diaforikos logismos
TRANSCRIPT
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1. f xx
x( ) , .=-
1
11
) 1 f;
) f(-3).
2. f(x)=x2+2x f(0), f(1), f(-4), f 12
.
3. f()=2-2+1 f(0)/ f(1), f().4. ()=3-2 (0), j p
2
, (), jp32
.
5. f(t)=lnt+lnt2 f(e), f(1), f(e2).6. ()=e-e2 (ln2), (0), (ln4).7. f x x
x( )=
-+1
42 f
12
aa
,f ( ) 0.
8. f(x)=4x+6, g(x)=x+1 : f(x)+g(x), f(x)-g(x),f(x)g(x),
f x
g x
g x
f x
( )
( ),
( )
( ).
9. f(x)=10x2
-29x+10, g(x)=1-8x
f x
g x
g x
f x
( )
( )
( )
( ) .
10. :) f x
x
x( ) ,=
2
) g xx
x( )=
++
2 1
1 ) h x
x x( )=
+ +1
7 102
11. :) f x
x x
x( ) ,=
+-
2 5
4
2
) ( )g x x x( )= + -5 7 24 ) h x x x( ) ln( )= - +hm5 1
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12.) x2+2x-8
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) ( )limx
1
4
-x x2 1 ) limx0 2
x
x +1
19. ) lim ( )
x
x xx
--1
11
) limx
-0
2 2x xx
) limx 2
--
x x
x
22
2 ) lim
2
-5
+5- 2
20. ) lim
h
h
h
--5
225
5 ) lim
x-
-+3
22 18
3
x
x
) limx - -+1
2
11
xx
) limx4
2
2x -16x -4x
21. ) lim
x
x x
x-
+ ++1
22 1
1 ) lim
x
+ --2
22 8
2
x x
x
) limt -
+ ++5
2
2
7 10
5
t t
t t ) lim
2
2
+5-6
2 +14+12- 6
22. ) lim
x
x
x
--7 2
7
7 ) lim
t
-+ -2 2
2
6
t
t t
) lim -4
2
+4
++
3 12
5 ) lim
x
2
5x-7
5x -2x-7 75
23. y=f(x), f(x)=x(x-1)+(x-2)(x-1) xx. yy.
24. f xx
x( )=-
+
24
2 .
lim ( ), lim ( ), lim ( )x x x
f x f x f x -0 2 2
.
25. :) y=x2-1, ) y=(x+1)2 ) y=(x-1)2
26. :) y x= , ) y x= - ) y x= +1
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1.
A te
t( ) ,= ++
10
11 t[0,5] ( ).
t=ln5.
2,5.
2. (t)=e
t
+ +
-
-355
1
1ee e
t ( .), t
.
) , t=0;
) ;
ln16
7
62
75=
.
3. : 1.500.000
20%.
) 4.000.000;
) x (x>1.500.000) y
y=f(x).
.
4. AIDS A t
e e
ee
t t
t
t
( )=-
+
-
-
2
(
) t .
100 AIDS, t=0
1986. ln425
18=
.
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5. f(x)=2x, f(x+3)-f(x-1)=152
f x( ) .
6. f(x)=x2-4x+6, f(2-h)=f(2+h).7. f x
x
x( )=-+
1
1 f x f x1
= - ( ) f x f x-
= -
1 1
( ) .
8. y=f(x)= 5 34 5
x
x
+-
x=f(y).
9. 500m. x
x.
;
10.
.
x
4
x
4
)
x.
)
.
11. 8 A t t( ) ,= +25
162002
B t t( )= +50
161502 ( .), t[0,8] (
).
) .
)
.
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12. t : Y t
e e
e e
t t
t t( )=-+
-
-5
l l
l l ( t
(t) ).
i) , t=1, 1 .
ii) , ,
t=5.
