themata ejetasevn
43
ΜΑΙΟΣ 2000 ΜΑΘΗΜΑΤΙΚΑ ΤΕΧΝΟΛΟΓΙΚΗΣ ΚΑΤΕΥΘΥΝΣΗΣ ΘΕΜΑ 2 ο Α. Δίνεται ο μιγαδικός αριθμός 5 + i z = . 2 + 3i α) Να γράψετε τον z στη μορφή α + βi, α, β ∈ IR. Στην παρακάτω ερώτηση να γράψετε τη σωστή απάντηση. δ) Το z 4 είναι ίσο με : Α. 4, Β. 4i, Γ. -4i, Δ. -4. Β. Να βρεθούν τα σημεία του επιπέδου που είναι εικόνες των μιγαδικών z, για τους οποίους ισχύει z - 1 = 1 z - i . ΘΕΜΑ 3 ο Δίνεται η συνάρτηση f, με Α. Να βρεθούν τα Β. Να βρεθούν τα α, β ∈ IR, ώστε η συνάρτηση f να είναι συνεχής στο x 0 = 5. Γ. Για τις τιμές των α, β του ερωτήματος Β, να βρείτε το ΘΕΜΑ 4 ο Φάρμακο χορηγείται σε ασθενή για πρώτη φορά. Έστω f (t) η συνάρτηση που περιγράφει τη συγκέντρωση του φαρμάκου στον οργανισμό του ασθενούς μετά από χρόνο t από τη χορήγησή του, όπου t ≥ 0. Αν ο ρυθμός μεταβολής της f (t) είναι 2 2 2 5 - x x - 8x + 16, 0 < x < 5 f (x) = (α + β ) (x - 5 + e) + 2(α + 1)e , x 5 n ⎧ ⎪ ⎨ ≥ ⎪ ⎩ A - + x 5 x 5 f (x) , f (x) im im → → A A x + f (x). im →∞ A 8 - 2 t + 1 . α) Να βρείτε τη συνάρτηση f (t). β) Σε ποια χρονική στιγμή t, μετά τη χορήγηση του φαρμάκου, η συγκέντρωσή του στον οργανισμό γίνεται μέγιστη; γ) Να δείξετε ότι κατά τη χρονική στιγμή t = 8, υπάρχει ακόμα επίδραση του φαρμάκου στον οργανισμό, ενώ πριν τη χρονική στιγμή t = 10, η επίδρασή του στον οργανισμό έχει μηδενιστεί. (Δίνεται ℓn11 2,4). ≅ 176 ΜΑΘΗΜΑΤΙΚΑ ΚΑΤΕΥΘΥΝΣΗΣ Γ΄ ΛΥΚΕΙΟΥ - ΠΑΝΕΛΛΑΔΙΚΕΣ
Transcript of themata ejetasevn
2000 2 . z =
5+i . 2 + 3i
) z + i, , IR. . ) z4 : . 4, . 4i, . -4i, . -4. . z,
z-1 = 1. z-i
3 f, x 2 - 8x + 16, 0 2, x0 (1 , 2) , f x0 xx. 4 , 1000 . . 4 25 . 200 . 4000 . , 10000 . ) (x)
K (x) = 10 x +
16 + 40 , x
x . ) , ; ) () .
2000 1 . f, . . f(x) > 0 x , f . . f(x) < 0 x , f; . () (). . f (x) = e1 - x . 1 + 3, x , . f, f(x) = -2x + 2 x 2 . . f(x) = g(x) + 3, x , h (x) = f (x) - g (x) . 3 f, ,
f (x) - e2x + 1 im = 5. x 0 2x . f (0). . f x0 = 0. . h (x) = e-x . f (x), f h (0 , f (0)) (0 , h (0)) . 4 , t ,
P (t) = 4 +
t-6 . 25 t2 + 4
. . . . . . . , .
1 A.1. z1, z2.
o
2001
z1 z2 = z1z2. .2. . z : . z2 = z z . z2 = z2 . z = - z . z = z . i z = z.1. z1 = 3 + 4i z2 = 1 + 3 i, , . . 4 . 2 1. z z 1
2. z1 3. z22
2
2
. 25 . - 5 . - 2 . 5 . 10
4. - z1 5. i z2
.2. z z = 1, z = 2
1 . z
x 2, x 3 f f (x) = 1 - e x - 3 ,x>3 x-3. f , = -1/9. . Cf f (4 , f (4)). . f, xx x = 1 x = 2. 3 f, R, : f3(x) + f2(x) + f(x) = x3 - 2x2 + 6x -1 xR, , 2 < 3. . f . . f . . f (x) = 0 (0 , 1).
