Ένα φυλλάδιο με τη θεωρία για τα Μαθηματικά της Γ' Λυκείου για τις Πανελλήνιες Εξετάσεις. Περιέχει, εκτός από τη θεωρία, ερωτήσεις συμπλήρωσης, πολλαπλής επιλογής, τύπου σωστό-λάθος.
0.2 [email protected] , 10 2006 a L TEX2
1 1.1 - . . . . . 1.2 . . . . . . . . . . . . 1.3 - . 1.4 1.5 .
. . . 1.6 . . . . 2 - 2.1 - . . . . . 2.2 . . . . . . . . . . . .
2.3 - . 2.4 2.5 . . . . 2.6 . . . . 3 3.1 - . . . . . 3.2 . . . . .
. . . . . . . 3.3 - . 3.4 3.5 . . . . 3.6 . . . . 4 4.1 - . . . . .
4.2 . . . . . . . . . . . . 4.3 - . 4.4 4.5 . . . .
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3 3 10 11 13 15 17 18 18 35 37 41 44 44 49 49 62 68 73 76 77 79
79 84 86 89 90
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2
1 1.1 - , : : : : / : :
( + ) + = + ( + ) + =+ +0 =
( ) = ( ) = 1= 1 + () = 0 = 1, = 0 ( + ) = +
1. . C R , : () , R, (0) (1) , () i ( ) , i2 = 1 ( i =
1),
() z C z = + i, , R.
3
1.
4
; ; + i, i2 = 1 , R. i, R ( = 0) . z = + i , z Re(z), z Im(z). +
i i.
2.
+ i
. z1 < z2 , z1 z1 z2 Im(z1 ) = Im(z2 ) = 0.
z2 , z1 > z2
+ i . , .
3. ; + i + i , = = . : + i = + i = = . : . . . ; ; . +i M(, )
M(, ) + i. M(, ) + i. x x , M(, 0)
4.
1.
5
= + 0i, y y , M(0, ) i = 0 + i. .
M(, ) () ()
5. z1 = + i z2 = + i, z1 z2 . z1 = + i z2 = + i
( + i) + ( + i) = ( + ) + ( + )i. M1 (, ) M2 (, ) z1 z2 , M( + ,
+ ). , + i + i . z1 = + i z2 = + i
( + i) ( + i) = ( ) + ( )i. M1 (, ) M2 (, ) z1 z2 , M( , ). , +
i + i .
6. z1 = x1 + y1i z2 = x2 + y2i, z1 z2 . z1 = + i z2 = + i
( + i)( + i) = ( ) + ( + )i. z1 = + i z2 = + i, z2 = 0
+ i + = 2 + 2 i. + i + 2 + 2
1.
6
, :
Re
+ i + i1 2
=
+ 2 2 + 21
Im3
+ i + i4
=
4 2 + 23
. i . ,
7.
z1 = z z = z1 z, N > 1., z = 0,
z0 = 1 z = i :
1 , N . z
i = i4+
i0 = 1, i1 = i, i2 = 1 i3 = i. i ( i) 90 ( ). , i, .
1, i, 4 4 = i i = (i ) i = 1 i = 1, i,
= 0 = 1 = 2 = 3
y(i) (i)
O(i) (i)
x
8.
; ;
. ,
1.
7
, .
y
M(z)
O
x
M(z) z = + i z = i
z + z = 2 = 2Re(z),
z z = 2i = 2Im(z)i zz = 2 + 2 . i
:
z1 + z2 + . . . + z = z1 + z2 + . . . + z z1 z2 . . . z = z1 z2
. . . z z = z z=z
9. = 2 4 < 0, z2 + z + = 0 , , R = 0 :
z1,2 :
i = 2
z2 + z + = 0 z2 +
z + = 0 z2 + 2 z + = 0 2 2 2 z2 + 2 z + 2 = 2 4 42 2 2 4 = z+ 2
42=0
z+ 2
2
=
42
1.
82
i2 (1)() < 0 = = 42 4a2 (2)2 z+ 22
=
i 2
2
=
i 2
2
i i z+ = z= 2 2 2
10. z2 + z + = 0, , , R = 0 C.
z2 + z + = 0, = 0 = 2 4 . : > 0, : z1,2
= . 2
= 0, : z = < 0, : z1,2
. 2 i = . 2
z2 + z + = 0, , , R = 0 Vieta, , z1 , z2 ,
z1 + z2 =
z1 z2 = .
, Vieta z2 + z + = 0, = 0. . z2 3z + 2 = 0 . z = + i, . z2 3z +
2 = 0 , z = w, w2 3w + 2 = 0 . ;
11.
1.
9
M(x, y) z = x + yi , z ( |z|) M . :
|z| = |OM| = y
x2 + y2 . M(x, y)
|z|x
12. z0 , z1, z2 C, |z z0 | = , > 0 |z z1 | = |z z2 |. |z z0 |
= , > 0 M(z) M(z0 ), K(z0 ) . |z z1 | = |z z2 | M(z) A(z1 ) B(z2
), A(z1 ) B(z2 ).
M(z) K(z0) A(z1)
M(z)
B(z2 )
:
|z1 z2 . . . z | = |z1 | |z2 | . . . |z | |z | = |z| ||z1 | |z2
|| |z1 + z2 | |z1 | + |z2 |
1.
10
||z1 | |z2 || = |z1 + z2 | z1 z2 . |z1 + z2 | = |z1 | + |z2 | z1
z2 .
, M1 z1 M2 z2 ,
y M1 (z1 )|z1 z2 | M2 (z2 )
(M1 M2 ) = |z1 z2 |.
O
x
1.2
1. () z1 + z2 = z1 + z2 () z1 z2 = z1 z2 () z1 z2 = z1 z2 ()
z1 z2
=
z1 z2
. () z1 = + i z2 = + i,
z1 + z2 = ( + i) + ( + i) = ( + ) + ( + )i = ( + ) ( + )i = ( i)
+ ( i) = z1 + z2 .() z1 = + i z2 = + i,
z1 z2 = ( + i) ( + i) = ( ) + ( )i = ( ) ( )i = ( i) ( i) = z1
z2 .() z1 = + i z2 = + i,
z1 z2 = ( + i) ( + i) = ( ) + ( + )i = ( ) ( + )i
z1 z2 = ( i) ( i) = ( ) + ( )i = ( ) ( + )i z1 z2 = z1 z2 ()
.
1.
11
2. :() |z| = |z| = | z| () |z|2 = z z () |z1 z2 | = |z1 | |z2 |
()
z1 |z1 | = z2 |z2 |
. () z = x + yi, z = x yi z = x yi. |z| = |z| = | z| = 2 x + y2
. 2 2 x + y2 = x2 +y2 z z = (x+yi) (xyi) = x2 +y2 . () z = x+yi,
|z|2 = |z|2 = z z. () , :
|z1 z2 | = |z1 | |z2 | |z1 z2 |2 = |z1 |2 |z2 |2
(z1 z2 )(z1 z2 ) = z1 z1 z2 z2 z1 z2 z1 z2 = z1 z1 z2 z2
, . () , :
|z1 | z1 z1 = z2 |z2 | z2
2
=
|z1 |2 z1 |z2 |2 z2
z1 z2
=
z1 z1 z1 z1 z1 z1 = z2 z2 z2 z2 z2 z2
, .
1.3 - .
1. z z2
0.
2. C : z2 + w2 = 0 z = w = 0. 3. C : z w = 0 z = 0 w = 0. 4. z C
|z| = |z| = | z|. 5. z C |z|2 = z2 . 6. |z|1, 1 z 1.
7. |z| = |w|, z = w. 8. z, w C, |z| + |w| = 0 z = w = 0. 9. z C
|z|Re(z).
1.
12
10. z1 , z2 C, |z1 + z2 | = |z1 | + |z2 |. 11. w = |z1 |2 +
2Re(z1 z2 ) + |z2 |2 w 12. z C |z1 + z2 |2 = z2 + 2z1 z2 + z2 . 1 2
13. z C |z1 z2 | < |z1 + z2 | 14. z1 , z2 C z1 , |z1 + z2 | =
|z1 | + |z2 |. z2 z 0.