13. 12 x.
x
A
x
14. :) lim
x
x
x
-
-3
3
3 ) lim
x
x
x
-
-4
2
2
16
) limx
x
x
-
-11
1 ) lim
x
x
x
--7 2
7
49
15. ) lim
x
x
x
+ -0
25 5 ) lim
x
x
x
-
+ -22
2 2
16. .
1
2 3
1, 2, 3 :
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12=23=31=
a
2.
, 1, 2, 3 .
i) .
ii) .17.
:
)
x x+, >0.
)
.
) 2 .
i) S V x.
ii) () x=.
iii) ().18.
, .
, 3 1
2
-
1
2 ,
. =60
;
E A
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1. :: 2 : 4 :
1
2 :
1
4 : 2
y y=x2
A B
4
0 x
2. x 1 :: f x( )1 : f x( )1 : f x( )0 : f x( )2 : f x( )2
3. ;
: f x( )1 : f x( )1 : f x( )0 : f x( )2 : f x( )2
4. f;: 1 : 2 : 3 : 4 :
-1
0 1 2 3 x
-1
1
3
2
y=f(x)
y
x
y
0
y=f(x)
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5. f;.: 3 : -2 :-1 :-4 : 4
6. , , , f;
: : : : :
7. ;
A: B:
y=f(x)
-4 -2 0 3
3
2
1
-2
-3
-3-1
-1 1 2 4
y
y=f(x)
x
x
y
O
y
O x
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: :
E:
8. f(x)-g(x);: 4 : 3 : 2 : 1 :
9. f x( )0;
: [-2,0] : [0,5] : [0,+ ] : [-1,+ ] : [-1,4]
x
y
O
x
y
O
O x
y
x
yy=f(x)
y=g(x)
y
-1 O 1 2 3 4 5x
3
2
1
-1
-2
-3
y=f(x)
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10. :
: [6,12] : [-1,12] : [3,12] : [3,9] : (- ,6]
1. , , , 1 8
.
1: f(x)=-x2 2: f(x)=(x+1)2
3: f(x)=-(x+1)2 4: f(x)=x2+1
5: f(x)=(x-1)2 6: f(x)=-(x+1)2
7: f(x)=-x2+1 8: f(x)=x2-1
A B
y=f(x)
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2. , , , , 1 3
.: f(x) = -x2 1.
: g(x) = ex
: h(x)=x2 2.
: (x) = x
: (x) = x3 3.
A B
3. f. , , , 1
6
.
: f(-1) 1: 0
: f(0) 2: 4
: f(4) 3: -1
: f(3) 4: 1
5: -3
6: 2
A B
4. f. , , , , , 1 3
.
: [-1,1] 1. f .
: [5,+ ]
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: (- ,1] 2. f
: [1,2]
: [3,4] 3. f
: [4,5]A B
5. , , , 1 6
.
: f x x( )= -1 1: x, x1
B: g xx
( )=-1
12: x(1,+ )
: h xx
( )=-
1
13: x (- ,1)
: (x)=x-1 4: x [-1,+ )
5: x
6: x [1,+ )A B
-1 O 1 5
2 3 4 x
y
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1. ) f x1,x2
x1 x2 f(x1) f(x2).) f x1,x2
x1 x2 f(x1) > f(x2).
2. ) f xx
( )=1
) f(x)=ln(x-2)
3. :
f x=
.
f . , f x=
x= .
4.
:
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) , x=-4, x=0 x=2.
) (-,-2] [1,+ )
[-2,1].
) f(-2)=2 f(1)=-3.
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1-14 .