186
-
4 f, R, : i) f (x) 0, xR
ii) f (x) = 1 - 2x 2 t f 2 (xt) dt , xR.0
1
g 1 g (x) = - x 2 , xR. f (x) . f(x) = -2xf2(x). . g .. f f (x) = . x +
im
( x f (x) 2x ) .
1 1 + x2
2001 1 1. 3 , 7 2. . , . , . , . , . . 1. 1 - , 2 - , 3 - , 4 - , 5 - .
2. z = 1 z
2
= 1 zz = 1 z =
2 . f , x0 = 3, im f (x) = im f (x) = f (3)x 3x 3+
1 . z
x 3
im f (x) = im x 2 = 9 x 3 0 0 L'Hospital
1 - ex - 3 im f (x) = im x 3+ x 3+ x-3 f (3) = 9 9 = -1 = - 1/9. 1 - e4 - 3 . f (4) = =1-e 4-3
=
x 3
im +
- ex - 3 = -1 1
1 - e x - 3 (1 - ex - 3 ) (x - 3) - (1 - ex - 3 ) (x - 3) x > 3 : f(x) = = (x - 3)2 x-3 - e x - 3 (x - 3) - (1 - e x - 3 ) = (x - 3)2 f(4) = - e4 - 3 (4 - 3) - (1 - e4 - 3 ) - e - (1 - e) = = -1 2 (4 - 3) 12
() : y - f (4) = f(4) . (x - 4) () : y - (1 - e) = -1 . (x - 4) () : y - 1 + e = -x + 4 () : y = -x + 5 - e.
1 A.1. f . F f , : G (x) = F (x) + c , c IR f f G (x) = F (x) + c, c IR. .2. , .. . .
o
2001
f (x) dx = ..........................................................
[ f (x) + g (x)] dx = ..............................................
[ f (x) + g (x)] dx = ..........................................
, R f, g [ , ]. .1. f f(x) = 6x + 4, x IR (0 , 3) 2. .2. : 2 1 4 3x . (e x + x) dx, . dx, . 2 (2x + 3x) dx. 0 1 0 x 2o . z + 16 = 4z + 1. . z - 1 = z - i. 3o x 1 x + , , IR. f (x) = -x + 1 ) n(x - 1), x (1 , 2] (1 - e -x + 1 . im 1 - e . x 1 x-1 . IR, f x0 = 1. . = -1, (1 , 2), f ( , f ()) xx. 4o f (0 ,+) x tf (t) 1 + dt, x > 0 . f (x) = 1 x2 x . f (0 ,+). . f (x) = 1 + nx , x > 0. x . f. . f. . f, xx x = 1 x = e.
190
-
1 A. f ' [ , ]. G
o
2002
f [ , ],
f (t) dt = G () - G ().
.1. f (x) = x. f R f(x) = x. .2. . . f [ , ] ( , ], f [ , ] . . , 1 - 1 , . . f x0 im f (x) = 0 , x x0x x0
im f (x) = 0 .
. f R, f (x) dx = x f (x) - x f(x) dx . . im f (x) > 0 , f (x) > 0 x0.x x0
2 z f () = i z, IN*. . f (3) + f (8) + f (13) + f (18) = 0. 3 f, g R . fog 1 - 1. . g 1 - 1. . g (f (x) + x3 - x) = g (f (x) + 2x - 1) . 4 . h, g [ , ].
h (x) > g (x) x [ , ],
h (x) dx >
g (x) dx .
. R f, : f (x) - e-f (x) = x - 1, x R f (0) = 0. i) f f.
x < f (x) < x . f(x), x > 0. 2 iii) f, x = 0, x = 1 xx, 1 1 < E < f (1) . 4 2ii)
2002 1o () ().
.
f (x) dx 0 , f (x) 0, x [ , ].
