# Ασεπ θεματα με λυσεις 2009

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2009

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• 2009 03

-

1

)

f(x) =

{x+ x2 sin

2

x, x 6= 0

0 , x = 0

i. pi f (x) x R f (x) x = 0.

ii. > 0 f(x) (, ).

) pi AB pi a. K N pi pi AB A KN pi pi B M(. ). AKN,NM pi KBN , :

i. pi K pi pi AM .ii. pi AK,AN M a.

) pi A = [aij], 1 i, j n aij R. pi 1, 2 R 1 6= 2 ~x Rn ~y Rn : A~x = 1~x A~y = 2~y, pi ~x ~y .

1 pi 23

• pi

( ). , .

( ). 1 2 pi pi 1 2 A,B E,Z . H

1 2 , AB

B=EZ

ZH,

H pi pi 1 2.

) i. x 6= 0 : f (x) = 1 + 2x sin 2x 2 cos 2

x. pipi

x 6= 0 f (0) = f(x) f(x0)x 0 = 1 + x sin

2

x. x > 0

x x sin 2x x lim

x0+

(x sin

2

x

)= 0.

x < 0

x x sin 2x x lim

x0

(x sin

2

x

)= 0.

,

limx0

(x sin

2

x

)= 0

f (0) = lim

x0

(1 + x sin

2

x

)= 1.

f (x) =

{1 + 2x sin

2

x 2 cos 2

x, x 6= 0

1 , x = 0

f (x) x = 0, - (xn)n xn 0, pipi limn+ f (xn) =f (0) = 1. (xn)n pi xn =

1

pin. xn 0

f ( 1pin

) = 1 +2

pinsin(2pin) 2 cos(2pin) = 1.

limn+ f (xn) = 1 f x = 0.

2 pi 23

• ii. f(x) (, ) pi > 0. f (x) 0 x (, ). pi n0 N 1

pin< n > n0. pi pi

f (xn0) = 1, pi f (, ) . pi - pi f . pi 0 f pipi pi.

) i. (AKN) = (BKN) = (NM) (AKN)+(BKN) =a2

3

4. (AKN) = (BKN) = (NM) = a

2

3

8

(BKM) = (BKN) = (NM) =a2

3

4. pi -

(BK)(B)

=

(AB)

(BM), K AM . (BK)

(B)=

(AB)

(BM)

3 pi 23

• (BK)

a=

a

(BM) (BK)(BM) = a2.

(BK)(BM) = a2. (BKM) = a2

3

4 1

2(K)(BM) =

a2

3

4 (K)(BM) = a

2

3

2. pipi (K)

(BK)= sin 60

(K) = (BK) sin 60 = (BK)

3

2(1). pi pipi

(BK)

3

2(BM) =

a2

3

2 (BK)(BM) = a2.

pi K AM .

ii. (AKN) = (AB)2

12

(AK)(AN) sin 60 =a2

3

8

(AK)(AN) =a2

2(2). pi (NM) = (AB)

2

pipi (N)(M) = a2

2 (aAN)(M) = a

2

2(3). pi

(1), (2) (3) :

(AK) =2a

3, (AN) =

3a

4 (M) = 2a.

) ~x ~y . pi R 6= 0 ~x = ~y. : A~x = 1~x A~y =1~y A~y = 1~y 2~y = 1~y (2 1)~y = ~0. pi 6= 0, 1 6= 2 ~y 6= ~0.

4 pi 23

• 2

) f(x) =(x+

1

x

)a, 0 < x < 1 a > 1.

i. pi f(x) (0, 1).

ii. pi x, y (0, 1) a > 1 f(x+ y

2

)

f(x) + f(y)

2.

iii. pi , x, y > 0, a > 1 x+ y = 1, :(x+

1

x

)a+

(y +

1

y

)a 5

a

2a1.

) - pi x = 5 pi pi s =

5.