15. z, w C |z w|2 = |z|2 + |w|2 2Re(zw). 16. z C > 0 |z| = z
= . 17. z C |z| > |z 1| Re(z) >1 2
18. z : |z | = , (, > 0)M(z) O
19. z1 z2 , |z1 z2 |.
||z1 | |z2 ||.
20. z , z 21. N, = i + i2 + . . . + i .
22. z z2
0.
23. . 24. z C R |z + i| = |z i|. 25. + i + i (, R) O(0, 0).
26. 4 , i = i . 27. + i = 0 = 0 = 0.
1.
13
28. z + 2i z 2i. 29. , . 30. : z1 = z2 z1 = z2 . 31. z : |z|2 =
zz. 32. z : |z| = |z|. 33. z : |iz| = |z|. 34. f : [, ] R , z = x +
if(x), x [, ], .
1.4 1. :. (z z)i . (z + z)2 . (z z)2 . z2 z2
2. z = (3 + 5i)10 (3 5i)10 :. . . .
3. : x2 4x + 2 = 0, R() , .
2
.
2
. 2 < < 2
. < 2 > 2
() : . 2 + 4i . 4 + 3i . 2 + 3i . 2 + 2i
() 2 + i, : 1. : . i 2 .
1 2+i
.
1 2i
.
5 2+i
.
5 2i
2. . = 2 . = 4 . = 5 . = 5
4. |z|2 = z2 , . z < 0 . z I . z2 I .
1.
14
5. |z1 | = 3 |z2 |. 5
6, : |z1 + z2 | :. 8 . 9 . 12 . 14
6. |z| = 1 = 2z + i, (K, ) : . K(0, 1) = 2 . K(1, 0) = 2 . K(0,
1) = 2 . . K(0, 1) =
1 2
7. |z|
1 = f(z). ||. =
2, :. =
. = 2z . =
1 z 2
2 z
iz 2
.
8. |z 3| = 1, = iz (K, ) : . K(0, 3) = 2 . K(0, 6) = 2 . K(0, 6)
= 1 . K(3, 0) = 1 . K(0, 6) = 2
9. z = 1 + 2i, = 3 4i, R Im(z + ) = 0, =. 1 . 0 . 2 . 3 . 2
10. |z| < 1, :. |1 + z|
1 + |z|
. |1 + z|
2
. Re(z) + Im(z) >
2
.
11. |z| = 1 =.
2z + 2 , || : z3. 2 .
1 4
13 10
.
5 4
.
5 2
12. w = , z = 0, :. Re(z) = 0 Re(w) = 0 . z R w R . |z| = 1 |w|
= 1
1 z
. z = w
1.
15
1.5 1. , .
1 ) () |z z0 | = |z + z0 | () |z z0 | = |z0 | () |z| = |z0 |
2 x x) y y ) )
z0 C ()
2. , z = 0.
1 () |z + 1| = |z 1| () |z i| = |z + i| () |z 1| = |z i| () |z +
1| = |z i|
2 1. Re(z) = Im(z) 2. Re(z) = Im(z) 3. Re(z) = 0 4. Im(z) =
0
3. z1 z2 z1 = z2 . .
i) ) |zz1 | > |zz2 | ) |z z1 | < ) |z z1 | >
ii) iii) iv)
v)
4. z, -
1.
16
, .
i) x + y = 0) |z| = 2 ) |z i + 1| < 2 ) |z i| = |z + 1| ) z +
z = 0 ) z z = 0 ) |z1|+|z+1| =
ii) x = 0 iii) x2 + y2 = 4 iv) y = 0 v) (x+1)2 +(y1)2 4
vi) (x1)2 +(y+1)2 = 16 vii) x2 y2 + =1 4 3
4
viii) x2 y2 = 1
5. . ) z2 2z + 2 = 0 ) (z + 2)2 = 9 ) z2 (2)z + 1 =
i) +i i ii) 1 i 1 + i iii) 1 + i 1 i 1 1 iv) 2 (1 + i 3) 2 (1 i
3) v) + i + i vi) 2 + 3i 2 3i
0) z +
1 =1 z
6. z1 = 3 + 4i z2 = 1
3i,
,
1.
17
.
a) 4 1) |z1 z2 | 2) |z2 | 1 3) |z2 |2 4) |z1 | 5) |iz2 | b) 2 c)
25 d) 5 e) 2 f) 5 g) 10
1.6 1. z z . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
2. z1 , z2 C, ||z1 | |z2 ||
|z1 + z2 |
|z1 | + |z2 |.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
2 - 2.1 - N
N = {0, 1, 2, 3, . . .} Z
Z = {. . . , 3, 2, 1, 0, 1, 2, 3, . . .} Q
, , Z = 0, Q=
| , Z = 0 2,
e . R , . :
= 3.1415926535897932384626433832795028841971693993751 . . .
18
2. -
19
e :
e = 2.7182818284590452353602874713526624977572470936999 . .
.
N, Z, Q, R
NZQR N \ {0}, Z \ {0}, Q \ {0} R \ {0} N , Z , Q R . :
1.
,
. + . , .
2. :
+
3. > 0, < 0, 4. 5. ,
.
, +
+ .
> 0
> 0,
6. :
0 ( 0 = 0). : > 0 > 0. 1
0 N ,
.
7. > 0,
1 .
, R < , , : (, ) = {x R | < x < } [, ] = {x R | x } [,
) = {x R | x < } - (, ] = {x R | < x } - R, :
2. -
20
(, +) = {x R | x > } [, +) = {x R | x } (, ) = {x R | x <
} (, ] = {x R | x } , R (, +). . , || :
|| =
0 , , < 0
, , . :
1) ||2 = 2 3) || = |||| 5) ||| ||| | + | || + ||
2 = || || = 4) || 6) |x x0 | < x0 < x < x0 + , > 0
2)
1. . ; ; A R. A () f x A y. y f x f(x). x, A , y, f x, . f x A,
f f(A).
f(A) = {y|y = f(x) x A}. .
2. -
21
f B, B . f(B) f x B. :
f(B) = {y|y = f(x) x B}. f, : , f(x), x .
f , f x, f(x) .
2. ; ; () , () () x0 ; f A Oxy , M(x, y) y = f(x), M(x, f(x)), x
A, f Cf . x A y R. . , . Cf f, : f A Cf . f f(A) Cf . f x0 A x = x0
Cf . Cf , f , , f |f|.
2. -
22
f , x x, f, M (x, f(x)) M(x, f(x)), x x. |f| Cf x x , x x, Cf .
0 < = 1, x, x1 , x2 > 0 k R :
. log x = y y = x . log
. log ax = x log x = x . log (x1 x2 ) = log x1 + log x2
. log = 1 log 1 = 0
x1 . log xk = k log x1 = log x1 log x2 1 x2 . > 1, log x1
< log x2 x1 < x2 . 0 < < 1, log x1 < log x2 x1 >
x2 .. x = ex ln , = eln x .
3. ; f g x A f(x) = g(x).
4.
f, g A B ; , . f, g A B A B. x f(x) = g(x), f g . ,
f(x) =
x2 1 x2 + x g(x) = x1 x
A = R {1} B = R {0} , = R {0, 1}, x
f(x) = g(x) = x + 1.
2. -
23
5. , , . f+g, fg, fg
f g
f, g (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (fg)(x) =
f(x)g(x) f f(x) (x) = . g g(x) f + g, f g fg A B A B f g ,
f A B, g
x g(x), {x|x A B g(x) = 0}. , , f, g f g . , , .
6. . , , f g , g f = f g. f, g A, B , f g, g f, (g f)(x) =
g(f(x)) x A f(x) B, g f f(A) B = . f(x) = ln x g(x) = x. f Df = (0,
+), g Dg = [0, +). g(f(x))
Dgf = {x A|f(x) B}.
x Df f(x) Dg
x>0 f(x) 0 x
x>0 ln x 0
x>0 x x 1
1,
1. , g f (g f)(x) = g(f(x)) = g(ln x) = ln x, x [1, +).
2. -
24
f(g(x)) :
x Dg g(x) Df ,
x 0 g(x) > 0
x 0 x>0
x 0 x > 0, x>0
x > 0. , f g
(f g)(x) = f(g(x)) = f( x) = ln x, x (0, +).
f g = g f. , f, g, h f (g f), (h g) f
h (g f) = (h g) f.
