ŤƊ�ƆƀŹƁŶ ŦŹΌſ�ŹƉƀƁŶ ŠƉŹƀƆŽŷŹŬƉŹƈŷƃƅƊ 36 şƆŹƋ� 102 ŬƉƆƑźƅƂƅƇ 2003 ťŽƊƁƎƈŷŹ

ŭſƂŵƋƎƃƅ� 357! 22378101 ůŹƄ� 357! 22379122

BLR�BLR�NQF�BX VVV�BLR�NQF�BX

ūŝşťůūŬŤŠŭ ŠũŠŮŝŭŠŤŭ ����ŦВЙИНВФКЛВ ŤВФЖХЙХОУИУ

Ţ�ŽƆƅ�ſƃŷŹ� ����������

ūŬŪŮŠŤŨŪŧŠŨŠŭ ŦůŭŠŤŭ

ŧŠŬŪŭ ŝ!� Ũź ƄƔƊžƋž ƃźƂ ƋƂƉ �� źƊƃŷƊžƂƉ ƋƇƌ ŧŶƈƇƌƉ ŝ!�ťŵƁž ŵƊƃƀƊƀ ŻźƁΐƇƄƇżžŸƋźƂ ΐž � ΐƇƅŵŽžƉ�

�� ΊŷƃƅƃƉŹƀ ƉŹ ƈſ�ŽŷŹ A(1, 1) ƁŹƀ B(7, 9)� ŧŹ źƆŽŷƉŽ Ɖſƃ ŽƄŷƈƎƈſ ƉƅƊ ƁƒƁƂƅƊ �ƅƊ ŵƌŽƀżƀŴ�ŽƉƆƅ Ɖƅ ŽƊΌƒŻƆŹ��ƅ Ɖ�Ŷ�Ź AB�ŦƔƊƀũƀ ƈƊƃƉŽƉŹŻ�ŵƃŽƇ ƉƅƊ ƁŵƃƉƆƅƊ K ƉƅƊ ƁƒƁƂƅƊ ŽŷƃŹƀ

xK =xA + xB

2=

1 + 7

2= 4 ƁŹƀ yK =

yA + yB2

=1 + 9

2= 5 .

Ţ ŹƁƉŷƃŹ R ƉƅƊ ƁƒƁƂƅƊ ƀƈƅƒƉŹƀ �Ž

R = (AK) =!

(xA ! xK)2 + (yA ! yK)2 =!

(1! 4)2 + (1! 5)2 ="25 = 5 .

ŝ�Ƒ Ɖƅƃ Ɖƒ�ƅ (x! xK)2 + (y ! yK)2 = R2 ƁŹƉŹƂŶŻƅƊ�Ž ƈƉſƃ ŽƄŷƈƎƈſ

(x! 4)2 + (y ! 5)2 = 25 .

�� ŧŹ źƆŽŷƉŽ ƈƊƃŴƆƉſƈſ f : R # R ŻƀŹ Ɖſƃ ƅ�ƅŷŹ ƀƈƌƒŽƀ f !(x) = 3x $x % R ƁŹƀ ƉſƇƅ�ƅŷŹƇ ſ ŻƆŹƋƀƁŶ �ŹƆŴƈƉŹƈſ �ŽƆƃŴ Ź�Ƒ Ɖƅ ƈſ�Žŷƅ A(2, 6)�ŦƔƊƀŖƌƅƊ�Ž

f(x) =

"f !(x)Cx =

"3xCx =

3x2

2+ c .

ŝƋƅƒ ſ ŻƆŹƋƀƁŶ �ŹƆŴƈƉŹƈſ ƉſƇ ƈƊƃŴƆƉſƈſƇ �ŽƆƃŴ Ź�Ƒ Ɖƅ A(2, 6) ƉƑƉŽ f(2) = 6� ŔƆŹ

3 · 22

2+ c = 6 ù& 6 + c = 6 ù& c = 0 .

ŤŹƉŹƂŶŻƅƊ�Ž Ƃƅƀ�Ƒƃ ƈƉſ ƈƊƃŴƆƉſƈſ �Ž Ɖƒ�ƅ f(x) = 3x2

2 �

�� ŧŹ źƆŽŷƉŽ Ɖƅ ŹƑƆƀƈƉƅ ƅƂƅƁƂŶƆƎ�Ź"

(ſ�x+ ƈƊƃx)2 Cx .

�

ŦƔƊƀŖƌƅƊ�Ž

"(ſ�x+ ƈƊƃx)2 Cx =

" #ſ�2 x+ ƈƊƃ2 x+ 2 ſ�xƈƊƃx

$Cx

=

"(1 + 2 ſ�xƈƊƃx)Cx

=

"(1 + ſ� 2x)Cx

= x! 1

2ƈƊƃ 2x+ c .

�� Ţ ŻƆŹƋƀƁŶ �ŹƆŴƈƉŹƈſ Cf ƉſƇ ƈƊƃŴƆƉſƈſƇ f ŵƌŽƀ �ƂŴŻƀŹ Źƈƒ��ƉƎƉſ Ɖſƃ ŽƊΌŽŷŹ

y = 2x+ KM 2, ƈƉƅ !'.

ŧŹ źƆŽŷƉŽ ƉŹ ƑƆƀŹ�

ź� KHLx"#$

f(x) + 3x

x

Ż� KHLx"#$

(f(x)! 2x) �

ŦƔƊƀŝƆƌƀƁŴ �ŹƆŹƉſƆƅƒ�Ž ƑƉƀ ŹƋƅƒ ſ ŽƊΌŽŷŹ y = 2x + KM 2 ŵƌŽƀ �ƂŴŻƀŹ Źƈƒ��ƉƎƉſ ƈƉƅ!' ƉƑƉŽ Ź�Ƒ Ɖƅ ΌŽƓƆſ�Ź ŻƀŹ Ɖſƃ �ƂŴŻƀŹ Źƈƒ��ƉƎƉſ ΌŹ ƀƈƌƒŽƀ

KHLx"#$

f(x)

x= 2 ƁŹƀ KHL

x"#$[f(x)! 2x] = KM 2 .

ź� ŖƌƅƊ�Ž

KHLx"#$

f(x) + 3x

x= KHL

x"#$

%f(x)

x+

3x

x

&= KHL

x"#$

%f(x)

x+ 3

&= 2 + 3 = 5 .

Ż� ŪƆƅƁƒ�ƉŽƀ Ŵ�ŽƈŹ Ź�Ƒ Ɖƅ ΌŽƓƆſ�Ź ƑƉƀ

KHLx"#$

[f(x)! 2x] = KM 2 .

�� ΊŷƃŽƉŹƀ ſ ŽƄŷƈƎƈſ� x2 + y2 ! 4!x+ 2!y + 16 + !2 = 0, ! % R�

ź� ŧŹ źƆŽŷƉŽ ƑƂŽƇ ƉƀƇ Ɖƀ�ŵƇ ƉƅƊ ! ŵƉƈƀ ƓƈƉŽ ſ �ƀƅ �ŴƃƎ ŽƄŷƈƎƈſ ƃŹ �ŹƆƀƈƉŴƃŽƀƁƒƁƂƅ�

Ż� şƀŹ ƉƀƇ Ɖƀ�ŵƇ ƉƅƊ ! ŻƀŹ ƉƀƇ ƅ�ƅŷŽƇ ſ �ƀƅ �ŴƃƎ ŽƄŷƈƎƈſ �ŹƆƀƈƉŴƃŽƀ ƁƒƁƂƅ C! ƃŹźƆŽŷƉŽ Ɖſ ƈƊŻƁŽƁƆƀ�ŵƃſ Ɖƀ�Ŷ ƉƅƊ ! ŵƉƈƀ ƓƈƉŽ ſ żƒƃŹ�ſ ƉƅƊ ƈſ�ŽŷƅƊ A(4, 3) ƎƇ�ƆƅƇ Ɖƅƃ ƁƒƁƂƅ C! ƃŹ ŽŷƃŹƀ ŷƈſ �Ž !C!(A) = 32�

ŦƔƊƀ

ź� ŦŽ ƉƅƊƇ ƈƊƃŶΌŽƀƇ ƈƊ�źƅƂƀƈ�ƅƒƇ ŵƌƅƊ�Ž g = !2!, f = ! ƁŹƀ c = 16 + !2� şƀŹ ƃŹ

�

�ŹƆŹƈƉŴƃŽƀ ſ �ƀƅ �ŴƃƎ ŽƄŷƈƎƈſ ƁƒƁƂƅ �Ɔŵ�Žƀ

g2 + f2 ! c > 0 (& 4!2 + !2 ! (16 + !2) > 0

(& 4!2 ! 16 > 0

(& !2 > 4

(& ! % (!',!2) ) (2,+') .