. f () f . . f IR , [ , ] f Rolle. . f [ , ] x0 [ , ] f . f(x0) = 0. . f [ , ] x0 ( , ), f (x0) = 0, f () . f () < 0. 2oex - 1 , x IR. ex + 1 . f f-1. . f-1(x) = 0 .
f (x) =
. 3o
1 2 -1 1 2
f (x) dx.
f, IR, f (x) =
x-z
2
- x+z2
2
x2 + z
,
z z = + i, , IR 0. . im f (x) im f (x).x + x -
. f, z + 1 > z - 1. . f. 4o f, IR , f(x)f (x) + (f(x))2 = f (x)f(x), x IR f (0) = 2f(0) = 1. . f. . g x g (t) dt = 1 [0 , 1], 2x - 0 1 + f 2 (t) [0 , 1].
. f , f . . f x0 . f x0 f(x0) = 0, f x0. 2 z = + i, , IR w = 3z - iz + 4, z z. . Re(w) = 3 - + 4 m(w) = 3 - . . , w y = x -12, z y = x - 2. . z, y = x - 2, . 3 f (x) = x5 + x3 + x. . f f . . f (ex) f (1 + x), x IR. . f (0 , 0) f f -1. . f -1, x x = 3. 4 f [ , ] ( , ). f () = f () = 0 ( , ), ( , ), f () f () < 0, : . f (x) = 0 ( , ). . 1, 2 ( , ) f(1) < 0 f(2) > 0. . f.
1 . , f x0, . . ; . , . . z z , z = z = -z. . f . f(x) > 0 x , f . . f, , f(x) dx = f (x) + c , c IR.
o
2003
1o A. f . F f , : . G(x) = F (x) + c, c R, f . G f , G(x) = F (x) + c, c R. . , . . z1, z2 , z1 - z2 z1 + z2 z1 + z2 . f ' ( , ), x0, f . f(x) > 0 (, x0) f(x) < 0 (x0 , ), f (x0) f. . f : R 1 - 1, x1, x2 A : x1 = x2, f (x1) = f (x2). . f, g , : f (x) g(x) dx = f (x) g (x) - f(x) g (x) dx . x = x0 f; 2 . () z : z = 2 m(z) 0. . , z (), w = 1 z + 4 2 z xx. 3 f (x) = x 2 + 1 - x. . im f (x) = 0 .x +
2003
. Cf, x -. . f(x) x 2 + 1 + f (x) = 0. 1 1 . 0 x2 + 1 dx = n( 2 + 1). 4 f IR , : f (x) = - f (2 - x) f (x) 0, x IR. . f . . f (x) = 0 . . g (x) = f (x) . Cg f(x) xx, 450 .
1 . f ' x0 . f x0 , f(x0) = 0. . f x0 ; . . . . . im f (x) = , im f (x) = im+ f (x) = .x x0
o
2004
x x0
x x0
. f, g x0, f . g x0 : (f . g)(x0) = f(x0) . g(x0). . f, . f(x) > 0 x , f . . f [ , ]. G
f [ , ],
2 f f (x) =x2 . nx. . f, . . f . . f. 3 g (x) = ex . f (x), f IR f (0) = f (3/2) = 0. . 0 ,
f (t) dt = G() - G().
3 , 20
f() = -f ().. f (x) = 2x2 - 3x, I () = g (x) dx ,
IR. . im () . 4 f: IR IR, f (1) = 1. -
x IR, g (x) =
x3
1
z f (t) dt - 3 z +
1 (x - 1) 0 , z = + i C, z
, IR*, : . g IR g.. z = z +
1 . z
. Re(z2) = - . . f (2) = > 0, f (3) = > , x0 (2 , 3), f (x0) = 0.
