2 x 1 = x + 2 x 1 x x x x 2 = x + 2x 1 x x x = ( x + 2 x 1 )' ' ' '
(
)
10'
= ( x ) + ( 2) ( x 1 ) = 1 + 0 + -2 = 1 + -2 .
8.i) f(x) = 8x3 x + 5 ii) f(x) = 6x 8(x2 + x) i) f(x) = (8x3 x + 5) = (8x3) (x) + (5) = 24x2 x ii) f(x) = (6x 8(x2+x)) = (6x 8x2 8x) = (6x) (8x2) (8x) = 6x 16x 8
9.i) f() = (3 + 1)(4 + 1) ii) f() = (1 ) i) f(x) = (3 + 1)(4 + 1) = 7 + 4 + 3 + 1 f() = (7 + 4 + 3 + 1) = 76 + 43 + 32
ii) f() = (1 ) = f () = ( )= () () = [() + ()] = ( ) = (2 2) = 2
10.i) f() = + 3( + 1)( 1) ii) f() = 42 32
11 i) f() = + 3( + 1)( 1) = + 32 3 f() = ( 32 3) = () (3
