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