208
-
1o A. f . f f(x) = 0 x , f . . , . . f x0 , . . . . f, g IR fog gof, . . C C f f -1 y = x xOy xOy. im im . f x0, x x k f (x) = k x x f (x) , 0 0
2004
f (x) 0 x0, k k 2. . f (, ) [, ]. 2 f: IR IR f (x) = 2x + mx - 4x - 5x, mIR, m > 0. . m f (x) 0, x IR . . m = 10, f, xx x = 0 x = 1. 3 f: [, ] IR [, ] f (x) 0 x [, ] z Re(z) 0, m(z) 0 Re(z) > Im(z). z + 1 = f () z2 + 1 = f 2 () , : z z2 . z = 1 . f2() < f2() . x3 f () + f () = 0 (-1, 1). 4 f [0, +) IR , 1 x2 f (x) = + 2 2x f (2xt) dt. 0 2 . f (0, +). . f (x) = ex - (x + 1). . f (x) [0, +). . im f (x) im f (x).x + x-
212
-
2005 A.1 f, [ , ]. f [ , ] f () f (), f () f () x0 ( , ), f (x0) = . .2 y = x + Cf +; . , . . f [ , ] f () < 0 ( , ), f () = 0, f () > 0. . im [ f (x) + g (x)] , xx0
. f f -1 Cf y = x, f -1. . im f (x) = 0 f (x) > 0 x0, imx x0
x x0
im f (x)
x x0
im g (x).
x x0
1 = +. f (x)
. f
,
(
x
f (t) dt
)
= f (x) - f () ,
x . . f , x x , . 2 z1, z2, z3, z1=z2=z3= 3.. : z1 =
9. z1
z1 z + 2 . z2 z1 1 . : z1 + z2 + z3= z1 z2 + z2 z3 + z3 z1. 3.
3 f, f (x) = ex, > 0. . f . . f, , y = ex. . . () , f,
e-2 . 2 2 () . . im + 2 + yy, () =
4 f IR , 2 f(x) = ex - f (x), x IR f (0) = 0. x . : f (x) = n 1 + e . 2 . N :x 0
im
x
0
f (x - t) dt x
.
. : h (x) =
x
-x
t
2005
h (x) = g (x), x IR. x 1 . t 2005 f (t) dt = -x 2008 (0 , 1). z z 9 2
x 2007 . f (t) dt g (x) = 2007
Re(z1 z2 ) IR 9
z z
9 9 9 z z z z z z z1 z2 z1 + z2= z1 + z1 z1 z1 + = + = + = + = 2 Re 1 IR 9 z2 z1 z2 z2 z2 z2 z2 z2 2 1 z1 w-w=0 2Im(w) = 0 w IR
3 w=w
z1 z z + 2 = 1 z2 z1
216
-
1 A.1 f, f (x) =
o
2005
x . f (0, +) f(x) = 1 . 2 x .2 f : A IR 1 - 1; . , . . , f 0, f . . f ( , ) xo. f ( , xo) (xo , ) , (x0 , f (x0) f. . . . f , g fog gof, fog gof. . z, z xx. . f IR*, f (x) dx = f (x) dx . 2 . z1, z2 z1 + z2 = 4 + 4i 2z1 - z2 = 5 + 5i, z1, z2.
. z , w z - 1 - 3i 2 w - 3 - i 2 : i. z, w , z = w ii. z - w. 3 f, IR f(x) 0, x IR. . f 1 - 1. . Cf f (1 , 2005) (-2 , 1), f -1(- 2004 + f (x2 - 8)) = -2. . Cf, 1 x +2005. Cf (): y = 668 4 f : IR IR,
f (x) - x = 2005 . x 0 x2 . : i. f (0) = 0, im
ii. f(0) = 1. 2 x 2 + ( f (x)) . IR, im =3 2 x 0 2x 2 + ( f (x))
220
-
. f IR f(x) > f (x), x IR, : i. x . f (x) > 0, x 0. ii.
f (x) dx < f (1).0
1
1 A.1 10 , 10. .2 , 21. . . , . , . , . , 2 . z1 = x + yi z2 = + i.
o
2005
. ,
. .
x + = 4 (1) z1 + z 2 = 4 + 4i (x + ) + (y + ) = 4 + 4 i y + = 4 (2) 2x - = 5 (3) 2z1 - z 2 = 5 + 5i (2x - ) + (2y + ) = 5 + 5 i 2y + = 5 (4) x = 3 y = 1 (2) , (4) (1) , (3) = 1 = 3 z1 = 3 + i z2 = 1 + 3i. . i. z (1 , 3) = 2 , w (3 , 1) R = 2 .y
A 3 K M 1 O y 1 3
() = (3 - 1) + (1 - 3) = 2 2 = R + , , x z , w, z = w ( ).