, ; f , x1 , x2 x1 < x2 f(x1 ) < f(x2 ). , x1 , x2 x1 <
x2 f(x1 ) > f(x2 ). f , f . f ( ) , f ( f ). , , , x1 , x2 x1
< x2 f(x1 ) f(x2 ). , x1 , x2 x1 < x2 f(x1 ) f(x2 ).
7.
2. -
25
() , () () ; f A : x0 A () , f(x0 ), f(x)
8.
f(x0 ) x A. f(x0 )
x0 A () , f(x0 ), f(x) x A.
() () f f. . f(x) = x3 . ( ) . , . , f(x) = x , y = 1, 2k + , k
Z , y = 1, 2k , k Z. 2 2 -- ; -- ; f : A R --, x1 , x2 A : x1 = x2
f(x1 ) = f(x2 ). f -- f . f . : f : A R 1-1, x1 , x2 A f(x1 ) =
f(x2 ), x1 = x2 .
9.
2. -
26
f : A R 1-1 x1 , x2 A x1 = x2 f(x1 ) = f(x2 ). , f(x) = x2 1-1
f(1) = f(1) = 1. f : A R 1-1, y f(A) y = f(x) x. , 1-1.
10. ; f : A R 1-1. , y , f(A), f x , f(x) = y. f1 : f(A) R f y
f(A) x A f(x) = y. :
f(x) = y f1 (y) = x,
f1 (f(x)) = x, x A f(f1 (y)) = y, y f(A). C C f f1 y = x xOy x
Oy .
11. f x0 R, . f x0 . f , x x0 , f x0 xx0
lim f(x) = .
f x0 , f x0 , f (, x0 ) (x0 , ) (, x0) (x0 , ). x0 (. ) (.
).
2. -
27
f x0 , , x0 (. ) (. ).
()
()
()
O
x0
O
x0
O
x0
f 1 , x x0 , :xx 0
lim f(x) = 1
f(x), x x0 1 . f 2 , x x0 , :xx+ 0
lim f(x) = 2
f(x), x x0 2 . f (, x0 ) (x0 , ), :xx0
lim f(x) = lim f(x) = lim+ f(x) = xx0 xx0
f (x0 , ), (, x0 ), :xx0
lim f(x) = lim+ f(x).xx0
f (, x0 ),
2. -
28
(x0 , ), :xx0
lim f(x) = lim f(x).xx0
f x0 , . :xx0
lim f(x) = lim (f(x) ) = 0xx0
xx0
lim f(x) = lim f(x0 + h) = h0
lim f(x) > 0, f(x) > 0 x0 .xx0
lim f(x) < 0, f(x) < 0 x0 .xx0
f, g x0 f(x)xx0
g(x) x0 ,
lim f(x)
xx0
lim g(x)
f g x0 , : lim (f(x) + g(x)) = lim f(x) + lim g(x).xx0 xx0
xx0
lim (kf(x)) = k lim f(x).xx0 xx0
lim (f(x) g(x)) = lim f(x) lim g(x).xx0 xx0 xx0
lim
lim f(x) f(x) xx0 = , lim g(x) = 0. xx0 g(x) xx0 lim g(x)xx0
lim |f(x)| =xx0
xx0
lim f(x) .k
lim
k
xx0
f(x) =
xx0
lim f(x), f(x)
0 x0 .
2. -
29
xx0
lim [f(x)] =
xx0
lim f(x)
, N
12. . f, g, h (, x0 ) (x0 , ). h(x)xx0
f(x)
g(x) x0 xx0
lim h(x) = lim g(x) = lim f(x) = .xx0
f. , f x0 . h(x)xx0
f(x)
g(x) x0 lim h(x) = 1 xx0 xx0
lim g(x) = 2 1
lim f(x)
2
f x0 . x R,
|x| :
|x|
x0
lim
x x 1 = 1 lim =0 x0 x x
13. f x0 R, . f x0 . f M, x
2. -
30
x0 , f x0 + xx0
lim f(x) = +.
, f M (M > 0), x x0 , f x0 xx0
lim f(x) = .
(, x0 ) (x0 , ) :xx0
lim f(x) = + lim f(x) = lim+ f(x) = +xx0 xx0
xx0
lim f(x) = lim f(x) = lim+ f(x) = xx0 xx0
: lim f(x) = +, f(x) > 0 x0 , lim f(x) = , xx0 xx0
f(x) < 0 x0 . lim f(x) = +, lim (f(x)) = , lim f(x) = , xx0
xx0 xx0 xx0
lim (f(x)) = +. 1 = 0. xx0 f(x) 1 = +, lim f(x) = xx0 f(x)
xx0
lim f(x) = + , limxx0
lim f(x) = 0 f(x) > 0 x0 , limxx0
0 f(x) < 0 x0 , limxx0
1 = . xx0 f(x)
lim f(x) = + , lim |f(x)| = +.xx0
lim f(x) = +, limxx0
k
xx0
f(x) = +.
( ) x0 R f : g : f + g :
R + +
R
+ + +
+ ;
+;
2. -
31
( ) x0 R f : g : f g :
>0 + +
0
0x0 x0 x0
x0
lim f(x) = lim (x2 + 1) = 1 lim+ f(x) = lim+ (2 x) = 2,
f 0.
2 x 1 , x = 1 1, ) f(x) = x1 3, x = 0x1
lim f(x) = lim
(x 1)(x + 1) = lim (x + 1) = 2, f(1) = 3. x1 x1 x1
, .
2. -
33
f x0 , (x0 , f(x0 )) Cf Cf (x0 , f(x0 )). P R.
P . Q
f(x) =
. , . x, x, x (0 < = 1), log x(0 < = 1) .
1 , x
f g x0 , x0 :
f + g,
c f (c R),
f g,
f , g
|f|
f
x0 . f x0 g f(x0 ), g f x0 .
17. Bolzano. f, [, ]. : f [, ] , f() f() < 0, , , x0 (, ) ,
f(x0 ) = 0. , , , f(x) = 0 (, ).
2. - f [, ], , A(, f()) B(, f()) x x, x x .
34
f() x0 x0
B(, f())
O f()
x0
A(, f())
f , f f ( ) f, [, ]. : f [, ] f() = f() , f() f() , , x0 (, ) ,
f(x0 ) = f() f .
18. . f [, ], f [, ] M m, x1 , x2 [, ] , , m = f(x1 ) M = f(x2
),
m
f(x)
M, x [, ].
2. -
35
f [, ] [m, M], m M . f (, ), (A, B),
A = lim+ f(x) B = lim f(x).x x
, , f (, ), (B, A). A, B + . f [, ], [f(), f()]. , , f [, ],
[f(), f()].
2.2
1. , --. ; . . f : A R . f x1 , x2 A x1 = x2 , x1 > x2 x1
< x2 , f f(x1 ) > f(x2 ) f(x1 ) < f(x2 ), f(x1 ) = f(x2 ).
f 1-1. , f . , 1-1 .
y y = g(x) O x
g(x) =
x, x 0 1 , x>0 x
1-1 .
2. -
36
2. f f1 y = x. . C C f f1 ,
f(x) = y f1 (y) = x, M(, ) C f, M (, ) C f1 . , , xOy x Oy . , C
C f f1 y = x xOy x Oy .
3. f, g xx f(x) = lim0
+ lim g(x) = , xx0 xx0
lim (f(x) + g(x)). 1 1 g(x) = 2 , 2 x x 1 = + x0 x2 =0
. f(x) =
x0
lim f(x) = lim
x0
1 x2
= , lim g(x) = limx0
x0
lim (f(x) + g(x)) = lim
x0
1 1 + 2 x2 x
f(x) =
1 1 + 1 g(x) = 2 , 2 x xx0
x0
lim f(x) = lim
x0
1 +1 x2
= , lim g(x) = lim 1 1 +1+ 2 x2 xxx0
1 = + x0 x2 = 1.
x0
lim (f(x) + g(x)) = limxx0
x0
lim f(x) = + lim g(x) = , lim (f(x) + g(x)).xx0
4. f [, ]. f [, ] f() = f() , f() f() , x0 (, ) ,
f(x0 ) =
2. -
37
. f() < f(). f() < < f(). g(x) = f(x) , x [, ], : g [,
] g() g() < 0, g() = f() < 0 g() = f() > 0. , Bolzano, x0
(, ) , g(x0 ) = f(x0 ) = 0 f(x0 ) = .