ŠƃŹƂƂŹƁƉƀƁŴ ��ƅƆƅƒ�Ž ƃŹ ƈƊ��ƂſƆƓƈƅƊ�Ž ƉŹ ƉŽƉƆŴŻƎƃŹ ŻƀŹ ƃŹ �ŴƆƅƊ�Ž

x2+y2!4!x+2!y+16+!2 = 0 (& (x!2!)2+(y+!2) = 4!2+!2!(16+!2) = 4(!2!4)

ƁŹƀ �ŽƉŴ ƃŹ ƈƊƃŽƌŷƈƅƊ�Ž Ƒ�ƎƇ ƈƉſƃ �ƆƅſŻƅƒ�Žƃſ Ƃƒƈſ�Ż� ŖƌƅƊ�Ž

!C!(A) = 32 (& 42 + 32 ! 4 · ! · 4 + 2 · ! · 3 + 16 + !2 = 32

(& 16 + 9! 16!+ 6!+ 16 + !2 = 32

(& !2 ! 10!+ 9 = 0

(& (!! 1)(!! 9) = 0 .

ŔƆŹ ! = 1 Ŷ ! = 9�ř�ƎƇ ſ ! = 1 Ź�ƅƆƆŷ�ƉŽƉŹƀ Ź�Ƒ Ɖƅ Ź �Ɔŵ�Žƀ ! % (!',!2) ) (2,+') ŴƆŹŽŷƃŹƀ ! = 9�

�� ź� ŧŹ Ź�ƅżŽŷƄŽƉŽ ƑƉƀ ſ �ŹƆŹźƅƂŶ �Ž ŽƈƉŷŹ Ɖƅ ƈſ�Žŷƅ E(", 0), " > 0 ƁŹƀ żƀŽƊΌŽзƉƅƒƈŹ Ɖſƃ ŽƊΌŽŷŹ x+ " = 0 ŵƌŽƀ ŽƄŷƈƎƈſ y2 = 4"x�

Ż� ŬŽ ƉƊƌŹŷƅ ƈſ�Žŷƅ M("t2, 2"t), t *= 0 ƉſƇ �ŹƆŹźƅƂŶƇ y2 = 4"x, " > 0 �Ž ŽƈƉŷŹ E ƋŵƆƅƊ�Ž Ɖſƃ ŽƋŹ�Ɖƅ�ŵƃſ ſ ƅ�ƅŷŹ Ɖŵ�ƃŽƀ Ɖƅƃ ŴƄƅƃŹ y!y ƈƉƅ ƈſ�Žŷƅ A� ŧŹ żŽŷƄŽƉŽƑƉƀ ſ ŻƎƃŷŹ !EAM = 90%�

ŦƔƊƀ

ź�

�

ŝ�Ƒ Ɖƅƃ ƅƆƀƈ�Ƒ ƉſƇ �ŹƆŹźƅƂŶƇ ŻƃƎƆŷžƅƊ�Ž �ƎƇ ſ Ź�ƑƈƉŹƈſ ƉƊƌŹŷƅƊ ƈſ�ŽŷƅƊ A

Ź�Ƒ Ɖſƃ ŽƈƉŷŹ ƉſƇ �ŹƆŹźƅƂŶƇ ŽŷƃŹƀ ŷƈſ �Ž Ɖſƃ Ź�ƑƈƉŹƈſ ƉƅƊ ƈſ�ŽŷƅƊ A Ź�Ƒ ƉſżƀŽƊΌŽƉƅƒƈŹ ƉſƇ �ŹƆŹźƅƂŶƇ�Š�ƅ�ŵƃƎƇ

(AE) = d ù&!

(x! ")2 + (y ! 0)2 =|x+ "|"12 + 02

= |x+ "|

ù& (x! "2) + y2 = (x+ ")2

ù& x2 ! 2"x+ "2 + y2 = x2 + 2"x+ "2

ù& y2 = 4"x .

Ż�

ŖƌƅƊ�Ž2y

CyCx = 4" ù& Cy

Cx =2"

yù& !ŽƋ =

2"

2"t=

1

t.

Ţ ŽƄŷƈƎƈſ ƉſƇ ŽƋŹ�Ɖƅ�ŵƃſƇ ŽŷƃŹƀ

y!yM = !(x!xM ) ù& y!2"t =1

t(x!"t2) ù& ty!2"t2 = x!"t2 ù& ty!x = "t2 .

şƀŹ Ɖƅ ƈſ�Žŷƅ A� x = 0 ù& y = "t ù& A(0,"t)� ŭƑƉŽ

!ŽƋ!AE =1

t· at! 0

0! a=

1

t· (!t) = !1 .

ŔƆŹ ſ ŽƋŹ�Ɖƅ�ŵƃſ ŽŷƃŹƀ ƁŴΌŽƉſ ƈƉſƃ AE� $�ƅ�ŵƃƎƇ !EAM = 90%�

�� ΊŷƃŽƉŹƀ ſ ƈƊƃŴƆƉſƈſ f : [",#] # R Ƒ�ƅƊ " > 0� ŝƃ ſ f ŽŷƃŹƀ ƈƊƃŽƌŶƇ ƈƉƅ [",#] �ŹƆŹŻƎŻŷƈƀ�ſ ƈƉƅ (",#) ƁŹƀ ƀƈƌƒŽƀ f(") = f(#) = 0 ƃŹ żŽŷƄŽƉŽ ƑƉƀ�

ź� Ţ ƈƊƃŴƆƉſƈſ g(x) = f(x)x ƀƁŹƃƅ�ƅƀŽŷ ƉƀƇ �ƆƅƐ�ƅΌŵƈŽƀƇ ƉƅƊ ΌŽƎƆŶ�ŹƉƅƇ 1NKKD ƈƉƅ

żƀŴƈƉſ�Ź [",#]�Ż� Ů�ŴƆƌŽƀ $ % (",#) ƉŵƉƅƀƅ ƓƈƉŽ $f !($) = f($)�

�

ŦƔƊƀ

ź� Ţ ƈƊƃŴƆƉſƈſ g ŽŷƃŹƀ ƈƊƃŽƌŶƇ ƈƉƅ [",#] ƁŹƀ �ŹƆŹŻƎŻŷƈƀ�ſ ƈƉƅ (",#) ƎƇ �ſƂŷƁƅ�ŹƆŹŻƎŻŷƈƀ�Ǝƃ ƈƊƃŹƆƉŶƈŽƎƃ� ΊƀƑƉƀ " > 0 Ž�ƅ�ŵƃƎƇ ƅ �ŹƆƅƃƅ�ŹƈƉŶƇ żŽƃ �ſзżŽƃŷžŽƉŹƀ� ŖƌƅƊ�Ž

g(") =f(")

"= 0 ƁŹƀ g(#) =

f(#)

#= 0 ,

ŴƆŹ g(") = g(#) = 0�Š�ƅ�ŵƃƎƇ ſ g ƀƁŹƃƅ�ƅƀŽŷ ƉƀƇ �ƆƅƐ�ƅΌŵƈŽƀƇ ƉƅƊ ŽƎƆŶ�ŹƉƅƇ 1NKKD ƈƉƅ [",#]�

Ż� ŝ�Ƒ Ɖƅ ŽƓƆſ�Ź 1NKKD Ɗ�ŴƆƌŽƀ ƉƅƊƂŴƌƀƈƉƅƃ ŵƃŹ $ % (",#) ƉŵƉƅƀƅ ƓƈƉŽ g!($) = 0�ŠŷƃŹƀ

g!($) = 0 ù& $f !($)! f($)

$2= 0 ù& $f !($)! f($) = 0 ù& $f !($) = f($) .