1. ) f(x)=-3 ) f(x)=-3x2 ) f(x)=-3x-3 ) f(x)=4x4
) f(x)= 25x
) f(x)= x ) f(x)= x
2
3 ) f(x)= x-4
5
. ) f(x)=1
3 x ) f(x)= -
4
43 x
2. ) f(x)=x5-4x3+2x-1 ) f(x)= x
x3
337
+ +
) f(x)= 3 2
2
3
5
2 3x x x- + - ) f(x)= x x x23 2 1- +
3. ) f(x)=x4
-lnx ) f(x)=x2
lnx2
) f(x)= x x ) f(x)=(1-4x3)(1+2x2)
4. ) f(x)=x7ex ) f(x)=-4x5e-x
) f(x)=3x2e ) f(x)=exx
5. ) f(x)=1
x ) f(x)=
2
2
2x
x -
) f(x)=
2 4
2 2
x
xa - ) f(x)=
x
x
53
3 2-
6. ) f(x)=
1+ ) f(x)=
x
x
2
2
1
1
+-
) f(x)=e
e
x
x
+-
1
1 ) f(x)=
ln
ln
x
x
++
2
4
7. ) f(x)=x3(x2+x) ) f(x)=x3 x -1
) f(x)= ( )x x -3 2 ) f(x)= ( )( )x x- -2 3 5
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8. ) f(x)=2x ) f(x)= 3
13x x
x+ +
) f(x)=2x+3x ) f(x)=2x+53x
9. ) f(x)= ( )xx-3
2
) f(x)= ( )xx+
1
2
) f(x)=( )1 2 3
3
- xx
) f(x)=e
x
x2
10. ) f(x)=log2x ) f(x)=
x x
x x
+-
) f(x)= x2 2+ a ) f(x)=x+x+1
11. ) f(x)=(x2+3x+2) ) f(x)= ( )13
+ x
) f(x)= x x+ ) f(x)=e e
e e
x x
x x
-+
12. ) f(x)= ( )ex 33x-3x ) f(x)= ( )+x a - x
) f(x)=x-x
5 ) f(x)= 3x3x
13.) f(x)= x+x+x x
1 1
+ ) f(x)=
1
3
1
x3x -
) f(x)= ln(x) ) f(x)= 3x+x
5 x+
14. ) f(x)= 21
1
3
2
xx
x
-
+
) f(x)=1
1
-+
x
x
2
2
) f(x)=1
1
+-
x
x ) f(x)=
2 1
1
2
2
x
x x
-
+
15. :
) f(x)=xex ) f(x)=5x+1 ) f(x)=lnx
16. :
) f(x)=x ) f(x)=x
x +1 ) f(x)= x +7
17. :
) i) f(x)=e-5x ii) f(x)=ex- +5 12
) i) f(x)=ln(x+2) ii) f(x)=ln(x+2)
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) i) f(x)=7x ii) f(x)= x +1
18. f(x)=x g(x)=f(2x2-3) g(1).
19. f(x)=x g(x)=2x+1. :
f(x)g(x) f x
g x
( )
( )
20. y=f(x) x=x0 :
) f(x)=x, x0 =
2 ) f(x)=2x-x2, x0=-
) f(x)=x-x, x0 =
2 ) f(x)=x+x, x0=
21. f(x2)=x5 x>0 f(9).22. f(x)=x2(3x2+8x+6)
.
23. y=2x3+9x2-24x-7 xx.
24. y=- =
2.
25. y
e e
e e
x x
x x=-+
-
- x=0.
26. t0 ( sec)
25cm.
4 t
10
27. m1, m2 F G m m
r= 1 2
2
, r
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G . r
F r=1.
28. f(x)=x x= x+y=0.
29. f(x)=ln(x2+4) y x= +
1
23.
30. y=+3 =
2.
31. , y=x f(x)=x2+x+ (1,1).
32. y=x2-7x+3 x-y+2=0.
33. x f(x)=0, f(x)=x+2x.
34. =2.
y y=x2
A
0 x
35. xx t0 : S(t)=ln(t+1)+ 1
2t +.
t=1.
36. , xx xA, xB t0 :
xA(t)=100+5t, xB(t)=1
2
2t
. ( cm sec).
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37. t[0,2] :
(t)=1
3
5
2
223
6e t
t -
+
(C)
t=1
. ( ).