2.3 - .
1. y2 = 4x y x. 2. f f(x) = 1x+ x 5.
3. f : R R , :f2 (x2 ) + f(2x ) + 1 = 0 x R
4. , x x .
5. , A(2, 1) B(2, 3).
6. .y
O
x
7. f , Cf x x . 8. Cf y y.
9. f 1 2 , f 1 2 . 10. f , f 1-1. 11. 1-1 .
2. -
38
12. 1-1. 13. 1-1. 14. f : (, ) R , Cf .
15. lim f(x) f.xx0
16. lim f(x) x0 xx0
f.
17. lim f(x) = lim f(y).x y
18. f(2) = 3, lim f(x) = 3.x2
19. f(x) =
x2 7x + 12 +
x 3, lim f(x) = 0.x3 x1 x1
20. f (1, +) lim+ f(x) = 2, lim f(x) = 2. 21. limf(x) = 1
lim+f(x) = 2, lim f(x) .xx0 xx0 xx0
22. lim [f(x) + g(x)], lim [f(x) + g(x)] = lim f(x) + lim
g(x).xx0 xx0 xx0 xx0
23. lim [f(x) + g(x)] , lim f(x), lim g(x) .xx0 xx0 xx0
24. lim |f(x)| = 5, lim f(x) = 5.xx0 xx0
25. 1 x
f(x)
x2 + |x| x R, lim f(x) .x0
26. x x2
5 , 6 , g(x) 2
f(x)
h(x) lim g(x) = lim h(x) = , x2 x2
lim f(x) = . |x 1|, lim f(x) = 0.x1
27. x R, |f(x)|
28. x (1, 0) (0, 1), f(x)
|x|, lim f(x) = 0.x0
29. x (1, 0) (0, 2), 0 < f(x) 30. < 0 f(x) =.
x2 , lim f(x) = 0.x0 x
x2 + x +
x2 + 2 lim f(x)
2. -
39
31. lim
x = + lim f(x) = 0 x+ f(x) x+ f(x) = lim f(x) = + x x x
32. lim
33. f x0 , x0 . 34. f x0 , lim f(x).xx0
35. f [, ] f(x) = 0 x (, ), f() f() 36. f : R Q.
0.
37. f : [0, 4] R f([0, 4]) = (0, 4) [0, 4].
38. f : R R |f(x)| x0 = 0.
|x| x R
39. f = [0, 1] f(0) = 1 f(1) = 0. ;
f(x) x [0, 1] 1 . x0 [0, 1] , f(x0 ) = . 2 3 . [0, 1] f() = . 2
. f() = [0, 1]. [0, 1] f(x) = x (0, 1).
. [0, 1] , f()
40. f() f .
41. f + g f g, f g .
42. f g R, g R f2 (x) = g2 (x) x R, f R.
43. f, g h h(x)xx0 xx0
f(x) lim h(x) = 1 lim g(x) = 2 , 1 , 2 R, 1
xx0
g(x) x0 , lim f(x) 2 .
44. f() f .
45. f : R x0 , g, g(x) = (f(x))2 , x , x0 .
2. -
40
46. f R, f()f()f() < 0, , , R, x0 R, f(x0 ) = 0.
47. f [, ], x0 [, ], f(x0 ) =x0
f(x)
+
2
.
x [, ],
48. lim+ f(x) = 0, lim R.
1 = +. x0+ f2 (x)
49. f R, f f 50. f1 , f f1 y = x.
51. f g R f g R, f R.
52. f , f f .
53. f(x) > 0 x R lim+ f(x), x0 x0
lim+ f(x) > 0.
54. lim+f(x) = 0, lim+xx0
1 = +. xx0 f(x)
55. f [, ], f [, ].
56. f : R R f1 , f1 f.
57. f : R R R, f1 R.
58. f (0, +) f(x) lim f(x) = 0.x+
1 x (0, +), x2
59. f . 60. f : R R R f , .
61. f [, ] f(x0 ) = 0,x0 (, ), f()f() < 0.
2. -
41
62. lim f2 (x) = +, lim f(x) = + lim f(x) = .x0 x0 x0
63. f , f() .
64. f : R R R . 65. f : [, ] R , .
66. f , f f .
2.4 1. f : A R, g : B R. g f :. A B = . f(A) B = . f(A) B . f(A)
B = . A f(B) =
2. f(x) =
3, x > 14 (f f)(15) = 5, x 14. 3 . 5 . 8 . 9
. 15
3. f Df = R , y = Cf :. . .
4. f(1 x2 ) =. 3
1 x2 , x = 0, f x2.
2
1 = 2 2 . 2
.
3 4
. 1
5. f(x) = ex g(x) = ln x, (f g)(x) =. e ln x . 1 . x . ex .
e
6. f(x) = 1 + , x = 0 (f g)(x) = x, g(x) =.
1 x
x2 x+1
.
x x1
.
x x+1
.
1 1x
.
1 x1
2. -
42
7. f(x 1) = 2x2 3x + 1, f(x) =. 2(x 1)2 3(x 1) + 1 . 2x2 + 5x +
3 . 2(x + 1)2 3(x + 1) + 1 . . 2x2 + 3
8. f(x) [0, 2], g(x) = f(ln x) . [e, 2 ln e] . [1, ln 2] .
[1,
e]
. [1, e2 ]
. [1, ln 2e ]
9. f(x) = 2ex f1 (x) =. ln
2 x
. 2ex
.
1 ln x 2
.
ln x
. ln
x 2
10. f(x) = ln1 . 2
x2 9 x2
, 0 < x < 3, f1 (0) = 3 2 . 2
3 . 4
. 3
.
3
11. f(x) = x5 + 2 f1 (x) =1 5 +2 x 1 . 5 x2. .
5
x+2
.
5
x2
.
12. f(x) =
x2 + x
x2 x, lim |f(x)| =x0
. 0
. 1
.
.
13. lim f(x) = 5, lim fx4 x2
x2 4 x1
=.
. 4
. 2
. 5
5 2
. 10
14. f(x) = : .
x, .
x x+
lim f(x)
x+ x
lim f(x)
. lim f(x)x
. lim f(x)
. lim f(x)
15. .
2. - . f x0 , lim f(x).xx0
43
. . . f [, ] f() f()
0, f (, ).
. x0 (, ) f(x0 ) = 0, f() f() < 0.
. f 0 f(), f(x) = 0 .
x+43 4 x = 5 16. f(x) = . f x0 = 5, c = x5 c x = 5.
1 6
. 0
.
1 6
. 1
. 6
17. lim. 0
x1 x
ln(x) ln x = 1. lim = x0 2 x 1. 1 .
1 2
.
1 2
.
18. lim
x4 = x+ 4 3 x.
1 3
. 1
19. f : R R, x3 f(x) f(0) . 0 . 1 . + .
. +
. 0
.
1 3
x x x R.
1 6
.
2 3
20. f(x) = x2 + x + , , R . = 0 . < 0 . > 0 . = 0 .
2. -
44
2.5 1. .
ex ex ) lim x0 x) lim
i) 1 ii) 2 iii) 3 iv) v)1 3 1 2
x + x x0 2x 2x
ln x ) lim 3 x1 x 1) lim
ex1 x x1 (x 1)2
vi) 0 vii) +
2.6
1. f.. : ) Df1 = . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . ) Rf1 = . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . ) f1 (1) = . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . ) f1 (. . . . . . .
. .) = 3 . Cf Cf1 .
2 1
O3 2 1 1 2 1 2 3
2. f g , g f, . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . = .
3. N f(x) =
1 , x
2. -
45
4. f f(x) =
ln x , N , , . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . A = [1, 3] {4}. i.
1
5. f . :x1+ x1
lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
ii. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
. iii. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
. . .x0 2 1
vi. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
.x1
v. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
.x1+
1
O
1
2
3
4
vi. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . .
. . .x2
vii. f(2) = . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . i. lim f(x) x1
ii. lim f(x) = f(2)x2
iii. lim f(x) = 2x4
vi. lim f(x) (1, 3).x
6. :i. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . .
. .x1
ii. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . .
.x1+ 3 2 1 1 O 1
iii. lim f(x + 1) = . . . . . . . . . . . . . . . . . . . . .
.x2
vi. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . x
2
v. lim f(x) = . . . . . . . . . . . . . . . . . . . . . .
.x2
2
1
2
3
vi. lim f(3x 7) = . . . . . . . . . . . . . . . . . . . .
.x3
7. :
2. -
46
. :x1
3 2 1 1 O 1
i. lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . ii.
lim f(x) = . . . . . . . . . . . . . . . . . . . . . . .x2
iii. lim f(x) = . . . . . . . . . . . . . . . . . . . . . .
.x4
2
1
2
3
4
. : i. lim
1 x1 f(x). + . . 1 . 0 .
ii. lim
1 x2 f(x)
iii. lim
1 x4 f(x)
. +
.
.
1 2
. 0
.
. 0
8.
. +
.
.
1 4
.
:
x2 1 =........................ x1 f(x) x2 3 ii. lim =. . . . . .
. . . . . . . . . . . . . . . x2+ f(x) x + 4 f(x) =..............
iii. lim x2+ |x| + 2i. lim vi.x2
3 2 1
4
3
2
1
O
1
2
3
lim [(x 1) f(x) + x2 ] = . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9. :
2. - . . . .
47
x
lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . .
1 =........................ x f(x) limx1
lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . . lim
f(x) = . . . . . . . . . . . . . . . . . . . . . . . .1
2
x1+
1 =. . . . . . . . . . . . . . . . . . . . . . . . . . lim x1
f(x). .x+
O
lim f(x) = . . . . . . . . . . . . . . . . . . . . . . . .
1 =........................ x+ f(x) lim
10. f , . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
11. .) lim ) ) ) )x0
x =
..................................................................
x3 1 =
............................................................... lim
x x x 1 lim+ x =
............................................................ x0 x 1
lim x2 3 = . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x+ x 1 lim x =
.................................................................
x0 x
12. f(x) = ex f1 (x) = . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 13. f f(x0 ), x0 14. [, ] f , . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
15. f : R f1 , f(f1 (x)) = x x . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
16. f 2 x + x + , x < 2 f(x) = x+2 x+2 e + , x 2
2. -
48
,
i) = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . ii) = . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
3 3.1 -
1. f A(x0, f(x0)) Cf lim ;
f(x) f(x0 ) xx0 x x0
f(x) f(x0 ) xx0 x x0 , Cf A , f A(x0 , f(x0 )) Cf . lim :
y f(x0 ) = (x x0 ). Cf f x0 . A(x0 , f(x0 )) f , x0 ( ), .
2.
x0 ; f x0 ;
f x0 ,
49
3.
50
f(x) f(x0 ) xx0 x x0 lim . f x0 f (x0 ). :
f (x0 ) = lim
xx0
f(x) f(x0 ) . x x0
f (x0 ) = lim
h0
f(x0 + h) f(x0 ) . h
x0 f, f x0 , R xx0
lim
f(x) f(x0 ) f(x) f(x0 ) lim x x0 x x0 xx+ 0
. , t0 , x = S(t) t0 ,
(t0 ) = S (t0 ). Cf f A(x0 , f(x0 )) f x0 .
y f(x0 ) = f (x0 )(x x0 ) f (x0 ) A(x0 , f(x0 )) Cf A f x0 . f
x0 , . f x0 , x0 .
3.
51
3. ; ; f A. : f , x0 A. f ( A), x0 .
4. ; f 1 . x 1 f (x), f : 1 R x f x. f f
df f (x). dx .
1. (c) = 0 2. (x) = 1 3. 4. (x ) = x-1 1 x = 2 x
1 2 x 1 8. (x) = 2 x x x 9. (e ) = e 7. (x) = 10. (x ) = x ln
11. (ln |x|) = 1 x
5. (x) = x 6. (x) = x
f, g x0 , f + g x0
(f + g) (x0 ) = f (x0 ) + g (x0 ) f, g , x
(f + g) (x) = f (x) + g (x)
3.
52
f, g x0 , f g x0
(f g) (x0 ) = f (x0 )g(x0 ) + f(x0 )g (x0 ) f, g , x :
(f g) (x) = f (x)g(x) + f(x)g (x) f c R, (c) = 0, :
(cf(x)) = cf (x) f, g x0 g(x0 ) = 0,
f x0 g f g
(x0 ) =
f (x0 )g(x0 ) f(x0 )g (x0 ) g2 (x0 )
f, g , x :
f g
f (x)g(x) f(x)g (x) (x) = g2 x
g x0 f g(x0 ), f g x0
(f g) (x0 ) = f (g(x0 )) g (x0 ) g f g(), x :
((f g)(x)) = f (g(x)) g (x)
3.
53
, . , .
5.
y x
x0 , x, y y = f(x); x, y y = f(x), f x0 , y x x0 f (x0 ).
6. Rolle. f : [, ] (, ) f() = f()
M(, f()) B(, f()) A(, f()) O
, , (, ) , f () = 0. , (, ) , f M(, f()) x.
7.
. f : [, ] (, ) , , (, ) f() f() , f () = .
M(, f())
B(, f())
A(, f()) O
3.
54
, (, ) , Cf M(, f()) A(, f()) B(, f()). f . f f (x) = 0 x , f .
f, g f, g f (x) = g (x) , c x :
f(x) = g(x) + c . f, . f (x) > 0 x , f . f (x) < 0 x , f .
f : R R f (x) > 0 x R, f R. , f : R R f (x) < 0 x R, f R. R,
f (x) 0 ( f(x) > 0). , R, f (x) 0 ( f(x) < 0).
; f, A, x0 A
8.
3. , > 0 ,
55
f(x)
f(x0 ) x A (x0 , x0 + ).
x0 , f(x0 ) f.
9.
;
f, A, x0 A , > 0 ,
f(x)
f(x0 ) x A (x0 , x0 + ).
x0 , f(x0 ) f.
() . () , , , . () . , .
10. ; f f. f . (Fermat1 ) f x0 . f x0 ,
f (x0 ) = 0 f : R R x0 , f (x0 ) = 0.
3.
56
. , f x0 x0 , . f(x) = x3 f (0) = 0, x0 = 0 .
11.
f ; f : 1. f . 2. f ( ) f . 3. ( )
() f : R R f. () f : R R f (x) = 0 x R, f .
12. f ; f f . f (, ), x0 , f .
() f (x) > 0 (, x0 ) f (x) < 0 (x0 , ), f(x0 ) f. () f (x)
< 0 (, x0 ) f (x) > 0 (x0 , ), f(x0 ) f. () f (x) (, x0 ) (x0
, ), f(x0 ) f (, ).
3.
57
() f (, ) x0 (, ). f x0 f A(x0 , f(x0 )), A(x0 , f(x0 )) . () f
(, ), (, ).
0 () f . f (x) (. f (x) 0) f(x) = 0 x0 , f (. ) .
13. f : [, ] R; f , . : 1. f. 2. f . 3. f. ; ; f . : f , f . f ,
f . , f (. ) , f (. ) , .
14.
3.
58
. f(x) = ex , g(x) = ex h(x) = x2 R. , f , g h .
f(x) = x2 f(x) = ex O O f(x) = ex O
f . f (x) > 0 x , f . f (x) < 0 x , f . . , f(x) = x4 . f
(x) = 4x3 R, f(x) = x4 R. , f (x) R, f (0) = 0. f : R R f (x) >
0 x R, f R. , f : R R f (x) < 0 x R, f R.
15.
f;
f (, ), x0 . f (, x0 ) (x0 , ), , Cf A(x0 , f(x0 )),
3.
59
A(x0 , f(x0 )) f. A(x0 , f(x0 )) Cf , f x0 x0 . Cf . A(x0 , f(x0
)) f f , f (x0 ) = 0. f . f ; f : 1. f . 2. f f (, ) x0 (, ). f x0
Cf A(x0 , f(x0 )), A(x0 , f(x0 )) .