�� ŧŹ Ź�ƅżŽŷƄŽƉŽ Ɖſƃ ŹƃƀƈƑƉſƉŹ

2! e

%< KM% <

%

e.

ŦƔƊƀŽƎƆƅƒ�Ž Ɖſ ƈƊƃŴƆƉſƈſ

f(x) = KMx, x % [e,%] .

' f ŽŷƃŹƀ ƈƊƃŽƌŶƇ ƈƉƅ [e,%] ƁŹƀ �ŹƆŹŻƎŻŷƈƀ�ſ ƈƉƅ (e,%) ƎƇ ƂƅŻŹƆƀΌ�ƀƁŶ ƈƊƃŴƆƉſƈſ�ţƁŹƃƅ�ƅƀƅƒƃƉŹƀ ƅƀ �ƆƅƐ�ƅΌŵƈŽƀƇ ƉƅƊ ŽƎƆŶ�ŹƉƅƇ ŦŵƈſƇ ŭƀ�ŶƇ� ŔƆŹ Ɗ�ŴƆƌŽƀ ƉƅƊƂŴзƌƀƈƉƅƃ ŵƃŹ $ % (",#) ƓƈƉŽ

f !($) =f(%)! f(e)

% ! eù& 1

$=

KM% ! KM e

% ! e=

KM% ! 1

% ! e.

ŝƋƅƒ e < $ < % ƉƑƉŽ

1

%<

1

$<

1

eù& 1

%<

KM% ! 1

% ! e<

1

eù& 1! e

%< KM%!1 <

%

e!1 ù& 2! e

%< KM% <

%

e.

�� ΊŷƃŽƉŹƀ ſ ƈƊƃŴƆƉſƈſ f(x) = x ƉƅƄŽƋx, x % R�

ź� ŧŹ żŽŷƄŽƉŽ ƑƉƀ ſ f ŽŷƃŹƀ ƁƊƆƉŶ ƈƉƅ R�Ż� ŝƃ ",# % R �Ž " > # ƃŹ żŽŷƄŽƉŽ ƑƉƀ ƀƈƌƒŽƀ ƉƅƄŽƋ"! ƉƅƄŽƋ# > "

1+"2 ! #1+#2 �

ŦƔƊƀ

ź� ' f ŽŷƃŹƀ żƒƅ ƋƅƆŵƇ �ŹƆŹŻƎŻŷƈƀ�ſ ƈƉƅ R ƎƇ �ƆŴƄſ żƒƅ ƋƅƆŵƇ �ŹƆŹŻƎŻŷƈƀ�ƎƃƈƊƃŹƆƉŶƈŽƎƃ� ŖƌƅƊ�Ž

f !(x) =x

1 + x2+ ƉƅƄŽƋx ù& f !!(x) =

1 + x2 ! x · 2x(1 + x2)2

+1

1 + x2

=1! x2 + 1 + x2

(1 + x2)2=

2

(1 + x2)2> 0 $x % R .

�

ŔƆŹ ſ f ŽŷƃŹƀ ƁƊƆƉŶ ƈƉƅ R�Ż� Š�ŽƀżŶ f !!(x) > 0 ŻƀŹ ƁŴΌŽ x % R ƉƑƉŽ ſ f ! ŽŷƃŹƀ ŻƃſƈŷƎƇ ŹƒƄƅƊƈŹ ƈƉƅ R�

ŔƆŹ ŻƀŹ ",# % R �Ž " > # ƀƈƌƒŽƀ ƑƉƀ f !(") > f !(#)� Š�ƅ�ŵƃƎƇ

"

1 + "2+ ƉƅƄŽƋ" >

#

1 + #2+ ƉƅƄŽƋ# ù& ƉƅƄŽƋ"! ƉƅƄŽƋ# >

#

1 + #2! "

1 + "2.

��� ΊŷƃŽƉŹƀ ſ ƈƊƃŴƆƉſƈſ f : A # R ſ ƅ�ƅŷŹ ŵƌŽƀ ƈƊƃŽƌŶ żŽƒƉŽƆſ �ŹƆŴŻƎŻƅ ƈƉƅ ŹƃƅƀƁƉƑżƀŴƈƉſ�Ź A�

ź� ŧŹ Ź�ƅżŽŷƄŽƉŽ ƑƉƀ ƈƉƅ A ƀƈƌƒŽƀ"

[f(x) + f !!(x)] ſ�xCx = f !(x) ſ�x! f(x)ƈƊƃx+ c .

Ż� ŧŹ źƆŽŷƉŽ Ɖƅ ŹƑƆƀƈƉƅ ƅƂƅƁƂŶƆƎ�Ź ƈƉƅ (0,+') " '

KMx! 1

x2

(ſ�xCx .

ŦƔƊƀ

ź� ŠŷƃŹƀ"

[f(x) + f !!(x)] ſ�xCx =

"f(x) ſ�xCx+

"f !!(x) ſ�xCx

=

"f(x) ſ�xCx+ f !(x) ſ�x!

"f !(x)ƈƊƃxCx

=

"f(x) ſ�xCx+ f !(x) ſ�x! f(x)ƈƊƃx!

"f(x) ſ�xCx

= f !(x) ſ�x! f(x)ƈƊƃx+ c .

Ż� ŠƋŹƆ�ƑžƅƊ�Ž Ɖƅ Ź ŻƀŹ Ɖſ ƈƊƃŴƆƉſƈſ f(x) = KMx, x > 0� ŪŹŷƆƃƅƊ�Ž" '

KMx! 1

x2

(ſ�xCx =

"[KMx+ (KMx)!!] ſ�xCx

= (KMx)! ſ�x! KMxƈƊƃx+ c

=ſ�x

x! KMxƈƊƃx+ c .

�

ŧŠŬŪŭ Ş!� Ũź ƄƔƊžƋž ƃźƂ ƋƂƉ � źƊƃŷƊžƂƉ ƋƇƌ ŧŶƈƇƌƉ Ş!�ťŵƁž ŵƊƃƀƊƀ ŻźƁΐƇƄƇżžŸƋźƂ ΐž �� ΐƇƅŵŽžƉ�

�� ΊŷƃŽƉŹƀ ſ ƈƊƃŴƆƉſƈſ f(x) = ex

(x#2)2 � ŧŹ źƆŽŷƉŽ Ɖƅ �Žżŷƅ ƅƆƀƈ�ƅƒ ƉŹ ƈſ�ŽŷŹ Ɖƅ�ŶƇ ƉſƇŻƆŹƋƀƁŶƇ �ŹƆŴƈƉŹƈſƇ �Ž ƉƅƊƇ ŴƄƅƃŽƇ ƉƎƃ ƈƊƃƉŽƉŹŻ�ŵƃƎƃ ƉŹ żƀŹƈƉŶ�ŹƉŹ �ƅƃƅƉƅƃŷŹƇ ƉŹ Ɖƅ�ƀƁŴ ŹƁƆƑƉŹƉŹ ƉƀƇ Źƈƒ��ƉƎƉŽƇ ƉſƇ ŻƆŹƋƀƁŶƇ �ŹƆŴƈƉŹƈſƇ ƉſƇ ƈƊƃŴƆƉſƈſƇ ƁŹƀƃŹ Ɖſƃ �ŹƆŹƈƉŶƈŽƉŽ ŻƆŹƋƀƁŴ�ŦƔƊƀūžŽŸƇ ŪƈƂƊΐƇƔ� A = R! {2}�ŭƀΐžŸź ƋƇΐŷƉ ΐž ŵƆƇƅžƉ�

� ŝƃ x = 0 ƉƑƉŽ y = e0

(0#2)2 = 14 �

� şƀŹ y = 0 �Ɔŵ�Žƀ ex

(x! 2)2= 0 Ɖƅ ƅ�ƅŷƅ ŽŷƃŹƀ ŹżƒƃŹƉƅ�

ŔƆŹ Ɖƅ �ƅƃŹżƀƁƑ ƈſ�Žŷƅ Ɖƅ�ŶƇ �Ž ƉƅƊƇ ŴƄƅƃŽƇ ŽŷƃŹƀ Ɖƅ (0, 14)�ŧƇƅƇƋƇƅŸź � ŮƇ�Ƃƃŵ ŝƃƈƓƋźƋź�ŠŷƃŹƀ

f !(x) =ex(x! 2)2 ! ex(2(x! 2))

(x! 2)4=

ex(x! 4)

(x! 2)3.

ŭƑƉŽf !(x) = 0 (& ex(x! 4)

(x! 2)3= 0 (& x = 4 .

ŤŹƉŹƈƁŽƊŴžƅƊ�Ž Ɖƅƃ �ŷƃŹƁŹ �ƆƅƈŶ�ƅƊ�

x

f !(x)

f

!' 2 4 +'

+ ! 0 +

3�$�

ŪŹƆŹƉſƆƅƒ�Ž ƑƉƀ

� Ţ f ŽŷƃŹƀ ŻƃſƈŷƎƇ ŹƒƄƅƊƈŹ ƈƉŹ (!', 2) ƁŹƀ [4,+')�� Ţ f ŽŷƃŹƀ ŻƃſƈŷƎƇ ƋΌŷƃƅƊƈŹ ƈƉƅ (2, 4]�� Ţ f �ŹƆƅƊƈƀŴžŽƀ Ɖƅ�ƀƁƑ ŽƂŴƌƀƈƉƅ ƈƉſ Όŵƈſ x = 4 Ɖƅ f(4) = e4

4 �

ťźƋźƃƓƈƌƍžƉ ŝƊƔΐ�ƋƐƋžƉ�ŠŷƃŹƀ

KHLx"2

f(x) = KHLx"2

ex

(x! 2)2= +' ,

ŹƋƅƒKHLx"2

ex = e2, KHLx"2

(x! 2)2 = 0 ƁŹƀ (x! 2)2 > 0, $x % A .

ŔƆŹ ſ ŽƊΌŽŷŹ x = 2 ŽŷƃŹƀ ƁŹƉŹƁƑƆƊƋſ Źƈƒ��ƉƎƉſ ƉſƇ ŻƆŹƋƀƁŶƇ �ŹƆŴƈƉŹƈſƇ ƉſƇ f �ŪƈƂſƓƅƋƂžƉ ŝƊƔΐ�ƋƐƋžƉ�

�

Š�ŽƀżŶKHL

x"+$ex = +' ƁŹƀ KHL

x"+$(x! 2)2 = +'

ŵƌƅƊ�Ž Ź�ƆƅƈżƀƅƆƀƈƉŷŹ ƉſƇ �ƅƆƋŶƇ +$+$ � ţƁŹƃƅ�ƅƀƅƒƃƉŹƀ ƅƀ �ƆƅƐ�ƅΌŵƈŽƀƇ ƉƅƊ ΌŽƎƆŶз

�ŹƉƅƇ ƉƅƊ CD K²'wOHS@K ŴƆŹ

KHLx"+$

f(x) = KHLx"+$

ex

(x! 2)2= KHL

x"+$

(ex)!

((x! 2)2)!= KHL

x"+$

ex

2(x! 2).

ŖƌƅƊ�Ž �ŴƂƀ Ź�ƆƅƈżƀƅƆƀƈƉŷŹ ƉſƇ �ƅƆƋŶƇ +$+$ ƁŹƀ ŽƋŹƆ�ƑžƅƃƉŹƇ �ŴƂƀ Ɖƅ ΌŽƓƆſ�Ź ƉƅƊ

CD K²'wOHS@K �ŹŷƆƃƅƊ�Ž

KHLx"+$

f(x) = KHLx"+$

(ex)!

(2(x! 2))!= KHL

x"+$

ex

2= +' .

Š�ƅ�ŵƃƎƇ ƀƈƌƒŽƀ ƑƉƀ KHLx"+$

f(x) = +'�ŠƄŴƂƂƅƊ

KHLx"#$

f(x) = KHLx"#$

ex

(x! 2)2= KHL

x"#$ex · KHL

x"#$

1

(x! 2)2= 0 · 0 = 0 .

ŔƆŹ ſ ŽƊΌŽŷŹ y = 0 ŽŷƃŹƀ ƅƆƀžƑƃƉƀŹ Źƈƒ��ƉƎƉſ ƉſƇ ŻƆŹƋƀƁŶƇ �ŹƆŴƈƉŹƈſƇ ƉſƇ f ƈƉſƃ�ŽƆƀƅƌŶ ƉƅƊ !'�ūƄŵżƂžƉ ŝƊƔΐ�ƋƐƋžƉ�ŠƋŹƆ�ƑžƅƊ�Ž żƀŹżƅƌƀƁŴ ƉƆŽƀƇ ƋƅƆŵƇ ƉƅƊ ΌŽƓƆſ�Ź ƉƅƊ CD K²'wOHS@K� ŬŽ ƁŴΌŽ ŽƋŹƆ�ƅŻŶƀƁŹƃƅ�ƅƀƅƒƃƉŹƀ ƅƀ �ƆƅƐ�ƅΌŵƈŽƀƇ żƀƑƉƀ ŵƌƅƊ�Ž Ź�ƆƅƈżƀƅƆƀƈƉŷŹ ƉſƇ �ƅƆƋŶƇ +$

+$ ƁŹƀ ƑƂŽƇƅƀ ƈƊƃŹƆƉŶƈŽƀƇ ŽŷƃŹƀ �ŹƆŹŻƎŻŷƈƀ�ŽƇ�ŖƌƅƊ�Ž

KHLx"+$

f(x)

x= KHL

x"+$

ex

x(x! 2)2

DLH= KHL

x"$

(ex)!

(x(x! 2)2)!= KHL

x"$

ex

3x2 ! 8x+ 4

DLH= KHL

x"$

(ex)!

(3x2 ! 8x+ 4)!= KHL

x"$

ex

6x! 8

DLH= KHL

x"$

(ex)!

(6x! 8)!= KHL

x"$

ex

6= +' .