38. ( ) t :
f t t t t( ) (%)= - + -
+
10
223
8
63
100
2132 10
3 2
t[0,24] t=0 . 8 ( t=20
).
39. t0 x(t), y(t). y x=
1
182
.
40.) u(t) (t) .
)
( xx).
)
.
) t=4 t=10.
) [0,10].
( ).
41. . t[0,7] ( sec)
h(t)=35t-5t2.
:
) .
) t=3 t=4.) .
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( ).
42. . t[0,4] ( sec)
h(t)=35t-5t
2
( ) 15
, .
43. f x e ex x( )= + - , x>0 xf x f x f x + - =( ) ( ) ( )
1
2
1
40
44. y=f(x).y
x
O
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1. t x(t)=5t2-3t.
:
: 5 : 10t-3 : -3 : 10 :5
2
2. f(5)=x-4 f(x)=: -4x5 : -
45x
: -4x3 : -4
3x :
13x
3. y=lnx+2x x=1;
: 2 : 1 : 3 : ln2 :
4. t x(t)= t t+ . t=4;
:1
2 :
5
4 :
1
4 : 6 : -
1
32
5. 2 Q( )t t t= - +39 (C), t[0,2] . t=1:
: 1
2
o C .
: 1
2
o C .
: 1oC .
: 1
4o C .
: 1
4o C .
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6. f x ex( )= 2 f(x)=: x e
x2 12 - : 2
2 32
x ex -
: 2ex : 22
ex
: 22
xex
7. f(x)=x f (x)=: x : x : -x : -x :
8. t0 u(t)=lnt. t=e :
:1
e : 1 : -1 : -
12e
: -e
9. f(x)=(x) f(x)=: (x)x B: -(x)x : (x)x
: -(x)x E:
10. f(x)= 1x
f (x)=
:1
3x :
12x
:2
3x : -
23x
: lnx
1. , , 1 4
.
: (x) 1: x
B: (x) 2: (x): (x) 3: -x
4: x
A B
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2. , , , 1 6
.
: x-1 1: 12
12x
B: x3 2: -1
33
4x
: x 3: -x-2
:1
3 x4:
1
21
2x-
5: - 13
43x
6: 3x2
A B
3. , , , 1 6
x=1.
: y=ex 2 1-
1: 3
B: y=x+1
x2: 0
: y=3xlnx 3: 0
: yx
x=
ln4: 2
5: 1
6: -2
A B
4. t0 S(t).
, , , 1 6 S(t)
.
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: 10t-5t2 1: -10
B: 5t-10t2 2: 5
: -5t+10t2 3: -20
: 10t+5t
2
4: 105: 20
6: -5
A B
1. ) 8
t0 : S(t)=3t+ t2.
S(t)=
t=1
)
t0 S(t)=-5t2
-2t. S(t)=
t=2 .
2. ) f(x)= ex 4 3+
= +f x ex( )4 3 =
) f(x)=(lnx+1) f(x)=(lnx+1) =
) f(x)=1821x f(x)=18211820x =
) f(x)=1999x f(x)=
3. ) t0
S(t)=ln(t2+1).
S(t) =
S (t)=
t=1 .
) t0
S(t)= et t2 3 2- -
.
S(t) =
S (t)=
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t=1 .
4.
f.
x
y
O
y
x
O
y=f(x)
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1. f(x)=2x+1.) x=-2, ) x=0, ) x=5
2. , , , ., y=f(x) f.
) , ) , ) .
.
3. ) f(x)=x2-x x=2 ) ()=
5
w =1
4. ) f(x)=2x2+x+1 x=-1 ) g x
x( )=
+2
3 x=0
5. f(x)=x3) x=8, ) x=0
6. f(x)=-x2
) x=-1, ) x=0, ) x=1
7. =f(x) x=x0 :) f x
x
( )=1
x0=2 ) f(x)=x3+1 x0=-1 ) f(x)=x
2+2x+4 x0=0
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8. x x=8.