16.
x = x0 f; lim f(x), lim f(x) + , xx+ 0 xx 0
17.
x = x0 f.
18.
y = f +; ;x+
lim f(x) = , ( lim f(x) = ), y = x
f + ( ). y = x + f +; ; y = x + f +, , x+
19.
lim [f(x) (x + )] = 0,
3.
60
x
lim [f(x) (x + )] = 0.
y = x + f +, ,
f(x) = R lim [f(x) x] = R, x+ x x+ lim
f(x) = R lim [f(x) x] = R x x x lim 2 .
P(x) , Q(x)
, .
20. f; f : f . , f . +, , (, +), (, ). x = x0 , .
3. . f : R R
61
f(x) =
1 , x
0,
x=0 x=0
O
x = 0 (0, 0) Cf . . , , .
lim f(x) = 0, lim g(x) = 0, x0 R {, +} limxx0 xx0
f (x) xx0 g (x)
( ),
f(x) f (x) = lim . xx0 g(x) xx0 g (x) lim lim f(x) = +, lim g(x)
= +, x0 R {, +} xx0 xx0 xx0 g (x)
lim
f (x)
( ),
f(x) f (x) = lim . xx0 g(x) xx0 g (x) lim
+ , , . +
L Hospital2 -
3.
62
, . : 1. f. 2. f . 3. f f . f f, f f . 4. ( , .) 5. f f. Cf f. ,
f A , Cf y y, , Cf O. , x A, x 0. T , Cf T .
3.2
1. f x0, . . x = x0 :
f(x) f(x0 ) =
f(x) f(x0 ) (x x0 ), x x0
3.
63
xx0
lim [f(x) f(x0 )] = lim
f(x) f(x0 ) (x x0 ) x x0 f(x) f(x0 ) lim (x x0 ) = lim xx0 xx0 x
x0 = f (x0 ) 0 = 0,xx0 xx0
f x0 . , lim f(x) = f(x0 ), f x0 .
2. f(x) = c, c R R f (x) = 0. . x0 R, x = x0
f(x) f(x0 ) cc = = 0. x x0 x x0,
f(x) f(x0 ) = lim 0 = 0, xx0 xx0 x x0 lim f (x) = 0.
3. f(x) = x R f (x) = 1.. x0 R, x = x0
f(x) f(x0 ) x x0 = = 1. x x0 x x0,
f(x) f(x0 ) = lim 1 = 1, xx0 xx0 x x0 lim f (x) = 1.
f (x) = x1 .
4. f(x) = x, N R. x0 R, x = x0
x x (x x0 )(x1 + x2 x0 + . . . + x1 ) f(x) f(x0 ) 0 0 = = = x1
+x2 x0 +. . .+x1 , 0 x x0 x x0 x x0
3.
64
,
f(x) f(x0 ) = lim (x1 +x2 x0 +. . .+x1 ) = x1 +x1 +. . .+x1 = x1
, 0 0 0 0 0 xx0 xx0 x x0 lim f (x) = x1 .
5.
f(x) =
x (0, +)
1 f (x) = . 2 x x x0 x + x0 x x0 (x x0 ) = = x x0 (x x0 )( x +
x0 ) (x x0 ) x + x0 1 = , x + x0 1 1 f(x) f(x0 ) = lim = , xx0 xx0
x x0 2 x0 x + x0 lim
. x0 (0, +), x = x0
f(x) f(x0 ) = x x0
,
f (x) =
1 . 2 x0
f(x) = x R f (x) = x. . x R h = 0
6.
f(x + h) f(x) (x + h) x x h + x h x = = h h h h h 1 + h . = x h
h,
h = 1 h0 h lim
h 1 =0 h0 h lim
f(x + h) f(x) = x 0 + x 1 = x, h0 h (x) = x. lim f(x) = x R f
(x) = x.
7.
3. . x R h = 0
65
f(x + h) f(x) (x + h) x x h x h x = = h h h h 1 h = x h . h
h,h0
lim
h = 1 h
h0
lim
h 1 =0 h
f(x + h) f(x) = x 0 x 1 = x, h0 h (x) = x. lim
8.
f, g x0 , f + g x0 (f + g) (x0 ) = f (x0 ) + g (x0 ). . x = x0 ,
:
(f + g)(x) (f + g)(x0 ) f(x) + g(x) f(x0 ) g(x0 ) f(x) f(x0 )
g(x) g(x0 ) = = + . x x0 x x0 x x0 x x0 f, g x0 , :
xx0
lim
f(x) f(x0 ) g(x) g(x0 ) (f + g)(x) (f + g)(x0 ) = lim + lim xx0
xx0 x x0 x x0 x x0 = f (x0 ) + g (x0 ), (f + g) (x0 ) = f (x0 ) + g
(x0 ).
9.
f(x) = x , N R f (x) = x1 .
. x R :
(x ) =
1 x
=
x1 (1) x 1 (x ) = = x1 . (x )2 x2
3.
66
10.
f(x) = x R
{x|x = 0} f (x) =
12 x
.
. x R {x|x = 0} :
(x) =
x (x) x x (x) x x + x x = = x 2 x 2 x 2 2 1 x + x = = 2x 2 x
11. 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 )
f ( ) =
f(x2 ) f(x1 ) . x2 x1
(3.1)
, f () = 0, , 3.1, f(x1 ) = f(x2 ). x2 < x1 , f(x1 ) = f(x2
). , , f(x1 ) = f(x2 ). . ,
f(x) =
1, x < 0 1, x>0
f (x) = 0 x (, 0) (0, +), f (, 0) (0, +).
12. f, g . f, g , f (x) = g (x) x , c x : f(x) = g(x)+c.
3.
67
. 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.
13.
f, . f (x) > 0 x , f . . . x1 , x2 x1 < x2 . f(x1 ) <
f(x2 ). , [x1 , x2 ] f ... , (x1 , x2 ) , f () =
f(x2 ) f(x1 ) , x2 x1
f(x2 ) f(x1 ) = f ()(x2 x1 ). f () > 0 x2 x1 > 0, f(x2 )
f(x1 ) > 0, f(x2 ) > f(x1 ).
14.
f x0 . f x0 ,
f (x0) = 0
. x0 f , > 0 , (x0 , x0 + )
f(x)
f(x0 ), x (x0 , x0 + ).
(3.2)
, , f x0 ,
f (x0 ) = limxx0
f(x) f(x0 ) f(x) f(x0 ) = lim+ . x x0 x x0 xx0
, x (x0 , x0 ), , 3.2,
f(x) f(x0 ) x x0 0
0, (3.3)
f (x0 ) = limxx0
f(x) f(x0 ) x x0
3.
68
x (x0 , x0 + ), , 3.2,
f(x) f(x0 ) x x0 0
0, (3.4)
f (x0 ) = lim+xx0
f(x) f(x0 ) x x0
, 3.3 3.4 f (x0 ) = 0.
3.3 -1. f A(x0 , f(x0 )) Cf . : ) Cf A(x0 , f(x0 )) Cf . ) f x0
, Cf A(x0 , f(x0 )). ) f x0 f x0 . ) f(x) = x + Cf .
2. f x0 , x0 . 3. f x0 , x0 . 4. f x0 , x0 . 5. f + g f x0 , g
x0 .
6. (f + g) (x0 ), f (x0 ) g (x0 ). 7. 8. f(x) = 7
= . 7
1 1 , f (x) = . g(x) g (x)
9. f Rolle, 1-1.
10. Rolle, .
11. f, g f (x) = g (x) x , Cf Cg .
3.
69
12. f (x) = 0 x (, 0) (0, +), c R f(x) = c x R .
13. f, g f (x) = g (x), x [, ], g (x) = f (x) + c,x [, ].
14. f, g f (x) = g(x), x R, f (x) g(x).
15. f . . ) f , f (x) < 0 x . ) f (x) = g (x) + 3, x , h(x) =
f(x) g(x) . ) f = [, +) f() = f () = 0, . f ) , f R.
y Cf O x
) f [, ], f() f() < 0 f (x) < 0 x [, ], f(x) = 0 (, ).
16. f f (x0 ) = 0 x0 , f x0 .