2
2
x
ii. H z - w (), 2R + 2 = 4 2 . 3 . f 1 - 1. , IR, < f () = f (). . Rolle [ , ], ( , ) f() = 0. f(x) 0, x IR. f 1 - 1.
1 A.1 f, . : f(x) > 0 x , f . f(x) < 0 x , f . .2 f . f ; . , . . z, z2 = z2. . im f (x) > 0 , f (x) > 0 x0.x x0
o
2006
. f () f . . (3x) = x . 3x - 1, x IR. .
f (x) g(x) dx = [ f (x) g (x)]
f(x) g (x) dx ,
f, g [ , ]. 2 f (x) = 2 + (x - 2)2, x 2. . f 1 - 1. . f -1 f . . i. f f -1 y = x. ii. f f -1. 3 z1, z2, z3 z1 = z2 = z3 = 1 z1 + z2 + z3 = 0. . : i) z1 - z 2 = z 2 - z3 = z3 - z1
4 Re(z1z2 ) -1 . z1, z2, z3 , . 4 x+1 f (x) = - nx . x-1 . f. . f (x) = 0 2 . . g (x) = nx ( , n), > 0 h (x) = ex ( , e), IR , f (x) = 0. . g h .ii) z1 - z2
2
1 A.1 (x)= - x, x IR . .2 f . f ; B. , . . z1, z2 , z1 - z2 z1 + z2. . f, g x0 g (x0) 0, f x0 : g f f (x 0 )g(x 0 ) - f(x 0 )g (x 0 ) . (x 0 ) = [g (x 0 )]2 g
o
2006
. x 0 [ nx] = 1 . x . f : IR 1 - 1, y f (x) = y x. . f [ , ]. G
f [ , ], 2
f (t) dt = G () - G ().
1 + e x , x IR. 1 + ex + 1 . f IR. 1 . f (x) dx . . x < 0 f (5x) + f (7x) < f (6x) + f (8x). 3 z, (4 - z)10 = z10 f f (x) = x2 + x + , IR. . z x = 2. . () f x = 2 yy y0 = -3, : i. (). ii. f, (), f (x) = xx x = 3 . 5 4 f (x) = x . n(x + 1) - (x + 1) . nx, x > 0. . i. : n(x + 1) - nx < 1 , x > 0. x ii. f (0 , +). . im x n 1 + 1 . x + x . (0 , +), ( + 1) = + 1.
228
-
) ( + 1) = + 1
>0
n( + 1) = n + 1 n( + 1) = ( + 1) n
n( + 1) - ( + 1) n = 0 f () = 0 f. im f (x) = im [ x n(x + 1) - (x + 1) nx ] = 0 - (- ) = + + +x 0 x +
im f (x) = im [ x n(x + 1) - (x + 1) nx ] = im [ x n(x + 1) - x nx - nx ]x + x +
x 0
x+1 - nx = im x [ n(x + 1) - nx ] - nx = im x n x + x + x .
= im xx +
() 1 - nx = 1 - (+ ) = n 1+ x
2007 1o A.1 z1 , z2 , z1 . z2=z1.z2. .2 f, g ; .3 y = Cf +; B. , . . f [ , ] x[ , ] f (x) 0 f (x) dx > 0 .
. f x . f f(x) > 0 x . . f x0 g x0, gof x0. . f
,
(
g (x)
f (t) dt
) = f (g (x)) g(x)
2
. im . > 1 x - x = 0.
z = 2 + i , IR. + 2i . z (0 , 0) = 1. . z1, z2 z = 2 + i + 2i = 0 = 2 . i. z1 z2. ii. (z1)2v = (-z2)v .
3 f (x) = x3 - 3x - 22, IR + /2 , Z . . f , . . f (x) = 0 . . x1, x2 x3 f, (x1, f (x1)), B(x2, f (x2)) (x3, f (x3)) y = -2x - 22. . f y = -2x - 22. 4 f [0 , 1] f (0) > 0. g [0 , 1] g (x) > 0, x [0 , 1]. :
F (x) = G (x) =
x
0
f (t) g (t) dt, x [0 , 1] g (t) dt, x [0 , 1]
x
0
. F(x) > 0, x (0 , 1]. . N f (x) . G(x) > F(x), x (0 , 1]. . N F (x) F (1) x (0 , 1].
x 2 0 f (t) g (t) dt 0 t dt . im x x 0+ 5 g (t) dt xx2
(
G (x)
G (1)
(
)
0
)
232
-
1 A.1. f x0, . .2. Rolle ; B. , . . f () f . . f, g, g [ , ],
o
2007
f (x) g(x) dx =
f (x) dx g(x) dx.