1
x+2
9. >0. =4
1. k 2k+1, k>0.
k : ) k=1 ) k=2
) k=3
;
2. f x x( )= x=4.
3. x>0 1
1
x
+ .
x x=5
11. ( .
=22
7).
4. y= x -2
x=11.
5.
f(x)=-x2+x+.
) f.
) f x=0.
f x=8 .
) f x0=4.
;
y
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S
0 4 x
6. (oC) :
Q( ) ( )t t t= - - +2
2510 37
t[0,10] ( ). t=7
.
1. , , , , ;
: : : : :
2. f(5)=-4 y=f(x) x=5 :
y
x
H
ZO
I
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: -4 : 5 : - -4
5 : -
5
4 :
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3. x1, x2, x3, x4 f g.
: x1 : x2 : x3
: x4 : y y=g(x)
y=f(x)
0 x1 x2 x3 x4 x
4. f(x)=2x x=1000 :: 2 : 1000 :
1
500 : -2 : 1
y y=2x
2
0 1 x
5. , , , , f 0;
: : y : : :
0 x
6. y=f(x) :y B y=f(x)
A
0 x1 x2 x
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: lim( ) ( )
h
f x h f x
h
+ -0
1 1 : lim
( ) ( )
h
f x h f x
h
+ -0
1 2 :
lim( ) ( )
h
f x h f h
h
+ -0
1
: lim( ) ( )
h
f x h f x
h
+ -0
2 1 :
7. y=f(x) x=2 y=7x-5 :
A: f(2)=5 : f(2)=-5 : f(-2)=7 : f(2)=7 :
f (7)=2
8. xx S(t). t=3;
: .
: .
: .
: .
:
9. y=f(x). f(1)=A: 1 : -1 : 2 : -2 :
1 2 3 4 t
4
3
2
1
S
-5 -4 -3 -2 -1 0 1 2 3 4 5
x K x
y
y=f(x)2
1
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10. f(0)=5 f(0)=0 y=f(x) x=0 :
A: y=5 : y=x : y=1
5x : x=5 : y=5x
1. , , 1 6
.
: f(3) 1: lim( ) ( )
h
f h f
h
- - -0
3 3
: f(-3) 2: lim( ) ( )
h
f h f
h
-0
0
: f(0) 3: lim( ) ( )
h
f h f
h
+ -0
3 3
4: lim( ) ( )
h
f h f
h
- -0
3 3
5: lim( ) ( )
h
f h f
h
- -0
0
6: lim( ) ( )
h
f h f h
h
+ -0
3
2. , , , , , , 1 4
:
1:
2:
3:
O 1 2x
y
A
B
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4:
x
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1. ) f x0 = f x h( ) lim0 0
) y=f(x)
x=x0 .
2.
S(t).
t=t0.
xx
3. 4 , , , y=f(x) f , ,
.
4. y=f(x) :) f 0 2.
) f (1,+)
(-,1).
) f x=1 .
) f(1)=-2
3
2
1
0
-3 -2 -1
-1
1 2 3 4
-2
-3
-4
y
xO
y
x
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1. :) f(x)=x2+2x+3 ) g(x)=3+2x-x2
) h(x)=(2-x)3 ) (x)=(x2-4)2
2. :
) f(x)=(x-4)4
(x+3)3
) g(x)=x2
(1-4x2
)2
) h(x)=x3+2x2-4x-8 ) (x)=(x-1)1/3(x+2)2/3
3. :
) f(x)=1
3
1
26 13 2x x x+ - + ) g(x)=x3-6x2+9x-8
) h(x)=x4-8x3+18x2-27 ) (x)= - - + +1
4
1
354 3 2x x x
4. f(x)=-2x2+16x+ 39.
5. 120 .