ŔƆŹ ƈƉſƃ �ŽƆƀƅƌŶ ƉƅƊ +' żŽƃ Ɗ�ŴƆƌƅƊƃ �ƂŴŻƀŽƇ Źƈƒ��ƉƎƉŽƇ�ΊŽƃ ƌƆŽƀŴžŽƉŹƀ ŵƂŽŻƌƅƇ ŻƀŹ �ƂŴŻƀŹ Źƈƒ��ƉƎƉſ ƈƉſƃ �ŽƆƀƅƌŶ ƉƅƊ !' ŹƋƅƒ ŶżſŵƌƅƊ�Ž żŽƀ ƑƉƀ ŽƁŽŷ Ɗ�ŴƆƌŽƀ ƅƆƀžƑƃƉƀŹ Źƈƒ��ƉƎƉſ�

�

şƈźƍƂƃŷ �źƈŵƊƋźƊƀ�

�� ΊŷƃƅƃƉŹƀ ƅƀ ƁƒƁƂƅƀ C1 : x2 + y2 = 5 ƁŹƀ C2 : x2 + y2 ! 8x! 4y + 15 = 0�

ź� ŧŹ źƆŽŷƉŽ ƉƀƇ ƈƊƃƉŽƉŹŻ�ŵƃŽƇ ƉƅƊ ƁŵƃƉƆƅƊ ƉƅƊ ƁŴΌŽ ƁƒƁƂƅƊ ƁŹƀ ƃŹ Ɗ�ƅƂƅŻŷƈŽƉŽƉƅ �ŶƁƅƇ ƉſƇ ŹƁƉŷƃŹƇ ƉƅƊ ƁŴΌŽ ƁƒƁƂƅƊ�

Ż� ŧŹ żŽŷƄŽƉŽ ƑƉƀ ƅƀ ƁƒƁƂƅƀ ŽƋŴ�ƉƅƃƉŹƀ ŽƄƎƉŽƆƀƁŴ ƁŹƀ ƃŹ źƆŽŷƉŽ ƉƀƇ ƈƊƃƉŽƉŹŻ�ŵƃŽƇƉƅƊ Ɓƅƀƃƅƒ ƉƅƊƇ ƈſ�ŽŷƅƊ�

ż� ŧŹ źƆŽŷƉŽ ƉƀƇ ƈƊƃƉŽƉŹŻ�ŵƃŽƇ ƉƅƊ ƈſ�ŽŷƅƊ A �ƅƊ ŹƃŶƁŽƀ ƈƉƅƃ ƁƒƁƂƅ C1 ŻƀŹ Ɖƅƅ�ƅŷƅ ƀƈƌƒŽƀ ƑƉƀ !C1(A) = !C2(A)�

Ž� ŧŹ źƆŽŷƉŽ Ɖſƃ ŽƄŷƈƎƈſ ƉſƇ ŽƋŹ�Ɖƅ�ŵƃſƇ (&) ƉƅƊ ƁƒƁƂƅƊ C1 ƈƉƅ ƈſ�Žŷƅ ƉƅƊB(1,!2) ƁŹƀ ƃŹ Ź�ƅżŽŷƄŽƉŽ ƑƉƀ ſ (&) ŽƋŴ�ƉŽƉŹƀ ƁŹƀ ƈƉƅƃ ƁƒƁƂƅ C2�

ŦƔƊƀ

ź� ũ C1 ŵƌŽƀ ƁŵƃƉƆƅ Ɖƅ O(0, 0) ƁŹƀ ŹƁƉŷƃŹ R ="5 ƅ C2 ŵƌŽƀ ƁŵƃƉƆƅ Ɖƅ K(4, 2) ƁŹƀ

ŹƁƉŷƃŹ ' ="5�

Ż� Ţ żƀŴƁŽƃƉƆƅƇ ŵƌŽƀ �ŶƁƅƇ (OK) ="42 + 22 =

"20 = 2

"5� Š�ŽƀżŶ (OK) = R + '

ƅƀ ƁƒƁƂƅƀ ŽƋŴ�ƉƅƃƉŹƀ ŽƄƎƉŽƆƀƁŴ� şƀŹ Ɖƅ ƈſ�Žŷƅ Ž�ŹƋŶƇ Ž�ƀƂƒƅƊ�Ž Ɖƅ ƈƒƈƉſ�Ź)

x2 + y2 = 5

x2 + y2 ! 8x! 4y + 15 = 0

*(&

)x2 + y2 = 5

y = 5! 2x

*(&

)x2 + (5! 2x)2 = 5

y = 5! 2x

*

(&)x2 ! 4x+ 4 = 0

y = 5! 2x

*(&

)x = 2

y = 1

*.

ŔƆŹ ƅƀ ƈƊƃƉŽƉŹŻ�ŵƃŽƇ ƉƅƊ ƈſ�ŽŷƅƊ Ž�ŹƋŶƇ ŽŷƃŹƀ (2, 1)�ż� ŠƋƑƈƅƃ Ɖƅ ƈſ�Žŷƅ A ŹƃŶƁŽƀ ƈƉƅƃ ƁƒƁƂƅ C1 ŽŷƃŹƀ !C1(A) = 0� Š�ŽƀżŶ żŷƃŽƉŹƀ

!C1(A) = !C2(A) ŵƌƅƊ�Ž !C2(A) = 0� ŔƆŹ Ɖƅ ƈſ�Žŷƅ A ŹƃŶƁŽƀ ƈƉƅƃ ƁƒƁƂƅ C2�

�

ŬƊƃŽ�ƎƇ Ɖƅ ƈſ�Žŷƅ A ŽŷƃŹƀ Ɖƅ ƁƅƀƃƑ ƈſ�Žŷƅ ƈſ�Žŷƅ Ž�ŹƋŶƇ ƉƎƃ żƒƅ ƁƒƁƂƎƃ żſƂŹżŶ A(2, 1)�

Ž� ŖƌƅƊ�Ž

x2 + y2 = 5 ù& 2x+ 2yy! = 0 ù& y! = !x

yù& !ŽƋ� ƈƉƅ B = ! 1

!2=

1

2.

ŔƆŹ ſ ŽƄŷƈƎƈſ ƉſƇ ŽƋŹ�Ɖƅ�ŵƃſƇ ƈƉƅ B(1,!2) ŽŷƃŹƀ�

y + 2 =1

2(x! 1) ù& 2y + 4 = x! 1 ù& (&) : x! 2y ! 5 = 0 .

ŠŷƃŹƀd(K, (&)) =

|4! 2 · 2! 5|!12 + (!2)2

=5"5=

"5 = ' .

ŔƆŹ ſ ŽƊΌŽŷŹ (&) ŽƋŴ�ƉŽƉŹƀ ƉƅƊ C2�

�� ΊŷƃŽƉŹƀ ſ ƈƊƃŴƆƉſƈſ f(x) = e#x, x % R ƁŹƀ ƈſ�Žŷƅ ƉſƇ B((,!) Ƒ�ƅƊ ( > 0� Ţ ŽƋŹ�Ɖƅз�ŵƃſ ƉſƇ ŻƆŹƋƀƁŶƇ �ŹƆŴƈƉŹƈſƇ ƉſƇ ƈƊƃŴƆƉſƈſƇ ƈƉƅ B Ɖŵ�ƃŽƀ Ɖƅƃ ΌŽƉƀƁƑ ſ�ƀŴƄƅƃŹOy ƈƉƅ ƈſ�Žŷƅ A� ŬƉƅƃ ΌŽƉƀƁƑ ſ�ƀŴƄƅƃŹ Ox �ŹŷƆƃƅƊ�Ž ƈſ�Žŷƅ ) ƉŵƉƅƀƅ ƓƈƉŽ ƉƅƉŽƉƆŴ�ƂŽƊƆƅ OAB) ƃŹ ŽŷƃŹƀ ƉƆŹ�ŵžƀƅ�

ź� ŧŹ żŽŷƄŽƉŽ ƑƉƀ Ɖƅ Ž�źŹżƑƃ E(() ƉƅƊ ƉƆŹ�ŽžŷƅƊ OAB) żŷƃŽƉŹƀ Ź�Ƒ Ɖſ ƈƌŵƈſ

E(() =1

2(((+ 2)e#$ .

Ż� ŧŹ źƆŽŷƉŽ Ɖſƃ Ɖƀ�Ŷ ƉƅƊ ( ŻƀŹ Ɖſƃ ƅ�ƅŷŹ Ɖƅ Ž�źŹżƑƃ E(() ŻŷƃŽƉŹƀ �ŵŻƀƈƉƅ�

ŦƔƊƀ

ź� Š�ŽƀżŶ y = e#x, x % R ƉƑƉŽ y! = !e#x, x % R� ŔƆŹ !ŽƋ� ƈƉƅ B = !e#$�ŭƅ ƈſ�Žŷƅ B((,!) ŹƃŶƁŽƀ ƈƉſƃ ƁŹ��ƒƂſ ƁŹƀ ŵƉƈƀ ! = e#$ ŴƆŹ B((, e#$)�

Ţ ŽƄŷƈƎƈſ ƉſƇ ŽƋŹ�Ɖƅ�ŵƃſƇ ƉſƇ ƁŹ��ƒƂſƇ ƈƉƅ B ŽŷƃŹƀ�

y ! e#$ = !e#$(x! k) ù& y ! e#$ = !e#$x+ ke#$ .

