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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�
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