. f
,
(
X
f (t) dt
) = f (x) , x.x
. f ( , ), ( , ) A = im f (x) = im f (x). + x
. f, g . f, g f(x) = g(x) x , f (x) = g (x), x. 2
3x , x 0. . f . (1, +). . f (x) e, x > 0. .
0
f (x) dx.
x2 + 22
x +1
f (t) dt =
x2 + 22
x +3
f (t) dt +
4
2
f (t) dt
(0, +). 4 z1 = + i z 2 =2 - z1 , , IR 2 + z1
0. z2 - z1 IR. . z2 - z1 = 1. . z1 . . z12 > 0, z1 (z1 + 1 + i)20 - ( z 1 + 1 - i)20 = 0.
1 A.1. f (x) = nx, xIR* 1 IR* ( n x ) = . x .2. f [ , ]; B. , . . f : A IR 11, f -1 f -1(f (x)) = x, xA f (f -1(y)) = y, yf (A). . f f Df. . z2 + z + = 0, , , IR 0 , C. . f / R , f(x)>0, xIR. . A f , , ,
2008
f (x) dx =
f (x) dx +
f (x) dx.
2 z w (i + 2 2)z = 6 w - (1 - i) = w - (3 - 3i) , : . z. . w. . w. . z - w. 3 x nx, x > 0 f (x) = x=0 0, . f 0. . f . . x = e x , . . f(x + 1) > f (x + 1) - f (x), x>0. 4 f R . f (x) = 20x3 + 6x - 45. . g R.
f (x) = (10x 3 + 3x) f (t) dt - 45.0
2
g(x) = imh 0
g(x) - g(x - h) . h
. f () g g (x + h) - 2 g (x) + g (x - h) () im = f (x) + 45 h0 h2 g (0) = g(0) = 1, i. g (x) = x5 + x3 + x + 1, ii. g 11.
2008 1 A. [ , ]. G f [ , ],
f (t) dt = G () - G ().
. ; . , . . 1 - 1, . . f , f , . .
f (x) dx
xx xx. . , , : + i = 0 = 0 = 0. . ( , x0) (x0 , ) . im (f (x) - ) = 0. : im f (x) =x x0 x x0
21+i 3 2 z2 + z + = 0, . . = -1 = 1. . z13 = -1. . w, w = z1 - z1 .
z1 =
3 f (x) = x2 - 2 nx, x > 0. . f (x) 1, x > 0. . f. nx ,x>0 . g (x) = f (x) k, x = 0 i. k g . 1 ii. k = - , g , , 2 (0 , e).
4 f [0 , +) f (x) > 0, x 0. : x F (x) F (x) = f (t) dt, x [0 , + ) h (x) = x , x (0 , + ). 0 t f (t) dt 0
.
e0
1
t-1
[f (t) + F (t)] dt = F (1).
. h (0, +). . h (1) = 2, : i. ii.
2
0
f (t) dt < 2 t f (t) dt.0
2
F (t) dt =0
1
1 F (1). 2
244
-
1
9 2009 : : (5) 1 o A. f(x) = x . f (0 , + ) :
f ( x ) =
1 2 x 9
B.
f x o . f x o ; 6
.
, , , , . . z ( z ) = ( z ) 2 . f 1-1, f . 2
1 5
2
. lim f(x) = 0 f(x) < 0 x o xx o
xx o
lim
1 = + f (x)
2 . f(x) = x. H f 1 = { x x = 0 }
f ( x ) = -
1 2 x
2 . , f, x
f (x) dx = f(x) + c,
c . 2 2 z : (2 i ) z + (2 + i ) z 8 = 0
. N z = x+yi . 10
2 5
3
. N z1 z 2 . 8 . z1, z 2 z1 + z 22
+ z1 z 2
2
= 40
7 3
f(x) = ln[(+1)x 2 +x+1] - ln(x+2), -1
x > -1
. , lim f(x) .x +
5 . = -1 . f . 10 f 6
.