��

ŖƌƅƊ�ŽxA = 0 ù& yA ! e#$ = ke#$ ù& yA = (k + 1)e#$ .

Ţ �ƅƃŹżƀƁŶ Όŵƈſ ƉƅƊ ) ƈƉƅƃ ſ�ƀŴƄƅƃŹ Ox ƓƈƉŽ Ɖƅ OAB" ƃŹ ŽŷƃŹƀ ƉƆŹ�ŵžƀƅŽŷƃŹƀ ) ((, 0)� ŭƑƉŽ

EOAB! =(AO +B) )(O) )

2=

(yA + yB)x%2

=[((+ 1)e#$ + e#$](

2.

ŔƆŹE(() =

1

2(((+ 2)e#$ .

Ż� ŖƌƅƊ�Ž

E!(() =(2(+ 2)e#$ ! e#$(k2 + 2k)

2=

(2(+ 2! (2 ! 2()e#$

2=

(2! (2)e#$

2.

Š�ƅ�ŵƃƎƇ Źƃ E!(() = 0 ƉƑƉŽ ( ="2 żŽƁƉŶ ŹƋƅƒ ( > 0 Ŷ ( = !

"2 Ź�ƅƆƆŷ�ƉŽз

ƉŹƀ�ŞƆŷƈƁƅƊ�Ž Ɖƅ ŽŷżƅƇ ƉƅƊ ŹƁƆƅƉŴƉƅƊ �Ž Ɖƅ ƁƆƀƉŶƆƀƅ ƉſƇ żŽƒƉŽƆſƇ �ŹƆŹŻƓŻƅƊ�ŠŷƃŹƀ

E!!(() =!2(e#$ ! (2! (2)e#$

2= !(2(+ 2! (2)e#$

2.

ŭƑƉŽE!!(

"2) = !(2

"2 + 2! 2)e#

&2

2= !

"2e#

&2 < 0 .

ŔƆŹ ŻƀŹ ( ="2 ŵƌƅƊ�Ž �ŵŻƀƈƉƅ Ž�źŹżƑƃ ƉƅƊ ƉƆŹ�ŽžŷƅƊ�

�� ź� ŧŹ żŽŷƄŽƉŽ ƑƉƀ Ɖƅ ƁƂŴƈ�Ź 1x(x+1)2 ŻƆŴƋŽƉŹƀ ƎƇ ŴΌƆƅƀƈ�Ź Ź�ƂƓƃ ƁƂŹƈ�ŴƉƎƃ ƎƇ

ŽƄŶƇ1

x(x+ 1)2=

1

x! 1

x+ 1! 1

(x+ 1)2.

Ż� ŧŹ źƆŽŷƉŽ Ɖƅƃ Ɖƒ�ƅ ƉſƇ ƈƊƃŴƆƉſƈſƇ f : (0,+') # R ſ ƅ�ƅŷŹ ŵƌŽƀ ƈƊƃŽƌŶ �ƆƓƉſ�ŹƆŴŻƎŻƅ ƈƉƅ �Žżŷƅ ƅƆƀƈ�ƅƒ ƉſƇ ƁŹƀ ŻƀŹ Ɖſƃ ƅ�ƅŷŹ ƀƈƌƒƅƊƃ�

f !(x) =KMx

(x+ 1)2, x % (0,+') ƁŹƀ f(1) = !1

2KM 2 +

1

4.

ż� ŧŹ źƆŽŷƉŽ Ɖƅ ƑƆƀƅ KHLx"+$

f(x)�

ŦƔƊƀ

ź� ŭƅ ƁƂŴƈ�Ź ŹƃŹƂƒŽƉŹƀ ƎƇ ŽƄŶƇ�

1

x(x+ 1)2=

A

x+

B

x+ 1+

)

(x+ 1)2.

Š�ƅ�ŵƃƎƇ ŵƌƅƊ�Ž

1 + A(x+ 1)2 +Bx(x+ 1) + )x (& (A+B)x2 + (2A+B + ) )x+A + 1 .

��

ŝ�Ƒ Ɖſƃ ƀƈƑƉſƉŹ ƉƎƃ �ƅƂƊƎƃƒ�Ǝƃ �ƆƅƁƒ�ƉŽƀ ƑƉƀ+,,-

,,.

A = 1

2A+B + ) = 0

A+B = 0

/,,0

,,1(&

+,,-

,,.

A = 1

2A+B + ) = 0

B = !A = !1

/,,0

,,1(&

+,,-

,,.

A = 1

B = !1

) = !2A!B = !1

/,,0

,,1.

ŔƆŹ1

x(x+ 1)2=

1

x! 1

x+ 1! 1

(x+ 1)2.

ΊƀŹƋƅƆŽƉƀƁŴ ��ƅƆƅƒ�Ž ƃŹ ŽƂŵŻƄƅƊ�Ž Ɖſƃ ƅƆΌƑƉſƉŹ ƉſƇ żƅΌŽŷƈŹƇ ƈƌŵƈſƇ ƁŴƃƅзƃƉŹƇ �ƆŴƄŽƀƇ �Ž ƅ�ƓƃƊ�Ź ƈƉƅ żŽƄŷ �ŵƂƅƇ�

Ż� ũƂƅƁƂſƆƓƃƅƃƉŹƇ ƁŹƀ ƉŹ żƒƅ �ŵƂſ ƉſƇ f !(x) = KMx(x+1)2 ΌŹ ŵƌƅƊ�Ž żƀŹżƅƌƀƁŴ

f(x) =

"KMxC

'(x+ 1)#2

!2

(=

KMx

!2(x+ 1)2+

"1

2x(x+ 1)2Cx

=KMx

!2(x+ 1)2+

1

2

" '1

x! 1

x+ 1! 1

(x+ 1)2

(Cx

= ! KMx

2(x+ 1)2+

1

2

'KMx! KM(1 + x) +

1

x+ 1

(+ c, c % R .

ΊŽƃ ŽŷƃŹƀ Ź�ŹƆŹŷƉſƉƅ ƃŹ ƌƆſƈƀ�ƅ�ƅƀŶƈƅƊ�Ž ƉŹ Ź�ƑƂƊƉŹ �ŵƈŹ ƈƉƅƃ ƋƊƈƀƁƑ ƂƅзŻŴƆƀΌ�ƅ ŹƋƅƒ �ŹƇ ŵƌŽƀ żƅΌŽŷ ƑƉƀ x > 0�ŝ�Ƒ Ɖſ żŽżƅ�ŵƃſ ŹƆƌƀƁŶ ƈƊƃΌŶƁſ ŵƌƅƊ�Ž

f(x) = !1

2KM 2 +

1

4ù& !1

2KM 2 +

1

4= 0 +

1

2

'KM 1! KM 2 +

1

2

(+ c

ù& !1

2KM 2 +

1

4= !1

2KM 2 +

1

4+ c

ù& c = 0 .

ŔƆŹ ƅ Ɖƒ�ƅƇ ƉſƇ ƈƊƃŴƆƉſƈſƇ ΌŹ ŽŷƃŹƀ

f(x) = ! KMx

2(x+ 1)2+

1

2

'KMx+ KM(1 + x) +

1

x+ 1

(.