17. f x0 , f x0 .
18. f x0 , f x0 .
19. f .
20. f , f (x) = 0 x , .
21. f, g x0 , f + g x0 .
22. x0 , x0 .
3.
70
23. f f (x0 ) =0 x0 , Cf (x0 , f(x0 )).
24. (, ) f (x) = 0 x (, ), Cf .
25. . 26. 2 . 27. P(x) P(x) Q(x) Q(x) ..
28. f (, ], 29. +. 30. . 31. f g x0 , f g x0 ,
(f g) (x0 ) = f (x0 )g(x0 ) + f(x0 )g (x0 )
32. P(x), > 2, P (x) 0 x R, P(x) .
33. f Rolle [, ], f x = x0 (, ) f (x0 ) = 0.
34. Cf f R , f 1-1.
35. f [, ], f , f ()f () < 0, f 1-1.
36.
P(x) . Q(x)
37. f g f f g x0 , f(x0 ) = 0, g x0 .
38. f y = x + f , f(x) x + x .
3.
71
39. f R 1-1, x0 R, f (x0 ) = 0.
40. f R f (x) f .
0 x R,
41. Cf f , Cf .
42. f R R, f (x) > 0 x R.
43. f R, 1-1, f (x) = 0 x R.
44. f R, f2 R.
45. f Rolle [, ], f Rolle [, ].
46. f(x) = |x|, (f(0)) = 0. 47. f : R R 1-1 , f1 .
48. f(x) = x4 , (0, 0) f.
49. f R, f(x)f (x) > 0 x R, f2 .
50. f (x)
0 x R f (x) = 0 x R, f .
51. f , f .
52. Cf f x + y = x + , Cf .
53. f : R R f (x) = 0 x R, f 1-1 R.
54. f [, ] f (x) = 0 x (, ), f [, ].
55. f f (x) = 0 x , f .
3.
72
56. f .
x0 f (x0 ) = f (x0 ) = 0, f x0
57. f + g x0 , f g x0 .
58. f (x) > 0 x A, f A. 59. f 1-1 Rolle [, ].
60. f [0, 1], f (x) =f(1) f(0) (0, 1).
61. f . 62. f f(x) f(x0 ) , x0 R xx0 x x0 lim f x0 .
63. f x0 R, x0 f (x0 ) (f(x0 )) . x0 A, f (x0 ) = 0.
64. f : A R , 65. f(x) = x2 (1 x)2 . 66. f(x) = x2 , x [1, 1]
f(x0 ) = 1, f (x0 ) = 0. 67. limex 1 = +. x0 x2
68. f : R R, f (x) = 10 x R. 69. f . () (), . ) f , f (x) > 0
x . ) f x0 , f (x0 ) = 0. ) f .
) f [, ] .
3. ) f (x) = 0 x , f(x) = 0 x .
73
70. f , f (x) x .
0
71. f(x) = x + x R. 72. . 73. f x x, f .
74. f , f .
75. .
3.4 1. f f (x0 ) = 0 x0 , Cf A(x0 , f(x0 )) :. y = f(x0 ). . y =
x. . x = x0 . . y y.
2. f, g R. A(1, 2), :. f (1) = g (1) . f (1) = g (1) : . . . .
.
. f(1) = g(1)
. f, g x0 = 1
3. R f, g f (x) = g (2x), x R, : . f(x) = g(2x) + c . f(x) = .
f(x) = 2g(2x) + c . f(x) = g(x) + 2c . f(x) = g(2x) + 2c
1 g(2x) + c 2
4. . f : R R,
3.
74
) . ) x0 , f (x0 ) = f (x0 ). ) , f (x) = 0 x R, 1-1. ) R .
5. f(x) = e2x , f (0) :. 1 . 0 . e2 . 2 . e
6. f(x) = x3 4x, Cf A(1, 3) x x : . 30 . 45 . 120 . 135 .
150
7. f(x) = x3 A(1, 1) :. y = 3x + 2 . y = 3x 2 . 3x + 2 . y = 3x
2
8. . : f R, f R. : f : [, ] R , . : f : R R 1-1 , f1 . : f , f f
.
9. (), (), () (), f, (1), (2), (3), (4) (5), f .
O
O
O
O
3.
75
O 1
O 2
O 3
O 4
O 5
10. f : R R f(3x 5) = x2 + x x R, f (3x 5) . 2x + 1 . 3(2x + 1)
.
2x + 1 2
.
2x + 1 3
. 3
11. f(x) = ;
x2 + x + 2 , R. x1
. Cf x = 1. .x
lim f(x) =
. Cf + y = 1 .
f(x) =1 x x . lim (f(x) x) = 2 limx2
12. f(x) = ex , f (0) =. 2 . 2 .
2 e
. 0
. 4
13. lim
x0
1 ln x
2+x 2
=. ln 2 .
. 0
1 2
.
1 ln 2
. +
14. Cf : f(x) = ex ln x (1, f(1)) :. y = e x . y = ex + 1 . y =
e(x 1) . y = e x + 1 . y = x 1
15. f(x) = ln(ln x), x > 1, f (x) =.
1 x
.
1 ln x
.
ln x x
. x
.
1 x ln x
3.
76
16. y = 3x + k Cf : f(x) = x3 , k =. 1 . 0 . 3 . 4 . 2
17. f(x) = xx , x > 0, f (x) =. x xx1 . xx x ln x . xx + x .
xx .
x + x ln x x
x + x ln x 2
18. f (x) = (x 1)3 (x 2)4 x R, f :. (..) x0 = 1 . .. x1 = 1 ..
x2 = 2 . . (..) x0 = 1 . .. x1 = 1 x2 = 2
19. f(x) = x4 4x3 + 6, x [1, 4], f :. 1 . 0 . 3 . 6 .
20. f(x) =
3x 3x, x , , Cf : 2 2 . , 2 3 , 3 ,0 . , 3 . (0, 3) . . 3 6 6 3
1 ln x.
21. lim
x0
x + x 2.
= 2. ln + ln .
+ 2
ln() 2
.
3.5 1. f, R, + y = 3x + 4. . f(x) ) lim x+ x)x+
lim [f(x)4x] xf 2 x
i) ii) 6 iii) + iv) 3 v) 1
) lim
x0+
)
f(x) + 4x x+ 2f(x) + x lim
3.
77
3.6 1. , : . (f + g) (x0 ) = . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . (f g) (x0 ) = . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
f g
(x0 ) = . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
f g x0 () g(x0 ) = 0.
e2x 2x 1 2. lim =
...................................................................
x0 x2 9x2 + 4x + 3x = . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 3. limx
4. [3(5x)] = . . . . . . . . . . . . . , f, , , . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . f .
5. f, f(x) = 6. limx+
x, =
x
2 x
=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 1 x = 0 x
7. f : R R f (x) = 8. > 1, x
f f(x) = . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2x + 1 x R, = . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
9. , .. y = x + . . . . . . . . . . . . . . . y = x + Cf +, = .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
=..............................................................................
10. f, f(x) = x3 + x2 + x + . : ) f(x) = 0 . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . ) f . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . ) f . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.
78
) f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . , , , x1 .
x 8 Cf
O
x1
3
y
4 4.1 -
1.
f ; .
f F F (x) = f(x), x . . f . F f , G(x) = F(x) + c, c R, f G f
G(x) = F(x) + c, c R.
2. f ;
79
4.
80
f f , f(x)dx . ,
f(x)dx = F(x) + c, c R, F f . f(x)dx = F(x) + c, c R , . . f , ,
f.
f ,
f (x)dx = f(x) + c, c R. . c . . x .
1. 2. 3. 4. 5.
0dx = c 1dx = x + c 1 dx = ln |x| + c x x+1 x dx = + c, = 1 +1
xdx = x + c
6. 7. 8. 9. 10.
xdx = x + c 1 dx = x + c 2 x 1 dx = x + c 2 x ex dx = ex + c x
dx = x +c ln
4. f g ,
81
f(x)dx = f(x)dx, R (f(x) + g(x))dx = f(x)dx + g(x)dx
3. . . :
f(x)g (x)dx = f(x)g(x) f (x)g(x)dx :
P(x)ex dx P(x) ln(x)dx
P(x)(x)dx ex (x)dx
P(x)(x)dx ex (x)dx
P(x) x R . . . :
4.
f(g(x))g (x)dx =
f(u)du, u = g(x) du = g (x)dx
:
f(x + )dx, f x +
x2 x2
+ x +
dx, 2 4 > 0
P(x) x P(x) , 2 2 4 > 0. + x +
5. f [, ]. f [, ]. = x0 < x1 < x2 < . . . < x = [,
]
4.