. f(x) + 2 = 0 0 4
3 5
4
4
f: [0 , 2] f ( x ) 4 f ( x ) + 4 f ( x ) = k x e 2 x , 0 x 2
f (0) = 2 f (0) , f (2) = 2 f(2)+12 e 4 , f(1) = e 2 k . . f ( x ) 2 f ( x ) , 0 x 2 g(x) = 3x 2 e2x Rolle [0,2]. 4 (0,2) , f ( ) + 4 f ( ) = 6 e 2 + 4 f ( ) 6 . k = 6 g(x) = 0 x [0,2]. 6 f ( x ) = x 3 e 2 x , 0 x 2 5 .
.
.
1
2
f (x) x2
dx
4
4 5
1 8 2009 : : (4)
1 . . f, g x 0 , f+g x 0 , : ( f + g ) (x ) = f (x ) + g (x ) .0 0 0
10 . x=x 0 f; 5 . 1, 2, 3, 4 5 , , . 1. +i +i . 2 2. . 3. : 1-1
2
x =0 . x 0 x lim 2
1 4
2
4. f ( x ) = ln x , x * , 1 * (ln x ) = . x 2 x +1 + c , , c 5. : x dx = +1 -1. 2
2 z=1 i (i 3) . 1+ i 2
. : z = 1 + i ,
z 2 = 2i ,
z 3 = 2 + 2i .
9 . , , z, z 2 , z 3 , , . 9 . :z z3 2 2
= z +z + z +z
2
2
3
2
. 7
3 f (x) = x e x , . . , C f (0, f(0)) y=ex. 10
2 4
3
.
=-1, i. f , 10 ii. x x C f . 5
4 f(x) = x-1 g(x) = lnx, x>0. . : f(x) g(x), x>0. M 8 . h(x) = f(x)-g(x), : i. : 0 h(x) e-2, x [1, e] . 7 ii. h, x x x = 1 x = e. 5 iii. I=
e h(x) e 1
[h ( x ) + 1]h( x )dx . 5
3 4
1
( ) 20 2009 : : (5)
1 o . f . f x f ( x ) = 0 , f . 10 . f x 0 ; 5 . , , , , . . z 1 , z 2 , z1z 2 = z1 z 2
2 . f () x 0 A, f(x)f(x 0 ) xA 2 1 5
2
.
x 1 =1 x 0 x lim
2 . f . 2 . f [, ] f(x) 0 1 x > 1,
A. f ( x ) 1 x > 1, =e 8 . =e, . f . 5 . f (1, 0] [0, + ) 6 . , (1, 0) (0, + ) , f () 1 f ( ) 1 + =0 x 1 x2
(1, 2)
6
3 5
4
4 f [0, 2]
0 (t 2)f (t )dt = 0 H( x ) =
2
0 t f (t )dt,
x
x [0, 2],
x H( x ) f ( t )dt + 3, x 0 G(x) = 1 1 t2 , 6 lim t 0 t2
x (0, 2] x=0
. G [0, 2]. 5 . G (0, 2) G( x ) = H( x ) x2
,
0 0 1. f 5 2. f (0,1] [1, + ) 5 3. f(x) = 0 . 6 4. x 1 , x 2 3 x 1 < x 2 , (x1, x2) , f() f() = 0 f (, f () ) . 9
3 5
http://edu.klimaka.gr
4
f: f(0) = 1 f(0) = 0 1. f(x) 1 x 41
x f ( xt )dt + x 32. lim0 x0
x
3
= + 6
f(x) + 2x = 2x f ( x ) + x 2 , x, :
(
)
3.
f(x) = e x x 2 ,
2
x 8
4. x +2
h(x) =
f (t)dt , x 0x4
x 2 + 2x +3 x 2 + 2 x +1
f (t)dt + f (t)dt < 06
7
4 5
1
( ) 19 2010 : : (4) A1. f . F f , : G(x)=F(x)+c, c f G f G(x)=F(x)+c, c
6 A2. x=x 0 f ; 4 A3. f . f ; 5 4. , , , , . ) +i +i . 1 4
2
) f . f , . ) f (,), (,), A = lim +f ( x ) B = lim f ( x )x x
) (x)=x, x ) lim f ( x ) < 0 , f(x)