ż� ŠŷƃŹƀ

KHLx"+$

f(x) = KHLx"+$

'!1

2

KMx

(x+ 1)2+

1

2KM

'x

x+ 1

(+

1

2(x+ 1)

(

=1

2

'! KHL

x"+$

KMx

(x+ 1)2+ KHL

x"+$KM

'x

x+ 1

(+ KHL

x"+$

1

x+ 1

(.

Ů�ƅƂƅŻŷžƅƊ�Ž ƉŹ ƑƆƀŹ ƄŽƌƎƆƀƈƉŴ�Š�ŽƀżŶ

KHLx"+$

KMx = +' ƁŹƀ KHLx"+$

(x+ 1)2 = +'

ŵƌƅƊ�Ž Ź�ƆƅƈżƀƅƆƀƈƉŷŹ ƉſƇ �ƅƆƋŶƇ +$+$ ƁŹƀ Ź�Ƒ Ɖƅ ΌŽƓƆſ�Ź ƉƅƊ CD K²'wOHS@K

��

ŵƌƅƊ�Ž

KHLx"+$

KMx

(x+ 1)2= ! KHL

x"+$

1/x

2(x+ 1)= ! KHL

x"+$

1

2x(x+ 1)= 0.

Š�ŽƀżŶ ſ KMx ŽŷƃŹƀ ƈƊƃŽƌŶƇ ƈƉƅ (0,+') ƁŹƀ KHLx"+$

x

x+ 1= 1 ƉƑƉŽ

KHLx"+$

KM'

x

x+ 1

(= KM

'KHL

x"+$

x

x+ 1

(= KM 1 = 0 .

ŠƃŹƂƂŹƁƉƀƁŴ Źƃ ΌŵƈƅƊ�Ž u = xx+1 ƉƑƉŽ KHL

x"$u = KHL

x"$

x

x+ 1= 1� Š�ƅ�ŵƃƎƇ

KHLx"+$

KM'

x

x+ 1

(= KHL

u"1KMu = 0 .

ŭŵƂƅƇ ŽŷƃŹƀ KHLx"+$

1

x+ 1= 0� ŔƆŹ

KHLx"+$

f(x) =1

2(!0 + 0 + 0) = 0 .

�� ΊŷƃŽƉŹƀ ſ �ŹƆŹźƅƂŶ y2 = 4x ƁŹƀ ƉŹ ƈſ�ŽŷŹ ƉſƇ T (t2, 2t) ƁŹƀ P ('2, 2') Ƒ�ƅƊ t *= ' t *= 0

ƁŹƀ ' *= 0�

ź� ŧŹ żŽŷƄŽƉŽ ƑƉƀ ſ ŽƄŷƈƎƈſ ƉſƇ ƌƅƆżŶƇ TP ŽŷƃŹƀ 2x! (t+ ')y + 2t' = 0�Ż� ŝƃ ſ ƌƅƆżŶ TP ŽƋŴ�ƉŽƉŹƀ ƉſƇ �ŹƆŹźƅƂŶƇ y2 = 2x ƃŹ żŽŷƄŽƉŽ ƑƉƀ�

J� (t+ ')2 = 8t'�JJ� Ţ ƁŹƆƉŽƈƀŹƃŶ ŽƄŷƈƎƈſ ƉſƇ ƁŹ��ƒƂſƇ ƈƉſƃ ƅ�ƅŷŹ ŹƃŶƁŽƀ ƅ ŻŽƎ�ŽƉƆƀƁƑƇ ƉƑ�ƅƇ

ƉƅƊ ƈſ�ŽŷƅƊ Ɖƅ�ŶƇ M ƉƎƃ ŽƋŹ�ƉƑ�ŽƃƎƃ ƉſƇ y2 = 4x ƈƉŹ ƈſ�ŽŷŹ ƉſƇ T ƁŹƀP ŵƌŽƀ ŽƄŷƈƎƈſ y2 = 8x�

ż� ŝƃ * ŽŷƃŹƀ ƉƊƌŹŷƅ ƈſ�Žŷƅ ƉſƇ �ŹƆŹźƅƂŶƇ y2 = 8x żƀŹƋƅƆŽƉƀƁƑ Ź�Ƒ Ɖſƃ ƁƅƆƊƋŶƉſƇ ƁŹƀ E ŽŷƃŹƀ ſ ŽƈƉŷŹ ƉſƇ ƃŹ żŽŷƄŽƉŽ ƑƉƀ ƅ ƁƒƁƂƅƇ �Ž żƀŴ�ŽƉƆƅ *E ŽƋŴ�ƉŽƉŹƀƉƅƊ ŴƄƅƃŹ y!y�

ŦƔƊƀ

ź� şƀŹ Ɖſƃ ƁƂŷƈſ ƉſƇ ƌƅƆżŶƇ TP ŵƌƅƊ�Ž�

!TP =2t! 2'

t2 ! '2=

2(t! ')

(t! p)(t+ ')=

2

t+ '.

Š�ƅ�ŵƃƎƇ ſ ŽƄŷƈƎƈſƇ ƉſƇ ƌƅƆżŶƇ ŽŷƃŹƀ

TP : y ! 2t =2

t+ '(x! t2) (& (t+ ')(y ! 2t) = 2x! 2t2

(& (t+ ')y ! 2t2 ! 2t' = 2x! 2t2

(& 2x! (t+ ')y + 2t' = 0 �

��

Ż� J� ŖƌƅƊ�Ž Ɖƅ ƈƒƈƉſ�Ź ƉƎƃ ŽƄƀƈƓƈŽƎƃ)

y2 = 2x

2x! (t+ ')y + 2t' = 0

*(&

)y2 = 2x

y2 ! (t+ ')y + 2t' = 0

*. �

ŝƋƅƒ ſ ŽƊΌŽŷŹ TP ŽŷƃŹƀ ŽƋŹ�Ɖƅ�ŵƃſ ƉſƇ �ŹƆŹźƅƂŶƇ y2 = 2x Ɖƅ �ƆƅſŻƅƒ�ŽƃƅƈƒƈƉſ�Ź ŵƌŽƀ �ŷŹ �ƅƃŹżƀƁŶ �ƆŹŻ�ŹƉƀƁŶ ƆŷžŹ� Š�ƅ�ŵƃƎƇ ŻƀŹ Ɖſ żƀŹƁƆŷƃƅƊƈŹƉſƇ � ΌŹ ŵƌƅƊ�Ž

! = 0 (& (t+ ')2 ! 8t' = 0 (& (t+ ')2 = 8t' . �

JJ� Ź źƆƅƒ�Ž ƉƀƇ ŽƄƀƈƓƈŽƀƇ ƉƎƃ ŽƋŹ�Ɖƅ�ŵƃƎƃ (&1) ƁŹƀ (&2) ƉſƇ �ŹƆŹźƅƂŶƇ ƈƉŹƈſ�ŽŷŹ T ƁŹƀ P ŹƃƉŷƈƉƅƀƌŹ� ŪŹƆŹŻƎŻŷžƅƃƉŹƇ �Ž�ƂŽŻ�ŵƃŹ Ɖſƃ ŽƄŷƈƎƈſ ƉſƇ�ŹƆŹźƅƂŶƇ ŵƌƅƊ�Ž

2yCyCx = 4 ù& Cy

Cx =2

y.