82
x =
. k [xk1 , xk ], k {1, 2, . . . , }, ( Riemann3 )
S = f(1 )x + f(2 )x + . . . + f( )x =k=1
f(k )x
S + R, k f f(x)dx.
f(x)dx = lim
+
f(k )xk=1
f(x)dx , f(x)dx . :
f(x)dx = f(x)dx = 0
f(x)dx
f(x) 0 x [, ], f(x)dx f, x x x = x = . f(x) 0, x [, ], f, g [, ]
, R.
f(x)dx
0.
f(x)dx =
f(x)dx
[f(x)
+ g(x)]dx =
f(x)dx +
g(x)dx
[f(x)
+ g(x)]dx =
f(x)dx +
g(x)dx
4. f , , ,
83
f(x)dx =
f(x)dx +
f(x)dx
f [, ]. f(x) 0 x [, ] f ,
f(x)dx > 0
f [, ] f(x) :
0 x [, ],
f(x)dx = 0 f(x) = 0 x [, ]
f , x
F(x) =
f(t)dt, x ,
f . :x
f(t)dt
= f(x), x .
:g(x)
f(t)dt
= f(g(x)) g (x),
. :
x
f(t)dxx
=
f(t)dt
= f(x)
4.
84
F F(x) = h(x)
h(x) g(x)
f(t)dt : g(x) h(x)
F (x) =g(x)
f(t)dt +
f(t)dt
=
f(t)dt +
f(t)dt
= f(g(x))g (x) + f(h(x))h (x)
( ) f [, ]. G f [, ],
f(t)dt = G() G()
f(x)g (x)dx = [f(x)g(x)]
f (x)g(x)dx,
f , g [, ].
u2
f(g(x))g (x)dx =
f(u)du,u1
f, g , u = g(x), du = g (x)dx u1 = g(), u2 = g().
4.2 f . F f , G(x) = F(x) + c, c R, f G f G(x) = F(x) + c, c R.
. F f ,
1.
F (x) = f(x) x .
(4.1)
4.
85
G(x) = F(x) + c, c R x
G (x) = (F(x) + c) = F (x) + 0 = f(x), G f . G f
(4.1)
G (x) = f(x) x . 4.1 4.2
(4.2)
G (x) = F (x) x , c R ,
G(x) = F(x) + c x .
2.
f [, ] G f [, ],
f(t)dt = G() G()
. f [, ] x [, ], F(x) = f(t)dt f [, ]. G f [, ], c R ,
G(x) = F(x) + c 4.3, x = , G() = F() + c = , x = ,
(4.3)
f(t)dt + c = c, c = G().
G(x) = F(x) + G(),
G() = F() + G() =
f(t)dt + G()
f(t)dt = G() G().
4.
86
4.3 -1. :f (x)dx = f(x) + c 1 x R, (f(x) g(x)) dx = x + c. g(x)
f(x)dx
2. f (x) = 3. : 4. f(x) =
= f(x).
ln x [1, +). 1 + x2
5. f [, ] f(x) > 0 x [, ], f(x)dx > 0.
6. f R, f(x)dx = x f(x) x f (x)dx. 7. f [, ] f(x)
0, x [, ].
f(x)dx
0,
8. f [, ], :
f(x)dx
= f(x), x [, ].
9. f [, ], :
f(x)dx,
f(t)dt,
f(y)dt .
10. f , , , :
f(x)dx + f(t)dt
f(x)dx =
f(x)dx.
11.
x
= f(x). 0
12. f [, ] f(x)x [, ]
f(x)dx > 0, f(x) > 0 x [, ].
13. f [, ] f(x)dx (, ).
f(x)dx > 0, x
f(x)dx >
14. f g [, ]
f(t)dt > x (, ], f(x) > g(x) x (, ].
x
g(t)dt
15. f , f(x)dx = 0. 16. f [, ], f(x)
f(x)dx
0.x2 x1
0 x [, ],
17. f [, ] 1-1 , x1 , x2 [, ] x1 < x2 ,
f (x)dx = 0.
4.
87
18. f [, ] f (x) > 0 x [, ],
f(x)dx > ( )f().
19. f R = .
f(x)dx =
f(x)dx,
20. f , f(x)dx = f(x)dx R
21. f [, ],
f(x)dx = 2 0
f(x)dx.
22. f [, ], ,
f(x)dx.
23. f . 24. f [, ],
f(x)dx
f(x)dx
0.
0, x [1, 3], 2005 1
35. f(x) = ln x, x > 0 F(x) = 36. f [, ] x [, ] f(x) >
0.
ln tdt.
f(x)dx > 0,
37. , x x x = x = + 1. x x 10,
f(x)dx = 10.+1
38. f, g x x [, ],
E=
|f(x) g(x)|dx.
39. f [, ] f(x) = 0 x [, ] , Cf , x x x = x = ,
E=
f(x)dx .
4.
89
4.4 1. f [0, 2], 2 2 0 f (x)dx 1 2 0 f (x)dx
= ,
2 2 1 f (x)dx
= 2
= 6, :. 3 . 2 3 . 2 . 3 2
. 1
2. f(x)ex dx = f(x)ex x ex dx, f(x) :. x . x . x . ex . x
3. f , , , :
f(x)dx +
f(x)dx =
f(x)dx:
. < < . . < < . . , , . . . . .
4.
3 0
f(x + )dx = 4, R . 4 . 4
3+
f(x)dx =.
. 3
. 4 +
5.
3
0
2 x xdx =. 1 .
1 3
. 0
.
1 3
. 1
6. F(x) =.
2x 0
dt , F (x) = 1 t3.2
1 1 x3 0
1 1 2x3
.
2 1 2x3
.
1 1 8x3
.
2 1 8x3
7. I() =
xex dx, > 0, lim I() =+
. 1
.
1 2
. 2
. 1
.
1 2
8.
1 0 (x
+ )ex dx = e, =. e 1 . 1 . 1 e . e + 1 . 1
4.
90
4.5 1. f g [, ], , f g x = x = ,
E() = . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
2.
e 1
ln xdx = . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
3. f g R R g(x)
f(t)dt
=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .
4.
1 ex +1 0 ex +x dx
=
..........................................................................
91
Pierre de Fermat (16011669). . . Leibniz . ( , ) . Pascal . , .
Fermat . Fermat. 1
xn + yn = zn - n > 2. Fermat, ( Bachet), . , , x2 + y2 = z2 ,
Fermat : , , , : . , 1670 Bachet . 1993 Andrew Wiles, Wiles . 1994
Wiles . 300 . 2 Guillaume Franois Antoine de l Hospital (16611704).
. L Hospital . . , L Hospital , . , L Hospital . Johann Bernoulli
(16671748). , . , . L Hospital Johann Bernoulli. L Hospital
Bernoulli , , . 3 Gorg Friedrich Bernhard Riemann (18261866). . , .
Riemann Albert Einstein (18791955) . Riemann . . , . , , Riemann
Gottingen 1846 . , Riemann , . , . Gottingen 1849 . 1851 Riemann
Carl Friedrich Gauss (17771855), , . Riemann Gottingen, Gauss.
Gauss 1855, Dirichlet Gottingen Riemann. ,
92
, 1856 Riemann . Riemann Gottingen Dirichlet , . 1862 Riemann .
, . . Pisa, . , 1863, . , , 1866. Riemann, : .
[1] E. T. Bell. , : Cauchy. . , , 1992. [2] W. Rudin. . . . ,
Leader Books, 2000. [3] . , . , . , . , . . . . . . . . ., , (3 )
http://www.othisi.gr. [4] . . , . [5] . . , , , (2 ). [6] . . , , ,
. [7] . . , , , . [8] . . , , . [9] . , , , 2002. [10] . . , , , ,
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93