6. .7. .
t
[ ]h t t t( ) ,= - 10 2 52
t 0,2 2
. ;
30o
45o
h(t)
20m/sec
-
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38
8. t=0.
t :
Q( ) ( ) (%)t t t= - +
1
10 20 2
t[0,15] ( ).
.
;
9. x y 20cm.
) x y
.
) x, y .
;
10. x, x100
(x)=4x2+3x+6400
.
11. x (x100) :K x x x( )= + +30 3
1
1002
( .)
11000 .
12. 20. :)
)
) .
13. 60 :P( )t t t= - + +
1
200
2
5502 ( )
t[0,60] ( ) t=0
1930.
. ;
-
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39
14. Q( ) ,t
t
t=
++
6
336 62 (
oC)
t0 ( ).
.
;
15. x y.
x 3600 . y
1200 . .
4000m2
;
16. f x e ex x( )= --2
.
f(x) 1.
17. x y.) V S
V
xx= +4 2
) V=62,5cm3 S x .
1. f x x x x( ) ( ln ),= - 24
2 1 0 .
f -1
4.
x
x
y y
x
x
y
-
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40
2. r ( )
Fkx
x r=
+2 25
2
x
. F x r=2
.
3. .
V rA
r
B
r( )= -7 3 , r
, . r
. ;
4. , , .
1500m
;
5. .
.
-
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41
6. , , 6280cm3
. ;
( =3,14).
7. (. ) 125cm3. 32 . cm2
1 . cm2.
.
8. 5u2+75. ( . ).
500
.;
9. (3,4) .
10. 100 400 . .
5
10 .
.
11. y x= +12
12
5
20,
.
12. f(x)=(1-x)2+(2-x)++(-x)2 ( )x v=
1
+ +...+1 2 .
x
x
y
-
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42
13. f t
e t( )=
++- +
10000
11010 t[0,20] ( )
.
.
14. : h t
e
e e
t
t t( ) ,= +
-
- - +
2 10
10 10 0 t 20
)
.
) .
;
15. 400cm2. 4cm
3cm .
;
16. f x x( )= (4,0), (0,0). ,
y=f(x) , .
17. 2x-y+1=0 (2,3) (4,5) .
18. ,
3km.
10.000.000
. ,
2.000.000 .
5km,
.
-
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43
;
19. (8,0) (0,2) .
1=2cm/sec 2=1cm/sec .
.
20. 10 104 km . 16km/h
24 km/h. ;
;
21. 9 .
15 .
4 5
.
;
22. , ,
. (
).
O
A
9
15 B
-
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44
1. f(x)=x(x-1) f :: x[0,1] B: x[1,+) : x(-,0] : x(-,0] x[1,+ )
:x
2. f(x)=x2(x-2) f:: x=2 B: x=2
: x=0 x=2
: x=0 x=2
:
3. [,];
:
4. f, g, h, . f, g, h,
x=1;
-
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45
:
-
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46
5. f(x)=+ x = p4
:
: , fp
42
= : , fp
41
=
: , fp
42
=
. , fp
41
=
:
6. .A: f(x)=x2+1 B: g(x)=x2-5x : h(x)=2x+5
: (x)=1+2x2+x : ()=3-
7. f :: f(x0)>0, f (x0)>0
: f(x0)=0, f (x0)>0
: f(x0)0
: f(x0)>0, f (x0)
-
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47
-
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48
1. , , , 1 6 , , , .
1: f(x)0
4: f(x)=0 f (x)0 f (x)0
2. , , , 1 4
.
1: f(x)>0, f (x)>0
2: f(x)>0, f (x)
-
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49
1. f : >, 0, f (2)0
f(0)=2, f(-3)=-3, f(2)=-1
3
2
1
-2
-3
y=f(x)
y
x
x
y
-
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-3 -2 -1 O 1 2 3
-1
-2
-3
5. f(x)=x3- x=0 x=1. ,