( pi pi pi s2 = 1

i=1(ti x)2).

) pi A,B, pi AB = 200km, B = 400km ( A = 600km). - pi pi A, pi pi pi B pi 3 pi 6 . pi- pi pi 3 , pi- pi ( pi pi).

pi

( ). f :

[, ] pi (, )

pi (, ) , :

f () =f() f()

.

5 pi 23

• ( Rolle). f :

[, ] pi (, ) f() = f()

pi (, ) , :

f () = 0.

) i.

f (x) = a(x+

1

x

)a1(1 1

x2

)

f (x) = a(a 1)(x+

1

x

)a2(1 1

x2

)+ 2a

(x+

1

x

)a11

x3

x (0, 1) a > 1 f (x) > 0. f (0, 1).

ii.

pi pi f (x) > 0 x (0, 1) a > 1. f (0, 1). x < y. pi x1 (x, x+ y

2)

:

f (x1) =f

(x+ y

2

) f(x)

x+ y

2 x

=

2

(f

(x+ y

2

) f(x)

)y x . (1)

x2 (x+ y2

, y) :

f (x2) =2

(f(y) f

(x+ y

2

))y x . (2)

f , x1 < x2 f (x1) < f (x2) (3). pi (1), (2) (3) :

6 pi 23

• f(x+ y

2

) f(x) < f(y) f

(x+ y

2

)f(x+ y

2) 0, x < 1 y < 1. x, y (0, 1). a > 1 pi ii :

f

(x+ y

2

) f(x) + f(y)

2

f

(1

2

)

(x+

1

x

)a+

(y +

1

y

)a2

5a

2a

(x+

1

x

)a+

(y +

1

y

)a2

5a

2a1

(x+

1

x

)a+

(y +

1

y

)a.

) x1 = 3, x2 = , x3 = + x4 = + 3. :

x =x1 + x2 + x3 + x4

4

a = 5.

7 pi 23

• pi :

s2 =1

4

4i=1

(xi 5)2

5 =1

4

[(3)2 + ()2 + ()2 + (3)2]

20 = (92 + 2 + 2 + 92)

= 1.

= 1 x1 = 2, x2 = 4, x3 = 6 x4 = 8 = 1 x1 = 8, x2 = 6, x3 = 4 x4 = 2. pipi 2, 4, 6 8.

) S :[0, 6] R. pi , u. pi -:

f : [0, 3] Rt 7 S(t+ 3) 2S(t).

f [0, 3] pi (0, 3). pi f(0) =f(3) = 200. pi pi Rolle pit1 (0, 3) f (t1) = 0. ,

u(t1 + 3) 2u(t1) = 0

2u(t1) = u(t1 + 3)

pi pi .

3

1

pipi pi pi pi y = |x 1| y = 3 |x| :

8 pi 23

• ) 3.

) 4.

) 5.

) 6.

pi :

=

21

((3 |x|) |x 1|) dx

1 x 0 x 0 x1 0, 0 x 1 x 0 x1 0 1 x 2 x 0 x 1 0, pi :

=

01

(3+x(x+1))dx+ 10

(3x(x+1))dx+ 21

(3x(x1))dx

=

01

(2 + 2x)dx+

10

2dx+

21

(4 2x)dx

= 4.

9 pi 23

• 2 pi 1

4 . h

, :

) 43pi

5h3.

) 43pi(

5 1)h3.) 4

3pi(

5 2)h3.) 4

3pi(

5 3)h3.pi

( ). pi pi- = .

V , V , r h R .

V =1

4V

1

3pir2h =

1

4

4

3piR3

r2 =R3

h(1)

pi pi h(2R h) = r2 (2). pi (1) (2) :

10 pi 23

• h(2R h) = R3

hh2(2R h) = R3

2Rh2 h3 R3 = 0(hR)(h2 RhR2) = 0

h 6= R, pi h2RhR2 = 0. pi h. h = R1 +

5

2. pi :

R =2h

1 +

5=

(15)2h(1 +

5)(15) =

2h(15)15 +5 5

=h(15)2 =

h(

5 1)2

.