ŔƆŹ ƅƀ ƁƂŷƈŽƀƇ ƉƎƃ ŽƋŹ�Ɖƅ�ŵƃƎƃ ƉƎƃ (&1) ƁŹƀ (&2) ŽŷƃŹƀ

!1 =2

2t=

1

tƁŹƀ !2 =

2

2'=

1

'

ŹƃƉŷƈƉƅƀƌŹ� Š�ƅ�ŵƃƎƇ ƅƀ ŽƄƀƈƓƈŽƀƇ ƉƎƃ (&1) ƁŹƀ (&2) ŽŷƃŹƀ

(&1) : y ! 2t =1

t(x! t2) (& ty ! 2t2 = x! t2 (& ty = x+ t2 �

��

ƁŹƀ

(&1) : y ! 2' =1

'(x! '2) (& 'y ! 2'2 = x! '2 (& 'y = x+ '2 �

ťƒƃƅƃƉŹƇ Ɖƅ ƈƒƈƉſ�Ź ƉƎƃ ŽƄƀƈƓƈŽƎƃ � ƁŹƀ � ΌŹ ŵƌƅƊ�Ž)ty = x+ t2

'y = x+ '2

*(&

)ty = x+ t2

(t! ')y = (t! ')(t+ ') t *= p

*�

(&)t(t+ ') = x+ t2

y = (t+ ')

*(&

)x = t'

y = (t+ ')

*

ŝ�ŹƂŽŷƋƅƃƉŹƇ ƉƀƇ �ŹƆŹ�ŵƉƆƅƊƇ t, ' Ź�Ƒ ƉƀƇ � ƁŹƀ � ΌŹ ŵƌƅƊ�Ž y2 = 8x �ƅƊŽŷƃŹƀ ſ ŤŹƆƉŽƈƀŹƃŶ ŽƄŷƈƎƈſ ƉſƇ ƁŹ��ƒƂſƇ ƈƉſƃ ƅ�ƅŷŹ ŹƃŶƁŽƀ ƅ şŽƎ�ŽƉƆƀƁƑƇŭƑ�ƅƇ ƉƅƊ ƈſ�ŽŷƅƊ M �

ż� �ƇƉ ŮƈƓ�ƇƉ ūźƈźΐžƋƈƂƃŶƉ ŠƆƂƊƕƊžƂƉ�� ŖƈƉƎ *(2t2, 4t), t *= 0 ƁŹƀ K �ŵƈƅ ƉƅƊ*E� ŭƑƉŽ ƅƀ ƈƊƃƉŽƉŹŻ�ŵƃŽƇ ƉƅƊ ƁŵƃƉƆƅƊ ƉƅƊ ƁƒƁƂƅƊ ΌŹ ŽŷƃŹƀ

K

'2 + 2t2

2,0 + 4t

2

(Ŷ (1 + t2, 2t) .

ŖƈƉƎ Z(0, 2t) ſ ƅƆΌŶ �ƆƅźƅƂŶ ƉƅƊ K ƈƉƅƃ y!y� ŭƑƉŽ �ƆƅƋŹƃƓƇ (KZ) = 1 + t2�ř�ƎƇ ŻƀŹ Ɖſƃ ŹƁƉŷƃŹ ƉƅƊ ƁƒƁƂƅƊ ΌŹ ŵƌƅƊ�Ž

R = (K*) =!

(2t2 ! 1! t2)2 + (4t! 2t)2 =!

t4 ! 2t2 + 1 + 4t2 =!

(t2 + 1)2 = t2+1 .

ŔƆŹ (KZ) = R ƁŹƀ KZ ƁŴΌŽƉſ �ŴƃƎ ƈƉƅƃ ŴƄƅƃŹ y!y� Š�ƅ�ŵƃƎƇ Ɖƅ Z ŹƃŶƁŽƀ ƈƉƅƃƁƒƁƂƅ �Ž żƀŴ�ŽƉƆƅ Ɖƅ E* ƁŹƀ ƅ ƁƒƁƂƅƇ �Ž żƀŴ�ŽƉƆƅ Ɖƅ E* ŽƋŴ�ƉŽƉŹƀ ƉƅƊ ŴƄƅƃŹ y!y

ƈƉƅ Z��ƇƉ ŮƈƓ�ƇƉ ťźƈƋžƊƂźƅŶƉ ŭƌƅƋžƋźżΐŶƅžƉ�� ŖƈƉƎ *(x1, y1)� ũƃƅ�ŴžƅƊ�Ž B,),+ ƉƀƇƅƆΌŵƇ �ƆƅźƅƂŵƇ ƉƎƃ ƈſ�ŽŷƎƃ *,K,E Ƒ�ƅƊ K Ɖƅ ƁŵƃƉƆƅ ƉƅƊ ƁƒƁƂƅƊ żƀŹ�ŵƉƆƅƊ *E�ŴƃƎ ƈƉſ żƀŽƊΌŽƉƅƒƈŹ ƉſƇ �ŹƆŹźƅƂŶƇ

(,) : x+ 2 = 0 .

ŖƈƉƎ Z Ɖƅ ƈſ�Žŷƅ Ɖƅ�ŶƇ ƉſƇ ŽƊΌŽŷŹƇ K) �Ž Ɖƅƃ ŴƄƅƃŹ y!y� ŭƑƉŽ KZ , y!y� ŝ�Ƒ ƉƅƉƆŹ�ŵžƀƅ +B*E ΌŹ ŵƌƅƊ�Ž ƑƉƀ *B = x1 +2 ƁŹƀ E+ = 4 ƁŹƀ ŹƋƅƒ K) żƀŴ�ŽƈƅƇ ƉƅƊƉƆŹ�ŽžŷƅƊ ΌŹ ŵƌƅƊ�Ž ŻƀŹ Ɖƅ �ŶƁƅƇ ƉſƇ

K) =x1 + 2 + 4

2=

x12

+ 3 .

ŝƋƅƒ Ɖƅ * ŹƃŶƁŽƀ ƈƉſƃ �ŹƆŹźƅƂŶ ƀƈƌƒŽƀ *E = *B = x1+2� Š�ƅ�ŵƃƎƇ ſ ŹƁƉŷƃŹ ƉƅƊƁƒƁƂƅƊ żƀŹ�ŵƉƆƅƊ *E ŽŷƃŹƀ

R =*E

2=

x1 + 2

2=

x12

+ 1 .

��

ř�ƎƇK) = KZ + 2 ù& x1

2+ 3 = KZ + 2 ù& KZ =

x12

+ 1 .

ŔƆŹKZ = R Ź�Ƒ Ɖƅ ƅ�ƅŷƅ ƈƊ��ŽƆŹŷƃƅƊ�Ž ƑƉƀ ƅ ƁƒƁƂƅƇ �Ž żƀŴ�ŽƉƆƅ Ɖƅ E* ŽƋŴ�ƉŽƉŹƀƈƉƅƃ ŴƄƅƃŹ y!y ƈƉƅ ƈſ�Žŷƅ Z�ŭƀΛžˌƐƊƀ� ŬƉƅ ŷżƀƅ ƈƊ��ŵƆŹƈ�Ź ΌŹ ƁŹƉŹƂŶŻŹ�Ž Źƃ ƌƆſƈƀ�ƅ�ƅƀƅƒƈŹ�Ž Ɖƅ ƉƆŹ�ŵžƀƅON*E��ƇƉ ŮƈƓ�ƇƉ� Š�ŽƀżŶ

!&Z · !ZE =4t! 2t

2t2· 2t

!2=

4t2

!4t2= !1 ,

Ɖƅ ƉƆŷŻƎƃƅ *ZE ŽŷƃŹƀ ƅƆΌƅŻƓƃƀƅ� Š�ƅ�ŵƃƎƇ Ɖƅ ƈſ�Žŷƅ Z ŹƃŶƁŽƀ ƈƉƅƃ ƁƒƁƂƅ �Ž żƀŴз�ŽƉƆƅ Ɖƅ E* ƁŹƀ ŹƋƅƒ KZ żƀŴ�ŽƈƅƇ ƉƅƊ ƉƆƀŻƓƃƅƊ ΌŹ ŵƌƅƊ�Ž KZ = E*/2 = R�$�ŽƀżŶ KZ , (y!y) ƈƊ��ŽƆŹŷƃƅƊ�Ž ƑƉƀ ƅ ƁƒƁƂƅƇ �Ž żƀŴ�ŽƉƆƅ Ɖƅ E* ŽƋŴ�ƉŽƉŹƀ ƈƉƅƃŴƄƅƃŹ y!y ƈƉƅ Z�

��