:

V =4

3piR3

=4

3pih3(

5 1)38

=4

3pi(

5 2)h3.

3

pi K =(

cospi

8

)4+

(cos

3pi

8

)4 :

) 12.

) 25.

) 34.

)

3

2.

11 pi 23

• pi

pi :

2 cos cos = cos( ) + cos(+ ) (1)cos = sin(

pi

2 ) (2)

12 =

(sin2

(3pi

8

)+ cos2

(3pi

8

))2= sin4

(3pi

8

)+ cos4

(3pi

8

)+ 2 sin2

(3pi

8

)cos2

(3pi

8

)= cos4

(pi8

)+ cos4

(3pi

8

)+ 2 cos2

(pi8

)cos2

(3pi

8

)= K +

1

2

(2 cos

(pi8

)cos

(3pi

8

))2= K +

1

2

(cos

(3pi

8 pi

8

)+ cos

(3pi

8+pi

8

))2= K +

1

2

(cos(pi

4

)+ cos

(pi2

))2= K +

1

2

(2

2+ 0

)2= K +

1

4

K =3

4

4

2, 3 4 pi(x + y)n pi pi-, n :

) 6.

) 7.

) 9.

) 10. Pascal

pi n 7.

12 pi 23

• n = 0 1n = 1 1 1n = 2 1 2 1n = 3 1 3 3 1n = 4 1 4 6 4 1n = 5 1 5 10 10 5 1n = 6 1 6 15 20 15 6 1n = 7 1 7 21 35 35 21 7 1n = 8 1 8 28 56 70 56 28 8 1n = 9 1 9 36 84 126 126 84 36 9 1n = 10 1 10 45 120 210 252 210 120 45 10 1

5

a > 0, aa |x(x a)(x 2a)(x 3a)|dx :

) 7a5.

) 8a5.

) 9a5.

) 10a5.

pipi pi f(x) = x(xa)(x2a)(x3a) =x4 6ax3 + 11a2x2 6a3x pi . pi :

13 pi 23

• aa|f(x)|dx =

0af(x)dx

a0

f(x)dx

=

0a

(x4 6ax3 + 11a2x2 6a3x)dx

a0

(x4 6ax3 + 11a2x2 6a3x)dx

=

[x5

5 6ax

4

4+ 11a2

x3

3 6a3x

2

2

]0a

[x5

5 6ax

4

4+ 11a2

x3

3 6a3x

2

2

]a0

= (a5

5 6aa

4

4+ 11a2

a33 6a3a

2

2

)(a5

5 6aa

4

4+ 11a2

a3

3 6a3a

2

2

)= 9a5.

6

() y = 2x + 1 A(2, 1). - pi () :

) x = 32, y =

12

7.

) x = 12, y = 3

5.

) x = 87, y =

15

4.

) x = 65, y =

13

5.

y = 2x+1 pi pi A(2, 1) y = 1

2x+ 2.

pi pi .

{y = 2x+ 1

y = 12x+ 2

}

14 pi 23

• (25,9

5).

(x0, y0), :

x0 + 2

2=

2

5

y0 + 1

2=

9

5

x0 = 6

5

y0 =13

5.

pi (65,13

5).

4

7

pi pi pi 600 pi. pi 200 A, 250 B 150 . pi- pi 0, 93, 0, 96 0, 88 A,B , pi pi pi pi 600 :

) 0, 92.

) 0, 93.

15 pi 23

• ) 0, 94.

) 0, 95.

A,B pi- A,B . pi E - pi K . P (A) = 200

600=

1

3, P (B) =

250

600=

5

12

P () = 150600

=1

4. pi

pi pi - pi. P (K) =

1

30, 93 +

5

120, 96 +

1